TEST BANK For Statistics for Business and Economics 13th Edition By James McClave, George Benson, Te by ACADEMIAMILL

Statistics for Business and Economics, 13e James McClave, George Benson, Terry Sincich (Test Bank All Chapters, 100% Original Verified, A+ Grade) Answers At The End Of Each Chapter Chapter 1 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Which of the following is not the job of a statistician? A) determining whether the conclusions drawn from a study are to be trusted B) implementing new procedures based on the results of a study C) determining what information is relevant in a given problem D) collecting numerical information in the form of data

2) Which of the following is not an element of descriptive statistical problems? A) data are displayed visually in graphs B) information revealed in a data set is summarized C) patterns in a data set are identified D) predictions are made about a larger set of data

3) The way in which an interviewer asks a question about political party affiliation causes respondents to answer that they have no affiliation when they actually do. What type of problem has occurred? A) nonresponse bias B) measurement error C) selection bias

4) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. Identify the variable of interest to the university administration. A) the parking time, defined to be the amount of time the student spent finding a parking spot B) the entire set of students that park at the university C) a single student that parks at the university D) the 250 students that data was collected from

5) As part of an economics class project, students were asked to randomly select 500 New Your Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. Identify the variable of interest for this study. A) the current price (or closing price) of a NYSE stock B) a single stock traded on the NYSE C) the entire set of stocks that are traded on the NYSE D) the 500 NYSE stocks that current prices were collected from

6) When we study a process, what is generally the focus? A) the black box B) the subprocesses C) the output D) the input

7) A study in the state of Georgia was conducted to determine the percentage of all community college students who have taken at least one online class. 1500 community college students were contacted and asked if they had taken at least one online class during their time at their community college. These responses were then used to estimate the percentage of all community college students who have taken at least one online class. Identify the population of interest in this study. A) the response (Yes/No) to the question, "Have you taken at least one online class?" B) all community college students in the state of Georgia C) the number of online classes a student has taken D) the 1500 community college students contacted

8) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. Identify the population of interest to the university administration. A) a single student that parks at the university B) the parking time, defined to be the amount of time the student spent finding a parking spot C) the 250 students that data was collected from D) the entire set of students that park at the university

9) Does online teaching help or hinder student learning? To help answer this question, a statistics teacher decided to teach his three sections of a particular class using three different teaching models - a traditional face-to-face section, a completely online section, and a hybrid or blended section that incorporated both a face-to-face and online component in the section. Students were randomly assigned to the different sections, taught identical information using the different teaching formats, and given identical examinations to measure student learning. The goal was to identify if the teaching method used affected student learning performance. Identify the data collection method used in this study. A) data collected observationally B) data from a designed experiment C) data from a published source

10) Which data about paintings would not be qualitative? A) the value B) the theme C) the artist

10)

D) the style

11) Which of the following is not typically an element of inferential statistical problems? A) measure of reliability B) variable of interest C) sample D) census

11)

12) Because of the possible legal consequences, many people in a sample of the U.S. population chose not to participate in a survey regarding illegal drug use. What type of problem has occurred? A) selection bias B) measurement error C) nonresponse bias

12)

13) In the context of processes, what is a sample? A) any set of subprocesses C) any subset of the population

13)

B) any set of output D) any set of input

14) A watchdog group is investigating how people are treated during the foreclosure process. Surveys were mailed to a random sample of 300 people who had recently been threatened with foreclosure. 75 of the surveys were returned by the postal service because the intended recipients had moved and left no forwarding address. What type of problem has occurred? A) measurement error B) nonresponse bias C) selection bias 2

14)

15) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. The university is interested in using the information from the sample of 250 students collected to learn information about the entire student parking population. Would this be an application of descriptive or inferential statistics? A) Inferential statistics B) Descriptive statistics

15)

16) A county planning commission is attempted to survey 1500 households from the counties 400,000 households. A random sample was selected and surveys were mailed to the randomly selected households, but only 1075 were returned. The inability to collect data from the 425 households that didn't return the survey would be considered which type of sampling problem? A) measurement error B) selection bias C) nonresponse bias

16)

17) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. What type of variable is the administration interested in collecting? A) qualitative data B) quantitative data

17)

18) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. Identify the data collection method used by the administration in this study. A) data collected observationally B) data from a published source C) data from a designed experiment

18)

19) As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. Identify the population of interest for this study. A) the entire set of stocks that are traded on the NYSE B) a single stock traded on the NYSE C) the 500 NYSE stocks that current prices were collected from D) the current price (or closing price) of a NYSE stock

19)

20) As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. Identify the sample of interest for this study. A) a single stock traded on the NYSE B) the entire set of stocks that are traded on the NYSE C) the current price (or closing price) of a NYSE stock D) the 500 NYSE stocks that current prices were collected from

20)

21) A personnel director studied the eating habits of a company's employees. The director noted whether employees brought their own lunch to work, ate at the company cafeteria, or went out to eat lunch. This type of data collection would best be considered as a(n) __________. A) designed experiment B) observational study

21)

22) As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. Identify the experimental unit of interest for this study. A) a single stock traded on the NYSE B) the 500 NYSE stocks that current prices were collected from C) the current price (or closing price) of a NYSE stock D) the entire set of stocks that are traded on the NYSE

22)

23) A student completing a research project for a criminal justice class obtained a radar gun for determining automobile speeds and recorded the speeds of automobiles passing a fixed location over a period of several hours. The student was unaware that the device needed to be recharged after two hours of use and that the speeds recorded after two hours were not reliable. What type of problem has occurred? A) measurement error B) nonresponse bias C) selection bias

23)

24) A middle school was interested in surveying their students to find out opinions about the schools media center. To facilitate data collection, the homeroom period was extended 30 minutes to allow everyone in the school ample time to respond to a short questionnaire. Unfortunately, it was learned after the surveys had been completed that all honors students in the middle school were on an all-day field trip and away from school for the entire day. The exclusion of their input into the survey would be considered which type of sampling problem? A) selection bias B) measurement error C) nonresponse bias

24)

25) As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. Identify the data collection method used in this study. A) data from a designed experiment B) data from a published source C) data collected observationally

25)

26) A student worked on her statistics project in the library and found a reference book that contained the median family incomes for all 50 states. On her project, she would report her data as being collected __________. A) from a published source B) from a designed experiment C) from a survey D) observationally

26)

27) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. Identify the sample of interest to the university administration. A) the 250 students that data was collected from B) the parking time, defined to be the amount of time the student spent finding a parking spot C) the entire set of students that park at the university D) a single student that parks at the university

27)

28) A university was interested in student reaction to a proposal to spend more on athletic scholarships and28) less on academic scholarships. 35 student athletes were surveyed. What type of problem has occurred? A) measurement error B) selection bias C) nonresponse bias 29) A researcher studying malnutrition among children in a developing country collected weights of a random sample of children using a scale that she had set to give weights 2.5 kilograms less than the actual weight. Which statement best describes this situation? A) Measurement error has occurred, but the researcher is not guilty of unethical statistical practice. B) Measurement error has not occurred, but the researcher is guilty of unethical statistical practice. C) Measurement error has not occurred, and the researcher is not guilty of unethical statistical practice. D) Measurement error has occurred, and the researcher is guilty of unethical statistical practice.

29)

30) As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. Would this be an application of descriptive or inferential statistics? A) Inferential statistics B) Descriptive statistics

30)

31) A study in the state of Georgia was conducted to determine the percentage of all community college students who have taken at least one online class. 1500 community college students were contacted and asked if they had taken at least one online class during their time at their community college. These responses were then used to estimate the percentage of all community college students who have taken at least one online class. Identify the variable of interest in this study. A) all community college students in the state of Georgia B) the 1500 community college students contacted C) the number of online classes a student has taken D) the response (Yes/No) to the question, "Have you taken at least one online class?"

31)

32) As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. What type of variable is being collected? A) qualitative data B) quantitative data

32)

33) Parking at a large university has become a very big problem. University administrators are interested in determining the average parking time (e.g. the time it takes a student to find a parking spot) of its students. An administrator inconspicuously followed 250 students and carefully recorded their parking times. Identify the experimental unit of interest to the university administration. A) the entire set of students that park at the university B) a single student that parks at the university C) the 250 students that data was collected from D) the parking time, defined to be the amount of time the student spent finding a parking spot

33)

34) A study in the state of Georgia was conducted to determine the percentage of all community college students who have taken at least one online class. 1500 community college students were contacted and asked if they had taken at least one online class during their time at their community college. These responses were then used to estimate the percentage of all community college students who have taken at least one online class. What type of variable is being collected? A) qualitative data B) quantitative data

34)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 35) What is the most common way to satisfy the representative sample requirement?

35)

36) Three female students and two male students are to be chosen from a group of 30 female students and 20 male students. Does this sample of five students satisfy the conditions to be a random sample of the 50 students in the group? Explain.

36)

37) In a survey of 3000 high school students, 21% of those surveyed read at least one best-seller each month. Give an example of a descriptive statement and an inferential statement that could be made based on this information.

37)

38) Give an example of unethical statistical practice.

38)

39) In an attempt to determine the ages of its customers, one store asked every tenth customer who entered the store his or her age. Identify the sampling design used.

39)

40) What is meant by a representative sample?

40)

41) A chain of coffee shops has 45 stores in one metropolitan area. For liability reasons, the chain is interested in the average temperature of hot drinks served at the stores. Three stores were chosen and the temperature of every fifth hot drink served at each of these stores was recorded during a two-week period. At the end of the two-week period, the temperatures of 10,571 hot drinks had been recorded.

41)

a. b. c. d.

Identify the process of interest. Identify the variable of interest. Describe the sample. Describe the inference of interest.

42) A department store receives customer orders through its call center and website. These 42) orders as well as any special orders received in the stores are forwarded to a distribution center where workers pull the items on the orders from inventory, pack the items, and prepare the necessary paperwork for the shipping company that will pick the orders up and deliver them to the customers. In order to monitor the subprocess of pulling the items from inventory, every 15 minutes one order is checked to determine whether the worker has pulled the correct item. a. b. c. d.

Identify the process of interest. Identify the variable of interest. Describe the sample. Describe the inference of interest.

43) What is meant by selection bias?

43)

44) In one study of the moral of company's employees, 10 employees were randomly chosen from each of the departments within the company. Identify the sampling design used.

44)

45) Explain why it is not necessary to provide a measure of reliability when a census is used rather than a sample.

45)

46) A high school guidance counselor analyzed data from a sample of 300 community colleges throughout the United States. One of his goals was to estimate the annual tuition costs of community colleges in the United States. Describe the population and variable of interest to the guidance counselor.

46)

47) Define statistical thinking.

47)

48) A health food company has the following statement on their new product packaging: "Prevents all types of cancer!" (Fact: Past studies have shown that some ingredients in the new product have been know to possibly reduce the risk of many types of cancer). Discuss why it is unethical to make this statement.

48)

49) Parking at a university has become a problem. University administrators are interested in determining the average time it takes a student to find a parking spot. An administrator inconspicuously followed 110 students and recorded how long it took each of them to find a parking spot. Identify the population, sample, and variable of interest to the administrators.

49)

50) A quality inspector tested 33 copiers in an attempt to estimate the average failure rate of the copier model. His study indicated that the number of failures decreased from two years ago, indicating an increase in the reliability of the copiers. Describe the variable of interest to the inspector.

50)

51) What do we call a process whose operations are unknown or unspecified?

51)

52) Define business analytics.

52)

53) Gender is one variable of interest in a study of the effectiveness of a new medication. For data entry purposes, the researcher conducting the study assigns 1 for Male and 2 for Female. Is the gender data quantitative or qualitative?

53)

54) What is statistics?

54)

55) What term is used to describe the situation where sampling units contained in a sample do not produce sample observations?

55)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 56) A survey of high school teenagers reported that 93% of those sampled are interested in pursuing a college education. Does this statement describe descriptive or inferential statistics? A) Inferential statistics B) Descriptive statistics

56)

57) A fan observes the numbers on the shirts of a girl's soccer team. Identify the type of data collected. A) quantitative B) qualitative

57)

58) Parking at a university has become a problem. University administrators are interested in determining the average time it takes a student to find a parking spot. An administrator inconspicuously followed 220 students and recorded how long it took each of them to find a parking spot. Identify the variable of interest to the university administration. A) time to find a parking spot B) number of empty parking spots C) students who drive cars on campus D) number of students who cannot find a spot

58)

Answer the question True or False. 59) In an observational study, the researcher exerts strict control over the units in the study. A) True B) False Solve the problem. 60) An insurance company conducted a study to determine the percentage of cardiologists who had been sued for malpractice in the previous three years. The sample was randomly chosen from a national directory of doctors. What is the variable of interest in this study? A) all cardiologists in the directory B) the number of doctors who are cardiologists C) the responses: have been sued/have not been sued for malpractice in the last three years D) the doctor's area of expertise (i.e., cardiology, pediatrics, etc.) Answer the question True or False. 61) When using data from a published source, it is not important to know how the data were collected and whether randomization was used. A) True B) False 62) When we take data obtained from a sample and make generalizations or predictions about the entire population, we are utilizing inferential statistics. A) True B) False

59)

60)

61)

62)

Solve the problem. 63) A postal worker counts the number of complaint letters received by the United States Postal Service in a given day. Identify the type of data collected. A) qualitative B) quantitative

63)

64) A study attempted to estimate the proportion of Florida residents who were willing to spend more tax dollars on protecting the Florida beaches from environmental disasters. Twenty-six hundred Florida residents were surveyed.Which of the following describes the variable of interest in the study? A) the response to the question "Do you use the beach?" B) the response to the question "Do you live along the beach?" C) the response to the question, "Are you willing to spend more tax dollars on protecting the Florida beaches from environmental disasters?" D) the 2600 Florida residents surveyed

64)

65) An usher records the number of unoccupied seats in a movie theater during each viewing of a film. Identify the type of data collected. A) qualitative B) quantitative

65)

Answer the question True or False. 66) A census is feasible when the population of interest is small. A) True B) False

66)

Solve the problem. 67) Which type of problem has occurred when inaccuracies exist in the values of the data recorded? A) nonresponse bias B) selection bias C) measurement error

67)

Answer the question True or False. 68) Statistics involves two different processes, describing sets of data and drawing conclusions about the sets of data on the basis of sampling. A) True B) False 69) A variable is a characteristic or property of a population. A) True B) False Solve the problem. 70) An assembly line is operating satisfactorily if fewer than 2% of the phones produced per day are defective. To check the quality of a day's production, the company randomly samples 40 phones from a day's production to test for defects. Define the population of interest to the manufacturer. A) all the phones produced during the day in question B) the 40 responses: defective or not defective C) the 40 phones sampled and tested D) the 2% of the phones that are defective 71) The manager of a car dealership records the colors of automobiles on a used car lot. Identify the type of data collected. A) quantitative B) qualitative

68)

69)

70)

71)

72) The average age of the students in a statistics class is 22 years. Does this statement describe descriptive or inferential statistics? A) Descriptive statistics B) Inferential statistics

72)

73) From past figures, it is predicted that 47% of the registered voters will vote in the March primary. Does this statement describe descriptive or inferential statistics? A) Descriptive statistics B) Inferential statistics

73)

74) A study attempted to estimate the proportion of Florida residents who were willing to spend more tax dollars on protecting the Florida coastline from environmental disasters. Twenty-six hundred Florida residents were surveyed.Which of the following is the population used in the study? A) all Florida residents who lived along the coastline B) all Florida residents C) Florida residents willing to spend more tax dollars protecting the coastline from environmental disasters D) the 2600 Florida residents who were surveyed

74)

75) A recent report stated "Based on a sample of 130 truck drivers, there is evidence to indicate that, on average, independent truck drivers earn more than company-hired truck drivers." Does this statement describe descriptive or inferential statistics? A) Inferential statistics B) Descriptive statistics

75)

Answer the question True or False. 76) A measure of reliability is an important element of a descriptive statistical problem. A) True B) False Solve the problem. 77) The amount of television viewed by today's youth is of primary concern to Parents Against Watching Television (PAWT). 250 parents of elementary school-aged children were asked to estimate the number of hours per week that their child watches television. Identify the type of data collected by PAWT. A) quantitative B) qualitative

76)

77)

78) The amount of television viewed by today's youth is of primary concern to Parents Against Watching Television (PAWT). 330 parents of elementary school-aged children were asked to estimate the number of hours per week that their child watches television. Identify how the data were collected in this study. A) observationally B) from a survey C) from a published source D) from a designed experiment

78)

79) What method of data collection would you use to collect data for a study where a drug was given to 60 patients and a placebo to another group of 60 patients to determine if the drug has an effect on a patient's illness? A) survey B) observational study C) published source D) designed experiment

79)

80) What method of data collection would you use to collect data for a study where a political pollster wishes to determine if his candidate is leading in the polls? A) designed experiment B) survey C) published source D) observational study

80)

Answer the question True or False. 81) Measurement is the process of assigning numbers to variables of individual population units. A) True B) False 82) The process of using information from a sample to make generalizations about the larger population is called statistical inference. A) True B) False Solve the problem. 83) Parking at a university has become a problem. University administrators are interested in determining the average time it takes a student to find a parking spot. An administrator inconspicuously followed 90 students and recorded how long it took each of them to find a parking spot. Identify the population of interest to the university administration. A) the entire set of students who park at the university B) the 90 students about whom the data were collected C) the students who park at the university between 9 and 10 AM on Wednesdays D) the entire set of faculty, staff, and students who park at the university

81)

82)

83)

Answer Key Testname: CHAPTER 1 1) B 2) D 3) B 4) A 5) A 6) C 7) B 8) D 9) B 10) A 11) D 12) C 13) B 14) B 15) A 16) C 17) B 18) A 19) A 20) D 21) B 22) A 23) A 24) A 25) B 26) A 27) A 28) B 29) D 30) B 31) D 32) B 33) B 34) A 35) selecting a random sample 36) No; not every sample of 5 students from the group has an equal chance of selection; for example, a sample consisting of 5 males has no chance of being selected. 37) Descriptive: 21% of the students sampled (or 630) read at least one best-seller each month. Inferential: Based on the survey, we estimate that about 21% of all high school students read at least one best-seller each month. 38) Researchers select a biased sample, with the intention of misleading the public. 39) systematic sampling 40) a sample that exhibits characteristics typical of those possessed by the population of interest 41) a. serving of hot drinks at coffee shops in the chain b. temperature of hot drinks served c. 10,571 drinks whose temperatures recorded over the two-week period d. average temperature of all hot drinks served at all stores in the chain

Answer Key Testname: CHAPTER 1 42) a. fulfilling customers' orders from receiving the order to pick up by shipping company b. whether or not an order has been pulled correctly c. the set of all orders that are checked (one every 15 minutes) d. number or proportion of all orders that are pulled correctly (incorrectly) 43) Selection bias is when a subset of the experimental units in the population is excluded so that these units have no possibility of being selected in the sample. 44) stratified random sampling 45) When a census is used, there should be no error. 46) The population of interest to the guidance counselor is all community colleges in the United States. The variable of interest is the annual tuition cost of the community college. 47) Statistical thinking involves applying rational thought and the science of statistics to critically assess data and make inferences. 48) Answers may vary. One possible answer is that the past studies show that the ingredients only have possible cancer reducing effects on many, not all, types of cancer. 49) The population of interest are all students at the university who park. The sample is the parking times of the 110 students that were collected by the university administrator. The variable of interest to the administrators is the parking time variable. 50) The variable of interest to the researcher is the failure rate of the copiers. 51) a black box 52) Business analytics refers to methodologies (e.g., statistical methods) that extract useful information from data in order to make better business decisions. 53) Qualitative; The numbers are arbitrarily selected numerical codes for the categories and have no utility beyond that. 54) Statistics is the science of data that involves collecting, classifying, summarizing, organizing, analyzing, and interpreting numerical information. 55) nonresponse 56) B 57) B 58) A 59) B 60) C 61) B 62) A 63) B 64) C 65) B 66) A 67) C 68) A 69) B 70) A 71) B 72) A 73) B 74) B 75) A 76) B 77) A 78) B 79) D 80) B 13

Answer Key Testname: CHAPTER 1 81) A 82) A 83) A

Chapter 2 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Calculate the standard deviation of a sample for which n = 6, A) 6.19

B) 6.78

C) 46.00

2 x = 830,

x = 60.

D) 164.00

2) The amount spent on textbooks for the fall term was recorded for a sample of five university students - $400, $350, $600, $525, and $450. Calculate the value of the sample mean for the data. A) $600 B) $450 C) $465 D) $400

3) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed was 100 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. Assume that the statistician also gave us the information that the distribution of serve speeds was mound-shaped and symmetric. What percentage of the player's serves were between 115 mph and 145 mph? A) approximately 16% B) at most 34% C) at most 2.5% D) at most 13.5%

4) The amount spent on textbooks for the fall term was recorded for a sample of five hundred university students. It was determined that the 75th percentile was the value $500. Which of the following interpretations of the 75th percentile is correct? A) 75% of the students sampled had textbook costs that exceeded $500. B) The average of the 500 textbook costs was $500. C) 25% of the students sampled had textbook costs that exceeded $500. D) 75% of the students sampled had textbook costs equal to $500.

5) The total points scored by a basketball team for each game during its last season have been summarized in the table below. Which statement following the table must be true?

Score 41-60 61-80 81-100 101-120

Frequency 3 8 12 7

A) The range is at least 41 but at most 79. C) The range is 79.

B) The range is at least 41 but at most 120. D) The range is at least 81 but at most 100.

6) The amount spent on textbooks for the fall term was recorded for a sample of five university students - $400, $350, $600, $525, and $450. Calculate the value of the sample range for the data. A) $250 B) $450 C) $99.37 D) $98.75

7) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 103 miles per hour. Suppose that the statistician indicated that the serve speed distribution was skewed to the left. Which of the following values is most likely the value of the median serve speed? A) 85 mph B) 112 mph C) 103 mph D) 94 mph

8) The amount spent on textbooks for the fall term was recorded for a sample of five hundred university students. The mean expenditure was calculated to be $500 and the standard deviation of the expenditures was calculated to be $100. Suppose a randomly selected student reported that their textbook expenditure was $700. Calculate the z-score for this student's textbook expenditure. A) -3 B) +3 C) +2 D) -2

9) The payroll amounts for all teams in an international hockey league are shown below using a graphical9) technique from chapter 2 of the text. How many of the hockey team payrolls exceeded $20 million (Note: Assume that no payroll was exactly $20 million)?

A) 8 teams

B) 18 teams

C) 10 teams

D) 23 teams

10) Parking at a university has become a problem. University administrators are interested in determining the average time it takes a student to find a parking spot. An administrator inconspicuously followed 260 students and recorded how long it took each of them to find a parking spot. Which of the following types of graphs should not be used to display information concerning the students parking times? A) pie chart B) histogram C) box plot D) stem-and-leaf display

10)

11) In an eye color study, 25 out of 50 people in the sample had brown eyes. In this situation, what does the number .50 represent? A) a class B) a class relative frequency C) a class frequency D) a class percentage

11)

12) The amount spent on textbooks for the fall term was recorded for a sample of five hundred university students. The mean expenditure was calculated to be $500 and the median expenditure was calculated to be $425. Which of the following interpretations of the mean is correct? A) The most frequently occurring textbook cost in the sample was $500 B) 50% of the students sampled had textbook costs equal to $500 C) 50% of the students sampled had textbook costs that were less than $500 D) The average of the textbook costs sampled was $500

12)

13) The box plot shown below was constructed for the amount of soda that was poured by a filling machine into 12-ounce soda cans at a local soda bottling company.

13)

We see that one soda can received 12.15 ounces of soda on the plot above. Based on the box plot presented, how would you classify this observation? A) suspect outlier B) it has a lot of soda C) highly suspect outlier D) expected observation

14) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed was 100 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. If nothing is known about the shape of the distribution, what percentage of the player's serve speeds are less than 70 mph? A) at most 12.5% B) at most 25% C) approximately 2.5% D) approximately 5% E) at most 11%

14)

15) The box plot shown below was constructed for the amount of soda that was poured by a filling machine into 12-ounce soda cans at a local soda bottling company.

15)

We see that one soda can received 12.30 ounces of soda on the plot above. Based on the box plot presented, how would you classify this observation? A) it has a lot of soda B) expected observation C) highly suspect outlier D) suspect outlier

16) The amount of time workers spend commuting to their jobs each day in a large metropolitan city has a mean of 70 minutes and a standard deviation of 20 minutes. Assuming nothing is known about the shape of the distribution of commuting times, what percentage of these commuting times are between 30 and 110 minutes? A) at least 89% B) at least 95% C) at least 75% D) at least 0%

16)

17) The amount spent on textbooks for the fall term was recorded for a sample of five university students - $400, $350, $600, $525, and $450. Calculate the value of the sample median for the data. A) $465 B) $450 C) $600 D) $400

17)

18) Which of the following is a measure of the variability of a distribution? A) skewness B) median C) sample size

18)

D) range

19) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits 19) during the tournament. The statistician reported that the mean serve speed of a particular player was 100 miles per hour (mph) and the standard deviation of the serve speeds was 15 mph. Using the z-score approach for detecting outliers, which of the following serve speeds would represent outliers in the distribution of the player's serve speeds? Speeds: 50 mph, 80 mph, and 105 mph A) None of the three speeds are outliers. C) 50, 80, and 105 are all outliers.

B) 50 and 80 are both outliers, 105 is not. D) 50 is the only outlier.

20) A dot plot of the speeds of a sample of 50 cars passing a policeman with a radar gun is shown below. 20)

What proportion of the motorists were driving above the posted speed limit of 60 miles per hour? A) 2 B) 0.22 C) 0.04 D) 0.18

x 2 = 1320,

21) Calculate the variance of a sample for which n = 5, A) 326.00

B) 3.16

C) 10.00

x = 80.

21) D) 8.00

22) Which number on the screen below is the sample standard deviation of the data?

A) 2.67

B) 408

C) 5.8

D) 2.82

22)

23) One of the questions posed to a sample of 286 incoming freshmen at a large public university was, "Do 23) you have any tattoos?" Their responses are shown below in the pie chart. Please note that the values shown represent the number of responses in each category.

Based on the responses shown in the pie chart, what percentage of the freshmen responded with "Yes?" A) 73.4% B) 26.6% C) 76% D) 76

24) The amount spent on textbooks for the fall term was recorded for a sample of five hundred university students. The mean expenditure was calculated to be $500 and the median expenditure was calculated to be $425. Which of the following interpretations of the median is correct? A) The most frequently occurring textbook cost in the sample was $425 B) 50% of the students sampled had textbook costs equal to $425 C) The average of the textbook costs sampled was $425 D) 50% of the students sampled had textbook costs that were less than $425

24)

25) A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $118 and a standard deviation of $8. Three solar homes reported monthly utility bills of $84, $85, and $90. Which of the following statements is true? A) Homes using solar power always have lower utility bills than homes using only gas and electricity. B) Homes using solar power may actually have higher utility bills than homes using only gas and electricity. C) The utility bills for homes using solar power are about the same as those for homes using only gas and electricity. D) Homes using solar power may have lower utility bills than homes using only gas and electricity.

25)

26) The amount of time workers spend commuting to their jobs each day in a large metropolitan city has a mean of 70 minutes and a standard deviation of 20 minutes. Assuming the distribution of commuting times is known to be moundshaped and symmetric, what percentage of these commuting times are between 50 and 110 minutes? A) approximately 95% B) approximately 97.5% C) approximately 81.5% D) approximately 68%

26)

27) 260 randomly sampled college students were asked, among other things, to state their year in school 27) (freshman, sophomore, junior, or senior). The responses are shown in the bar graph below. How many of the students who responded would be classified as upperclassmen (e.g., juniors or seniors)?

A) Approximately 25 C) Approximately 100

B) Approximately 125 D) Approximately 10

28) The temperature fluctuated between a low of 73°F and a high of 89°F. Which of the following could be calculated using just this information? A) range B) median C) variance D) standard deviation

28)

29) The range of scores on a statistics test was 42. The lowest score was 57. What was the highest score? A) 99 B) 78 C) cannot be determined D) 70.5

29)

30) What class percentage corresponds to a class relative frequency of .37? A) .37% B) 37% C) .63%

30)

D) 63%

31) A survey was conducted to determine how people feel about the quality of programming available on television. Respondents were asked to rate the overall quality from 0 (no quality at all) to 100 (extremely good quality). The stem-and-leaf display of the data is shown below.

31)

Stem Leaf 33 6 40 3 4 7 8 9 9 9 50 1 1 2 3 4 5 61 2 5 6 6 70 6 8 93 What percentage of the respondents rated overall television quality as very good (regarded as ratings of 80 and above)? A) 12% B) 3% C) 1% D) 4%

32) The amount spent on textbooks for the fall term was recorded for a sample of five university students - $400, $350, $600, $525, and $450. Calculate the value of the sample standard deviation for the data. A) $98.75 B) $99.37 C) $450 D) $250

32)

33) 252 randomly sampled college students were asked, among other things, to estimate their college 33) grade point average (GPA). The responses are shown in the stem-and-leaf plot shown below. Notice that a GPA of 3.65 would be indicated with a stem of 36 and a leaf of 5 in the plot. How many of the students who responded had GPA's that exceeded 3.55? Stem and Leaf Plot of GPA Leaf Digit Unit = 0.01 19 9 represents 1.99 Stem 1 19 5 20 6 21 11 22 15 23 20 24 33 25 46 26 61 27 79 28 88 29 116 30 (19) 31 117 32 95 33 80 34 49 35 31 36 25 37 13 38 5 39 4 40

Minimum 1.9900 Median 3.1050 Maximum 4.0000

Leaves 9 0668 0 05567 0113 00005 0000000000067 0000005577789 000000134455578 000000000144667799 002356777 0000000000000000000011344559 0000000000112235666 0000000000000000345568 000000000025557 0000000000000000333444566677889 000003355566677899 000005 022235588899 00002579 7 0000

252 cases included

A) 39

B) 31

C) 19

D) 49

34) A sample of professional golfers was taken and their driving distance (measured as the average distance 34)as their drive off the tee) and driving accuracy (measured as the percentage of fairways that their drives landed in) were recorded. A scatterplot of the variables is shown below.

What relationship do these two variables exhibit? A) They exhibit a curvillinear relationship B) They exhibit no relationship C) They exhibit a positive linear relationship D) They exhibit a negative linear relationship

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 35) The calculator screens summarize a data set.

35)

a. Identify the mean and the median. b. Based only on the mean and the median, do you expect that the data set is skewed to the right, symmetric, or skewed to the left? Explain.

36) What is a time series plot?

36)

37) What is the primary advantage of a time series plot?

37)

38) Calculate the mean of a sample for which

38)

x = 196 and n = 8.

39) The total points scored by a basketball team for each game during its last season have been summarized in the table below. Score 41-60 61-80 81-100 101-120

39)

Frequency 3 8 12 7

a. Explain why you cannot use the information in the table to construct a stem-and-leaf display for the data. b. Construct a histogram for the scores.

40) Which is expressed in the same units as the original data, the variance or the standard deviation?

40)

41) In a summary of recent real estate sales, the median home price is given as $325,000. What percentile corresponds to a home price of $325,000?

41)

42) Parking at a university has become a problem. University administrators are interested in determining the average time it takes a student to find a parking spot. An administrator inconspicuously followed 280 students and recorded how long it took each of them to find a parking spot. The times had a distribution that was skewed to the left. Based on this information, discuss the relationship between the mean and the median for the 280 times collected.

42)

43) The calculator screens summarize a data set.

43)

a. Identify the smallest measurement in the data set. b. Identify the largest measurement in the data set. c. Calculate the range of the data set.

44) The calculator screens summarize a data set.

44)

a. Identify the mean and the sample standard deviation. Round to one place after the decimal, where necessary. b. Find the interval that corresponds to measurements within two standard deviations of the mean.

45) For a given data set, which is typically greater, the range or the standard deviation?

45)

46) The data show the total number of medals (gold, silver, and bronze) won by each country 46) winning at least one gold medal in the Winter Olympics. Find the mean, median, and mode of the numbers of medals won by these countries. 1

47) The z-score for a value x is -2.5. State whether the value of x lies above or below the mean and by how many standard deviations.

47)

48) The data below represent the numbers of absences and the final grades of 15 randomly selected48) students from a statistics class. Construct a scattergram for the data. Do you detect a trend? Student 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of Absences 5 6 2 12 9 5 8 15 0 1 9 3 10 3 11

Final Grade as a Percent 79 78 86 56 75 90 78 48 92 78 81 86 75 89 65

49) The amount of television viewed by today's youth is of primary concern to Parents Against Watching Television (PAWT). Three hundred parents of elementary school-aged children were asked to estimate the number of hours per week that their child watches television. The upper quartile for the distribution was given as 14 hours. Interpret this value.

49)

50) Test scores for a history class had a mean of 79 with a standard deviation of 4.5. Test scores for a physics class had a mean of 69 with a standard deviation of 3.7. One student earned a 87 on the history test and a 82 on the physics test. Calculate the z-score for each test. On which test did the student perform better?

50)

51) An annual survey sent to retail store managers contained the question "Did your store suffer any losses due to employee theft?" The responses are summarized in the table for two years. Compare the responses for the two years using side-by-side bar charts. What inferences can be made from the charts?

51)

Employee PercentagePercentage Theft in year 1 in year 2 Yes 34 23 No 51 68 Don't know 15 9 Totals

100

52) Explain how it can be misleading to draw the bars in a histogram so that the width of each bar is proportional to its height rather than have all bars the same width.

52)

53) The scores for a statistics test are as follows:

53)

87 76 90 77 92 94 88 85 66 89 79 98 50 98 83 88 82 56 15 69 Create a stem-and-leaf display for the data.

54) The scores of nine members of a women's golf team in two rounds of tournament play are listed 54) below. Player 1 2 3 4 5 6 7 8 9 Round 1 85 90 87 78 92 85 79 93 86 Round 2 90 87 85 84 86 78 77 91 82 Construct a scattergram for the data.

55) What characteristic of a Pareto diagram distinguishes it from other bar graphs?

55)

56) Explain how it can be misleading to report only the mean of a distribution without any measure of the variability.

56)

57) Suppose that 50 and 75 are two elements of a population data set and their z-scores are -3 and 2, respectively. Find the mean and standard deviation.

57)

58) The ages of five randomly chosen professors are 58, 56, 69, 70, and 59. Calculate the sample variance of these ages.

58)

59) Given the sample variance of a distribution, explain how to find the standard deviation.

59)

60) The calculator screens summarize a data set.

60)

a. How many data items are in the set? b. What is the sum of the data? c. Identify the mean, median, and mode, if possible.

61) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 103 miles per hour (mph) and the standard deviation of the serve speeds was 14 mph. Assume that the statistician also gave us the information that the distribution of serve speeds was mound-shaped and symmetric. Find the percentage of serves that were hit faster than 75 mph.

61)

62) The following data represent the scores of 50 students on a statistics exam. The mean score is 62) 80.02, and the standard deviation is 11.9. 39 71 79 85 90

51 71 79 86 90

59 73 79 86 91

63 74 80 88 91

66 76 80 88 92

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 98

Use the z-score method to identify potential outliers among the scores.

63) The total points scored by a basketball team for each game during its last season have been summarized in the table below. Identify the modal class of the distribution of scores. Score 41-60 61-80 81-100 101-120

Frequency 3 8 12 7

63)

64) The data show the total number of medals (gold, silver, and bronze) won by each country 64) winning at least one gold medal in the Winter Olympics. Find the range, sample variance, and sample standard deviation of the numbers of medals won by these countries. 1

65) The data show the total number of medals (gold, silver, and bronze) won by each country winning at least one gold medal in the Winter Olympics. 1

65)

11 11

11 14 14 19 22 23 24 25 29 a.

Complete the class frequency table for the data. Total Medals 1-5 6-10 11-15 16-20 21-25 26-30

Frequency

b. Using the classes from the frequency table, construct a histogram for the data.

66) A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $102.00 and a standard deviation of $12.00. Assuming the distribution is mound-shaped and symmetric, would you expect to see a 3-bedroom house using gas or electric energy with a monthly utility bill of $168.00? Explain.

66)

67) For a given data set, the lower quartile is 45, the median is 50, and the upper quartile is 57. The67) minimum value in the data set is 32, and the maximum is 81. a. b. c. d.

Find the interquartile range. Find the inner fences. Find the outer fences. Is either of the minimum or maximum values considered an outlier? Explain.

68) Which measures variability about the mean, the range or the standard deviation?

68)

69) Use a graphing calculator or software to construct a box plot for the following data set.

69)

12 13 12

18 14 16

14 11 17

17 16

19 18

16 15

14 13

18 17

15 15

17 14

11 19

70) The following data represent the scores of 50 students on a statistics exam. 39 71 79 85 90

51 71 79 86 90

59 73 79 86 91

63 74 80 88 91

66 76 80 88 92

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

70)

71 79 85 89 98

a. Find the lower quartile, the upper quartile, and the median of the scores. b. Find the interquartile range of the data and use it to identify potential outliers. c. In a box plot for the data, which scores, if any, would be outside the outer fences? Which scores, if any, would be outside the inner fences but inside the outer fences?

71) Various state and national automobile associations regularly survey gasoline stations to determine the current retail price of gasoline. Suppose one such national association contacts 200 stations in the United States to determine the price of regular unleaded gasoline at each station. In the context of this problem, define the following descriptive

71)

measures: µ, , x, s.

72) A small computing center has found that the number of jobs submitted per day to its computers has a distribution that is approximately mound-shaped and symmetric, with a mean of 88 jobs and a standard deviation of 8. On what percentage of days do the number of jobs submitted exceed 96?

72)

73) The mean x of a data set is 18, and the sample standard deviation s is 2. Explain what the interval (12, 24) represents.

73)

74) The data below show the types of medals won by athletes representing the United States in the74) Winter Olympics. gold bronze gold gold

gold gold silver gold

silver silver silver bronze

gold silver bronze bronze

bronze bronze bronze

silver silver gold

silver gold silver

a. Construct a frequency table for the data. b. Construct a relative frequency table for the data. c. Construct a frequency bar graph for the data.

75) Each year advertisers spend billions of dollars purchasing commercial time on network television. 75) In the first 6 months of one year, advertisers spent $1.1 billion. Who were the largest spenders? In a recent article, the top 10 leading spenders and how much each spent (in million of dollars) were listed: Company A $72.6 Company B 61.7 Company C 57.5 Company D 54.9 Company E 30.5

Company F $27.8 Company G 26.3 Company H 21.2 Company I 21.3 Company J 20.1

Calculate the mean and median for the data.

76) A retail store's customer satisfaction rating is at the 88th percentile. What percentage of retail stores has higher customer satisfaction ratings than this store?

76)

77) Explain how using a scale break on the vertical axis of a histogram can be misleading.

77)

78) By law, a box of cereal labeled as containing 36 ounces must contain at least 36 ounces of cereal. The machine filling the boxes produces a distribution of fill weights that is mound-shaped and symmetric, with a mean equal to the setting on the machine and with a standard deviation equal to 0.02 ounce. To ensure that most of the boxes contain at least 36 ounces, the machine is set so that the mean fill per box is 36.06 ounces. What percentage of the boxes do, in fact, contain at least 36 ounces?

78)

79) A sample of 100 e-mail users were asked whether their primary e-mail account was a free account, an institutional (school or work) account, or an account that they pay for personally. Identify the classes for the resulting data.

79)

80) Explain how stretching the vertical axis of a histogram can be misleading.

80)

81) The calculator screens summarize a data set.

81)

a. Identify the lower and upper quartiles of the data set. b. Find the interquartile range. c. Is there reason to suspect that the data may contain an outlier? Explain.

82) Complete the frequency table for the data shown below. green brown blue blue

blue orange brown brown

brown blue green blue

orange red red blue

blue green brown red

Color Frequency Green Blue Brown Orange

82)

83) The following data represent the scores of 50 students on a statistics exam. The mean score is 83) 80.02, and the standard deviation is 11.9. 39 71 79 85 90

51 71 79 86 90

59 73 79 86 91

63 74 80 88 91

66 76 80 88 92

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 98

Find the z-scores for the highest and lowest exam scores. 1 84) The output below displays the mean and median for the state high school dropout rates in year84) and in year 5.

N MEAN MEDIAN

Year 1 51 28.82 27.66

Year 5 51 26.23 25.38

Use the information to determine the shape of the distributions of the high school dropout rates in year 1 and year 5.

85) A radio station claims that the amount of advertising each hour has an a mean of 15 minutes and a standard deviation of 2.9 minutes. You listen to the radio station for 1 hour and observe that the amount of advertising time is 11.23 minutes. Based on your observation, what would you infer about the radio station's claim?

85)

86) The following data represent the scores of 50 students on a statistics exam. The mean score is 86) 80.02, and the standard deviation is 11.9. 39 71 79 85 90

51 71 79 86 90

59 73 79 86 91

63 74 80 88 91

66 76 80 88 92

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 98

What percentage of the scores lies within one standard deviation of the mean? two standard deviations of the mean? three standard deviations of the mean? Based on these percentages, do you believe that the distribution of scores is mound-shaped and symmetric? Explain.

87) The table shows the number of each type of car sold in June. Car compact sedan small SUV large SUV minivan truck Total

87)

Number 7,204 9,089 20,418 13,691 15,837 15,350 81,589

a. Construct a relative frequency table for the car sales. b. Construct a Pareto diagram for the car sales using the class percentages as the heights of the bars.

88) The table shows the number of each type of book found at an online auction site during a recent 88) search. Type of Book Children's Fiction Nonfiction Educational

Number 51,033 141,114 253,074 67,252

a. Construct a relative frequency table for the book data. b. Construct a pie chart for the book data.

89) A study was designed to investigate the effects of two variables (1) a student's level of mathematical anxiety and (2) teaching method on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 400 and a standard deviation of 50 on a standardized test. Find and interpret the z-score of a student who scored 560 on the standardized test.

89)

90) Many firms use on-the-job training to teach their employees computer programming. Suppose you work in the personnel department of a firm that just finished training a group of its employees to program, and you have been requested to review the performance of one of the trainees on the final test that was given to all trainees. The mean and standard deviation of the test scores are 80 and 3, respectively, and the distribution of scores is mound-shaped and symmetric. If a firm wanted to give the best 2.5% of the trainees a big promotion, what test score would be used to identify the trainees in question?

90)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question True or False. 91) Class relative frequencies must be used, rather than class frequencies or class percentages, when constructing a Pareto diagram. A) True B) False

91)

Solve the problem. 92) Which of the graphical techniques below can be used to summarize qualitative data? A) box plot B) dot plot C) bar graph D) stem-and-leaf plot

92)

93) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits 93) during the tournament. The statistician reported that the mean serve speed of a particular player was 101 miles per hour (mph) and the standard deviation of the serve speeds was 13 mph. Using the z-score approach for detecting outliers, which of the following serve speeds would represent outliers in the distribution of the player's serve speeds? Speeds: 56 mph, 114 mph, and 127 mph

A) None of the three speeds is an outlier. B) 56 is the only outlier. C) 56, 114, and 127 are all outliers. D) 56 and 114 are both outliers, but 127 is not. Answer the question True or False. 94) In a symmetric and mound shaped distribution, we expect the values of the mean, median, and mode to differ greatly from one another. A) True B) False 95) The sample variance and standard deviation can be calculated using only the sum of the data, and the sample size, n. A) True

94)

95)

B) False

Solve the problem.

4 2 4 9 1 1 , , . 96) Compute s2 and s for the data set: , , , 5 5 5 10 2 10

A) 9.367; 3.061

B) 0.01; 0.1

96) C) 0.094; 0.306

D) 2.366; 1.538

to 97) A sociologist recently conducted a survey of citizens over 60 years of age who have net worths too high97) qualify for Medicaid but have no private health insurance. The ages of the 25 uninsured senior citizens were as follows: 68 73 66 76 86 74 61 89 65 90 69 92 76 62 81 63 68 81 70 73 60 87 75 64 82 Suppose the mean and standard deviation are 74.04 and 9.75, respectively. If we assume that the distribution of ages is mound-shaped and symmetric, what percentage of the respondents will be between 64.29 and 93.54 years old? A) approximately 81.5% B) approximately 95% C) approximately 68% D) approximately 84%

Answer the question True or False. 98) If 25% of your statistics class is sophomores, then in a pie chart representing classifications of the students in your statistics class the slice assigned to sophomores is 90°. A) True B) False

98)

Solve the problem. 99) If nothing is known about the shape of a distribution, what percentage of the observations fall within 2 standard deviations of the mean? A) approximately 5% B) at least 75% C) at most 25% D) approximately 95%

99)

100) Many firms use on-the-job training to teach their employees computer programming. Suppose you work in the personnel department of a firm that just finished training a group of its employees to program, and you have been requested to review the performance of one of the trainees on the final test that was given to all trainees. The mean of the test scores is 76. Additional information indicated that the median of the test scores was 86. What type of distribution most likely describes the shape of the test scores? A) skewed to the right B) skewed to the left C) unable to determine with the information given D) symmetric

100)

101)

The bar graph shows the political affiliation of 1000 registered U.S. voters. What percentage of the voters belonged to one of the traditional two parties (Democratic or Republican)? A) 25% B) 75% C) 40% D) 35%

Answer the question True or False. 102) Chebyshev's rule applies to large data sets, while the empirical rule applies to small data sets. A) True B) False

102)

Solve the problem. 103) A study was designed to investigate the effects of two variables (1) a student's level of mathematical anxiety and (2) teaching method on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 500 with a standard deviation of 20 on a standardized test. Assuming no information concerning the shape of the distribution is known, what percentage of the students scored between 460 and 540? A) approximately 68% B) approximately 95% C) at least 89% D) at least 75% Answer the question True or False. 104) Your teacher announces that the scores on a test have a mean of 83 points with a standard deviation of 4 points, so it is reasonable to expect that you scored at least 70 on the test. A) True B) False

103)

104)

105) Both Chebyshev's rule and the empirical rule guarantee that no data item will be more than four standard deviations from the mean. A) True B) False

105)

106) The process for finding a percentile is similar to the process for finding the median. A) True B) False

106)

Solve the problem. 107) Calculate the range of the following data set: 5, 4, 6, 1, 6, 14, 5, 6, 9 A) 15

107)

B) 1

C) 13

D) 14

108) The mean x of a data set is 36.71, and the sample standard deviation s is 3.22. Find the interval representing measurements within one standard deviation of the mean. A) (30.27, 43.15) B) (35.71, 37.71) C) (33.49, 39.93) D) (27.05, 46.37)

108)

109) A shoe retailer keeps track of all types of information about sales of newly released shoe styles. One newly 109) released style was marketed to tall people. Listed below are the shoe sizes of 12 randomly selected customers who purchased the new style. Find the mode of the shoe sizes. 9

1 2

A) 10

1 2

B) 9

1 2

C) 10

1 4

D) 11

110) Which of the following statements concerning the box plot and z-score methods for detecting outliers is false? A) The z-score method uses the mean and standard deviation as a basis for detecting outliers. B) The z-score method is less affected by an extreme observation in the data set. C) The box plot method is less affected by an extreme observation in the data set. D) The box plot method uses the quartiles as a basis for detecting outliers.

110)

111) Fill in the blank. __________ is a method of interpreting the standard deviation of data that have a mound-shaped, symmetric distribution. A) The Empirical Rule B) Chebyshev's Rule C) both A and B D) neither A nor B

111)

112) A study was designed to investigate the effects of two variables (1) a student's level of mathematical anxiety and (2) teaching method on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 450 with a standard deviation of 30 on a standardized test. Assuming a mound-shaped and symmetric distribution, in what range would approximately 99.7% of the students score? A) below 360 and above 540 B) above 540 C) below 540 D) between 360 and 540

112)

113) Many firms use on-the-job training to teach their employees computer programming. Suppose you work in the personnel department of a firm that just finished training a group of its employees to program, and you have been requested to review the performance of one of the trainees on the final test that was given to all trainees. The mean and standard deviation of the test scores are 83 and 4, respectively. Assuming nothing is known about the distribution, what percentage of test-takers scored above 91? A) at most 25% B) at least 75% C) approximately 97.5% D) approximately 2.5%

113)

Answer the question True or False. 114) An outlier may be caused by accidentally including the height of a six-year-old boy in a set of data representing the heights of 12-year-old boys. A) True B) False Solve the problem. 115) The speeds of the fastballs thrown by major league baseball pitchers were measured by radar gun. The mean speed was 87 miles per hour. The standard deviation of the speeds was 3 mph. Which of the following speeds would be classified as an outlier? A) 84 mph B) 92 mph C) 81 mph D) 97 mph 116) By law, a box of cereal labeled as containing 20 ounces must contain at least 20 ounces of cereal. The machine filling the boxes produces a distribution of fill weights with a mean equal to the setting on the machine and with a standard deviation equal to 0.02 ounce. To ensure that most of the boxes contain at least 20 ounces, the machine is set so that the mean fill per box is 20.06 ounces. Assuming nothing is known about the shape of the distribution, what can be said about the proportion of cereal boxes that contain less than 20 ounces. A) The proportion is less than 2.5%. B) The proportion is at least 89%. C) The proportion is at most 11%. D) The proportion is at most 5.5%.

114)

115)

116)

117) Given a data set, which of the following is most likely to be the percentage of data within three standard deviations of the mean? A) 70% B) 85% C) 65% D) 95%

117)

118) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The lower quartile of a particular player's serve speeds was reported to be 98 mph. Which of the following interpretations of this information is correct? A) 25% of the player's serves were hit at 98 mph. B) 98 serves traveled faster than the lower quartile. C) 75% of the player's serves were hit at speeds less than 98 mph. D) 75% of the player's serves were hit at speeds greater than 98 mph.

118)

119) Fill in the blank. __________ gives us a method of interpreting the standard deviation of any data set, regardless of the shape of the distribution. A) Chebyshev's Rule B) The Empirical Rule C) both A and B D) neither A nor B

119)

120) A radio station claims that the amount of advertising each hour has a mean of 12 minutes and a standard deviation of 1.5 minutes. You listen to the radio station for 1 hour and observe that the amount of advertising time is 5 minutes. Calculate the z-score for this amount of advertising time. A) z = -4.67 B) z = 4.67 C) z = 0.29 D) z = -10.5

120)

Answer the question True or False. 121) In general, the sample mean is a better estimator of the population mean for larger sample sizes. A) True B) False Solve the problem. 122) Many firms use on-the-job training to teach their employees computer programming. Suppose you work in the personnel department of a firm that just finished training a group of its employees to program, and you have been requested to review the performance of one of the trainees on the final test that was given to all trainees. The mean and standard deviation of the test scores are 76 and 2, respectively, and the distribution of scores is mound-shaped and symmetric. What percentage of test-takers scored better than a trainee who scored 70? A) approximately 84% B) approximately 100% C) approximately 97.5% D) approximately 95%

121)

122)

Answer the question True or False. 123) In practice, the population mean µ is used to estimate the sample mean x. A) True B) False Solve the problem. 124) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 98 miles per hour (mph) and the standard deviation of the serve speeds was 13 mph. If nothing is known about the shape of the distribution, give an interval that will contain the speeds of at least three-fourths of the player's serves. A) 72 mph to 124 mph B) 124 mph to 150 mph C) 85 mph to 111 mph D) 59 mph to 137 mph

123)

124)

Answer the question True or False. 125) Box plots are used to detect outliers in qualitative data sets, while z-scores are used to detect outliers in quantitative data sets. A) True B) False

125)

126) An outlier in a data set may have a simple explanation such as a scale was not working properly or the researcher inverted the digits of a number when recording a measurement. A) True B) False

126)

127) The slices of a pie chart must be arranged from largest to smallest in a clockwise direction. A) True B) False

127)

Solve the problem. 128) The output below displays the mean and median for the state high school dropout rates in year 1 and in128) year 5.

N MEAN MEDIAN

Year 1 51 28.02 27.57

Year 5 51 26.66 25.87

Interpret the year 5 median dropout rate of 25.87. A) The most frequently observed dropout rate of the 51 states was 25.87%. B) Most of the 51 states had a dropout rate close to 25.87%. C) Half of the 51 states had a dropout rate below 25.87%. D) Half of the 51 states had a dropout rate of 25.87%.

Answer the question True or False. 129) The outer fences of a box plot are three standard deviations from the mean. A) True B) False 130) The mean of a data set is at the 50th percentile. A) True

B) False

Solve the problem. 131) A small computing center has found that the number of jobs submitted per day to its computers has a distribution that is approximately mound-shaped and symmetric, with a mean of 86 jobs and a standard deviation of 12. Where do we expect approximately 95% of the distribution to fall? A) between 62 and 110 jobs per day B) between 50 and 122 jobs per day C) between 74 and 98 jobs per day D) between 110 and 122 jobs per day Answer the question True or False. 132) In symmetric distributions, the mean and the median will be approximately equal. A) True B) False Solve the problem. 133) In a distribution that is skewed to the right, what is the relationship of the mean, median, and mode? A) mode > mean > median B) mean > median > mode C) median > mean > mode D) mode > median > mode 25

129)

130)

131)

132)

133)

Answer the question True or False. 134) A larger standard deviation means greater variability in the data. A) True B) False

134)

Solve the problem. 135) A study was designed to investigate the effects of two variables (1) a student's level of mathematical anxiety and (2) teaching method on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 450 with a standard deviation of 40 on a standardized test. Assuming a mound-shaped and symmetric distribution, what percentage of scores exceeded 370? A) approximately 97.5% B) approximately 100% C) approximately 84% D) approximately 95%

135)

136) Summary information is given for the weights (in pounds) of 1000 randomly sampled tractor trailers. 136) MIN: MAX: AVE:

4005 10,605 7005

25%: 75%: Std. Dev.:

5605 8605 1400

Find the percentage of tractor trailers with weights between 5605 and 8605 pounds. A) 75% B) 25% C) 50% D) 100%

137) The distribution of salaries of professional basketball players is skewed to the right. Which measure of central tendency would be the best measure to determine the location of the center of the distribution? A) range B) median C) mode D) mean Answer the question True or False. 138) Either vertical or horizontal bars can be used when constructing a bar graph. A) True B) False

137)

138)

139) According to the empirical rule, z-scores of less than -3 or greater than 3 occur very infrequently for data from a mounded and symmetric distribution A) True B) False Solve the problem. 140) The scores for a statistics test are as follows:

139)

140)

82 76 81 77 65 92 95 85 86 89 79 70 50 75 85 61 85 77 18 78 Compute the mean score. A) 66.25

B) 78.50

C) 75

D) 75.30

Answer the question True or False. 141) The bars in a bar graph can be arranged by height in ascending order from left to right. A) True B) False

141)

142) All class intervals in a histogram have the same width. A) True B) False

142)

Solve the problem. 143) A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $97 and a standard deviation of $10. If the distribution can be considered mound-shaped and symmetric, what percentage of homes will have a monthly utility bill of more than $87? A) approximately 34% B) approximately 84% C) approximately 95% D) approximately 16%

143)

144) Each year advertisers spend billions of dollars purchasing commercial time on network television. In the 144) first 6 months of one year, advertisers spent $1.1 billion. Who were the largest spenders? In a recent article, the top 10 leading spenders and how much each spent (in million of dollars) were listed: Company A $73.5 Company B 63.4 Company C 57.6 Company D 57.5 Company E 28.2

Company F $27.7 Company G 27.6 Company H 21.9 Company I 23.5 Company J 20.2

Calculate the sample variance. A) 1976.544 B) 3988.681

C) 413.543

D) 2205.569

Answer the question True or False. 145) The range is an insensitive measure of data variation for large data sets because two data sets can have the same range but be vastly different with respect to data variation. A) True B) False Solve the problem. 146) Many firms use on-the-job training to teach their employees computer programming. Suppose you work in the personnel department of a firm that just finished training a group of its employees to program, and you have been requested to review the performance of one of the trainees on the final test that was given to all trainees. The mean and standard deviation of the test scores are 74 and 4, respectively, and the distribution of scores is mound-shaped and symmetric. Suppose the trainee in question received a score of 65. Compute the trainee's z-score. A) z = 0.82 B) z = -36 C) z = -2.25 D) z = -9 147) Compute s2 and s for the data set: -4, -1, -4, -4, -1, -2 A) 49.4; 7.03 B) 2.27; 1.51 C) 0.56; 0.75

D) 0.47; 0.68

148) During one recent year, U.S. consumers redeemed 6.48 billion manufacturers' coupons and saved themselves $2.41 billion. Calculate and interpret the mean savings per coupon. A) The average savings was 268.9 cents per coupon. B) Half of all coupons were worth more than 268.9 cents in savings. C) Half of all coupons were worth more than $0.37 in savings. D) The average savings was $0.37 per coupon.

145)

146)

147)

148)

149)

The manager of a store conducted a customer survey to determine why customers shopped at the store. The results are shown in the figure. What proportion of customers responded that merchandise was the reason they shopped at the store? 3 2 1 A) B) 30 C) D) 7 7 2

150) Which of the following is a measure of relative standing? A) variance B) pie chart C) mean

D) z-score

Answer the question True or False. 151) A histogram can be constructed using either class frequencies or class relative frequencies as the heights of the bars. A) True B) False 152) In skewed distributions, the mean is the best measure of the center of the distribution since it is least affected by extreme observations. A) True B) False

150)

151)

152)

Solve the problem. 153) A standardized test has a mean score of 500 points with a standard deviation of 100 points. Five students' 153) scores are shown below. Adam: 575

Beth: 690

Carlos: 750

Doug: 280

Ella: 440

Which of the students have scores within two standard deviations of the mean? A) Adam, Beth, Carlos, Ella B) Carlos, Doug C) Adam, Beth, Ella D) Adam, Beth

154) On a given day, the price of a gallon of milk had a mean price of $2.09 with a standard deviation of $0.07. A particular food store sold milk for $2.02/gallon. Interpret the z-score for this gas station. A) The milk price of this food store falls 1 standard deviation below the milk gas price of all food stores. B) The milk price of this food store falls 7 standard deviations below the mean milk price of all food stores. C) The milk price of this food store falls 7 standard deviations above the mean milk price of all food stores. D) The milk price of this food store falls 1 standard deviation above the mean milk price of all food stores. Answer the question True or False. 155) Scatterplots are useful for both qualitative and quantitative data. A) True B) False

155)

Solve the problem. 156) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 97 miles per hour (mph) and the standard deviation of the serve speeds was 12 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was mound-shaped and symmetric. What proportion of the player's serves was between 109 mph and 133 mph? A) 133 B) 0.317 C) 0.997 D) 0.1585 157) What number is missing from the table? Year in College Freshman Sophomore Junior Senior

Frequency 600 560 400

A) 220

154)

156)

157)

Relative Frequency .30 .28 .22 .20

B) 480

C) 520

D) 440

158) A sociologist recently conducted a survey of senior citizens who have net worths too high to qualify for158) Medicaid but have no private health insurance. The ages of the 25 uninsured senior citizens were as follows: 71 77 72 66 63

76 64 95 71 90

69 92 79 84 78

79 68 65 73 67

89 93 84 76 85

Find the median of the observations. A) 73 B) 77

C) 76.5

Answer the question True or False. 159) For large data sets, a stem-and-leaf display is a better choice than a histogram. A) True B) False 29

D) 76

159)

Solve the problem. 160) Fill in the blank. One advantage of the __________ is that the actual data values are retained in the graphical summarization of the data. A) histogram B) pie chart C) stem-and-leaf plot Answer the question True or False. 161) An outlier is defined as any observation that falls within the outer fences of a box plot. A) True B) False

160)

161)

162) The sample variance is always greater than the sample standard deviation. A) True B) False

162)

163) A Pareto diagram is a pie chart where the slices are arranged from largest to smallest in a counterclockwise direction. A) True B) False

163)

Solve the problem. 164) The distribution of scores on a test is mound-shaped and symmetric with a mean score of 78. If 68% of the scores fall between 72 and 84, which of the following is most likely to be the standard deviation of the distribution? A) 6 B) 3 C) 12 D) 2

164)

Answer the question True or False. 165) Percentile rankings are of practical value only with large data sets. A) True B) False

165)

Solve the problem. 166) Find the z-score for the value 52, when the mean is 58 and the standard deviation is 9. A) z = 0.74 B) z = -0.78 C) z = -0.74 D) z = -0.67

166)

167) Which of the following statements could be an explanation for the presence of an outlier in the data? A) The measurement belongs to a population different from that from which the rest of the sample was drawn. B) The measurement may be correct and from the same population as the rest but represents a rare event. Generally, we accept this explanation only after carefully ruling out all others. C) The measurement is incorrect. It may have been observed, recorded, or entered into the computer incorrectly. D) All of the above are explanations for outliers.

167)

168) The amount of television viewed by today's youth is of primary concern to Parents Against Watching Television (PAWT). 300 parents of elementary school-aged children were asked to estimate the number of hours per week that their child watches television. The mean and the standard deviation for their responses were 12 and 2, respectively. PAWT constructed a stem-and-leaf display for the data that showed that the distribution of times was a symmetric, mound-shaped distribution. Give an interval where you believe approximately 95% of the television viewing times fell in the distribution. A) between 8 and 16 hours per week B) less than 16 C) less than 10 and more than 14 hours per week D) between 6 and 18 hours per week

168)

169) The box plot shown below displays the amount of soda that was poured by a filling machine into 12-ounce soda cans at a local bottling company.

169)

Based on the box plot, what shape do you believe the distribution of the data to have? A) skewed to the center B) skewed to the left C) approximately symmetric D) skewed to the right

170) When Scholastic Achievement Test scores (SATs) are sent to test-takers, the percentiles associated with scores are also given. Suppose a test-taker scored at the 98th percentile on the verbal part of the test and at the 29th percentile on the quantitative part. Interpret these results. A) This student performed better than 98% of the other test-takers on the verbal part and better than 29% on the quantitative part. B) This student performed better than 98% of the other test-takers on the verbal part and better than 71% on the quantitative part. C) This student performed better than 2% of the other test-takers on the verbal part and better than 29% on the quantitative part. D) This student performed better than 2% of the other test-takers on the verbal part and better than 71% on the quantitative part. Answer the question True or False. 171) The bars in a histogram should be arranged by height in descending order from left to right. A) True B) False

170)

171)

Solve the problem. 172)

172)

The pie chart shows the classifications of students in a statistics class. What percentage of the class consists of freshman, sophomores, and juniors? A) 44% B) 14% C) 86%

D) 54%

Answer the question True or False. 173) A frequency table displays the proportion of observations falling into each class. A) True B) False

173)

174) If a z-score is 0 or near 0, the measurement is located at or near the mean. A) True B) False

174)

175) The z-score uses the quartiles to identify outliers in a data set. A) True B) False

175)

Solve the problem. 176) A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $111 and a standard deviation of $15. If nothing is known about the shape of the distribution, what percentage of homes will have a monthly utility bill of less than $81? A) at most 25% B) at least 88.9% C) at least 75% D) at most 11.1%

176)

of 177) The top speeds for a sample of five new automobiles are listed below. Calculate the standard deviation177) the speeds. Round to four decimal places. 170, 160, 130, 165, 125 A) 238.0914

B) 168.8935

C) 20.9165

178) Which of the following is not a measure of central tendency? A) range B) mode C) mean

D) 133.23

D) median

178)

to 179) A sociologist recently conducted a survey of citizens over 60 years of age who have net worths too high179) qualify for Medicaid but have no private health insurance. The ages of the 25 uninsured senior citizens were as follows: 68 73 66 76 86 74 61 89 65 90 69 92 76 62 81 63 68 81 70 73 60 87 75 64 82 Find the upper quartile of the data. A) 92 B) 81.5

C) 65.5

D) 73

180)

For the distribution drawn here, identify the mean, median, and mode. A) A = mode, B = median, C = mean B) A = mode, B = mean, C = median C) A = median, B = mode, C = mean D) A = mean, B = mode, C = median

Answer the question True or False. 181) If a sample has mean 0 and standard deviation 1, then for every measurement x in the sample the z-score of x is x itself. A) True B) False Solve the problem. 182) The test scores of 30 students are listed below. Which number could be the 30th percentile? 31 41 45 48 52 55 56 56 63 65 67 67 69 70 70 74 75 78 79 79 80 81 83 85 85 87 90 92 95 99 A) 64 B) 90

C) 67

182)

D) 56

Answer the question True or False. 183) The mean and the median are useful measures of central tendency for both qualitative and quantitative data. A) True B) False

181)

183)

Solve the problem. 184) The following is a list of 25 measurements: 12 13 12

18 14 16

14 11 17

17 16

19 18

16 15

14 13

18 17

15 15

184) 17 14

11 19

How many of the measurements fall within one standard deviation of the mean? A) 18 B) 25 C) 13 D) 16

Answer the question True or False. 185) The scatterplot below shows a negative relationship between two variables.

A) True

185)

B) False

Solve the problem. 186) A shoe company reports the mode for the shoe sizes of men's shoes is 12. Interpret this result. A) Most men have shoe sizes between 11 and 13. B) Half of all men's shoe sizes are size 12 C) Half of the shoes sold to men are larger than a size 12 D) The most frequently occurring shoe size for men is size 12 187) What number is missing from the table? Grades on Test A B C D F

A) .07

Frequency 6 7 9 2 1

186)

187)

Relative Frequency .24 .36 .08 .04

B) .70

C) .72

D) .28

Answer the question True or False. 188) Chebyshev's rule applies to qualitative data sets, while the empirical rule applies to quantitative data sets. A) True B) False

188)

Solve the problem. 189) A study was designed to investigate the effects of two variables (1) a student's level of mathematical anxiety and (2) teaching method on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 300 with a standard deviation of 30 on a standardized test. Assuming a non-mound-shaped distribution, what percentage of the students scored over 390? A) approximately 2.5% B) at most 11% C) at least 89% D) at most 5.5%

189)

Answer the question True or False. 190) For any quantitative data set, A) True

(x - x ) = 0.

190) B) False

Answer Key Testname: CHAPTER 2 1) B 2) C 3) A 4) C 5) A 6) A 7) B 8) C 9) D 10) A 11) B 12) D 13) A 14) B 15) C 16) C 17) B 18) D 19) D 20) D 21) C 22) D 23) B 24) D 25) D 26) C 27) B 28) A 29) A 30) B 31) D 32) B 33) A 34) D 35) a. mean: x 73.65; median: Med=81 b. We expect the data to be skewed to the left because the mean is less than the median. 36) A scatterplot with the measurements on the vertical axis and time (or the order in which the measurements were made) on the horizontal axis. 37) A time series plot describes behavior over time and reveals movement (trend) and changes (variation) in the variable being monitored. 38) The mean is divided by n: x n

196 = 24.5. 8

Answer Key Testname: CHAPTER 2 39) a. The exact scores would be needed to construct a stem-and-leaf display but the exact scores are not available in the table given. b.

40) standard deviation 41) 50th percentile

42) Since the distribution is skewed to the left, we know that the median time will exceed the mean time. 43) a. minX=30 b. maxX=97 c. 97 - 30 = 67 44) a. mean: x = 5.5; sample standard deviation: Sx 3.0 b. (5.5 - 2 × 3.0, 5.5 + 2 × 3.0) = (-.5, 11.5) 45) range 46) The mean is the sum of the numbers divided by 18:

1 + 2 + 3 + 3 + 4 + 9 + 9 + 11 + 11 + 11 + 14 + 14 + 19 + 22 + 23 + 24 + 25 + 29 18 =

234 = 13 medals. 18

The median is the mean of the two middle numbers:

11 + 11 = 11 medals. 2

The mode is the most frequent number of medals: 11 medals. 47) The value of x lies 2.5 standard deviations below the mean.

Answer Key Testname: CHAPTER 2 48)

There appears to be a trend in the data. As the number of absences increases, the final grade decreases.

49) 75% of the TV viewing times are less than 14 hours per week. 25% of the times exceed 14 hours per week. 50) history z-score = 1.78; physics z-score = 3.51; The student performed better on the physics test. 51)

Losses due to employee theft have decreased from year 1 to year 2. 52) The reader may think that the area of the bar represents the quantity rather than the height of the bar, giving a disproportionate emphasis on the taller bars. 53) Stem Leaf 15 2 3 4 50 6 66 9 76 7 9 8 2357889 90 2 4 8 8

Answer Key Testname: CHAPTER 2 54)

55) In a Pareto diagram, the bars are arranged by height in a descending order from left to right. 56) When comparing means from two different distributions, the difference between them may be insignificant if the variability in one or both of the distributions is large. 57) mean: 65; standard deviation: 5 (x - x)2 2 = s 58) n -1 x=

x n

58 + 56 + 69 + 70 + 59 = 62.4 5

(58 - 62.4)2 + (56 - 62.4)2 + (69 - 62.4)2 + (70 - 62.4)2 + (59 - 62.4)2 s2 = 5-1 = 43.30

59) Take the square root of the sample variance to find the sample standard deviation. 60) a. n = 21 b.

x = 1679

c. mean: x 79.95; median: Med=82; mode: not possible 61) We use the Empirical Rule to determine the percentage of serves with speeds faster than 75 mph. We do this by first finding the percentage of serves with speeds between 75 and 103 mph. The Empirical Rule states that approximately 34.0% (68%/2) fall between 75 and 103 mph. Because the distribution is symmetric about the mean speed of 103 mph, we know 50% of the serve speeds were faster than 103 mph. We add these findings together to determine that 34.0% + 50% = 84.0% of the serves were hit faster than 75 mph. 62) The z-score of 39 is -3.46. Since this z-score is less than -3, the score of 39 is an outlier. All other scores have z-scores between -3 and 3, so there are no other outliers. 63) The modal class is the class with the greatest frequency: 81-100 points.

Answer Key Testname: CHAPTER 2 64) The range is 29 - 1 = 28 medals.

The variance is s2 =

x2 -

n- 1

The standard deviation is s =

s2 =

4372 =

(234)2 18

17 1330 17

1330 17

78.24

8.85

65) a. Total Medals 1-5 6-10 11-15 16-20 21-25 26-30

Frequency 5 2 5 1 4 1

66) The z-score for the value $168.00 is: z=

x - x 168 - 102 = = 5.5 s 12

An observation that falls 5.5 standard deviations above the mean is very unlikely. We would not expect to see a monthly utility bill of $168.00 for this home. 67) a. The interquartile range is 57 - 45 = 12. b. The inner fences are 45 - 1.5(12) = 27 and 57 + 1.5(12) = 75. c. The outer fences are 45 - 3(12) = 9 and 57 + 3(12) = 93. d. The maximum of 81 is a potential outlier since it lies outside the inner fences. The minimum is within the inner fence and is not considered to be an outlier. 68) standard deviation

Answer Key Testname: CHAPTER 2 69) The horizontal axis extends from 10 to 20, with each tick mark representing one unit.

70) a. The lower quartile is 73, the upper quartile is 89, and the median is 81. b. The interquartile range is 89 - 73 = 16. The score of 39 is a potential outlier since it is less than 73 - 1.5(16) = 49. c. No scores fall outside the outer fences, 25 and 137. Only the score of 39 lies between the inner and outer fences. 71) µ is the mean price of the regular unleaded gasoline prices of all retail gas stations in the United States. is the standard deviation of the regular unleaded gasoline prices of all retail gas stations in the United States. x is the mean price of the regular unleaded gasoline prices collected from the 200 stations sampled. s is the standard deviation of the regular unleaded gasoline prices collected from the 200 stations sampled. 72) The value 96 falls one standard deviation above the mean in the distribution. Using the Empirical Rule, 68% of the days will have between 80 and 96 jobs submitted. Of the remaining 32% of the days, half, or 32%/2 = 16%, of the days will have more than 96 jobs submitted. 73) measurements within three standard deviations of the mean 74) a. Medal Frequency Gold 9 Silver 9 Bronze 7 b. Medal Gold Silver Bronze

Relative Frequency .36 .36 .28

Answer Key Testname: CHAPTER 2

75) The mean of the data is x =

x n

72.6 + 61.7 + 57.5 + 54.9 + 30.5 + 27.8 + 26.3 + 21.2 + 21.3 + 20.1 10 =

393.9 10

= 39.39

$39.39 million

The median is the average of the middle two observations. M=

30.5 + 27.8 = 29.15 $29.15 million 2

76) 12% 77) Using a scale break on the vertical axis may make the shorter bars look disproportionately shorter than the taller bars. 78) The value of 36 ounces falls three standard deviations below the mean. The Empirical Rule states that approximately all of the boxes will contain cereal amounts between 36.00 ounces and 36.12 ounces. Therefore, approximately 100% of the boxes contain at least 36 ounces. 79) free account, institutional account, account paid for personally 80) Stretching the vertical axis may overemphasize the differences in the heights of the bars making the taller bars look much taller than the shorter bars. 81) a. lower quartile: Q1=75; upper quartile: Q3=90 b. interquartile range: 90 - 75 = 15 c. Yes; the smallest measurement, 30, is three times the interquartile range less than the lower quartile, so it is a suspected outlier. 82) Color Frequency Green 3 Blue 7 Brown 5 Orange 2 Red 3 83) highest: z = 1.51; lowest: z = -3.45 84) In both year 1 and year 5, the mean dropout rates exceed the median dropout rates. This indicates that both the year 1 and year 5 high school dropout rates have distributions that are skewed to the right. 85) The z-score for the value 11.23 is -1.3 Since the z-score would not indicate that 11.23 minutes represents an outlier, there is no evidence that the station's claim is incorrect. 86) 74% of the scores lie within one standard deviation of the mean, 96% within two standard deviations, and 98% within three standard deviations. These percentages are close to those given in the Empirical Rule, so the distribution is roughly mound-shaped and symmetric, though obviously skewed slightly to the left.

Answer Key Testname: CHAPTER 2 87) a. Car compact sedan small SUV large SUV minivan truck

Relative Frequency 0.09 0.11 0.25 0.17 0.19 0.19

Answer Key Testname: CHAPTER 2 88) a. Type of Book

Relative Frequency .10 .28 .49 .13

Children's Fiction Nonfiction Educational b.

89) The z-score is z =

x-µ

For a score of 56, z =

560 - 400 = 3.20. 50

This student's score falls 3.20 standard deviations above the mean score of 400. 90) The Empirical Rule states that 95% of the data will fall between 74 and 86. Because the distribution is symmetric, half of the remaining 5%, or 2.5%, will have test scores above 86. Thus, 86 is the cutoff point that will identify the trainees who will receive the promotion. 91) B 92) C 93) B 94) B 95) B 96) C 97) A 98) A 99) B 100) B 101) B 102) B 103) D 104) A 105) B 106) A 107) C 108) C 109) D 110) B 111) A 44

Answer Key Testname: CHAPTER 2 112) D 113) A 114) A 115) D 116) C 117) D 118) D 119) A 120) A 121) A 122) B 123) B 124) A 125) B 126) A 127) B 128) C 129) B 130) B 131) A 132) A 133) B 134) A 135) A 136) C 137) B 138) A 139) A 140) D 141) A 142) A 143) B 144) C 145) A 146) C 147) B 148) D 149) A 150) D 151) A 152) B 153) C 154) A 155) B 156) D 157) D 158) D 159) B 160) C 161) B 45

Answer Key Testname: CHAPTER 2 162) B 163) B 164) A 165) A 166) D 167) D 168) A 169) B 170) A 171) B 172) C 173) B 174) A 175) B 176) A 177) C 178) A 179) B 180) A 181) A 182) A 183) B 184) D 185) A 186) D 187) D 188) B 189) B 190) A

Chapter 3 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) An insurance company looks at many factors when determining how much insurance will cost for a home. 1) Two of the factors are listed below: A: {The home's roof is less than 10 years old} B: {The home has a security system} In the words of the problem, define the event Bc. A) The home is less than 10 years old B) The home does not have a security system C) The home has a security system D) The home is not less than 10 years old

2) An experiment consists of rolling two dice and summing the resulting values. Which of the following is not a sample point for this experiment? A) 6 B) 2 C) 7 D) 1

3) A basketball player has an 80% chance of making the first free-throw he shoots. If he makes the first free-throw shot, then he has a 90% chance of making the second free-throw he shoots. If he misses the first free-throw shot, then he only has a 70% chance of making the second free-throw he shoots. Suppose this player has been awarded two free-throw shots. Find the probability that he makes at least one of the two shots. A) 0.86 B) 0.80 C) 0.72 D) 0.94

4) A bag of candy was opened and the number of pieces was counted. The results are shown in the table 4) below: Color Number Red 25 Brown 20 Green 20 Blue 15 Yellow 10 Orange 10 Find the probability that a randomly chosen piece of candy is not blue or red. A) 0.40 B) 0.15 C) 0.85

D) 0.60

5) A medium-sized company characterized their employees based on the sex of the employee and their 5) length of service to the company. The results are summarized in the table below.

What proportion of the employees are male or have been employed for less than 11 years? A) 42/65 B) 45/130 C) 120/130 D) 165/130

6) A basketball player has an 80% chance of making the first free-throw he shoots. If he makes the first free-throw shot, then he has a 90% chance of making the second free-throw he shoots. If he misses the first free-throw shot, then he only has a 70% chance of making the second free-throw he shoots. Suppose this player has been awarded two free-throw shots. Are the events, A - the player makes the first shot, and B - the player makes the second shot, independent events? A) Yes B) No

7) A sample of 350 students was selected and each was asked the make of their automobile (foreign or 7) domestic) and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.

Given that you know the selected student is in the senior class, find the probability they drive a domestic automobile. A) 15/205 B) 25/35 C) 10/35 D) 15/350

8) In a particular town, 20% of the homes have monitored security systems. If an alarm is triggered, the security system company will contact the local police to alert them of the alarm. Of all the alarm calls that the local police receive, they only have the manpower to answer 30% of the calls. Suppose we randomly sample one home that was broken into over the last month from this town. What is the probability that this home has a monitored security system and that the police answered the alarm call? A) 0.0600 B) 0.9400 C) 0.2000 D) 0.3000

9) Which quantity is represented on the screen below?

A) The number of sample points when a die is rolled and a coin is flipped B) The number of ways two coins can be chosen from six coins C) The number of sample points when a coin is flipped six times D) The number of ways two dice can be rolled 10) A bag of candy was opened and the number of pieces was counted. The results are shown in the table 10) below: Color Number Red 25 Brown 20 Green 20 Blue 15 Yellow 10 Orange 10 Find the probability that a randomly selected piece is either yellow or orange in color. A) 20 B) 0.10 C) 0.20 D) 10

11) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.

11)

A: {The number is even} B: {The number is less than 7} Which expression represents the event that the number is even or less than 7 or both? A) Bc B) Ac C) A B D) A B

12) Which of the following assignments of probabilities to the sample points A, B, and C is valid if A, B, and C are the only sample points in the experiment? 1 2 1 1 1 A) P(A) = 0, P(B) = , P(C) = B) P(A) = , P(B) = , P(C) = 3 3 4 4 4 C) P(A) = -

1 1 3 , P(B) = , P(C) = 4 2 4

D) P(A) =

1 1 1 , P(B) = , P(C) = 9 3 10

13) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows. A: {The number is even} B: {The number is less than 7} Which expression represents the event that the number is both even and less than 7? A) Ac B) A B C) Bc D) A B

12)

13)

14) Suppose two dice, one blue and one red, are rolled and the outcomes of each are recorded. We define the 14) following two events: A: sum of the roll is 7 B. the result of the blue die is a number greater than 4 Are the two events, A and B, independent events? A) No

B) Yes

15) A medium-sized company characterized their employees based on the sex of the employee and their 15) length of service to the company. The results are summarized in the table below.

What proportion of the employees are female or have been employed for more than 10 years? A) 85/130 B) 25/130 C) 25/65 D) 110/130

16) Which number could be the probability of an event that occurs about as often as it does not occur? A) 0 B) 1 C) .51 D) -.51

16)

17) Suppose the probability of an athlete taking a certain illegal steroid is 10%. A test has been developed to detect this type of steroid and will yield either a positive or negative result. Given that the athlete has taken this steroid, the probability of a positive test result is 0.995. Given that the athlete has not taken this steroid, the probability of a negative test result is 0.992. Given that a positive test result has been observed for an athlete, what is the probability that they have taken this steroid? A) 0.9928 B) 0.0995 C) 0.9325 D) 0.9552

17)

18) Which number could be the probability of an event that rarely occurs? A) .51 B) .99 C) -.01

18)

19) Which expression is equal to A)

N! N!(N - n)!

D) .01

N ? n

19)

N! (N - n)!

N! n!

N! n!(N - n)!

20) A sample of 350 students was selected and each was asked the make of their automobile (foreign or 20) domestic) and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.

What is the probability of randomly selecting a student who is in the freshman class or drives a foreign automobile? A) 15/350 B) 15/205 C) 230/350 D) 215/350

21) A bag of candy was opened and the number of pieces was counted. The results are shown in the table 21) below: Color Number Red 25 Brown 20 Green 20 Blue 15 Yellow 10 Orange 10 List the sample space for this problem. A) {25, 20, 20, 15, 10, 10} B) {Red, Brown, Green, Blue, Yellow, Orange} C) {0.25, 0.20, 0.20, 0.15, 0.10, 0.10} D) {Red}

22) A sample of 350 students was selected and each was asked the make of their automobile (foreign or 22) domestic) and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.

Which of the following events listed would be considered mutually exclusive events? A) The student is a freshman and the student drives a foreign automobile B) The student is a senior and the student drives a domestic automobile. C) The student is a junior and the student drives a domestic automobile D) The student is a junior and the student is a freshman

23) A medium-sized company characterized their employees based on the sex of the employee and their 23) length of service to the company. The results are summarized in the table below.

Suppose an employee has been randomly selected from this company. Given that the employee is male, find the probability that they have worked for the company for more than 10 years? A) 20/65 B) 20/130 C) 20/30 D) 75/130

24) A sample of 350 students was selected and each was asked the make of their automobile (foreign or 24) domestic) and their year in college (freshman, sophomore, junior, or senior). The results are shown in the table below.

Find the probability that a randomly selected student is both a sophomore and drives a foreign automobile. A) 65/205 B) 45/350 C) 65/350 D) 65/110

25) Which number could be the probability of an event that is almost certain to occur? A) .51 B) 1.01 C) .01 D) .99

25)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 26) A certain game has a deck of numbered cards of various colors. The probability of drawing a green card from a well-shuffled deck is .25 and the probability of drawing a card numbered 3 is .1. Assuming that "green" and "3" are independent events, find the probability of drawing a green card numbered 3.

26)

27) A pair of fair dice is tossed. Events A and B are defined as follows.

27)

A: {The two numbers rolled are different} B: {At least one of the numbers is greater than 2} a. b. c. d. e. f.

Identify the sample points in the event Ac. Identify the sample points in the event Bc.

Identify the sample points in the event Ac Bc. Identify the sample points in the event Ac Bc. Find P(Ac Bc). Find P(Ac Bc).

28) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the number rolled is odd. Outcome Probability

1 .1

2 .1

3 .1

4 .2

5 .2

28)

6 .3

Find P(A | B).

29) A company evaluates its potential new employees using three criteria.

29)

A: The applicant has a minimum college GPA of 3.0. B: The applicant has relevant work experience. D: The applicant has a sufficient score on an aptitude test. Describe an applicant represented by A Bc D.

30) A college has 85 male and 75 female fulltime faculty members. Suppose one fulltime faculty member is selected at random and the faculty member's gender is observed.

30)

a. List the sample points for this experiment. b. Assign probabilities to the sample points.

31) Suppose that for a certain experiment P(A) =

1 1 and P(B) = , and events A and B are 3 4

31)

mutually exclusive. Find P(A B).

32) A hospital reports that two patients have been admitted who have contracted Crohn's disease. Suppose our experiment consists of observing whether each patient survives or dies as a result of the disease. The simple events and probabilities of their occurrences are shown in the table (where S in the first position means that patient 1 survives, D in the first position means that patient 1 dies, etc.). Simple Events SS SD DS DD

32)

Probabilities 0.52 0.19 0.14 0.15

Find the probability that neither patient survives.

33) At a small private college with 800 students, 240 students receive some form of government-sponsored financial aid. Find the probability that a randomly selected student receives some form of government-sponsored financial aid.

33)

34) In the game of Parcheesi each player rolls a pair of dice on each turn. In order to begin the game, you must roll a five on at least one die, or a total of five on both dice. Find the probability that the player does not get to begin the game on either the first or the second rolls.

34)

35) If 80% of a website's visitors are teenagers and 60% of those teenaged visitors are male, find the percentage of the website's visitors that are teenaged males.

35)

36) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red chips. Events A and B are defined as follows.

36)

A: {Both chips are red} B: {At least one of the chips is blue} a. b.

Identify the sample points in the event A B. Find P(A B).

37) Suppose that 62% of the employees at a company are male and that 35% of the employees just received merit raises. If 20% of the employees are male and received a merit raise, what is the probability that a randomly chosen employee is male given that the employee received a merit raise?

37)

38) Suppose that 80% of the employees of a company received cash or company stock as a bonus at the end of the year. If 60% of the employees received a cash bonus and 30% received stock, what is the probability that a randomly chosen employee received both cash and stock as a bonus?

38)

39) Based on past experience, Josh believes that the probability of catching a red snapper is .21 and the probability of catching a grouper is .19. Is enough information available to find the probability of catching a red snapper or a grouper? Explain. If possible, find the probability of catching a red snapper or a grouper.

39)

40) Three fair coins are tossed and either heads (H) or tails (T) is observed for each coin.

40)

a. b. c. d. e.

List the sample points for this experiment. Assign probabilities to the sample points. Find the probability of the event A = {Three heads are observed}. Find the probability of the event B = {Exactly two heads are observed}. Find the probability of the event C = {At least two heads are observed}.

41) At a certain university, 70% of the students own cars. However, only 45% of the residence hall students own cars. Are the events owning a car and living in a residence hall independent? Explain.

41)

42) The data show the total number of medals (gold, silver, and bronze) won by each country 42) winning at least one gold medal in the Winter Olympics. Suppose that one of the countries represented is chosen at random and the total numbers of medals won by that country is noted. 1

a. b. c.

List the sample points for this experiment. Find the probability of each sample point. What is the probability that the country won at least 20 total medals?

43) The table shows the number of each Ford car sold in the United States in June. Suppose the sales 43) record for one of these cars is randomly selected and the type of car is identified. Type of Car Sedan Convertible Wagon SUV Van Hatchback Total

Number 7,204 9,089 20,418 13,691 15,837 15,350 81,589

Events A and B are defined as follows. A: {Convertible, SUV, Van} B: {Fewer than 10,000 of the type of car were sold in June} a. b. c. d.

Identify the sample points in the event A B. Identify the sample points in the event A B. Find P(A B). Find P(A B).

44) A local country club has a membership of 600 and operates facilities that include an 18-hole championship golf course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to know how many members regularly use each facility. A survey of the membership indicates that 61% regularly use the golf course, 50% regularly use the tennis courts, and 21% use both of these facilities regularly. Find the probability that a randomly selected member uses the golf or tennis facilities regularly.

44)

45) An exit poll during a recent election revealed that 55% of those voting were women and that 65% of the women voting favored Democratic candidates. What is the probability that a randomly chosen participant of the exit poll would be a woman who favored Democratic candidates?

45)

46) A human gene carries a certain disease from a mother to her child with a probability rate of 0.20. That is, there is a 20% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has four children. Assume that the infections, or lack thereof, are independent of one another. Find the probability that none of the children get the disease from their mother.

46)

47) A pair of fair dice is tossed. Events A and B are defined as follows.

47)

A: {The two numbers rolled are different} B: {At least one of the numbers is greater than 2} Are the events A and B mutually exclusive? Explain.

48) In a sample of 750 of its online customers, a department store found that 420 were men. Use this information to estimate the probability that a randomly selected online customer is not a man.

48)

49) The table shows the number of each Ford car sold in the United States in June 2006. Suppose the 49) sales record for one of these cars is randomly selected and the type of car is identified. Type of Car Sedan Convertible Wagon SUV Van Hatchback Total

Number 7,204 9,089 20,418 13,691 15,837 15,350 81,589

Events A and B are defined as follows. A: {Convertible, SUV, Van} B: {Fewer than 10,000 of the type of car were sold in June 2006} Find P(A | B) and P(B | A).

50) Suppose that an experiment has five sample points, E1 , E2 , E3 , E4 , E5 , and that P(E1 ) = .4,

50)

P(E2 ) = .1, P(E3 ) = .1, P(E4 ) = .2, and P(E5 ) = .2. If the events A and B are defined as A = {E1 , E2 , E5 } and B = {E2 , E3 , E5 } find P(A B).

51) Suppose that an experiment has five sample points, E1 , E2 , E3 , E4 , E5 , and that P(E1 ) = .2,

51)

P(E2 ) = .3, P(E3 ) = .1, P(E4 ) = .1, and P(E5 ) = .3. If event A is defined as A = {E1 , E2 , E3 }, find P(Ac).

52) A clothing vendor estimates that 78 out of every 100 of its online customers do not live within 50 miles of one of its physical stores. It further estimates that 39 out of every 100 of its online customers is a man who does not live within 50 miles of one of its physical stores. Using this estimate, what is the probability that a randomly selected online customer is a man given that the customer does not live within 50 miles of a physical store?

52)

53) A fast-food restaurant chain with 700 outlets in the United States has recorded the geographic 53) location of its restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to test market a new chicken sandwich. Region NE SE SW NW 1% 6% 3% 0% <10,000 Population of City 10,000 - 100,000 15% 10% 12% 5% 20% 4% 4% 20% >100,000 What is the probability that the restaurant is located in a city with a population over 100,000, given that it is located in the southwestern United States?

54) The following data represent the scores of 50 students on a statistics exam. Suppose that one of54) the 50 students is chosen at random and that student's score is noted. 39 71 79 85 90 a. b. c.

51 71 79 86 90

59 73 79 86 91

63 74 80 88 91

66 76 80 88 92

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 98

What is the probability that the student's score is 88? What is the probability that the student's score is less than 60? What is the probability that the student's score is between 70 and 79, inclusive?

55) Suppose that an experiment has eight equally likely outcomes. What probability is assigned to each of the sample points?

55)

56) In how many ways can a manager choose 3 of his 8 employees to work overtime helping with inventory?

56)

57) A consumer advocacy group rates the quality of a cellular service provider using three criteria.57) A: Service is available at least 99% of the time. B: Reception is clear at least 95% of the time. C: Fewer than 5% of its customers have complaints about the quality of service. a. b.

Describe the event represented by A B C. Describe the event represented by A B C.

58) Compute

10 . 6

58)

59) Suppose that for a certain experiment P(A) = .8 and P(B) = .9. Use the Additive Rule to explain why the events A and B can not be mutually exclusive.

59)

60) An economy pack of highlighters contains 12 yellow, 6 blue, 4 green, and 3 orange highlighters. An experiment consists of randomly selecting one of the highlighters. Find the probability that a blue highlighter is chosen.

60)

61) A pair of fair dice is tossed. Events A and B are defined as follows.

61)

A: {The sum of the numbers on the dice is 6} B: {At least one of the numbers 3} a. b. c. d.

Identify the sample points in the event A B. Identify the sample points in the event A B. Find P(A B). Find P(A B).

62) The table shows the political affiliations and types of job for workers in a particular state. Suppose 62) a worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 13% 7% 19% Type of job Blue Collar 17% 18% 26% Given that a worker is a blue collar worker, what is the probability that the worker is a Democrat?

63) Suppose that for a certain experiment P(A) = .15 and P(B A) = .8. Find P(A B). 64) Compute

5 . 1

63) 64)

65) In a box of 50 markers, 30 markers are either red or black, 20 are missing their caps, and 12 markers are either red or black and are missing their caps. Are the events "red or black" and "missing cap" dependent or independent? Explain.

65)

66) The accompanying Venn diagram describes the sample space of a particular experiment 1 and events A and B. Suppose P(1) = P(2) = P(3) = P(4) = and P(5) = P(6) = P(7) = P(8) = 16

66)

P(9) = P(10) =

1 . Find P(A) and P(B). 8

67) A consumer advocacy group rates the quality of a cellular service provider using three criteria.67) A: Service is available at least 99% of the time. B: Reception is clear at least 95% of the time. D: Fewer than 5% of its customers have complaints about the quality of service. Describe a cellular service provider represented by Ac Bc Dc.

68) A pair of fair dice is tossed. Events A and B are defined as follows.

68)

A: {The sum of the dice is 7} B: {At least one of the numbers is 3} Find P(A | B) and P(B | A).

69) Suppose there is a 31% chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in three independent risky stocks. What is the probability that all three stocks end up in total losses?

69)

70) Two chips are drawn at random and without replacement from a bag containing three blue chips 70) and one red chip. a. b. c. d. e.

List the sample points for this experiment. Assign probabilities to the sample points. Find the probability of the event A = {Two blue chips are drawn}. Find the probability of the event B = {A blue chip and a red chip are drawn}. Find the probability of the event C = {Two red chips are drawn}.

71) The data show the total number of medals (gold, silver, and bronze) won by each country winning at least one gold medal in the Winter Olympics. 1

71)

Suppose that one of the countries represented is selected at random and the total number of medals won by that country is noted. What is the probability that the country won at least 25 medals given that the country did not win fewer than 10 medals?

72) An exit poll during a recent election revealed that 52% of those voting were women and 48% 72) were men. The results also showed that 70% of the women voting favored Democratic candidates while only 40% of the men favored Democratic candidates. These poll results may be summarized as follows: P(woman) = .52 P(favored Democrats | woman) =.70 a. b. c. d. e.

P(man) = .48 P(favored Democrats | man) = .40

Find P(woman and favored Democrats). Find P(man and favored Democrats). Find P(favored Democrats). Find P(woman | favored Democrats). Find P(man | favored Democrats).

73) The data below show the types of medals won by athletes representing the United States in the73) Winter Olympics. Suppose that one medal is chosen at random and the type of medal noted. gold bronze gold gold a. b. c.

gold gold silver gold

silver silver silver bronze

gold silver bronze bronze

bronze bronze bronze

silver silver gold

silver gold silver

List the sample points for this experiment. Find the probability of each sample point. What is the probability that the medal was not bronze?

74) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.

74)

A: {The number is even} B: {The number is less than 7} Find P(A | B) and P(B | A).

75) The table shows the number of each Ford car sold in the United States in June. Suppose the sales 75) record for one of these cars is randomly selected and the type of car is identified. Type of Car Sedan Convertible Wagon SUV Van Hatchback Total

Number 7,204 9,089 20,418 13,691 15,837 15,350 81,589

Events A and B are defined as follows. A: {Convertible, SUV, Van} B: {Fewer than 10,000 of the type of car were sold in June} Is P(A B) equal to the sum of P(A) and P(B)? Explain.

76) A package of self-sticking notepads contains 6 yellow, 6 blue, 6 green, and 6 pink notepads. An experiment consists of randomly selecting one of the notepads and recording its color. Find the sample space for the experiment.

76)

77) Suppose that for a certain experiment the probability of a particular event occurring is .21. Find the probability that this event does not occur.

77)

78) On a certain statistics test, 20% of the students earned a score of 90 or above. It was also true that 20% of the male students earned a score of 90 or above. Are the events earning a score of 90 or above and being male independent? Explain.

78)

79) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red chips. Events A and B are defined as follows.

79)

A: {Both chips are the same color} B: {At least one of the chips is blue} Are A and B independent events? Explain.

80) A fast-food restaurant chain with 700 outlets in the United States has recorded the geographic 80) location of its restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to test market a chicken sandwich. Region NE SE SW NW 6% 6% 3% 0% <10,000 Population of City 10,000 - 100,000 15% 10% 12% 5% >100,000 20% 4% 4% 15% What is the probability that the restaurant is located in the western portion of the United States?

81) A fair die is rolled one time. Let A be the event that an odd number is rolled. Describe the event Ac.

81)

82) Three companies (A, B, and C) are to be ranked first, second, and third in a list of companies with 82) the highest customer satisfaction. a. List all the possible sets of rankings for these top three companies. b. Assuming that all sets of rankings are equally likely, what is the probability that Company A will be ranked first, Company B second, and Company C third? c. Assuming that all sets of rankings are equally likely, what is the probability that Company B will be ranked first?

83) The table shows the number of each Ford car sold in the United States in June. Suppose the sales 83) record for one of these cars is randomly selected and the type of car is identified. Type of Car Sedan Convertible Wagon SUV Van Hatchback Total

Number 7,204 9,089 20,418 13,691 15,837 15,350 81,589

Event A is defined as follows. A: {Convertible, SUV, Van} a. b.

Identify the sample points in the event Ac. Find P(Ac).

84) Suppose that for a certain experiment P(A) = .32 and P(B) = .55. If A and B are independent events, find P(A B).

84)

85) A pair of fair dice is tossed. Events A and B are defined as follows.

85)

A: {The sum of the numbers showing is odd} B: {The sum of the numbers showing is 2, 11, or 12} Are A and B independent events? Explain.

86) A fair die is rolled one time. Let B be the event {1, 2, 5}. List the sample points in the event Bc.

86)

87) The table shows the number of each car sold in the United States in June. Suppose the sales record 87) for one of these cars is randomly selected and the type of car is identified. Type of Car Sedan Convertible Wagon SUV Van Hatchback Total a. b. c.

Number 7,204 9,089 20,418 13,691 15,837 15,350 81,589

List the sample points for this experiment. Find the probability of each sample point. What is the probability that the car was a Van or an SUV?

88) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red chips. Events A and B are defined as follows.

88)

A: {Both chips are red} B: {At least one of the chips is blue} Are the events A and B mutually exclusive? Explain.

89) The manager of a warehouse club estimates that 7 out of 10 customers will donate a dollar to help a children's hospital during an annual drive to benefit the hospital. Using the manager's estimate, what is the probability that a randomly selected customer will not donate a dollar?

89)

90) At a small private college with 800 students, 240 students receive some form of government-sponsored financial aid. Find the probability that a randomly selected student does not receive some form of government-sponsored financial aid.

90)

91) Suppose that an experiment has five sample points, E1 , E2 , E3 , E4 , E5 , and that P(E1 ) = .2,

91)

P(E2 ) = .3, P(E3 ) = .1, P(E4 ) = .1, and P(E5 ) = .3. If the events A and B are defined as A = {E1 , E2 , E3 } and B = {E2 , E3 , E4 } find P(A B).

92) A number between 1 and 10, inclusive, is randomly chosen. Events A, B, C, and D are defined as follows.

92)

A: {The number is even} B: {The number is less than 7} C: {The number is odd} D: {The number is greater than 5} Identify one pair of mutually exclusive events.

93) In a sample of 750 of its online customers, a department store found that 420 were men. Use this information to estimate the probability that a randomly selected online customer is a man.

93)

94) The manager of an advertising department has asked her creative team to propose six new ideas 94) for an advertising campaign for a major client. She will choose three of the six proposals to present to the client. (We will refer to the six proposals as A, B, C, D, E, and F.) a. In how many ways can the manager select the three of the six proposals? List the possibilities. b. It is unlikely that the manager will randomly select three of the six proposals, but if she does what is the probability that she selects proposals A, D, and E?

95) Three fair coins are tossed and either heads or tails is observed for each coin. Events A and B are defined as follows.

95)

A: {Three heads are observed}. B: {Exactly two heads are observed}. Is P(A B) equal to the sum of P(A) and P(B)? Explain.

96) Suppose that 62% of the employees at a company are male and that 35% of the employees just received merit raises. If 20% of the employees are male and received a merit raise, what is the probability that a randomly chosen employee is male or received a merit raise?

96)

97) Suppose that for a certain experiment P(A) = .37. Find P(Ac).

97)

98) Compute the number of ways you can select n elements from N elements for n = 6 and N = 15.

98)

99) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red chips. Event A is defined as follows.

99)

A: {Both chips are red} a. b. c.

Describe the event Ac.

Identify the sample points in the event Ac. Find P(Ac).

100) The accompanying Venn diagram describes the sample space of a particular experiment and events A and B. Suppose the sample points are equally likely. Find P(A) and P(B).

100)

101) The manager of a warehouse club estimates that 7 out of 10 customers will donate a dollar to help a children's hospital during an annual drive to benefit the hospital. Using the manager's estimate, what is the probability that a randomly selected customer will donate a dollar?

101)

102) In an exit poll, 45% of voters said that the main issue affecting their choices of candidates was the 102) economy, 35% said national security, and the remaining 20% were not sure. Suppose we select one of the voters who participated in the exit poll at random and ask for the main issue affecting his or her choices of candidates. a. List the sample points for this experiment. b. Assign reasonable probabilities to the sample points. c. Find the probability that the main issue affecting his or her choices was either the economy or national security.

103) A company evaluates its potential new employees using three criteria.

103)

A: The applicant has a minimum college GPA of 3.0. B: The applicant has relevant work experience. C: The applicant has a sufficient score on an aptitude test. a. Write the event that an applicant meets all three criteria as a union or intersection of A, B, and C. b. Write the event that an applicant meets at least one of the three criteria as a union or intersection of A, B, and C.

104) The table shows the political affiliations and types of jobs for workers in a particular state. 104) Suppose a worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 10% 19% 17% Type of job Blue Collar 18% 15% 21% What is the probability that the worker is a white collar Republican?

105) An experiment consists of randomly choosing a number between 1 and 10. Let A be the event that the number chosen is less than or equal to 7. List the sample points in A.

105)

106) The data below show the types of medals won by athletes representing the United States in the106) Winter Olympics. Suppose that one medal is chosen at random and the type of medal noted. gold bronze gold gold

gold gold silver gold

silver silver silver bronze

gold silver bronze bronze

bronze bronze bronze

silver silver gold

silver gold silver

Given that the medal is not bronze, what is the probability that the medal is gold?

107) Based on past experience, Josh believes that the probability of catching a red snapper is .21 and the probability of catching a fish that weighs less than 5 pounds is .45. Is enough information available to find the probability of catching a red snapper or a fish that weighs less than 5 pounds? Explain. If possible, find the probability of catching a red snapper or a fish that weighs less than 5 pounds.

107)

108) The table shows the number of each type of book found at an online auction site during a recent 108) search. Suppose that Juanita randomly chose one book to bid on and then noted its type. Type of Book Children's Fiction Nonfiction Educational a. b. c.

Number 51,033 141,114 253,074 67,252

List the sample points for this experiment. Find the probability of each sample point. What is the probability that the book was nonfiction or educational?

109) Suppose that for a certain experiment P(A) = .37, P(B) = .69, and P(A B) = .23. Find P(A B).

109)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 110) A human gene carries a certain disease from a mother to her child with a probability rate of 0.41. That is, there is a 41% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has five children. Assume that the infections, or lack thereof, are independent of one another. Find the probability that all five of the children get the disease from their mother. A) 0.988 B) 0.012 C) 0.071 D) 0.05 Answer the question True or False. 111) The quantity 0! is defined to be equal to 0. A) True

B) False

110)

111)

112) A pair of fair dice is tossed. Events A and B are defined as follows.

112)

A: {The sum of the numbers on the dice is 3} B: {At least one of the dice shows a 2} True or False: A B = B. A) True

B) False

Solve the problem. 113) At a community college with 500 students, 120 students are age 30 or older. Find the probability that a randomly selected student is age 30 or older. A) .76 B) .30 C) .24 D) .12

113)

114) In a class of 30 students, 18 are men, 6 are earning a B, and no men are earning a B. If a student is randomly selected from the class, find the probability that the student is a man or earning a B. A) .24 B) .4 C) .54 D) .8

114)

115) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.

115)

A: {The number is even} B: {The number is less than 7} Identify the sample points in the event A B. A) {1, 2, 3, 4, 5, 6, 7, 9} C) {1, 2, 3, 4, 5, 6, 7, 8, 10}

B) {1, 2, 3, 4, 5, 6, 8, 10} D) {2, 4, 6}

116) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a company's success is customer service. A study was conducted to determine the customer satisfaction levels for one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how often they used overnight shipping. The results are shown below in the following table:

Frequency of Use < 2 per month 2 - 5 per month > 5 per month TOTAL

High 250 140 70 460

Satisfaction level Medium Low 140 10 55 5 25 5 220 20

116)

TOTAL 400 200 100 700

Suppose that one customer who participated in the study is chosen at random. What is the probability that the customer did not have a medium level of satisfaction with the company? 24 2 11 5 A) B) C) D) 35 7 35 7

Answer the question True or False. 117) Two events, A and B, are independent if P(A and B) = P(A) × P(B). A) True B) False 118) Unions and intersections of events are examples of compound events. A) True B) False

117)

118)

Solve the problem. 119) A one-week study revealed that 60% of a warehouse store's customers are women and that 30% of women customers spend at least $250 on a single visit to the store. Find the probability that a randomly chosen customer will be a woman who spends at least $250. A) 0.50 B) 0.90 C) 0.36 D) 0.18 120) Evaluate A) 4

8 . 2

119)

120) B) 16

C) 28

D) 56

121) A number between 1 and 10, inclusive, is randomly chosen. Events A and B are defined as follows.

121)

A: {The number is even} B: {The number is less than 7} Identify the sample points in the event A B. A) {1, 2, 3, 4, 5, 6, 8, 10} C) {2, 4, 6}

B) {1, 2, 3, 4, 5, 6, 7, 8, 10} D) {1, 2, 3, 4, 5, 6, 7, 9}

122) Suppose that for a certain experiment P(A) = .33 and P(B) = .29. If A and B are mutually exclusive events, find P(A B). A) .31 B) .62 C) .38 D) .03

122)

of 123) A fast-food restaurant chain with 700 outlets in the United States has recorded the geographic location 123) its restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to test market a new chicken sandwich. Region NE SE SW NW 5% 6% 3% 0% <10,000 Population of City 10,000 - 100,000 15% 8% 12% 5% 20% 4% 1% 21% >100,000 What is the probability that the restaurant is located in a city with a population over 100,000 and in the southern portion of the United States? A) 0.05 B) 0.01 C) 0.04 D) 0.34

124) For two events, A and B, P(A) = .4, P(B) = .7, and P(A B) = .2. Find P(A | B). A) .5 B) .29 C) .14

D) .08

124)

125) Suppose that for a certain experiment P(A) = 0.6 and P(B) = 0.3. If A and B are independent events, find P(A B). A) 0.90 B) 0.30 C) 0.18 D) 0.50

125)

126) Suppose a basketball player is an excellent free throw shooter and makes 93% of his free throws (i.e., he has a 93% chance of making a single free throw). Assume that free throw shots are independent of one another. Find the probability that the player misses three consecutive free throws. A) 0.9997 B) 0.8044 C) 0.0003 D) 0.1956

126)

127) Classify the events as dependent or independent: Events A and B where P(A) = 0.5, P(B) = 0.2, and P(A and B) = 0.09. A) independent B) dependent

127)

128) Four hundred accidents that occurred on a Saturday night were analyzed. The number of vehicles involved and whether alcohol played a role in the accident were recorded. The results are shown below:

128)

Number of Vehicles Involved Did Alcohol Play a Role? 1 2 3 or more Totals Yes 58 98 14 170 No 26 174 30 230 Totals 84 272 44 400 Given that an accident involved multiple vehicles, what is the probability that it involved alcohol? 28 7 7 7 A) B) C) D) 79 22 25 200

129) At a certain university, one out of every 20 students is enrolled in a statistics course. If one student at the university is chosen at random, what is the probability that the student is enrolled in a statistics course? 1 1 1 1 A) B) C) D) 2 21 19 20

129)

130) Each manager of a Fortune 500 company was rated as being either a good, fair, or poor manager by his/her boss. The manager's educational background was also noted. The data appear below:

130)

Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Total Good 2 5 27 5 39 Fair 9 11 46 21 87 Poor 6 8 3 17 34 Total 17 24 76 43 160 What is the probability that a randomly chosen manager has earned at least one college degree? 43 41 19 119 A) B) C) D) 160 160 40 160

131) Fill in the blank. A(n) ______ is a process that leads to a single outcome that cannot be predicted with certainty. A) sample point B) experiment C) event D) sample space

131)

132) Fill in the blank. The __________ of two events A and B is the event that both A and B occur. A) Venn diagram B) union C) complement D) intersection

132)

133) Four hundred accidents that occurred on a Saturday night were analyzed. The number of vehicles involved and whether alcohol played a role in the accident were recorded. The results are shown below:

Did Alcohol Play a Role? Yes No Totals

133)

Number of Vehicles Involved 1 2 3 or more Totals 60 93 17 170 28 172 30 230 88 265 47 400

Suppose that one of the 400 accidents is chosen at random. What is the probability that the accident involved more than a single vehicle? 11 47 17 39 A) B) C) D) 50 400 400 50

Answer the question True or False. 134) For all events A and B, the conditional probabilities P(A | B) and P(B | A) are equal. A) True B) False Compute. 135)

6 5

134)

135) A) 1

B) 6

C) 5

D) 720

Solve the problem. 136) A state energy agency mailed questionnaires on energy conservation to 1,000 homeowners in the state 136) capital. Five hundred questionnaires were returned. Suppose an experiment consists of randomly selecting one of the returned questionnaires. Consider the events: A: {The home is constructed of brick} B: {The home is more than 30 years old} D: {The home is heated with oil} Which of the following describes the event B Dc? A) homes that are not older than 30 years old and heated with oil B) homes more than 30 years old or homes that are not heated with oil C) homes more than 30 years old that are heated with oil D) homes more than 30 years old that are not heated with oil

137) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the number rolled is odd. Find P(A B). Outcome Probability

A) .5

1 .1

2 .1

3 .1

4 .2

5 .2

B) .3

C) .7

6 .3

D) .2

137)

Answer the question True or False. 138) In some experiments, we assign subjective probabilities, which can be interpreted as our degree of belief in the outcome. A) True B) False Solve the problem. 139) There are 10 movies that Greg would like to rent but the store only allows him to have 4 movies at one time. In how many ways can Greg choose 4 of the 10 movies? A) 40 B) 10,000 C) 210 D) 5040

138)

139)

140) An economy pack of highlighters contains 12 yellow, 6 blue, 4 green, and 3 orange highlighters. An experiment consists of randomly selecting one of the highlighters and recording its color. Find the probability that a blue or yellow highlighter is selected given that a yellow highlighter is selected. 1 1 A) B) 0 C) 1 D) 2 3

140)

141) An experiment consists of randomly choosing a number between 1 and 10. Let E be the event that the number chosen is even. List the sample points in E. A) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B) {1, 3, 5, 7, 9} C) {2, 4, 6, 8, 10} D) {5}

141)

Answer the question True or False. 142) The probability of an event can be calculated by finding the sum of the probabilities of the individual sample points in the event and dividing by the number of sample points in the event. A) True B) False

142)

Solve the problem. 143) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a 143) worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 8% 14% 18% Type of job Blue Collar 20% 11% 29% Find the probability that the worker is a white collar worker affiliated with the Democratic Party. A) 0.25 B) 0.14 C) 0.51 D) 0.40

144) Evaluate A) 0

6 . 0

144) B) 1

C) 6

D) undefined

Answer the question True or False. 145) Unions and intersections cannot be defined for more than two sets, so that A B C and A B C are meaningless. A) True B) False

145)

Solve the problem. 146) The manager of a used car lot took inventory of the automobiles on his lot and constructed the following table based on the age of each car and its make (foreign or domestic):

Make Foreign Domestic Total

Age of Car (in years) 3-5 6 - 10 29 14 24 10 53 24

0-2 39 45 84

over 10 18 21 39

146)

Total 100 100 200

A car was randomly selected from the lot. Given that the car selected was a foreign car, what is the probability that it was older than 2 years old? 61 39 61 39 A) B) C) D) 116 116 100 100

147) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Find 147) the probability that the number rolled on a single roll of this die is less than 4. Outcome Probability

1 .1

A) .2

2 .1

3 .1

4 .2

5 .2

B) .3

6 .3

C) .7

D) .5

148) If sample points A, B, C, and D are the only possible outcomes of an experiment, find the probability of D using the table below. Sample Point Probability 1 A) 4

A 1/14

B 1/14 3 B) 14

C 1/14

148)

11 14

1 14

149) A clothing vendor estimates that 78 out of every 100 of its online customers do not live within 50 miles of one of its physical stores. Using this estimate, what is that probability that a a randomly selected online customer lives within 50 miles of a physical store? A) .50 B) .22 C) .28 D) .78

149)

Answer the question True or False. 150) An event and its complement are mutually exclusive. A) True B) False

150)

Solve the problem. 151) If P(A) = .55, P(B A) = .4, P(A B) = .22, and A and B are independent events, find P(B). A) .4 B) .88 C) .55 D) .22

151)

152) For two events, A and B, P(A) = A)

9 10

3 2 5 , P(B) = , and P(B | A) = . Find P(A B). 4 3 6 5 8

5 9

152) D)

1 2

Answer the question True or False. 153) If A and B are independent events, then A and B are also mutually exclusive. A) True B) False

153)

Solve the problem. 154) An experiment consists of randomly choosing a number between 1 and 10. Let E be the event that the number chosen is even. Assuming that each of the numbers between 1 and 10 is equally likely to be chosen, find P(E). A) .8 B) .2 C) .1 D) .5

154)

155) A state energy agency mailed questionnaires on energy conservation to 1,000 homeowners in the state 155) capital. Five hundred questionnaires were returned. Suppose an experiment consists of randomly selecting one of the returned questionnaires. Consider the events: A: {The home is constructed of brick} B: {The home is more than 30 years old} In terms of A and B, describe a home that is constructed of brick and is less than or equal to 30 years old. A) (A B)c B) A B C) A Bc D) A B

156) Suppose that B1 and B2 are mutually exclusive and complementary events, such that P(B1 ) = .6 and

156)

P(B2 ) = .4. Consider another event A such that P(A | B1 ) = .2 and P(A | B2) = .5. Find P(A).

A) .88

B) .70

C) .32

D) .38

157) A pair of fair dice is tossed. Events A and B are defined as follows.

157)

A: {The sum of the numbers on the dice is 3} B: {At least one of the dice shows a 2} Identify the sample points in the event A B. A) {(1, 2), (2, 1)} B) {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)} C) {(2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)} D) {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}

Answer the question True or False. 158) If two events, A and B, are mutually exclusive, then P(A and B) = P(A) × P(B). A) True B) False

158)

Solve the problem. 159) In the game of Parcheesi each player rolls a pair of dice on each turn. In order to begin the game, you must roll a five on at least one die, or a total of five on both dice. Find the probability that a player begins the game on the first roll. 1 11 5 15 A) B) C) D) 6 36 18 36 Compute the number of ways you can select n elements from N elements. 160) n = 2, N = 10 A) 8 B) 90 C) 19 26

D) 45

159)

160)

Solve the problem. 161) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a 161) worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 17% 11% 6% Type of job Blue Collar 19% 12% 35% Find the probability the worker is not an Independent. A) 0.28 B) 0.59 C) 0.41

D) 0.31

162) In a class of 30 students, 18 are men, 6 are earning a B, and no men are earning a B. If a student is randomly selected from the class, find the probability that the student is a man given that the student earning a B. 1 3 A) 0 B) C) D) 1 3 5

162)

163) Suppose that an experiment has five equally likely outcomes. What probability is assigned to each of the sample points? A) .5 B) .2 C) .05 D) 1

163)

164) If P(A B) = 0 and P(A) 0, then which statement is false? A) Events A and B are mutually exclusive. B) Events A and B have no sample points in common. C) Events A and B are dependent. D) Events A and B are independent.

164)

165) A hospital reports that two patients have been admitted who have contracted Crohn's disease. Suppose our experiment consists of observing whether each patient survives or dies as a result of the disease. The simple events and probabilities of their occurrences are shown in the table (where S in the first position means that patient 1 survives, D in the first position means that patient 1 dies, etc.).

165)

Simple Events SS SD DS DD

Probabilities 0.53 0.17 0.16 0.14

Find the probability that both patients survive. A) 0.53 B) 0.33

C) 0.2809

D) 0.14

166) A pair of fair dice is tossed. Events A and B are defined as follows.

166)

A: {The sum of the numbers on the dice is 3} B: {At least one of the dice shows a 2} Identify the sample points in the event A B. A) {(2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)} B) {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)} C) {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)} D) {(1, 2), (2, 1)}

Answer the question True or False.

167) For any events A and B, P(A | B) + P(A | Bc) = 1, meaning given that A occurs either B occurs or B does not occur. A) True B) False

167)

168) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red chips. Events A and B are defined as follows.

168)

A: {Both chips are red} B: {At least one of the chips is blue} True or False: A = Bc. A) True

B) False

169) If events A and B are not mutually exclusive, then it is possible that P(A) + P(B) > 1. A) True B) False Solve the problem. 170) Fill in the blank. A(n) __________ is the most basic outcome of an experiment. A) sample point B) experiment C) sample space 171) For two events, A and B, P(A) = A)

1 2

169)

D) event

1 1 1 , P(B) = , and P(A B) = . Find P(B | A). 2 3 4 1 12

1 8

170)

171) D)

3 4

172) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. 172) The manager's educational background was also noted. The data appear below: Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals Good 8 3 24 4 39 Fair 1 17 47 22 87 Poor 7 4 9 14 34 Totals 16 24 80 40 160 If we randomly selected one manager from this company, find the probability that he or she has an advanced (Master's or Ph.D.) degree and is a good manager. 79 39 1 57 A) B) C) D) 160 40 40 80

173) Fill in the blank. A(n) __________ is a collection of sample points. A) sample space B) event C) Venn diagram Compute. 174)

D) experiment

4 0

173)

174) A) 6

B) 1

C) 3

D) 4

Solve the problem. 175) If P(A B) = 1 and P(A B) = 0, then which statement is true? A) A and B are supplementary events. B) A and B are both empty events. C) A and B are complementary events. D) A and B are reciprocal events. 176) In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a student is randomly selected from the class, find the probability that the student is a woman given that the student is earning an A. 11 7 5 7 A) B) C) D) 20 22 11 10 Compute the number of ways you can select n elements from N elements. 177) n = 4, N = 7 A) 840 B) 6 C) 2

D) 35

175)

176)

177)

Solve the problem. 178) Four hundred accidents that occurred on a Saturday night were analyzed. The number of vehicles involved and whether alcohol played a role in the accident were recorded. The results are shown below:

Did Alcohol Play a Role? Yes No Totals

178)

Number of Vehicles Involved 1 2 3 or more Totals 57 96 17 170 20 171 39 230 77 267 56 400

Suppose that one of the 400 accidents is chosen at random. What is the probability that the accident involved alcohol or a single car? 57 77 19 17 A) B) C) D) 400 400 40 40

179) A study revealed that 45% of college freshmen are male and that 18% of male freshmen earned college credits while still in high school. Find the probability that a randomly chosen college freshman will be male and have earned college credits while still in high school. A) 0.530 B) 0.027 C) 0.081 D) 0.400

179)

180) A hospital reports that two patients have been admitted who have contracted Crohn's disease. Suppose our experiment consists of observing whether each patient survives or dies as a result of the disease. The simple events and probabilities of their occurrences are shown in the table (where S in the first position means that patient 1 survives, D in the first position means that patient 1 dies, etc.).

180)

Simple Events SS SD DS DD

Probabilities 0.57 0.15 0.19 0.09

Find the probability that at least one of the patients does not survive. A) 0.43 B) 0.15 C) 0.34

D) 0.09

Answer the question True or False. 181) If every sample point in event B is also a sample point in event A, then P(A | B) = 1. A) True B) False 182) For any events A and B, P(A | B) + P(Ac | B) = 1, meaning given that B occurs either A occurs or A does not occur. A) True B) False

181)

182)

Solve the problem. 183) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. 183) The manager's educational background was also noted. The data appear below: Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals Good 3 4 28 4 39 Fair 9 16 49 13 87 Poor 5 6 1 22 34 Totals 17 26 78 39 160 What is the probability that a randomly chosen manager is either a good managers or has an advanced degree? 1 37 39 39 A) B) C) D) 40 80 80 40

Answer the question True or False. 184) If A and B,are independent events, then P(A) = P(B A). A) True B) False

184)

185) The probability of a sample point is usually taken to be the relative frequency of the occurrence of the sample point in a very long series of repetitions of the experiment. A) True B) False Solve the problem. 186) Classify the events as dependent or independent: Events A and B where P(A) = 0.9, P(B) = 0.7, and P(A and B) = 0.63. A) dependent B) independent Compute. 187)

6 6

185)

186)

187) A) 1

B) 5

C) 120

D) 6

Solve the problem. 188) Suppose that for a certain experiment P(A) = .47 and P(B) = .25 and P(A B) = .14. Find P(A B). A) .58 B) .36 C) .86 D) .72

188)

189) Fill in the blank. The __________ of two events A and B is the event that either A or B or both occur. A) union B) Venn diagram C) intersection D) complement

189)

190) Suppose that B1 and B2 are mutually exclusive and complementary events, such that P(B1 ) = .6 and

190)

P(B2 ) = .4. Consider another event A such that P(A | B1 ) =.2 and P(A | B2 ) = .5. Find P(B1 | A).

A) .800

B) .625

C) .375

D) .240

Compute. 191)

10 4

191)

A) 210

B) 6

C) 5040

D) 34

Solve the problem. 192) The table displays the probabilities for each of the outcomes when three fair coins are tossed and the 192) number of heads is counted. Find the probability that the number of heads on a single toss of the three coins is at most 2. Outcome Probability

0 .125

A) .875 193) Evaluate

1 .375

2 .375

3 .125

B) .750

C) .125

D) .500

7 . 7

193)

A) 49

B) 7

C) 14

D) 1

Answer the question True or False. 194) A statistical experiment can be almost any act of observation as long as the outcome is uncertain. A) True B) False

194)

Solve the problem. 195) For two events, A and B, P(A) = .6, P(B) = .8, and P(A | B) = .5. Find P(A B). A) .4 B) .625 C) .3

195)

D) .833

196) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a company's success is customer service. A study was conducted to determine the customer satisfaction levels for one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how often they used overnight shipping. The results are shown in the following table:

Frequency of Use < 2 per month 2 - 5 per month > 5 per month TOTAL

High 250 140 70 460

Satisfaction level Medium Low 140 10 55 5 25 5 220 20

196)

TOTAL 400 200 100 700

Suppose that one customer who participated in the study is chosen at random. What is the probability that the customer had a high level of satisfaction and used the company less than two times per month? 43 61 9 5 A) B) C) D) 35 70 70 14

Answer the question True or False. 197) If A and B are mutually exclusive events, then P(A | B) = 0. A) True B) False

197)

198) An event may contain sample points that are not in the original sample space of the experiment. For example, the experiment of rolling two dice has the following sample space:

198)

{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} However, the event of rolling a sum of at least 11 on the two dice is {11, 12}. A) True B) False

Solve the problem. 199) Suppose a basketball player is an excellent free throw shooter and makes 96% of his free throws (i.e., he has a 96% chance of making a single free throw). Assume that free throw shots are independent of one another. Find the probability that the player will make five consecutive free throws. A) 0.8154 B) 0 C) 1 D) 0.1846

199)

of 200) A fast-food restaurant chain with 700 outlets in the United States has recorded the geographic location 200) its restaurants in the accompanying table of percentages. One restaurant is to be chosen at random from the 700 to test market a new chicken sandwich. Region NE SE SW NW 9% 6% 3% 0% <10,000 Population of City 10,000 - 100,000 15% 2% 12% 5% >100,000 20% 4% 10% 14% What is the probability that the restaurant is located in the northern portion of the United States? A) 0.37 B) 0.44 C) 0.63 D) 0.19

201) The table shows the political affiliations and types of jobs for workers in a particular state. Suppose a 201) worker is selected at random within the state and the worker's political affiliation and type of job are noted. Political Affiliation Republican Democrat Independent White collar 18% 5% 13% Type of job Blue Collar 19% 6% 39% Given that the worker is a Democrat, what is the probability that the worker has a white collar job. A) 0.139 B) 0.262 C) 0.455 D) 0.119

202) A clothing vendor estimates that 78 out of every 100 of its online customers do not live within 50 miles of one of its physical stores. Using this estimate, what is the probability that a randomly selected online customer does not live within 50 miles of a physical store? A) .78 B) .28 C) .50 D) .22

202)

203) A researcher investigated whether a student's seat preference was related in any way to the gender 203) of the student. The researcher divided a lecture room into three sections (1-front, middle of the room, 2-front, sides of the classroom, and 3-back of the classroom, both middle and sides) and noted where each student sat on a particular day of the class. The researcher's summary table is provided below.

Male Female Total

Area 1 17 12 29

Area 2 5 13 18

Area 3 11 14 25

Total 33 39 72

Suppose a person sitting in the front, middle portion of the class is randomly selected to answer a question. Find the probability that the person selected is female. 4 12 29 1 A) B) C) D) 13 29 39 6

204) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. 204) The manager's educational background was also noted. The data appear below: Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals Good 2 6 24 7 39 Fair 7 19 42 19 87 Poor 4 3 9 18 34 Totals 13 28 75 44 160 Given that a manager is rated as fair, what is the probability that this manager has no college background? 93 7 7 7 A) B) C) D) 160 13 87 160

205) In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a student is randomly selected from the class, find the probability that the student is earning an A given that the student is a woman. 7 7 1 5 A) B) C) D) 22 40 4 11

205)

206) The outcome of an experiment is the number of resulting heads when a nickel and a dime are flipped simultaneously. What is the sample space for this experiment? A) {HH, HT, TT} B) {HH, HT, TH, TT} C) {nickel, dime} D) {0, 1, 2}

206)

Compute. 207)

8 4

207) A) 24

B) 70

C) 2

D) 1680

Solve the problem. 208) A package of self-sticking notepads contains 6 yellow, 6 blue, 6 green, and 6 pink notepads. An experiment consists of randomly selecting one of the notepads and recording its color. Find the probability that a yellow or pink notepad is selected given that it is either blue or green. 1 1 A) B) 1 C) D) 0 4 2

208)

209) Two chips are drawn at random and without replacement from a bag containing four blue chips and three red chips. Find the probability of drawing two red chips. 1 9 6 1 A) B) C) D) 12 49 7 7

209)

210) A machine has four components, A, B, C, and D, set up in such a manner that all four parts must work for the machine to work properly. Assume the probability of one part working does not depend on the functionality of any of the other parts. Also assume that the probabilities of the individual parts working are P(A) = P(B) = 0.95, P(C) = 0.97, and P(D) = 0.94. Find the probability that the machine works properly. A) 0.8754 B) 0.8662 C) 0.1771 D) 0.8229

210)

211) A bag of colored candies contains 20 red, 25 yellow, and 35 orange candies. An experiment consists of randomly choosing one candy from the bag and recording its color. What is the sample space for this experiment? A) {20, 25, 35} B) {1/4, 5/16, 7/16} C) {80} D) {red, yellow, orange}

211)

Answer the question True or False.

212) If an event A includes the entire sample space, then P(Ac) = 0. A) True B) False

Solve the problem. 213) A local country club has a membership of 600 and operates facilities that include an 18-hole championship golf course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to know how many members regularly use each facility. A survey of the membership indicates that 57% regularly use the golf course, 48% regularly use the tennis courts, and 9% use both of these facilities regularly. Given that a randomly selected member uses the tennis courts regularly, find the probability that they also use the golf course regularly. A) .1343 B) .7164 C) .4737 D) .1875 Answer the question True or False. 214) Two chips are drawn at random and without replacement from a bag containing two blue chips and two red chips. Events A and B are defined as follows.

212)

213)

214)

A: {Both chips are red} B: {At least one of the chips is blue} True or False: A B = B. A) True

B) False

215) The conditional probability of event A given that event B has occurred is written as P(B | A). A) True B) False 35

215)

Solve the problem. 216) Fill in the blank. The __________ is the collection of all the sample points in an experiment. A) sample space B) event C) Venn diagram D) union

216)

217) Suppose that for a certain experiment P(B) = 0.5 and P(A B) = 0.2. Find P(A B). A) 0.4 B) 0.7 C) 0.1 D) 0.3

217)

218) In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a student is randomly selected from the class, find the probability that the student is a woman or earning an A. A) .8 B) .975 C) .625 D) .25

218)

Answer the question True or False. 219) The combinations rule applies to situations in which the experiment calls for selecting n elements from a total of N elements, without replacing each element before the next is selected. A) True B) False Solve the problem. 220) At a community college with 500 students, 120 students are age 30 or older. Find the probability that a randomly selected student is less than 30 years old. A) .76 B) .24 C) .30 D) .12 221) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose that the die is rolled once. Let A be the event that the number rolled is less than 4, and let B be the event that the number rolled is odd. Find P(A B). Outcome Probability

1 .1

2 .1

A) .5

3 .1

4 .2

5 .2

B) .3

C) .2

Frequency of Use < 2 per month 2 - 5 per month > 5 per month TOTAL

High 250 140 70 460

221)

D) .7

TOTAL 400 200 100 700

A customer is chosen at random. Given that the customer uses the company two to five times per month, what is the probability that the customer expressed medium satisfaction with the company? 11 1 11 73 A) B) C) D) 140 4 40 140

220)

6 .3

222) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a company's success is customer service. A study was conducted to determine the customer satisfaction levels for one overnight shipping business. In addition to the customer's satisfaction level, the customers were asked how often they used overnight shipping. The results are shown below in the following table: Satisfaction level Medium Low 140 10 55 5 25 5 220 20

219)

222)

223) Probabilities of different types of vehicle-to-vehicle accidents are shown below:

223)

Accident Probability Car to Car 0.67 Car to Truck 0.15 Truck to Truck0.18 Find the probability that an accident involves a car. A) 0.18 B) 0.67

C) 0.82

D) 0.15

224) A number between 1 and 10, inclusive, is randomly chosen. Events A, B, C, and D are defined as follows.

224)

A: {The number is even} B: {The number is less than 7} C: {The number is less than or equal to 7} D: {The number is 5} Identify one pair of independent events. A) A and C B) A and D

C) B and D

D) A and B

225) In a box of 75 markers, 36 markers are either red or black and 15 are blue. Find the probability that a randomly selected marker is red or black or blue. A) .51 B) .32 C) .68 D) .24 Answer the question True or False. 226) If A and B are mutually exclusive events, then P(A B) = 0. A) True B) False Solve the problem. 227) A pair of fair dice is tossed. Events A and B are defined as follows.

225)

226)

227)

A: {The sum of the numbers on the dice is 4} B: {The sum of the numbers on the dice is 11} Identify the sample points in the event A B. A) {(1, 4), (2, 2), (4, 1), (5, 6), (6, 5)} B) {(1, 4), (2, 3), (3, 2), (4, 1), (5, 6), (6, 5)} C) {(1, 3), (2, 2), (3, 1), (5, 6), (6, 5)} D) There are no sample points in the event A B.

Answer the question True or False. 228) In any experiment with exactly four sample points in the sample space, the probability of each sample point is .25. A) True B) False Solve the problem. 229) In a box of 50 markers, 30 markers are either red or black and 20 are missing their caps. If 12 markers are either red or black and are missing their caps, find the probability that a randomly selected marker is red or black or is missing its cap. A) .24 B) .38 C) .76 D) 1 37

228)

229)

230) Fill in the blank. The __________ of an event A is the event that A does not occur. A) intersection B) Venn diagram C) union D) complement

230)

231) The table displays the probabilities for each of the six outcomes when rolling a particular unfair die. Suppose that the die is rolled once.

231)

Outcome Probability

1 .1

2 .1

3 .1

4 .2

5 .2

6 .3

Events A, B, C, and D are defined as follows. A: {The number is even} B: {The number is less than 4} C: {The number is less than or equal to 5} D: {The number is greater than or equal to 5} Identify one pair of independent events. A) A and B B) B and D

C) A and D

232) Compute the number of ways you can select 3 elements from 7 elements. A) 343 B) 10 C) 35

D) B and C

D) 21

232)

233) A package of self-sticking notepads contains 6 yellow, 6 blue, 6 green, and 6 pink notepads. An experiment consists of randomly selecting one of the notepads and recording its color. Find the probability that a green notepad is selected given that it is either blue or green. 1 1 1 1 A) B) C) D) 3 4 2 12

233)

234) Kim submitted a list of 12 movies to an online movie rental company. The company will choose 3 of the movies and ship them to her. If all movies are equally likely to be chosen, what is the probability that Kim will receive the three movies that she most wants to watch? Express the probability as a fraction. 1 1 1 1 A) B) C) D) 1320 1728 4 220

234)

235) The following Venn diagram shows the six possible outcomes when rolling a fair die. Let A be the event of rolling an even number and let B be the event of rolling a number greater than 1.

Which of the following expressions describes the event of rolling a 1? A) B B) A B C) Bc

235)

D) Ac

236) A music store has 8 male and 12 female employees. Suppose one employee is selected at random and the employee's gender is observed. List the sample points for this experiment, and assign probabilities to the sample points. A) {male, female}; P(male) = .4 and P(female) = .6 B) {8, 12}; P(8) = .5 and P(12) = .6 C) {8, 12}; P(8) = .8 and P(12) = .12 D) {male, female}; P(male) = .8 and P(female) = .12

236)

237) A local country club has a membership of 600 and operates facilities that include an 18-hole championship golf course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to know how many members regularly use each facility. A survey of the membership indicates that 63% regularly use the golf course, 50% regularly use the tennis courts, and 10% use neither of these facilities regularly. What is the probability that a member regularly uses at least one of the golf or tennis facilities? A) .23 B) .50 C) .10 D) .90

237)

Answer Key Testname: CHAPTER 3 1) B 2) D 3) D 4) D 5) C 6) B 7) C 8) A 9) B 10) C 11) C 12) A 13) B 14) B 15) A 16) C 17) C 18) D 19) D 20) D 21) B 22) D 23) A 24) C 25) D 26) P(green and 3) = P(green) P(3) = (.25)(.1) = .025 27) a. {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} b. {(1, 1), (1, 2), (2, 1), (2, 2)} c. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} d. {(1, 1), (2, 2)} 8 2 e. P(Ac Bc) = = 36 9 f.

2 1 = P(Ac Bc) = 36 18

28) P(A | B) =

.1 + .1 = .5 .1 + .1 + .2

29) The applicant has a minimum college GPA of 3.0 and a sufficient score on the aptitude test but does not have relevant work experience. 30) a. {male, female} 85 75 b. P(male) = = .53125; P(female) = = .46875 160 160 31) P(A B) =

1 1 7 + = 3 4 12

32) P(Neither patient survives) = P(DD) = 0.15 240 = .3 33) P(receives government aid) = 800

Answer Key Testname: CHAPTER 3 34) Let Bi = event that the player can begin on roll i: c P(cannot begin on first or second roll) = P( B 1

c c c B 2 ) = P( B 1 ) × ( B 2 )

c c 15 P( B 1 ) = P( B 2 ) = 36 Dice combinations: c P( B 2

(1, 4), (4, 1), (2, 3), (3, 2), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6)

c 21 21 × = .3403 B 2) = 36 36

35) P(teenager and male) = P(teenager) P(male teenager) = (.8)(.6) = .48 or 48% 36) a. Let b1 and b2 represent the blue chips and r1 and r2 the red chips. A B = { b1 b2 , b1 r1 , b1 r2 , b2 r1 , b2 r2 , r1 r2 }

P(A B) =

6 =1 6

37) P(male | merit raise) =

.20 4 = = .571 .35 7

38) The Additive Rule states that P(A B) = P(A) + P(B) - P(A B), so P(A B) = P(A) + P(B) - P(A B) = .60 + .30 - .80 =.10. 39) Yes, there is enough information, since catching a red snapper and catching a grouper are mutually exclusive events. The probability is .19 + .21 = .40. 40) a. {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} 1 b. Each sample point is assigned the probability . 8 1 8

P(A) = P({HHH}) =

P(B) = P({HHT, HTH, THH}) =

P(C) = P({HHH, HHT, HTH, THH}) =

1 1 1 3 + + = 8 8 8 8 1 1 1 1 1 + + + = 8 8 8 8 2

41) No; P(owning car) = .7 and P(owning car | residence hall) = .45; Since these probabilities are not equal, the events are not independent. 42) a. {1, 2, 3, 4, 9, 11, 14, 19, 22, 23, 24, 25, 29} 1 1 1 1 1 1 b. P(1) = , P(2) = , P(3) = , P(4) = , P(9) = , P(11) = , 18 18 18 18 18 18

43) a. b. c. d.

P(14) =

1 1 1 1 1 , P(19) = , P(22) = , P(23) = , P(24) = , 18 18 18 18 18

P(25) =

1 1 , P(29) = , 18 18

P(22) + P(23) + P(24) + P(25) + P(29) =

1 1 1 1 1 5 + + + + = 18 18 18 18 18 18

{Sedan, Convertible, SUV, Van} {Convertible} 7,204 + 9,089 + 13,691 + 15,387 45,821 = P(A B) = 81,589 81,589 P(A B) =

9.089 81,589

.56

.11

Answer Key Testname: CHAPTER 3 44) P(uses golf or tennis regularly) = P(golf) + P(tennis) - P(both tennis and golf) = .61 + .50 - .21 = .90 45) P(woman and favored Democrats) = P(woman) P(favored Democrats | woman) = .55 × .65 = .3575 46) Let D be the event of a single child getting the disease. P(none get the disease) = P(Dc Dc Dc Dc) = P(Dc)P(Dc)P(Dc)P(Dc) = (0.8)(0.8)(0.8)(0.8) = = 0.4096 47) No, the events are not mutually exclusive. They have at least one sample point in common, (3, 4), for example. 420 = 1 - .56 = .44 48) P(not a man) = 1 - P(man) = 1 750

49) P(A | B) =

9,089 7,204 + 9,089

.558; P(B | A) =

9.089 9,089 + 13,691 + 15,837

.235

50) A B = {E1 , E2 , E3 , E5 }; P(A B) = P(E1 ) + P(E2 ) + P(E3 ) + P(E5) = .4 + .1 + .1 + .2 = .8 51) Ac = {E4 , E5 }; P(Ac) = P(E4 ) + P(E5 ) = .1 + .3 = .4. 52) P(man | does not live within 50 miles) =

.39 = .5 .78

53) Let >100,000 = event that the city has a population over 100,000 and SW = event that the location is in the southwestern United States. P(>100,000 | SW) =

54) a.

P(88) =

P(>100,000 and SW) 4% = = 0.211 P(SW) 19%

4 2 = 50 25 3 50

P(less than 60) =

P(between 70 and 79, inclusive) =

55)

1 = .125 8

56)

8 8! 8! = = = 56 3 3!(8 - 3)! 3! 5!

15 3 = 50 10

57) a. All three of the criteria are met. b. At least one of the three criteria is met. 10 10! 10! = = = 210 58) 6 6!(10 - 6)! 6! 4! 59) If the events were mutually exclusive, then by the Additive Rule we would have P(A B) = .8 + .9 = 1.7, which is not a valid probability since it is greater than 1. So the events can not be mutually exclusive. 6 6 = = .24 60) P(blue) = 12 + 6 + 4 + 3 25

Answer Key Testname: CHAPTER 3 61) a. b. c. d.

{(1, 3), (1, 5), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5) (3, 6) (4, 2), (4, 3), (5, 1), (5, 3), (6, 3)} {(3, 3)} 15 5 P(A B) = = 36 12 P(A B) =

1 36

62) Let blue = event that the worker is a blue collar worker and Democrat = event the worker is a Democrat P(Democrat | blue) =

P(Democrat and blue) 18% = = 0.295 P(blue) 61%

63) P(A B) = P(A) P(B A) = (.15)(.8) = .12 5 5! 5! = = =5 64) 1 1!(5 - 1)! 1! 4! 65) independent; P(red or black) =

30 12 - .6; P(red or black missing cap) = = .6 50 20

66) P(A) = .3125; P(B) = .5 67) The service is available less than 99% of the time or the reception is clear less than 95% of the time or at least 5% of the company’s customers have complaints about the quality of service. P(A B) 2/36 2 P(A B) 2/36 1 ; P(B | A) = = = = = 68) P(A | B) = P(B) 11/36 11 P(B) 6/36 3 69) Let Li be the event that stock i ends up in a total loss. P(all three stocks end in total loss) = P(L1 L2 L3) = P(L1 ) × P(L2 ) × P(L3 ) = 0.31 × 0.31 × 0.31 = 0.030

70) a.

Let b1 , b2 , and b3 , represent the blue chips and r the red chip. The sample space is { b 1 b2 ,

b3 r}. b.

Each sample point is assigned the probability

P(A) = P({b1b2 , b1 b3 , b2 b3}) =

P(B) = P({b1 r, b2 r, b3 r}) =

P(C) = P( ) = 0

1 . 6

1 1 1 1 + + = 6 6 6 2

71) P(at least 25 | no fewer than 10) =

2 11

b1 b3 , b1 r, b2 b3 , b2 r,

Answer Key Testname: CHAPTER 3 72) a. b. c.

73) a. b. c.

P(woman and favored Democrats) = P(woman) × P(favored Democrats | woman) = .52 × .7 = .364 P(man and favored Democrats) = P(man) × P(favored Democrats | man) = .48 × .4 = .192 P(favored Democrats) = P(woman and favored Democrats) + P(man and favored Democrats) = .364 + .192 = .556 P(woman | favored Democrats) = P(woman and favored Democrats)/ P(favored Democrats) .364 .655 = .556 P(man | favored Democrats) = P(man and favored Democrats)/ P(favored Democrats) .192 .345 = .556

{gold, silver, bronze} 9 9 4 P(gold) = = .36, P(silver) = = .36, P(bronze) = = .28 25 25 25 P(gold) + P(silver) =

74) P(A | B) =

9 9 18 + = = .72 25 25 25

P(A B) .3 P(A B) .3 = = .5; P(B | A) = = = .6 P(B) .6 P(B) .5

75) No, P(A B) is not equal to the sum of P(A) and P(B) because events A and B are not mutually exclusive. 76) {yellow, blue, green, pink} 77) The probability that the event does occur is 1 - .21 = .79. 78) Yes; P(earning a 90 or above) = .2 and P(earning a 90 or above | male) =.2; Since these probabilities are equal, the events are independent. 2 1 1 79) P(A) = = and P(A | B) = ; Since these probabilities are not equal, A and B are not independent events. 6 3 5 80) P(Western US) = P(SW NW) = P(SW) + P(NW) = (3% + 12% + 4%) + (0% + 5% + 15%) = 19% + 20% = 39% = .39 81) Ac is the event that an even number is rolled. 82) a.

b. c.

83) a. b.

ABC, ACB, BAC, BCA, CAB, CBA 1 P({ABC}) = 6 P({BAC, BCA}) =

1 1 1 + = 6 6 3

{Sedan, Wagon, Hatchback} 42,972 P(Ac) = .53 81,589

84) P(A B) = P(A) P(B) = (.32)(.55) = .176 18 1 2 1 = and P(A | B) = = ; Since these probabilities are equal, A and B are independent events. 85) P(A) = 36 2 4 2 86) Bc = {3, 4, 6}

Answer Key Testname: CHAPTER 3 87) a. b.

{Sedan, Convertible, Wagon, SUV, Van, Hatchback}

Type of Car Sedan Convertible Wagon SUV Van Hatchback

Probability .09 .11 .25 .17 .19 .19

c. P(Van) + P(SUV) = .17 +.19 =.36 88) Yes, the events are mutually exclusive. If the chips are both red, then neither of the chips is blue, so the events have no sample points in common. 7 = 1 - .7 = .3 89) P(will not donate) = 1 - P(will donate) = 1 10

90) P(does not receive government aid) = 1 - P(receives government aid) = 1 -

240 = 1 - .3 = .7 800

91) A B = {E2 , E3 }; P(A B) = P(E2 ) + P(E3) = .3 + .1 = .4

92) Events A and C are mutually exclusive since a number can not be both even and odd. 420 = .56 93) P(man) = 750 94) a.

6 6! 6! = = = 20; {ABC, ABD, ABE, ABF, ACD, ACE, ACF, ADE, ADF, AEF, BCD, BCE, BCF, BDE, BDF, BEF, 3 3!(6 - 3)! 3! 3!

CDE, CDF, CEF, DEF} 1 b. P(ADE) = = .05 20

95) Yes, P(A B) is equal to the sum of P(A) and P(B) because events A and B are mutually exclusive. 96) Using the Additive Rule, the probability is .62 + .35 - .20 = .77. 97) P(Ac) = 1 - 0.37 = .63 98)

15 15! 15! = = = 5005 6 6!(15 - 6)! 6! 9!

99) a. b. c.

At least one chip is not red. {b1 b2 , b1 r1 , b1 r2 , b2 r1 , b2 r2 } 5 P(Ac) = 6

100) P(A) = .3; P(B) = .4 101) P(donates a dollar) =

7 = .7 10

102) a. {economy, national security, not sure} b. P(economy) = .45, P(national security) = .35, P(not sure) = .20 c. P(economy) + P(national security) =.45 + .35 =.80 103) a. A B C b. A B C 104) P(white collar Republican) = P(white collar Republican) = 10% = .10 105) {1, 2, 3, 4, 5, 6, 7} 9 = .5 106) P(gold | not bronze) = 18 45

Answer Key Testname: CHAPTER 3 107) No, there is not enough information, since catching a red snapper and catching a fish that weighs less than 5 pounds are probably not mutually exclusive events (It may be possible to catch a red snapper that weighs less than 5 pounds). 108) a. {children’s, fiction, nonfiction, educational} b. Type of Book Probability Children's .10 Fiction .28 Nonfiction .49 Educational .13 c. P(nonfiction) + P(educational) = .49 + .13 = .62 109) P(A B) = .37 + .69 - .23 = .83 110) B 111) B 112) A 113) C 114) D 115) D 116) A 117) A 118) A 119) D 120) C 121) A 122) B 123) A 124) B 125) C 126) C 127) B 128) A 129) D 130) D 131) B 132) D 133) D 134) B 135) B 136) B 137) D 138) A 139) C 140) C 141) C 142) B 143) B 144) B 145) B 146) C

Answer Key Testname: CHAPTER 3 147) B 148) C 149) B 150) A 151) A 152) B 153) B 154) D 155) C 156) C 157) D 158) B 159) D 160) D 161) B 162) A 163) B 164) D 165) A 166) D 167) B 168) A 169) A 170) A 171) A 172) C 173) B 174) B 175) C 176) D 177) D 178) C 179) C 180) A 181) A 182) A 183) B 184) B 185) A 186) B 187) A 188) A 189) A 190) C 191) A 192) A 193) D 194) A 195) A 196) D 47

Answer Key Testname: CHAPTER 3 197) A 198) B 199) A 200) C 201) C 202) A 203) B 204) C 205) A 206) D 207) B 208) D 209) D 210) D 211) D 212) A 213) D 214) B 215) B 216) A 217) C 218) C 219) A 220) A 221) A 222) C 223) C 224) D 225) C 226) B 227) C 228) B 229) C 230) D 231) C 232) C 233) C 234) D 235) C 236) A 237) D

Chapter 4 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit. A) 0.105 B) 0.219 C) 0.114 D) 0.895

2) The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth. A) 0.8333 B) 0.7619 C) 0.3600 D) 0.2381

3) The time between arrivals at an ATM machine follows an exponential distribution with = 10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals. A) 0.141045 B) 0.305215 C) 0.223130 D) 0.082085

4) Which binomial probability is represented on the screen below?

A) P(x > 4)

B) P(x = 4)

C) P(x 4)

D) P(x < 4)

5) Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the mean of this distribution. A) 12 B) 3 C) 4 D) 1

is 6) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution 6) shown below. Find the mean of the distribution. x 2 3 5 8 10 p(x) 0.10 0.20 0.30 0.30 0.10

A) 5.6

B) 5.7

C) 5.5

D) 5.0

7) Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the probability that none of the four cans selected contains an incorrect mix of paint. A) 0.21818 B) 0.01818 C) 0.50909 D) 0.25455

8) A statistician received some data to analyze. The sender of the data suggested that the data was normally 8) distributed. Which of the following methods can be used to determine if the data is, in fact, normally distributed? I. Construct a histogram and/or stem-and-leaf display of the data and check the shape. II. Compute the intervals x ± s, x ± 2s, and x ± 3s, and determine the percentage of measurements falling in each. Compare these percentages to 68%, 95%, and 100%. IQR III. Calculate a value of . If this value is approximately 1.3, then the data is normal. s IV. Construct a normal probability plot of the data. If the points fall on a straight line, then the data is normal. A) I, II, III, and IV B) III only C) II only D) IV only E) I only

9) A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information. What proportion of the paint cans contain less than 5.54 ml of the dye? A) 0.0885 B) 0.9115 C) 0.9885 D) 0.5885

10) The time between arrivals at an ATM machine follows an exponential distribution with = 10 minutes. Find the mean and standard deviation of this distribution. A) Mean = 10, Standard Deviation = 100 B) Mean = 3.16, Standard Deviation = 3.16 C) Mean = 10, Standard Deviation = 10 D) Mean = 10, Standard Deviation = 3.16

10)

11) The number of homeruns hit during a major league baseball game follows a Poisson distribution with = 3.2. Find the mean and standard deviation for this distribution. A) mean = 3.2, standard Deviation = 10.24 B) mean = 1.79, standard Deviation = 3.2 C) mean = 3.2, standard Deviation = 3.2 D) mean = 3.2, standard Deviation = 1.79

11)

12) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the probability for the value of x = 5.

12)

x 2 3 5 8 10 p(x) 0.10 0.20 ??? 0.30 0.10

A) 0.3

B) 0.1

C) 0.7

D) 0.2

is 13) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution 13) shown below. Find the standard deviation of the distribution. x 2 3 5 8 10 p(x) 0.10 0.20 0.30 0.30 0.10

A) 2.532

B) 5.7

C) 1.845

D) 6.41

14) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 10 but fewer than 16 of the 20 eligible voters sampled will vote in the next presidential election. A) 0.845 B) 0.780 C) 0.714 D) 0.649

14)

15) The school newspaper surveyed 100 commuter students and asked two questions. First, students were asked how many courses they were currently enrolled in. Second, the commuter students were asked to estimate how long it took them to drive to campus. Considering these two variables, number of courses would best be considered a _________ variable and drive time would be considered a _________ variable. A) discrete; continuous B) continuous; discrete C) discrete; discrete D) continuous; continuous

15)

16) It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they did have at least one tattoo. A) 0.218 B) 0.207 C) 0.515 D) 0.722

16)

17) After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts. A) 2.02 inches B) 4.08 inches C) 1.42 inches D) 18.5 inches

17)

18) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Use a binomial probability table to find the probability that more than 12 of the eligible voters sampled will vote in the next presidential election. A) 0.228 B) 0.392 C) 0.772 D) 0.887 E) 0.608

18)

19) After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the probability that a randomly selected location in this village had between 17 and 18 inches of snow. Round to the nearest ten-thousandth. A) 0.4286 B) 0.2857 C) 0.5714 D) 0.1429

19)

20) A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Can the normal approximation be used to estimate the binomial distribution in this problem? A) No B) Yes

20)

21) It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Find the standard deviation for this binomial random variable. Round to the nearest hundredth when necessary. A) 1.77 B) 10.5 C) 4.5 D) 3.15

21)

22) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 22) 20 eligible voters were randomly selected from the population of all eligible voters. Which of the following is necessary for this problem to be analyzed using the binomial random variable? I. There are two outcomes possible for each of the 20 voters sampled. II. The outcomes of the 20 voters must be considered independent of one another. III. The probability a voter will actually vote is 0.70, the probability they won't is 0.30. A) III only B) I, II, and III C) I only D) II only

23) The time between arrivals at an ATM machine follows an exponential distribution with = 10 minutes. Find the probability that more than 25 minutes will pass between arrivals. A) 0.082085 B) 0.670320 C) 0.329680 D) 0.917915

23)

24) A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information. Find the dye amount that represents the 9th percentile of the distribution. A) 4.464 ml B) 5.536 ml C) 4.936 ml D) 4.836 ml E) 4.964 ml

24)

25) Which of the following is not a method used for determining whether data are from an approximately normal distribution? A) Construct a histogram or stem-and-leaf display. The shape of the graph or display should be uniform (evenly distributed). B) Construct a normal probability plot. The points should fall approximately on a straight line.

25)

C) Compute the intervals x ± s, x ± 2s, and x ± 3s. The percentages of measurements falling in each should be approximately 68%, 95%, and 100% respectively. IQR 1.3. D) Find the interquartile range, IQR, and standard deviation, s, for the sample. Then s 26) For a standard normal random variable, find the point in the distribution in which 11.9% of the z-values fall below. A) -1.18 B) 1.18 C) -1.45 D) -0.30

26)

27) Which one of the following suggests that the data set is not approximately normal?

27)

A) A data set with 68% of the measurements within x ± 2s. B) Stem Leaves 3 0 3 9 4 2 4 7 7 5 1 3 4 8 8 9 9 9 6 0 0 5 6 6 7 8 7 1 1 5 8 2 7 C) A data set with IQR = 752 and s = 574. D)

28) A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 36% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled and asked if they believe their company president has low ethical standards and their years of experience at the company. Could the probability distribution for the number of years of experience be modelled by a binomial probability distribution? A) Yes, the sample size is n = 20. B) No, the employees would not be considered independent in the present sample. C) Yes, the sample is a random and independent sample. D) No, a binomial distribution requires only two possible outcomes for each experimental unit sampled.

28)

29) The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. What proportion of the milk vendors had prices that were less than $3.075 per gallon? A) 0.3944 B) 0.2112 C) 0.8944 D) 0.1056

29)

30) For a standard normal random variable, find the probability that z exceeds the value -1.65. A) 0.9505 B) 0.4505 C) 0.5495 D) 0.0495

30)

31) A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. How many of the sampled voters do we expect to vote in the next presidential election? A) 6 B) 0.7 C) 14 D) 0.3

31)

32) An alarm company reports that the number of alarms sent to their monitoring center from customers owning their system follow a Poisson distribution with = 4.7 alarms per year. Identify the mean and standard deviation for this distribution. A) mean = 4.7, standard Deviation = 4.7 B) mean = 2.17, standard Deviation = 2.17 C) mean = 2.17, standard Deviation = 4.7 D) mean = 4.7, standard Deviation = 2.17

32)

33) If a data set is normally distributed, what is the proportion of measurements you would expect to fall within µ ± ? A) 50% B) 68% C) 100% D) 95%

33)

34) Which binomial probability is represented on the screen below?

34)

A) The probability of 2 successes in 8 trials where the probability of success is .3. B) The probability of 2 successes in 8 trials where the probability of failure is .3. C) The probability of 8 successes in 2 trials where the probability of success is .3. D) The probability of 8 failures in 2 trials where the probability of failure is .3. 35) The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. Find the price for which 12.3% of milk vendors exceeded. A) $3.238 B) $3.084 C) $3.316 D) $3.215

35)

36) An alarm company reports that the number of alarms sent to their monitoring center from customers owning their system follow a Poisson distribution with = 4.6 alarms per year. Find the probability that a randomly selected customer had more than 7 alarms reported. A) 0.905 B) 0.818 C) 0.087 D) 0.095 E) 0.182

36)

37) Which of the following statements is not a property of the normal curve? A) P(µ - 3 < x < µ + 3 ) .997 B) P(µ - < x < µ + ) .95 C) symmetric about µ D) mound-shaped (or bell shaped)

37)

38) Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the standard deviation of this distribution. A) 1 B) 0.739 C) 0.297 D) 0.545

38)

39) Which one of the following suggests that the data set is approximately normal? A) A data set with Q1 = 14, Q3 = 68, and s = 41.

39)

B) A data set with Q1 = 105, Q3 = 270, and s = 33.

C) A data set with Q1 = 1330, Q3 = 2940, and s = 2440. D) A data set with Q1 = 2.2, Q3 = 7.3, and s = 2.1. 40) The time between arrivals at an ATM machine follows an exponential distribution with = 10 minutes. Find the probability that less than 25 minutes will pass between arrivals. A) 0.917915 B) 0.329680 C) 0.670320 D) 0.082085

40)

is 41) A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution 41) shown below. Find the probability that the random variable x is a value greater than 5. x 2 3 5 8 10 p(x) 0.10 0.20 0.30 0.30 0.10

A) 0.70

B) 0.60

C) 0.30

D) 0.40

42) The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the average age of customers to this hardware store. A) 39 years old B) 60 years old C) 45 years old D) 50 years old

42)

43) Suppose a man has ordered twelve 1-gallon paint cans of a particular color (lilac) from the local paint store in order to paint his mother's house. Unknown to the man, three of these cans contains an incorrect mix of paint. For this weekend's big project, the man randomly selects four of these 1-gallon cans to paint his mother's living room. Let x = the number of the paint cans selected that are defective. Unknown to the man, x follows a hypergeometric distribution. Find the probability that at least one of the four cans selected contains an incorrect mix of paint. A) 0.74545 B) 0.49091 C) 0.78182 D) 0.50909

43)

44) For a binomial distribution, which probability is not equal to the probability of 1 success in 5 trials where the probability of success is .4? A) the probability of 4 failures in 5 trials where the probability of failure is .6 B) the probability of 4 failures in 5 trials where the probability of success is .6 C) the probability of 4 failures in 5 trials where the probability of success is .4 D) the probability of 1 success in 5 trials where the probability of failure is .6

44)

45) 50 students were randomly sampled and asked questions about their exercise habits. One of the questions they were asked concerned the frequency of exercise, defined to be the number of times they exercised in a week. This variable would be characterized as which type of random variable? A) discrete B) continuous

45)

46) A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the mean and standard deviation for this binomial distribution. A) Mean = 80, Standard Deviation = 3.87 B) Mean = 20, Standard Deviation = 3.87 C) Mean = 20, Standard Deviation = 15 D) Mean = 80, Standard Deviation = 15

46)

47) Data has been collected and a normal probability plot for one of the variables is shown below. Based on47) your knowledge of normal probability plots, do you believe the variable in question is normally distributed? The data are represented by the"o" symbols in the plot.

A) Yes. The plot reveals a curve and this indicates the variable is normally distributed. B) No. The plot does not reveal a straight line and this indicates the variable is not normally distributed. C) Yes. The plot reveals a straight line and this indicates the variable is normally distributed. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 48) For a binomial distribution, if the probability of success is .48 on the first trial, what is the probability of failure on the second trial?

48)

49) Calculate the mean for the discrete probability distribution shown here.

49)

X 2 6 11 13 P(X) .2 .3 .3 .2

50) A bank offers online banking to its customers free of charge. While online, customers can also sign up for additional services that the bank offers. Let x be the number of customers who sign up for additional services online each day. Suppose the distribution of x is approximated well by a Poisson distribution with mean = 42.3. Find E(x) and interpret its value.

50)

51) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 4200 miles. If the manufacturer guarantees the tread life of the tires for the first 54,960 miles, what proportion of the tires will need to be replaced under warranty?

51)

52) Explain why the following is or is not a valid probability distribution for the discrete random variable x. x p(x)

1 .1

0 .2

1 .3

2 .3

52)

3 .1

53) The board of examiners that administers the real estate broker's examination in a certain state found that the mean score on the test was 595 and the standard deviation was 72. If the board wants to set the passing score so that only the best 10% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

53)

54) A local newspaper claims that 70% of the items advertised in its classifieds section are sold within 1 week of the first appearance of the ad. To check the validity of the claim, the newspaper randomly selected n = 25 advertisements from last year's classifieds and contacted the people who placed the ads. They found that 17 of the 25 items sold within a week. Based on the newspaper's claim, is it likely to observe x 17 who sold their item within a week? Use a binomial probability table.

54)

55) Compute

6 (.3)2 (.7)6-2 . 2

55)

56) Consider the given discrete probability distribution. Construct a graph for p(x). x p(x)

1 .1

2 .2

3 .2

4 .3

56)

5 .2

57) The length of time (in months) that a cashier works for a certain fast food restaurant is exponentially distributed with a mean of 7 months.

57)

a. Find the probability that a cashier works for the restaurant for at least 2 years. b. Find the probability that a cashier works for the restaurant for less than 1 month.

58) A small life insurance company has determined that on the average it receives 5 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.

58)

59) Suppose that x has an exponential distribution with

59)

a. P(x 2) b. P(x < 2)

= 1.75. Find each probability.

60) The printout below contains summary statistics of the heights of a sample of 200 adult men in the 60) United States. Descriptive Statistics: HT Variable HT

N 200

Mean 70.187

StDev 2.716

Minimum 62.375

Q1 67.875

Median 69.625

Q2 71.500

Maximum 91.125

Use the information in the printout to determine whether the distribution of heights is approximately normal. Explain your reasoning.

61) An automobile manufacturer has determined that 30% of all gas tanks that were installed on its 2002 compact model are defective. If 10 of these cars are independently sampled, what is the probability that more than half need new gas tanks?

61)

62) Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 459 tomatoes and a standard deviation of 30 tomatoes. If there are 417 tomatoes available to be sold at the roadside stand at the beginning of a day, what is the probability that they will all be sold?

62)

63) High temperatures in a certain city for the month of August follow a uniform distribution over the interval 75°F to 95°F. What is the probability that a random day in August has a high temperature that exceeds 80°F?

63)

64) The following data represent the scores of a sample of 50 students on a statistics exam. The

64)

mean score is x = 80.3, and the standard deviation is s = 11.37. 49 71 79 85 90

51 71 79 86 91

59 73 79 86 92

63 74 80 88 92

66 76 80 88 93

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 98

What percentage of the scores fall in each of the intervals x ± s, x ± 2s, and x ± 3s? Based on these percentages, do you believe that the distribution of scores is approximately normal? Explain.

65) A coin is flipped 6 times. The variable x represents the number of tails obtained. List the possible values of x. Is x discrete or continuous? Explain.

65)

66) An airline has requests for standby flights at half of the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $180 and the cost of flying first class is $370. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying standby.]

66)

67) Consider the given discrete probability distribution. Construct a graph for p(x). x p(x)

1 .30

2 .25

3 .20

4 .15

5 .05

6 .05

68) The hypergeometric random variable x counts the number of successes in the draw of 3 elements from a set of 8 elements containing 4 successes. List the possible values of x. 69) Compute

x ex!

for

67)

= 5 and x = 7.

68)

69)

70) Determine if it is appropriate to use the normal distribution to approximate a binomial distribution when n = 24 and p = 0.3.

70)

71) Determine if it is appropriate to use the normal distribution to approximate a binomial distribution when n = 10 and p = 0.2.

71)

72) The rate of return for an investment can be described by a normal distribution with mean 37% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 43%?

72)

73) It is against the law to discriminate against job applicants because of race, religion, sex, or age. Of the individuals who apply for an accountant's position in a large corporation, 42% are over 45 years old. If the company decides to choose 66 of a very large number of applicants for closer credential screening, claiming that the selection will be random and not age-biased, what is the z-value associated with fewer than 31 of those chosen being over 45 years old? (Assume that the applicant pool is large enough so that x, the number in the sample over 45 years old, has a binomial probability distribution.)

73)

74) Suppose x is a random variable for which a Poisson probability distribution with = 3 provides a good approximation.

74)

a. b. c.

Graph p(x) for x = 0, 1, 2, 3, 4, 5, 6. Find µ and for x. What is the probability that x will fall in the interval µ ± ?

75) Consider the given discrete probability distribution. x p(x) a.

1 .1

2 .2

3 .2

4 .3

75) 5 .2

Find µ = E(x).

b. Find = E[(x - µ)2 ]. c. Find the probability that the value of x falls within one standard deviation of the mean. Compare this result to the Empirical Rule.

76) The time between equipment failures (in days) at a particular factory is exponentially distributed 76) with a mean of 4.5 days. A machine just failed and was repaired today. a. Find the probability that another machine will fail within the next day. b. Find the probability that there will be no more equipment failures in the next week.

77) Consider the given discrete probability distribution. Find P(x = 1 or x = 2). x p(x)

0 .30

1 .25

2 .20

3 .15

4 .05

77)

5 .05

78) A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 5.5 gallons per minute. Find the probability that the machine pumps less than 4.75 gallons during a randomly selected minute.

78)

79) Explain why the following is or is not a valid probability distribution for the discrete random variable x.

79)

x p(x)

10 .3

20 .2

30 .2

40 .2

50 .2

80) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 5.3. Find the probability that fewer than two accidents will occur on this stretch of road during a month.

80)

81) Suppose x is a uniform random variable with c = 10 and d = 70. Find the mean of the random variable x.

81)

82) Explain why the following is or is not a valid probability distribution for the discrete random variable x.

82)

x p(x)

1 .1

3 .1

5 .2

7 .1

9 .2

83) You are performing a study about the weight of preschoolers. A previous study found the 83) weights to be normally distributed with a mean of 30 pounds and a standard deviation of 4 pounds. You randomly sample 30 preschool children and find their weights (in pounds) to be as follows. 25 29 33

25 29 33

26 30 34

26.5 30 34.5

27 30.5 35

27 31 35

27.5 31 37

28 32 37

28 32.5 38

28.5 32.5 38

Draw a histogram to display the data. Is it reasonable to assume that the weights are normally distributed? Why?

84) For a binomial distribution, if the probability of success is .63 on the first trial, what is the probability of success on the second trial?

84)

85) An automobile insurance company estimates the following loss probabilities for the next year on 85) a $25,000 sports car: Total loss: 50% loss: 25% loss: 10% loss: No loss:

0.001 0.01 0.05 0.10 0.839

Assuming the company will sell only a $500 deductible policy for this model (i.e., the owner covers the first $500 damage), how much annual premium should the company charge in order to average $405 profit per policy sold?

86) Given that x is a hypergeometric random variable with N = 10, n = 3, and r = 5:

86)

a. Display the probability distribution in tabular form. b. Compute µ and for x. c. What is the probability that x will fall within the interval µ ± 2

87) A bottle contains 16 ounces of water. The variable x represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of x? Is x discrete or continuous? Explain.

87)

88) About 40% of the general population donate time and energy to community projects. Suppose 15 people have been randomly selected from a community and each asked whether he or she donates time and energy to community projects. Let x be the number who donate time and energy to community projects. Use a binomial probability table to find the probability that more than five of the 15 donate time and energy to community projects.

88)

89) Suppose that the random variable x has an exponential distribution with

89)

= 3.

a. Find the probability that x assumes a value more than three standard deviations from µ. b. Find the probability that x assumes a value less than one standard deviation from µ. c. Find the probability that x assumes a value within a half standard deviation of µ.

90) The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 7.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 5.5 millimeters?

90)

91) You test 4 items from a lot of 15. What is the probability that you will test no defective items if the lot contains 3 defective items?

91)

92) Consider the given discrete probability distribution. Find P(x < 2 or x > 3). x p(x)

1 .1

2 .2

3 .2

4 .3

92)

5 .2

93) Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 6. Find the probability that exactly two babies are born during a randomly selected hour.

93)

94) Given that x is a hypergeometric random variable with N = 8, n = 4, and r = 3:

94)

a. Display the probability distribution in tabular form. b. Find P(x 2).

95) Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 112 tomatoes and a standard deviation of 30 tomatoes. How many tomatoes must be available on any given day so that there is only a 1.5% chance that all tomatoes will be sold?

95)

96) You test 3 items from a lot of 12. What is the probability that you will test no defective items if the lot contains 2 defective items?

96)

97) Explain why the following is or is not a valid probability distribution for the discrete random variable x.

97)

x p(x)

0 -.1

2 .1

4 .2

6 .3

8 .5

98) A new drug is designed to reduce a person's blood pressure. Eleven randomly selected hypertensive patients receive the new drug. Suppose the probability that a hypertensive patient's blood pressure drops if he or she is untreated is 0.5. Then what is the probability of observing 9 or more blood pressure drops in a random sample of 11 treated patients if the new drug is in fact ineffective in reducing blood pressure? Round to six decimal places.

98)

99) Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of x.

99)

100) You randomly select 7 students from a class with 15 male and 20 female students. What is the probability that you will choose exactly 4 females?

100)

101) Suppose that 88% of the stocks listed on a particular exchange increased in value yesterday. Let x be the number of stocks that increased in value yesterday in a random of 72 stocks listed on the exchange. Find the mean and standard deviation of x.

101)

102) Find the mean and standard deviation of the probability distribution for the random variable x, which represents the number of cars per household in a small town. x 0 1 2 3 4

102)

P(x) .125 .428 .256 .108 .083

103) The board of examiners that administers the real estate broker's examination in a certain state found that the mean score on the test was 466 and the standard deviation was 72. If the board wants to set the passing score so that only the best 80% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

103)

104) An online retailer reimburses a customer's shipping charges if the customer does not receive his order within one week. Delivery time (in days) is exponentially distributed with a mean of 3.2 days. What percentage of customers have their shipping charges reimbursed?

104)

105) Given that x is a hypergeometric random variable with N = 10, n = 5, and r = 6, find each probability.

105)

a. P(x = 0) b. P(x = 1) c. P(x 1) d. P(x 2)

106) A loan officer has 78 loan applications to screen during the next week. If past record indicates that she turns down 19% of the applicants, what is the z-value associated with 73 or more of the 78 applications being rejected?

106)

107) The rate of return for an investment can be described by a normal distribution with mean 47% and standard deviation 3%. What is the probability that the rate of return for the investment will be at least 42.5%?

107)

108) The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of x.

108)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the probability. 109) Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 30. Find P(x > 22). A) 0.6 B) 0.04 C) 0.4 D) 0.08 Solve the problem. 110) Suppose a random variable x is best described by a normal distribution with µ = 60 and Find the z-score that corresponds to the value x = 60. A) 12 B) 5 C) 0 D) 1

= 12.

109)

110)

111) The waiting time (in minutes) between ordering and receiving your meal at a certain restaurant is exponentially distributed with a mean of 10 minutes. The restaurant has a policy that your meal is free if you have to wait more than 25 minutes after ordering. What is the probability of receiving a free meal? A) 0.670320 B) 0.917915 C) 0.082085 D) 0.329680

111)

112) Consider the given discrete probability distribution. Find the probability that x equals 5.

112)

x 3 P(x) 0.26 A) 0.64

5 ?

7 0.28

9 0.1

B) 3.2

C) 1.8

D) 0.36

113) Classify the following random variable according to whether it is discrete or continuous. The speed of a car on a Los Angeles freeway during rush hour traffic A) discrete B) continuous

113)

Find the probability. 114) Assume that x is a binomial random variable with n = 400 and p = 0.30. Use a normal approximation to find P(x 110). A) 0.5517 B) 0.8508 C) 0.8749 D) 0.3749 Solve the problem. 115) Suppose a random variable x is best described by a normal distribution with µ = 60 and Find the z-score that corresponds to the value x = 0. A) -3.75 B) 3.75 C) 16 D) -16

114)

= 16.

116) Given that x is a hypergeometric random variable with N = 15, n = 6, and r = 10, compute P(x = 0). A) .001 B) 0 C) 1 D) .002 Answer the question True or False. 117) The probability density function for an exponential random variable x has a graph called a bell curve. A) True B) False Solve the problem. 118) Suppose a random variable x is best described by a normal distribution with µ = 60 and Find the z-score that corresponds to the value x = 70. A) 10 B) 70 C) 6 D) 1 Answer the question True or False. 119) The total area under a probability distribution equals 1. A) True B) False Solve the problem. 120) Classify the following random variable according to whether it is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch A) discrete B) continuous

= 10.

115)

116)

117)

118)

119)

120)

Answer the question True or False. 121) When the points on a normal probability plot lie approximately on a straight line, the data are approximately normally distributed. A) True B) False Solve the problem. 122) Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 6. Find the value of a that makes the following probability statement true: P(x a) = 0.4. A) 4 B) 4.4 C) 3.6 D) 1.6

121)

122)

123) The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 4. Find the probability that exactly six road construction projects are currently taking place in this city. A) 0.104196 B) 0.133853 C) 0.423040 D) 0.032968

123)

124) The volume of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a mean of 12.54 ounces and a standard deviation of 0.36 ounce. The company receives complaints from consumers who actually measure the amount of soda in the cans and claim that the volume is less than the advertised 12 ounces. What proportion of the soda cans contain less than the advertised 12 ounces of soda? A) .9332 B) .0668 C) .4332 D) .5668

124)

125) Determine the value of e-a for =2 and a = 3. A) 4.481689 B) 0.223130

125)

C) -4.481689

D) 0.513417

126) Transportation officials tell us that 90% of drivers wear seat belts while driving. What is the probability of observing 511 or fewer drivers wearing seat belts in a sample of 600 drivers? A) approximately 1 B) 0.1 C) approximately 0 D) 0.9

126)

127) If x is a binomial random variable, calculate µ for n = 50 and p = 0.7. A) 10.5 B) 35 C) 3.5

127)

D) 25

Answer the question True or False. 128) P(-1 < x < 0) = P(0 < x < 1) for any random variable x that is normally distributed. A) True B) False Solve the problem. 129) Suppose that the random variable x has an exponential distribution with probability that x will assume a value within the interval µ ± 2 . A) .950213 B) .716531 C) .049787

= 1.5. Find the

129)

D) .864665

130) The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 8.9. Find the probability that exactly four tickets are written on a randomly selected day. A) .964344 B) .035656 C) .058433 D) .941567

128)

130)

131) Calculate the mean for the discrete probability distribution shown here. X 1 5 8 12 P(X) 0.2 0.28 0.25 0.27 A) 1.71 B) 6.84

C) 26

131)

D) 6.5

132) Classify the following random variable according to whether it is discrete or continuous. The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete

132)

133) If x is a binomial random variable, calculate 2 for n = 40 and p = 0.2. A) 6.4 B) 2.53 C) 8

133)

134) Compute A) 20

D) 1.6

5 . 4

134) B) 10

C) 5

D) 1

135) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.5. Find the probability that fewer than three accidents will occur next month on this stretch of road. A) 0.990717 B) 0.969891 C) 0.009283 D) 0.030109 Find the probability. 136) Assume that x is a binomial random variable with n = 400 and p = 0.30. Use a normal approximation to find P(x > 100). A) 0.9875 B) 0.5854 C) 0.5910 D) 0.9834 Solve the problem. 137) Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Find the probability that exactly seven babies will be born during a particular 1-hour period at this hospital. A) 0.000026 B) 0.021604 C) 0.000000 D) 0.002701

135)

136)

137)

138) The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 3 goals per game. Find the probability that the team scored exactly six goals in each of four randomly selected games. A) 0.20163763 B) 0.00002251 C) 0.79836237 D) 0.00000646

138)

139) The preventable monthly loss at a company has a normal distribution with a mean of $4900 and a standard deviation of $30. A new policy was put into place, and the preventable loss the next month was $4720. What inference can you make about the new policy? A) Because the probability that the monthly loss would be as low as $4720 is small, the new policy is working. B) Because the probability that the monthly loss would be as low as $4720 is not very small, the new policy is not working. C) The new policy is probably less effective than the one it replaced. D) While the probability that the monthly loss would be as low as $4720 is small, it is not unexpected.

139)

140) Suppose the candidate pool for two appointed positions includes 6 women and 9 men. All candidates were told that the positions were randomly filled. Find the probability that two men are selected to fill the appointed positions. A) .160 B) .360 C) .143 D) .343

140)

141) Given that x is a hypergeometric random variable, compute p(x) for N = 8, n = 5, r = 3, and x = 2. A) .140 B) .536 C) .343 D) .464

141)

Find the probability. 142) Assume that x is a binomial random variable with n = 1000 and p = 0.80. Use a normal approximation to find P(800 < x 810). A) 0.2574 B) 0.0239 C) 0.2807 D) 0.2967 Solve the problem. 143) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 1800 miles. What is the probability a particular tire of this brand will last longer than 58,200 miles? A) .2266 B) .1587 C) .7266 D) .8413 144) The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 5.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 4.8 millimeters? A) 0.35 B) 0.8727 C) 0.5333 D) 2 Find the probability. 145) Assume that x is a binomial random variable with n = 100 and p = 0.60. Use a normal approximation to find P(x 48). A) 0.4906 B) 0.0054 C) 0.3156 D) 0.0094 Solve the problem. 146) In a pizza takeout restaurant, the following probability distribution was obtained for the number of toppings ordered on a large pizza. Find the mean and standard deviation for the random variable. x P(x) 0 .30 1 .40 2 .20 3 .06 4 .04 A) mean: 1.54; standard deviation: 1.30 C) mean: 1.30; standard deviation: 1.54

142)

143)

144)

145)

146)

B) mean: 1.14; standard deviation: 1.04 D) mean: 1.30; standard deviation: 2.38

Answer the question True or False. 147) For a continuous probability distribution, the probability that x is between a and b is the same regardless of whether or not you include the endpoints, a and b, of the interval. A) True B) False

147)

Find the probability. 148) Suppose x is a random variable best described by a uniform probability distribution with c = 40 and d = 100. Find P(40 x 64). A) 0.24 B) 0.04 C) 0.5 D) 0.4 Solve the problem. 149) A machine is set to pump cleanser into a process at the rate of 6 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 5.5 to 7.5 gallons per minute. Find the variance of the distribution. A) 0.08 B) 14.08 C) 3.00 D) 0.33 150) Transportation officials tell us that 60% of drivers wear seat belts while driving. Find the probability that more than 409 drivers in a sample of 650 drivers wear seat belts. A) 0.0594 B) 0.6 C) 0.9406 D) 0.4

148)

149)

150)

151) A local bakery has determined a probability distribution for the number of cheesecakes it sells in a given 151) day. The distribution is as follows: Number sold in a day Prob (Number sold)

0 0.16

5 0.17

10 0.24

15 0.22

20 0.21

Find the number of cheesecakes that this local bakery expects to sell in a day. A) 10.75 B) 10 C) 10.91

D) 20

152) A machine is set to pump cleanser into a process at the rate of 6 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 5.5 to 8.5 gallons per minute. What is the probability that at the time the machine is checked it is pumping more than 7.0 gallons per minute? A) .50 B) .667 C) .7692 D) .25 Answer the question True or False. 153) The exponential distribution is governed by two quantities, µ and , that determine its shape and location A) True B) False Solve the problem. 154) The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a standard deviation of 0.04 ounce. Every can that has more than 12.10 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the mean amount of soda the machine should dispense if the company wants to limit the percentage that must be cleaned because of spillage to 3%? A) 12.0132 ounces B) 12.1752 ounces C) 12.0248 ounces D) 12.1868 ounces

152)

153)

154)

155) Classify the following random variable according to whether it is discrete or continuous. The blood pressures of a group of students the day before the final exam A) discrete B) continuous

155)

156) Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 80. What is the mean outcome of this experiment? A) 40 B) 80 C) 60 D) 65

156)

157) We believe that 81% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 39 students from the population. How many of the sampled students do we expect to consider statistics to be an exciting subject? A) 32.16 B) 33.82 C) 39 D) 31.59

157)

158) A dice game involves rolling three dice and betting on one of the six numbers that are on the dice. 158) The game costs $3 to play, and you win if the number you bet appears on any of the dice. The distribution for the outcomes of the game (including the profit) is shown below: Number of dice with your number 0 1 2 3

Profit -$3 $3 $5 $9

Probability 125/216 75/216 15/216 1/216

Find your expected profit from playing this game. A) -$0.31 B) $3.16

159) Compute A) 840

C) $1.85

D) $0.50

7! . 3!(7 - 3)!

159) B) 70

C) 210

D) 35

160) A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 41% of all employees believe their company president possesses low ethical standards. Assume that responses were randomly and independently collected. A president of a local company that employs 1,000 people does not believe the paper's claim applies to her company. If the claim is true, how many of her company's employees believe that she possesses low ethical standards? A) 41 B) 590 C) 959 D) 410

160)

161) The number of road construction projects that take place at any one time in a certain city follows a Poisson distribution with a mean of 3. Find the probability that more than four road construction projects are currently taking place in the city. A) 0.352768 B) 0.815263 C) 0.647232 D) 0.184737

161)

162) A certain baseball player hits a home run in 6% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits more than 32 home runs in 750 at-bats? A) 0.0274 B) 0.94 C) 0.06 D) 0.9726

162)

163) The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 5.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing process? A) 4.5 millimeters B) 5.5 millimeters C) 5.0 millimeters D) 4.0 millimeters

163)

164) According to a published study, 1 in every 9 men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled twenty-five men and asked each whether he has been involved in a minor traffic accident. How many of the 25 men do we expect to have never been involved in a minor traffic accident? Round to the nearest whole number. A) 25 B) 3 C) 9 D) 22

164)

165) A recent survey found that 73% of all adults over 50 wear glasses for driving. In a random sample of 80 adults over 50, what is the mean and standard deviation of the number who wear glasses? Round to the nearest hundredth when necessary. A) mean: 58.4; standard deviation: 3.97 B) mean: 58.4; standard deviation: 7.64 C) mean: 21.6; standard deviation: 7.64 D) mean: 21.6; standard deviation: 3.97

165)

166) According to a recent study, 1 in every 7 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that more than 22 of the women sampled have not been the victim of domestic abuse. A) 0.286196 B) 0.774262 C) 0.067132 D) 0.176664

166)

167) A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below:

167)

Price Probability

$10.00 0.2

$12.50 0.25

$15.00 0.55

How much should the lab budget for next year's frog orders assuming this distribution does not change? (Hint: Find the expected price and assume 52 weeks per year.) A) $13.38 B) $3,616,600.00 C) $1338.00 D) $695.50

168) Suppose that x has an exponential distribution with A) 0.527633 B) 0.736403

= 2. Find P(x < 1.5). C) 0.472367

D) 0.263597

169) Suppose that x has an exponential distribution with A) 0.223130 B) 0.513417

=1.5. Find P(x > 1). C) 0.776870

D) 0.486583

170) Suppose a random variable x is best described by a normal distribution with µ = 60 and the z-score that corresponds to the value x = 84. 5 A) B) 4 C) 24 D) 6 2

= 6. Find

Answer the question True or False. 171) The exponential distribution has the property that its mean equals its standard deviation. A) True B) False Solve the problem. 172) Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Some people believe that the presence of a full moon increases the number of births that take place. Suppose during the presence of a full moon, the hospital experienced eight consecutive hours with more than four births each hour. Based on this fact, comment on the belief that the full moon increases the number of births. A) The belief is not supported as the probability of observing this many births is 0.00000137. B) The belief is supported as the probability of observing this many births would be 0.00000137. C) The belief is not supported as the probability of observing this many births is 0.185. D) The belief is supported as the probability of observing this many births would be 0.185.

168)

169)

170)

171)

172)

173) A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the approximate probability that more than 17 and less than 26 of the sampled students had at least one tattoo. A) 0.6644 B) 0.1800 C) 0.2422 D) 0.4222

173)

174) A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 7.5 gallons per minute. Find the probability that between 5.0 gallons and 6.0 gallons are pumped during a randomly selected minute. A) 1 B) 0 C) 0.67 D) 0.33

174)

175) Compute A) 6

4 . 4

175) B) 1

C) 4

D) 16

Answer the question True or False. 176) The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 12 elements containing 7 successes. The numbers 0, 1, 2, 3, 4, 5, 6, and 7 are all possible values of x. A) True B) False Find the probability. 177) Suppose x is a uniform random variable with c = 30 and d = 50. Find P(x > 42). A) 0.1 B) 0.6 C) 0.9 D) 0.4 Solve the problem. 178) A certain baseball player hits a home run in 4% of his at-bats. Consider his at-bats as independent events. How many home runs do we expect the baseball player to hit in 650 at-bats? A) 26 B) 4 C) 654 D) 24.96 179) If x is a binomial random variable, calculate when necessary. A) 4.648 B) 7.348

for n = 90 and p = 0.6. Round to three decimal places

C) 21.6

177)

178)

179)

D) 54

180) Classify the following random variable according to whether it is discrete or continuous. The number of pills in a container of vitamins A) discrete B) continuous Find the probability. 181) Suppose x is a uniform random variable with c = 10 and d = 80. Find P(13 < x < 75). Round to the nearest hundredth when necessary. A) 0.89 B) 0.5 C) 0.11 D) 1 Solve the problem. 182) Transportation officials tell us that 90% of drivers wear seat belts while driving. What is the probability that between 525 and 530 drivers in a sample of 600 drivers wear seat belts? A) 0.0811 B) 0.0985 C) 0.0174 D) 0.9015

176)

180)

181)

182)

183) High temperatures in a certain city for the month of August follow a uniform distribution over the interval 65°F to 88°F. What is the probability that the high temperature on a day in August exceeds 70°F? A) 0.7826 B) 0.4575 C) 0.0435 D) 0.2174

183)

184) Find a value of the standard normal random variable z, called z 0 , such that P(z z 0 ) = 0.70.

184)

185) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.6. Find the probability that exactly five accidents will occur on this stretch of road each of the next two months. A) 0.005209 B) 0.072174 C) 0.000425 D) 0.144347

185)

186) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 440 seconds and a standard deviation of 40 seconds. Between what times do we expect approximately 95% of the boys to run the mile? A) between 345 and 535 seconds B) between 0 and 505.824 seconds C) between 361.6 and 518.4 seconds D) between 374.2 and 505.824 seconds

186)

187) The Fresh Oven Bakery knows that the number of pies it can sell varies from day to day. The owner believes that on 50% of the days she sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200 pies on the remaining 25% of the days. To make sure she has enough product, the owner bakes 200 pies each day at a cost of $2 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies for $5 each, find the probability distribution for her daily profit. A) B) Profit P(profit) Profit P(profit) $500 .5 $100 .5 $750 .25 $350 .25 $1000 .25 $600 .25 C) D) Profit P(profit) Profit P(profit) $300 .5 $300 .5 $550 .25 $450 .25 $800 .25 $600 .25

187)

A) .53

B) .47

C) .98

D) .81

Answer the question True or False. 188) The expected value of a discrete random variable must be one of the values in which the random variable can result. A) True B) False Solve the problem. 189) The time between customer arrivals at a furniture store has an approximate exponential distribution with mean = 8.5 minutes. If a customer just arrived, find the probability that the next customer will arrive in the next 5 minutes. A) 0.182684 B) 0.444694 C) 0.817316 D) 0.555306

188)

189)

190) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 1200 miles. What is the probability a certain tire of this brand will last between 57,480 miles and 57,840 miles? A) .9813 B) .0180 C) .4920 D) .4649

190)

191) The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 7. Find the probability that fewer than six tickets are written on a randomly selected day. A) 0.449711 B) 0.699292 C) 0.550289 D) 0.300708

191)

192) The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a mean of 12.42 ounces and a standard deviation of 0.28 ounce. Each can holds a maximum of 12.70 ounces of soda. Every can that has more than 12.70 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the probability that a randomly selected can will need to go through this process? A) .1587 B) .6587 C) .8413 D) .3413

192)

193) A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the approximate probability that more than 30 of the sampled students had at least one tattoo. A) 0.9929 B) 0.4929 C) 0.0034 D) 0.0071

193)

194) Suppose that 4 out of 12 liver transplants done at a hospital will fail within a year. Consider a random sample of 3 of these 12 patients. What is the probability that all 3 patients will result in failed transplants? A) .296 B) .333 C) .037 D) .018

194)

195) Consider the given discrete probability distribution. Find the probability that x exceeds 5.

195)

x 3 P(x) 0.01 A) 0.37

5 ?

6 0.31

8 0.32

B) 0.63

C) 0.36

D) 0.99

196) The time between customer arrivals at a furniture store has an approximate exponential distribution with mean = 8.5 minutes. If a customer just arrived, find the probability that the next customer will not arrive for at least 20 minutes. A) 0.095089 B) 0.653770 C) 0.346230 D) 0.904911

196)

197) Use the standard normal distribution to find P(-2.25 < z < 0). A) .4878 B) .0122 C) .6831

197)

D) .5122

198) The weight of corn chips dispensed into a 24-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 24.5 ounces and a standard deviation of 0.2 ounce. What proportion of the 24-ounce bags contain more than the advertised 24 ounces of chips? A) .5062 B) .4938 C) .9938 D) .0062

198)

199) Which geometric shape is used to represent areas for a uniform distribution? A) Bell curve B) Circle C) Rectangle 200) Suppose that x has an exponential distribution with A) 0.464739 B) 0.201897

= 2.5. Find P(x 4). C) 0.798103

201) Use the standard normal distribution to find P(-2.50 < z < 1.50). A) .8822 B) .6167 C) .5496

D) Triangle

D) 0.535261

D) .9270

202) The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is less than 1 year. A) 0.745189 B) 0.966627 C) 0.033373 D) 0.254811 Answer the question True or False. 203) The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events. A) True B) False Solve the problem. 204) Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 9. Find the value of a that makes the following probability statement true: P(3.5 x a) = 0.5. A) 6 B) -1.2 C) 4 D) 6.5

199)

200)

201)

202)

203)

204)

205) Given that x is a hypergeometric random variable with N = 10, n = 5, and r = 6, compute the mean of x. A) 2 B) 4 C) 1 D) 3

205)

206) Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 9. Find the value of a that makes the following probability statement true: P(x a) = 1. A) a 9 B) a 3 C) a 3 D) a 9

206)

207) Management at a home improvement store randomly selected 100 customers and observed their shopping habits. They recorded the number of items each of the customers purchased as well as the total time the customers spent in the store. Identify the types of variables recorded by the managers of the home improvement store. A) number of items - discrete; total time - continuous B) number of items - continuous; total time - discrete C) number of items - discrete; total time - discrete D) number of items - continuous; total time - continuous

207)

Answer the question True or False. 208) The conditions for both the hypergeometric and the binomial random variables require that each trial results in one of two outcomes. A) True B) False Find the probability. 209) Assume that x is a binomial random variable with n = 100 and p = 0.60. Use a normal approximation to find P(x < 48). A) 0.3015 B) 0.0094 C) 0.4946 D) 0.0054 26

208)

209)

Solve the problem. 210) Suppose x is a random variable best described by a uniform probability distribution with c = 4 and d = 10. Find the value of a that makes the following probability statement true: P(x a) = 0.25. A) 5.1 B) 1.5 C) 8.5 D) 5.5 Answer the question True or False. 211) The mean of the standard normal distribution is 1 and the standard deviation is 0. A) True B) False Solve the problem. 212) Classify the following random variable according to whether it is discrete or continuous. The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska A) continuous B) discrete

210)

211)

212)

213) Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $500 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the probability that a randomly selected month had PCE's below $350. A) 0.3000 B) 0.7000 C) 0.9987 D) 0.0013

213)

214) Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $800 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the point in the distribution below which 2.5% of the PCE's fell. A) $20.00 B) $898.00 C) $702.00 D) $780.00

214)

215) Given that x is a hypergeometric random variable with N = 8, n = 4, and r = 3, compute the variance of x. A) .469 B) .732 C) .538 D) .700

215)

216) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 460 seconds and a standard deviation of 40 seconds. The fitness association wants to recognize the fastest 10% of the boys with certificates of recognition. What time would the boys need to beat in order to earn a certificate of recognition from the fitness association? A) 511.2 seconds B) 525.8 seconds C) 394.2 seconds D) 408.8 seconds

216)

217) Use the standard normal distribution to find P(0 < z < 2.25). A) .7888 B) .8817 C) .5122

D) .4878

217)

218) Suppose that x has an exponential distribution with A) 0.135335 B) 0.606531

D) 0.864665

= 5. Find P(x 10). C) 0.393469

219) Use the standard normal distribution to find P(-2.25 < z < 1.25). A) .0122 B) .8821 C) .4878

D) .8944

220) Classify the following random variable according to whether it is discrete or continuous. The height of a player on a basketball team A) continuous B) discrete

218)

219)

220)

221) We believe that 81% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 30 students from the population. If the true percentage is really 81%, find the probability of observing 29 or more students who consider statistics to be an exciting subject. Round to six decimal places. A) 0.001797 B) 0.012646 C) 0.985557 D) 0.014443

221)

222) Find a value of the standard normal random variable z, called z0 , such that P(-z0 z z0 ) = 0.98.

222)

A) 1.96

B) 1.645

C) .99

D) 2.33

Find the probability. 223) Suppose x is a random variable best described by a uniform probability distribution with c = 50 and d = 10. Find P(x 50). A) 0.4 B) 0.5 C) 1 D) 0 Solve the problem. 224) Given that x is a hypergeometric random variable with N = 10, n = 3, and r = 6, compute P(x = 0). A) .033 B) .200 C) 0 D) .216

223)

224)

225) According to a recent study, 1 in every 10 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that at least 2 of the women sampled have been the victim of domestic abuse. Round to six decimal places. A) 0.728794 B) 0.271206 C) 0.462906 D) 0.265888

225)

226) The probability that an individual is left-handed is 0.12. In a class of 30 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth when necessary. A) mean: 30; standard deviation: 1.9 B) mean: 3.6; standard deviation: 1.78 C) mean: 30; standard deviation: 1.78 D) mean: 3.6; standard deviation: 1.9

226)

227) A literature professor decides to give a 15-question true-false quiz. She wants to choose the passing grade such that the probability of passing a student who guesses on every question is less than .10. What score should be set as the lowest passing grade? A) 11 B) 12 C) 10 D) 9

227)

228) Given that x is a hypergeometric random variable with N = 9, n = 3, and r = 5, compute the standard deviation of x. A) .745 B) .208 C) .456 D) .556

228)

Find the probability. 229) Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d = 70. Find P(60 x 70). A) 0.25 B) 0.35 C) 0.1 D) 0.025 Solve the problem. 230) Suppose x is a uniform random variable with c = 30 and d = 80. Find the standard deviation of x. A) = 31.75 B) = 14.43 C) = 2.04 D) = 3.03

229)

230)

231) Consider the given discrete probability distribution. Find P(x 4). x p(x)

0 .30

1 .25

A) .95

2 .20

3 .15

4 .05

B) .10

231)

5 .05

C) .05

D) .90

232) Given that x is a hypergeometric random variable, compute p(x) for N = 6, n = 3, r = 3, and x = 1. A) .375 B) .125 C) .45 D) .55 Answer the question True or False. 233) Nearly 100% of the observed occurrences of a random variable x that is normally distributed will fall within three standard deviations of the mean of the distribution of x. A) True B) False 234) The continuity correction factor is the name given to the .5 adjustment necessary when estimating the binomial with the normal distribution. A) True B) False Solve the problem. 235) The university police department must write, on average, five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 9.5. Interpret the value of the mean. A) The mean has no interpretation since 0.5 ticket can never be written. B) On half of the days less than 9.5 tickets are written and on half of the days have more than 9.5 tickets are written. C) If we sampled all days, the arithmetic average number of tickets written would be 9.5 tickets per day. D) The number of tickets that is written most often is 9.5 tickets per day.

232)

233)

234)

235)

236) Find a value of the standard normal random variable z, called z0 , such that P(z z0 ) = 0.70. A) -.47 B) -.98 C) -.53 D) -.81

236)

237) A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below:

237)

Price Probability

$10.00 0.25

$12.50 0.3

$15.00 0.45

Suppose the mean cost of the frogs is $13.00 per week. Interpret this value. A) Most of the weeks resulted in frog costs of $13.00. B) The median cost for the distribution of frog costs is $13.00. C) The frog cost that occurs more often than any other is $13.00. D) The average cost for all weekly frog purchases is $13.00.

238) Suppose a Poisson probability distribution with = 0.7 provides a good approximation of the distribution of a random variable x.. Find µ for x. A) 0.7 B) 0.49 C) 0.7 D) 0.4

238)

239) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.8. Find the probability of observing exactly five accidents on this stretch of road next month. A) 2.963192 B) 1.778130 C) 0.066289 D) 79.484492 Find the probability. 240) Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d = 50. Find P(x < 42). A) 0.6 B) 0.7 C) 0.12 D) 0.06 Solve the problem. 241) IQ test scores are normally distributed with a mean of 104 and a standard deviation of 16. An individual's IQ score is found to be 115. Find the z-score corresponding to this value. A) -1.45 B) 0.69 C) -0.69 D) 1.45 Solve the problem. Round to four decimal places. 242) If x is a binomial random variable, compute p(x) for n = 6, x = 3, p = 0.4. A) 0.2903 B) 0.3152 C) 0.2765

D) 0.2599

Answer the question True or False. 243) A binomial random variable is defined to be the number of units sampled until x successes is observed. A) True B) False Solve the problem. 244) The random variable x represents the number of boys in a family with three children. Assuming that births of boys and girls are equally likely, find the mean and standard deviation for the random variable x. A) mean: 1.50; standard deviation: .87 B) mean: 2.25; standard deviation: .87 C) mean: 2.25; standard deviation: .76 D) mean: 1.50; standard deviation: .76

239)

240)

241)

242)

243)

244)

245) Suppose a Poisson probability distribution with = 7.9 provides a good approximation of the distribution of a random variable x. Find for x. A) 62.41 B) 4 C) 7.9 D) 7.9

245)

246) Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $900 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Using the distribution above, what is the probability that during a randomly selected month PCE's were between $775.00 and $990.00? A) .9579 B) .9999 C) .0421 D) .0001

246)

247) Consider the given discrete probability distribution. Find P(x > 3).

247)

x p(x)

A) .3

1 .1

2 .2

3 .2

4 .3

5 .2

B) .2

C) .5

D) .7

Answer the question True or False. 248) The number of children in a family can be modelled using a continuous random variable. A) True B) False Solve the problem. 249) As part of a promotion, both you and your roommate are given free cellular phones from a batch of 13 phones. Unknown to you, four of the phones are faulty and do not work. Find the probability that one of the two phones is faulty. A) .462 B) .231 C) .538 D) .077

248)

249)

250) Classify the following random variable according to whether it is discrete or continuous. The number of goals scored in a soccer game A) continuous B) discrete

250)

251) Which shape is used to represent areas for a normal distribution? A) Circle B) Triangle C) Bell curve

251)

252) Use the standard normal distribution to find P(z < -2.33 or z > 2.33). A) .7888 B) .0606 C) .0198

D) Rectangle

D) .9809

253) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 460 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 322 seconds. A) .4893 B) .0107 C) .5107 D) .9893 254) Compute

9 . 4

A) 15,120

252)

253)

254) B) 3024

C) 84

D) 126

255) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 1400 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty? A) 57,550 miles B) 61,400 miles C) 62,450 miles D) 58,600 miles

255)

256) We believe that 78% of the population of all Business Statistics students consider statistics to be an exciting subject. Suppose we randomly and independently selected 29 students from the population and observed fewer than five in our sample who consider statistics to be an exciting subject. Make an inference about the belief that 78% of the students consider statistics to be an exciting subject. A) The 78% number is too low. The real percentage is higher than 78%. B) The 78% number is too high. The real percentage is lower than 78%. C) The 78% number is exactly right. D) It is impossible to make any inferences about the 78% number based on this information.

256)

257) The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 8.5 to 10.5 millimeters. Any ball bearing with a diameter of over 10.25 millimeters or under 8.75 millimeters is considered defective. What is the probability that a randomly selected ball bearing is defective? A) .75 B) .25 C) 0 D) .50

257)

Solve the problem. Round to four decimal places. 258) If x is a binomial random variable, compute p(x) for n = 6, x = 2, q = 0.4. A) 0.1313 B) 0.1382 C) 0.3328

258)

D) 0.3110

Answer the question True or False. 259) The hypergeometric random variable x counts the number of successes in the draw of n elements from a set of N elements containing r successes. A) True B) False Solve the problem. 260) A certain baseball player hits a home run in 5% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits at most 23 home runs in 700 at-bats? A) 0.05 B) 0.95 C) 0.9767 D) 0.0233

259)

260)

261) A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 38% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's president possesses low ethical standards. Round to six decimal places. A) 0.497379 B) 0.295997 C) 0.426047 D) 0.198644

261)

262) High temperatures in a certain city for the month of August follow a uniform distribution over the interval 60°F to 90°F. Find the temperature which is exceeded by the high temperatures on 90% of the days in August. A) 90°F B) 87°F C) 70°F D) 63°F

262)

263) The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 years or more. A) 0.493383 B) 0.770210 C) 0.229790 D) 0.506617

263)

264) The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 5 goals per game. Find the probability that the team will score more than three goals during a game. A) 0.734974 B) 0.875348 C) 0.265026 D) 0.124652

264)

265) Suppose that the random variable x has an exponential distribution with and standard deviation of x. A) µ = 0; = 1 B) µ = 1.5; = 1 C) µ = 1.5; = 1.5

265)

= 1.5. Find the mean

D) µ = 0;

= 1.5

Answer the question True or False. 266) For any continuous probability distribution, P(x = c) = 0 for all values of c. A) True B) False 267) The conditions for both the hypergeometric and the binomial random variables require that the trials are independent. A) True B) False

266)

267)

Solve the problem. 268) Mamma Temte bakes six pies each day at a cost of $2 each. On 19% of the days she sells only two pies. On 30% of the days, she sells 4 pies, and on the remaining 51% of the days, she sells all six pies. If Mama Temte sells her pies for $6 each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.] A) $15.84 B) -$6.00 C) $27.84 D) -$7.36 269) Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 50 to 70. What is the probability that this experiment results in an outcome less than 60? Round to the nearest hundredth when necessary. A) 1 B) 0.35 C) 0.08 D) 0.5 Answer the question True or False. 270) The binomial distribution can be used to model the number of rare events that occur over a given time period. A) True B) False Solve the problem. 271) Compute A) 10

5 . 0

268)

269)

270)

271) B) 5

C) 1

D) undefined

Find the probability. 272) Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 50. Find P(x > 50). A) 1 B) 0.5 C) 0.4 D) 0 Solve the problem. 273) A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 5.0 to 8.0 gallons per minute. Would you expect the machine to pump more than 7.85 gallons per minute? A) Yes, since .05 is a high probability. B) Yes, since .95 is a high probability. C) No, since .05 is a low probability. D) No, since .95 is a high probability.

272)

273)

Answer Key Testname: CHAPTER 4 1) C 2) D 3) A 4) C 5) D 6) B 7) D 8) A 9) B 10) C 11) D 12) A 13) A 14) C 15) A 16) B 17) A 18) C 19) D 20) B 21) A 22) B 23) A 24) A 25) A 26) A 27) A 28) D 29) D 30) A 31) C 32) D 33) B 34) A 35) C 36) D 37) B 38) B 39) A 40) A 41) D 42) A 43) A 44) B 45) A 46) B 47) B 48) Since the probability of success remains the same from trial to trial, the probability of success on the second trial is .48, so the probability of failure on the second trial is 1 -.48 = .52. 34

Answer Key Testname: CHAPTER 4

49) µ =

x·p(x) = 2(.2) + 6(.3) + 11(.3) + 13(.2)

= 8.1 50) E(x) = = 42.3; If the daily numbers of customers who sign up for additional services online were averaged for all days, the result would be 42.3 customers per day. 51) Let x be the tread life of this brand of tire. Then x is a normal random variable with µ = 60,000 and = 4200. To determine what proportion of tires fail before reaching 54,960 miles, we need to find the z-value for x = 54,960. z=

x-µ

54,960 - 60,000 = -1.20 4200

P(x 54,960) = P(z -1.20) = .5 - P(-1.20 z 0) = .5 - .3849 = .1151 52) This is a valid probability distribution because the probabilities are all nonnegative and their sum is 1. 53) Let x be a score on this exam. Then x is a normally distributed random variable with µ = 595 and = 72. We want to find the value of x 0 , such that P(x > x 0 ) = .10. The z-score for the value x = x 0 is z=

x0 - µ

x 0 - 595 72

P(x > x 0) = P z >

We find

x0 - 595

x 0 - 595 72

x 0 - 595 = 1.28(72)

= .10

1.28. x 0 = 595 + 1.28(72) = 687.16

54) Let x = the number of the 25 ads that resulted in the item being sold within a week. Then X is a binomial random variable with n = 25 and p = 0.70. P(x 17) = 0.488 (from a binomial probability table) A value of x that is less than or equal to 17 will occur in about 48.8% of all such samples. 6 6! (.3)2 (.7)6-2 = (.3)2 (.7)4 = 15(.09)(.2401) .324 55) 2 2!(6 - 2)!

56)

Answer Key Testname: CHAPTER 4

57) a. P(x 24) = e-24/7 .032433 b. P(x < 1) = 1 - P(x 1) = 1 - e-1/7 1 - .866878 = .133122 58) Let x = the number of death claims received per day. Then x is a Poisson random variable with = 5.

P(x 7) = 1 - P(x 6) = 0.237816 59) a. P(x 2) = e-2/1.75 .318907 b. P(x < 2) = 1 - P(x 2) = 1 - e-2/1.75 1 - .318907 = .681093 IQR 71.500 - 67.875 1.33; Since this number is reasonably close to 1.3, the distribution of heights is approximately = 60) s 2.176

normal. 61) Let x = the number of the 10 cars with defective gas tanks. Then X is a binomial random variable with n = 10 and p = .30. P(more than half) = P(x > 5) = P(x 6) = 1 - P(x 5) = 1 - 0.953 = 0.047 62) Let x be the number of tomatoes sold per day. Then x is a normal random variable with µ = 459 and

= 30.

To determine the probability that all 417 tomatoes will be sold, we need to find the z-value for x = 417. z=

x-µ

417 - 459 = -1.4 30

P(x 417) = P(z -1.4) = .5 + P(-1.4 z 0) = .5 + .4192 = .9192 63) Let x = high temperature in August. Then x is a uniform random variable with c = 75°F and d = 95°F. P(x > a) =

d-a d-c

P(x > 80) =

95 - 80 15 = = .75 95 - 75 20

64) The percentages are 70%, 96%, and 100%, respectively. Since these percentages are reasonably close to 68%, 95%, and 100%, we conclude that the distribution of scores is approximately normal. 65) possible values of x: {0, 1, 2, 3, 4, 5, 6}; The variable x is discrete since it has a finite number of distinct possible values. 66) Let x = cost of fare paid by passenger. The probability distribution for x is: x p(x)

$90 1/5

$370 4/5

The expected cost is E(x) = µ =

x· p(x) = $90

1 4 + $370 = $314.00 5 5

Since the expected cost is more than the usual one-way air fare, the passenger should not opt to fly as a standby.

Answer Key Testname: CHAPTER 4 67)

68) 0, 1, 2, 3 x e5 7 e-5 = = .1044445 69) x! 7! 70) can use normal distribution 71) cannot use normal distribution 72) Let x be the rate of return. Then x is a normal random variable with µ = 37% and that x exceeds 43%, we need to find the z-value for x = 43%. z=

x-µ

43 - 37 =2 3

P(x > 43%) = P(z 2) = .5 - P(0 z 2) = .5 - .4772 = .0228 73) x is a binomial random variable with n = 66 and p = 0.42. z=

(x + .5) - np (31 + .5) - 66(0.42) = = 0.94 np(1 - p) 66(0.42)(1 - 0.42)

74) a.

b. c.

µ = = 3; = µ = 3 1.73 P(µ - < x < µ + ) = P(1.27 < x < 4.73) = .22 + .22 + .17 = .61

= 3%. To determine the probability

Answer Key Testname: CHAPTER 4 75) a. µ = E(x) = 1(.1) + 2(.2) + 3(.2) + 4(.3) + 5(.2) = 3.3 b.

2.32(.1) + 1.32 (.2) + 0.32 (.2) + 0.72(.3) + 1.72 (.2)

1.27.

c. P(µ - < x < µ + ) = P(2.03 < x < 4.57) = .2 + .3 = .5; The Empirical Rule states that about .68 of the data lie within one standard deviation of the mean for a mound-shaped symmetric distribution. For our distribution, this value is only .5, but it is not a surprise that these numbers aren't closer since our distribution is not symmetric. 76) a. P(x < 1) = 1 - P(x 1) = 1 - e-1/4.5 1 - .800737 = .199263 b. P(x > 7) = e-7/4.5 .211072 77) P(x = 1 or x = 2) = p(x = 1) + p(x = 2) = .25 + .20 = .45 78) Let x = gallons pumped per minute. Then x is a uniform random variable with c = 4.5 and d = 5.5. P(x < a) =

a-c d-c

P(x < 4.75) =

4.75 - 4.5 .25 = = .25 5.5 - 4.5 1

79) This is not a valid probability distribution because the sum of the probabilities is greater than 1. 80) Let x = the number of accidents that occur on the stretch of road during a month. Then x is a Poisson random variable with = 5.3. P(x < 2) = P(x = 0) + P(x = 1) = 0.031447 c + d 10 + 70 = = 40 81) µ = 2 2

82) This is not a valid probability distribution because the sum of the probabilities is less than 1. 83)

It is not reasonable to assume that the heights are normally distributed since the histogram is not mound-shaped and symmetric about the mean of 31 pounds. 84) Since the probability of success remains the same from trial to trial, the probability of success on the second trial is also .63.

Answer Key Testname: CHAPTER 4 85) To determine the premium, the insurance agency must first determine the average loss paid on the sports car. Let x = amount paid on the sports car loss. The probability distribution for x is: x p(x)

$24,500 .001

$12,000 .01

$5,750 .05

$2,000 .10

$0 .839

Note: These losses paid have already considered the $500 deductible paid by the owner. The expected loss paid is: µ=

xp(x) = $24,500(.001) + $12,000(.01) + $5,750(.05) + $2,000(.10) + $0(.839)

= $632 In order to average $405 profit per policy sold, the insurance company must charge an annual premium of $632 + $405 = $1037.00. 86) a. x p(x) 0 .083 1 .417 2 .417 3 .083

b. µ =

3(5) = 1.5; 10

2 = 5(10 - 5) · 3(10 - 3) = 7 ; 12 102(9)

7 12

.764

c. P(-.028 < x < 3.028) = 1 87) natural bounds for x: 0 ounces and 16 ounces; The variable x is continuous since the values of x correspond to the points in some interval. 88) X is a binomial random variable with n = 15 and p = 0.4.

P(x > 5) = 1 - P(x 5)

= 1 - 0.403 (from a binomial probability table) = 0.597 89) a. P(x < -6) + P(x > 12) = 0 + P(x > 12) = e-12/3 = e-4 .018316 b. P(0 < x < 6) = P(x < 6) = 1 - P(x 6) = 1 - e-6/3 = 1 - e-2 .864665 c. P(1.5 < x < 4.5) = P(x 1.5) - P(x 4.5)= e-1.5/3 - e-4.5/3 = e-.5 - e-1.5 .383400 90) Let x = ball bearing diameter. Then x is a uniform random variable with c = 3.5 and d = 7.5.

P(x < a) =

a-c d-c

P(x < 5.5) =

91) P(x = 0) =

5.5 - 3.5 2 = = 0.5 7.5 - 3.5 4

3 12 0 4 15 4

.363; P(x 1) = 1 - P(x = 0) 1 - .363 = .637

92) P(x < 2 or x > 3) = p(x = 1) + p(x = 4) + p(x = 5) = .1 + 03 + .2 = .6

Answer Key Testname: CHAPTER 4 93) Let x = the number of babies born during a one-hour period at this hospital. Then x is a Poisson random variable with = 6. P(x = 2) = 0.044618 94) a. x p(x) 0 .071 1 .429 2 .429 3 .071 b. P(x 2) = .071 + .429 + .429 = .929 95) Let x be the number of tomatoes sold per day. Then x is a normal random variable with µ = 112 and We want to find the value x 0 , such that P(x > x 0) = .015. The z-value for the point x = x 0 is z=

x-µ

x 0 - 112 30

P(x > x 0) = P(z >

We find

x 0 - 112 30

x0 - 112 30

96) P(x = 0) =

12 3

)= .015

= 2.17

x 0 - 112 = 2.17(30)

2 10 0 3

x 0 = 112 + 2.17(30) = 177

.545

97) This is not a valid probability distribution because one of the probabilities given is negative. 98) Let x = the number of the 11 hypertensive patients whose blood pressure drops. Then X is a binomial random variable with n = 11 and p = .5. P(x 9) = P(x = 9) + P(x = 10) + P(x = 11) = 0.032715 = = = 33.5; standard deviation = Mean µ .67(72) 99) 20 15 4 3 .328 100) P(x = 0) = 35 7

101) Mean = µ = .88(72) = 63.36; standard deviation = 102) µ = 1.596; = 1.098

50(.67)(.33) 3.32

72(.88)(.12) 2.76

= 30.

Answer Key Testname: CHAPTER 4 103) Let x be a score on this exam. Then x is a normally distributed random variable with µ = 466 and find the value of x0 , such that P(x > x 0 ) = .80. The z-score for the value x = x 0 is z=

x0 - µ

x 0 - 466 72

P(x > x 0) = P z >

We find

x2 - 466

x 0 - 466 72

= 72. We want to

= .80

-.84.

x 0 - 466 = -.84(72)

x 0 = 466 - .84(72) = 405.52 -7/3.2 .112197; about 11.2 percent 104) a. P(x 7) = e 105) a. P(x = 0) = 0 6 4 1 4 6(1) 1 = = b. P(x = 1) = .024 10 252 42 5 c. P(x 1) = P(x = 0) + P(x = 1) 0 + .024 = .024 d. P(x 2) = 1 - P(x 1) 1 - .024 = .976 106) Let x be the number of the 78 applications rejected. Then x is a binomial random variable with n = 78 and p = 0.19. z=

(x - .5) - np (73 - .5) - 78(0.19) = = 16.65 np(1 - p) (78)(0.19)(1 - 0.19)

107) Let x be the rate of return. Then x is a normal random variable with µ = 47% and that x is at least 42.5%, we need to find the z-value for x = 42.5%. z=

x-µ

42.5 - 47 = -1.5 3

P(x 42.5%) = P(-1.5 z) = .5 + P(-1.5 z 0) = .5 + .4332 = .9332 108) 0, 1, 2 109) C 110) C 111) C 112) D 113) B 114) C 115) A 116) B 117) B 118) D 119) A 120) A 121) A 122) B

= 3%. To determine the probability

Answer Key Testname: CHAPTER 4 123) A 124) B 125) B 126) C 127) B 128) B 129) A 130) B 131) B 132) B 133) A 134) C 135) C 136) D 137) B 138) D 139) A 140) D 141) B 142) C 143) D 144) A 145) D 146) B 147) A 148) D 149) D 150) A 151) A 152) A 153) B 154) C 155) B 156) C 157) D 158) A 159) D 160) D 161) D 162) D 163) A 164) D 165) A 166) A 167) D 168) A 169) B 170) B 171) A 172) B 42

Answer Key Testname: CHAPTER 4 173) A 174) D 175) B 176) B 177) D 178) A 179) A 180) A 181) A 182) A 183) A 184) A 185) A 186) C 187) B 188) B 189) B 190) B 191) D 192) A 193) C 194) D 195) B 196) A 197) A 198) C 199) C 200) B 201) D 202) D 203) A 204) D 205) D 206) A 207) A 208) A 209) D 210) D 211) B 212) A 213) D 214) C 215) C 216) D 217) D 218) D 219) B 220) A 221) D 222) D 43

Answer Key Testname: CHAPTER 4 223) C 224) A 225) A 226) B 227) A 228) A 229) A 230) B 231) A 232) C 233) A 234) A 235) C 236) C 237) D 238) A 239) C 240) A 241) B 242) C 243) B 244) A 245) D 246) A 247) C 248) B 249) A 250) B 251) C 252) C 253) B 254) D 255) A 256) B 257) B 258) B 259) A 260) D 261) B 262) D 263) C 264) A 265) C 266) A 267) B 268) A 269) D 270) B 271) C 272) D 44

Answer Key Testname: CHAPTER 4 273) C

Chapter 5 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The length of time a traffic signal stays green (nicknamed the "green time") at a particular intersection follows a normal probability distribution with a mean of 200 seconds and the standard deviation of 10 seconds. Use this information to answer the following questions. Which of the following describes the derivation of the sampling distribution of the sample mean? A) A single sample of sufficiently large size is randomly selected from the population of "green times" and its probability is determined. B) The means of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted. C) The standard deviations of a large number of samples of size n randomly selected from the population of "green times" are calculated and their probabilities are plotted. D) The mean and median of a large randomly selected sample of "green times" are calculated. Depending on whether or not the population of "green times" is normally distributed, either the mean or the median is chosen as the best measurement of center.

2) Which of the following describes what the property of unbiasedness means? A) The center of the sampling distribution is found at the population parameter that is being estimated. B) The sampling distribution in question has the smallest variation of all possible sampling distributions. C) The center of the sampling distribution is found at the population standard deviation. D) The shape of the sampling distribution is approximately normally distributed.

3) Suppose students' ages follow a skewed right distribution with a mean of 24 years old and a standard deviation of 5 years. If we randomly sample 350 students, which of the following statements about the sampling distribution of the sample mean age is incorrect? A) The shape of the sampling distribution is approximately normal. B) The mean of the sampling distribution is approximately 24 years old. C) The standard deviation of the sampling distribution is equal to 5 years. D) All of the above statements are correct.

4) The sampling distribution of the sample mean is shown below.

p(x) 1/9 2/9 3/9 2/9 1/9 Find the expected value of the sampling distribution of the sample mean. A) 4 B) 6 C) 7

D) 5

5) The probability distribution shown below describes a population of measurements.

x 0 2 4 p(x) 1/3 1/3 1/3 Suppose that we took repeated random samples of n = 2 observations from the population described above. Which of the following would represent the sampling distribution of the sample mean?

A) C)

p(x) 1/5 1/5 1/5 1/5 1/5 x

p(x) 2/9 2/9 1/9 2/9 2/9

p(x) 1/3 1/3 1/3 x

p(x) 1/9 2/9 3/9 2/9 1/9

6) The probability distribution shown below describes a population of measurements.

x 0 2 4 p(x) 1/3 1/3 1/3 Suppose that we took repeated random samples of n = 2 observations from the population described above. Find the expected value of the sampling distribution of the sample mean. A) 3 B) 0 C) 1 D) 4 E) 2

7) Which of the following describes what the property of minimum variance means? A) The center of the sampling distribution is found at the population parameter that is being estimated. B) The shape of the sampling distribution is approximately normally distributed. C) The sampling distribution in question has the smallest variation of all possible unbiased sampling distributions. D) The center of the sampling distribution is found at the population standard deviation. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8) Consider the population described by the probability distribution below. x p(x)

2 .2

5 .5

7 .3

The random variable x is observed twice. The observations are independent. The different samples of size 2 and their probabilities are shown below. Sample 2, 2 2, 5 2, 7

Probability .04 .10 .06

Sample 5, 2 5, 5 5, 7

Probability .10 .25 .15

Sample 7, 2 7, 5 7, 7

Find the sampling distribution of the sample mean x.

Probability .06 .15 .09

9) The probability distribution shown below describes a population of measurements that can assume values of 1, 5, 9, and 13, each of which occurs with the same frequency: x p(x)

1 1 4

5 1 4

9 1 4

13 1 4

Consider taking samples of n = 2 measurements and calculating x for each sample. Construct the probability histogram for the sampling distribution of x.

10) The probability distribution shown below describes a population of measurements that can assume values of 4, 9, 14, and 19, each of which occurs with the same frequency: x p(x)

4 1 4

9 1 4

14 1 4

10)

19 1 4

Find E(x) = µ. Then consider taking samples of n = 2 measurements and calculating x for each sample. Find the expected value, E(x), of x.

11) Suppose a random sample of n = 36 measurements is selected from a population with mean µ = 256 and variance 2 = 144. Find the mean and standard deviation of the sampling

11)

distribution of the sample mean x.

12) Consider the population described by the probability distribution below. x p(x) a.

3 .1

5 .7

12)

7 .2

Find µ.

b. Find the sampling distribution of the sample mean x for a random sample of n = 2 measurements from the distribution. c.

Show that x is an unbiased estimator of µ.

13) Suppose a random sample of n = 64 measurements is selected from a population with mean µ = 65 and standard deviation an 68.75.

13)

= 12. Find the probability that x falls between 65.75

14) A random sample of size n is to be drawn from a population with µ = 1500 and = 300. What size sample would be necessary in order to reduce the standard error to 20?

14)

15) The amount of time it takes a student to walk from her home to class has a skewed right distribution with a mean of 15 minutes and a standard deviation of 1.6 minutes. If times

15)

were collected from 30 randomly selected walks, describe the sampling distribution of x, the sample mean time.

16) Suppose a random sample of n = 64 measurements is selected from a population with mean µ = 65 and standard deviation = 12. Find the values of µx and x .

16)

17) Suppose a random sample of n = 64 measurements is selected from a population with

17)

mean µ = 65 and standard deviation = 68.

= 12. Find the z-score corresponding to a value of x

18) The weight of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of .2 ounce. Suppose 100 bags of chips are randomly selected. Find the probability that the mean weight of these 100 bags exceeds 10.45 ounces.

18)

19) Consider the probability distribution shown here.

19)

5 1 3

p(x)

7 1 3

9 1 3

Let x be the sample mean for random samples of n = 2 measurements from this distribution. Find E(x) and E(x).

20) Consider the population described by the probability distribution below. x p(x)

0 1 3

2 1 3

20)

4 1 3

a. Find µ. b. Find the sampling distribution of the sample mean for a random sample of n = 3 measurements from this distribution. c. Find the sampling distribution of the sample median for a random sample of n = 3 observations from this population. d. Show that both the mean and the median are unbiased estimators of µ for this population. e. Find the variances of the sampling distributions of the sample mean and the sample median. f. Which estimator would you use to estimate µ? Why?

21) Consider the population described by the probability distribution below. x p(x)

3 .1

5 .7

7 .2

a. Find µ. b. Find the sampling distribution of the sample median for a random sample of n = 2 observations from this population. c. Show that the median is an unbiased estimator of µ.

21)

22) Consider the population described by the probability distribution below. x p(x) a.

3 .1

5 .7

22)

7 .2

Find 2 .

b. Find the sampling distribution of the sample variance s2 for a random sample of n = 2 measurements from the distribution. c. Show that s2 is an unbiased estimator of 2 .

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 23) Suppose a random sample of n measurements is selected from a binomial population with probability of success p = .28. Given n = 350, describe the shape, and find the mean and the

23)

standard deviation of the sampling distribution of the sample proportion, p. A) skewed right; 0.28, 0.024 B) approximately normal; 0.28, 0.0006 C) skewed right; 98, 8.40 D) approximately normal; 0.28, 0.024

24) Which of the following does the Central Limit Theorem allow us to disregard when working with the sampling distribution of the sample mean? A) The standard deviation of the population distribution. B) The shape of the population distribution. C) The mean of the population distribution. D) All of the above can be disregarded when the Central Limit Theorem is used.

24)

Answer the question True or False. 25) The sample mean, x, is a statistic. A) True

B) False

25)

26) The minimum-variance unbiased estimator (MVUE) has the least variance among all unbiased estimators. A) True B) False

26)

27) The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic. A) True B) False

27)

Solve the problem. 28) The Central Limit Theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following is a necessary condition for the Central Limit Theorem to be used? A) The population from which we are sampling must be normally distributed. B) The population from which we are sampling must not be normally distributed. C) The population size must be large (e.g., at least 30). D) The sample size must be large (e.g., at least 30).

28)

29) A random sample of n = 300 measurements is drawn from a binomial population with probability of success .43. Give the mean and the standard deviation of the sampling distribution of the sample

29)

proportion, p. A) .43; .014

B) .57; .029

C) .43; .029

D) 57; .014

Answer the question True or False. 30) The term statistic refers to a population quantity, and the term parameter refers to a sample quantity. A) True B) False Solve the problem. 31) One year, the distribution of salaries for professional sports players had mean $1.6 million and standard deviation $0.8 million. Suppose a sample of 400 major league players was taken. Find the approximate probability that the average salary of the 400 players that year exceeded $1.1 million. A) approximately 0 B) .7357 C) .2357 D) approximately 1

30)

31)

32) Which of the following statements about the sampling distribution of the sample mean is incorrect? A) The standard deviation of the sampling distribution is . B) The sampling distribution is approximately normal whenever the sample size is sufficiently large (n 30). C) The mean of the sampling distribution is µ. D) The sampling distribution is generated by repeatedly taking samples of size n and computing the sample means.

32)

33) A random sample of n = 300 measurements is drawn from a binomial population with probability of success .26. Give the mean and the standard deviation of the sampling distribution of the sample

33)

proportion, p. A) .26; .025

B) .74; .025

C) .74; .011

D) .26; .011

Answer the question True or False. 34) In most situations, the true mean and standard deviation are unknown quantities that have to be estimated. A) True B) False 35) A statistic is biased if the mean of the sampling distribution is equal to the parameter it is intended to estimate. A) True B) False Solve the problem. 36) The daily revenue at a university snack bar has been recorded for the past five years. Records indicate that the mean daily revenue is $1200 and the standard deviation is $400. The distribution is skewed to the right due to several high volume days (football game days). Suppose that 100 days are randomly selected and the average daily revenue computed. Which of the following describes the sampling distribution of the sample mean? A) skewed to the right with a mean of $1200 and a standard deviation of $400 B) normally distributed with a mean of $1200 and a standard deviation of $40 C) normally distributed with a mean of $1200 and a standard deviation of $400 D) normally distributed with a mean of $120 and a standard deviation of $40

34)

35)

36)

Answer the question True or False. 37) The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, µ and , are statistics. A) True B) False Solve the problem. 38) The weight of corn chips dispensed into a 15-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 15.5 ounces and a standard deviation of 0.3 ounce. Suppose 400 bags of chips are randomly selected. Find the probability that the mean weight of these 400 bags exceeds 15.6 ounces. A) .3085 B) .1915 C) approximately 0 D) .6915 Answer the question True or False. 39) A point estimator of a population parameter is a rule or formula which tells us how to use sample data to calculate a single number that can be used as an estimate of the population parameter. A) True B) False Solve the problem. 40) The number of cars running a red light in a day, at a given intersection, possesses a distribution with a mean of 4.7 cars and a standard deviation of 5. The number of cars running the red light was observed on 100 randomly chosen days and the mean number of cars calculated. Describe the sampling distribution of the sample mean. A) shape unknown with mean = 4.7 and standard deviation = 0.5 B) shape unknown with mean = 4.7 and standard deviation = 5 C) approximately normal with mean = 4.7 and standard deviation = 0.5 D) approximately normal with mean = 4.7 and standard deviation = 5

37)

38)

39)

40)

41) The Central Limit Theorem is important in statistics because _____. A) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size B) for a large n, it says the population is approximately normal C) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the population D) for any size sample, it says the sampling distribution of the sample mean is approximately normal

41)

42) The average score of all golfers for a particular course has a mean of 69 and a standard deviation of 3. Suppose 36 golfers played the course today. Find the probability that the average score of the 36 golfers exceeded 70. A) .3707 B) .1293 C) .4772 D) .0228

42)

Answer the question True or False. 43) As the sample size gets larger, the standard error of the sampling distribution of the sample mean gets larger as well. A) True B) False

43)

Solve the problem. 44) The Central Limit Theorem is considered powerful in statistics because __________. A) it works for any sample size provided the population is normal B) it works for any sample provided the population distribution is known C) it works for any population distribution provided the sample size is sufficiently large D) it works for any population distribution provided the population mean is known Answer the question True or False. 45) Sample statistics are random variables, because different samples can lead to different values of the sample statistics. A) True B) False 46) When estimating the population mean, the sample mean is always a better estimate than the sample median. A) True B) False Solve the problem. 47) A random sample of n = 600 measurements is drawn from a binomial population with probability of success .08. Give the mean and the standard deviation of the sampling distribution of the sample

44)

45)

46)

47)

proportion, p. A) .08; .011

B) .92; .003

C) .08; .003

D) .92; .011

Answer the question True or False. 48) The Central Limit Theorem guarantees that the population is normal whenever n is sufficiently large. A) True B) False

48)

49) The standard error of the sampling distribution of the sample mean is equal to , the standard deviation of the population. A) True B) False

49)

50) If x is a good estimator for µ, then we expect the values of x to cluster around µ. A) True B) False

50)

51) The ideal estimator has the greatest variance among all unbiased estimators. A) True B) False

51)

Solve the problem. 52) A random sample of n = 400 measurements is drawn from a binomial population with probability of success .21. Give the mean and the standard deviation of the sampling distribution of the sample ^

proportion, p. A) .21; .008

B) .79; .02

C) .21; .02

D) .79; .008

52)

Answer Key Testname: CHAPTER 5 1) B 2) A 3) C 4) B 5) D 6) E 7) C 8) 9)

3.5

4.5

p(x)

.04

.20

.12

.25

.30

.09

1 1 1 1 10) E(x) = (4)( ) + (9)( ) + (14)( ) + (19)( ) = 11.5 4 4 4 4 E(x) = (4)(

1 13 2 3 11.5 4 3 33 2 1 ) + ( )( ) + (9)( ) + ( )( ) + (14)( ) + ( )( ) + (19)( ) = 11.5 16 2 16 16 2 16 16 2 16 16

11) µx = µ = 256; x = 12) a. b.

144 12 = =2 6 36

µ = E(x) = .1(3) + .7(5) + .2(7) = 5.2 x

p(x)

.01

.14

.53

.28

.04

E(x) = .01(3) + .14(4) + .53(5) + .28(6) + .04(7) = 5.2; Since E(x) = µ, x is an unbiased estimator of µ.

13) P(65.75 x 68.75) = P(.5 z 2.5) .3023 14) The standard error is x = 20 = / n =

300 n

. If the standard error is desired to be 20, we get:

n · 20 = 300

300 = 15 20

n = 225

15) By the Central Limit Theorem, the sampling distribution of x is approximately normal with µx = µ = 15 minutes and x=

1.6 = 0.2921 minutes. 30

Answer Key Testname: CHAPTER 5

16) µx = µ = 65; x = 17) z =

12 12 = = 1.5 8 64

68 - 65 =2 1.5

18) P(x > 10.45) = P z >

10.45 - 10.50 .2/ 100

= P (z > -2.5) = .5 + .4938 = .9938

1 1 1 19) E(x) = µ = (5)( ) + (7)( ) + (9)( ) = 7 3 3 3 1 2 3 2 1 E(x) = (5)( ) + (6)( ) + (7)( ) + (8)( ) + (9)( ) = 7 9 9 9 9 9

Answer Key Testname: CHAPTER 5

20) a.

µ = E(x) =

1 1 1 (0) + (2) + (4) = 2 3 3 3

2 3

4 3

8 3

10 3

p(x)

1 27

1 9

2 9

7 27

2 9

1 9

1 27

0 7 27

2 13 27

p(M)

E(x) =

4 7 27

1 1 2 2 4 7 2 8 1 10 1 (0) + (2) + (4) = 2; Since E(x) = µ, the sample + + + + 27 9 3 9 3 27 9 3 9 3 27

mean is an unbiased estimator of µ. 7 13 7 E(M) = (0) + (2) + (4) = 2; Since E(M) = µ, the sample median is an unbiased estimator 27 27 27 of µ. 2 2 4 2 7 2 2 1 2 1 2 2 2 8 x = 27 (0 - 2) + 9 3 - 2 + 9 3 - 2 + 27 (2 - 2) + 9 3 - 2

e. +

2 1 1 10 8 -2 + (4 - 2)2 = 9 3 27 9 2 7 2 13 2 7 2 56 M = 27 (0 - 2) + 27 (2 - 2) + 27 (4 - 2) = 27

f. 21) a. b.

sample mean; The variance is smaller. µ = E(x) = .1(3) + .7(5) + .2(7) = 5.2

M p(M)

22) a. b.

4 .14

5 .53

6 .28

7 .04

E(M) = .01(3) + .14(4) + .53(5) + .28(6) + .04(7) = 5.2; Since E(M) = µ, the sample median is an unbiased estimator of µ. 2 = .1(3 - 5.2)2 + .7(5 - 5.2)2 + .2(7 - 5.2)2 = 1.16

s2 p(s2 )

c. 23) D 24) B 25) A 26) A 27) A 28) D 29) C

3 .01

.54

.42

.04

E(s2 ) = .54(0) + .42(2) + .04(8) = 1.16; Since E(s2 ) = 2 , s2 is an unbiased estimator of 2 .

Answer Key Testname: CHAPTER 5 30) B 31) D 32) A 33) A 34) A 35) B 36) B 37) B 38) C 39) A 40) C 41) C 42) D 43) B 44) C 45) A 46) B 47) A 48) B 49) B 50) A 51) B 52) C

Chapter 6 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Suppose it is desired to estimate the average time a customer spends in a particular store to within 5 minutes (e.g., + 5 minutes) at 99% reliability. It is estimated that the standard deviation of the times is 15 minutes. How large a sample should be taken to get the desired interval? A) n = 35 B) n = 25 C) n = 299 D) n = 60

2) A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60-year-old male non-smoker. He randomly sampled 65 subjects (60-year-old male non-smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). What value of alpha was used to create this confidence interval? A) 0.025 B) 0.01 C) 0.10 D) 0.05

3) A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60-year-old male non-smoker. He randomly sampled 65 subjects (60-year-old male non-smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). Explain what the phrase "95 percent confident" means in this situation. A) 95 percent of all retired statisticians are underinsured. B) In repeated sampling, 95 percent of the intervals constructed would contain the value of the true population mean. C) In repeated sampling, the mean of the population will fall within the specified intervals 95 percent of the time. D) 95 percent of all the life insurance costs will fall within the specified interval.

4) It is desired to estimate the average time it takes Statistics students to finish a computer project to within two hours at 90% reliability. It is estimated that the standard deviation of the times is 14 hours. How large a sample should be taken to get the desired interval? A) n = 325 B) n = 189 C) n = 231 D) n = 133

5) How much money does the average professional football fan spend on food at a single football game? That question was posed to 60 randomly selected football fans. The sampled results show that the sample mean was $70.00 and prior sampling indicated that the population standard deviation was $17.50. Use this information to create a 95 percent confidence interval for the population mean. 17.50 17.50 A) 70 ± 1.671 B) 70 ± 1.833 60 60

C) 70 ± 1.960

17.50 60

D) 70 ± 1.645

17.50 60

6) Suppose you want to estimate a population proportion p where p = 0.61, n = 1000, and N = 7500. Find an approximate 95% confidence interval for p. A) 0.61 ± .036 B) 0.61 ± .046 C) 0.61 ± .056 D) 0.61 ± .029

7) A study was conducted to determine what proportion of all college students considered themselves as full-time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full-time students. A computer program was used to generate the following 95% confidence interval for the population proportion: (0.64814, 0.75186). The sample size that was used in this problem is considered a large sample. What criteria should be used to determine if n is large? A) If n > 30, then n is considered large. B) When working with proportions, any n is considered large. ^

C) Both np 15 and nq 15. D) If n > 25, then n is considered large. 8) Parking at a large university can be extremely difficult at times. One particular university is trying 8) to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval. Variable N Lo 95% CI Mean Up 95% CI Parking Time 338 9.1944 10.466 11.738

SD 11.885

Give a practical interpretation for the 95% confidence interval given above. A) We are 95% confident that the average parking time of all COBA students falls between 9.19 and 11.74 minutes. B) 95% of the COBA students had parking times that fell between 9.19 and 11.74 minutes. C) 95% of the COBA students had parking times of 10.466 minutes. D) We are 95% confident that the average parking time of the 338 COBA students surveyed falls between 9.19 and 11.74 minutes.

9) Suppose a large labor union wishes to estimate the mean number of hours per month a union member is absent from work. The union decides to sample 352 of its members at random and monitor the working time of each of them for 1 month. At the end of the month, the total number of hours absent from work is recorded for each employee. Which of the following should be used to estimate the parameter of interest for this problem? A) A large sample confidence interval for µ. B) A small sample confidence interval for p. C) A large sample confidence interval for p. D) A small sample confidence interval for µ.

10) After elections were held, it was desired to estimate the proportion of voters who regretted that they did not vote. How many voters must be sampled in order to estimate the true proportion to within 2% (e.g., + 0.02) at the 90% confidence level? Assume that we believe this proportion lies close to 30%. A) n = 2401 B) Cannot determine because no estimate of p or q exists in this problem. C) n = 1421 D) n = 1692 E) n = 2017

10)

11) Parking at a large university can be extremely difficult at times. One particular university is trying 11) to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval. Variable N Lo 95% CI Mean Up 95% CI Parking Time 338 9.1944 10.466 11.738

SD 11.885

Explain what the phrase "95% confident" means when working with a 95% confidence interval. A) In repeated sampling, 95% of the intervals created will contain the population mean. B) 95% of the observations in the population will fall within the endpoints of the interval. C) In repeated sampling, 95% of the sample means will fall within the interval created. D) In repeated sampling, 95% of the population means will fall within the interval created.

12) It is desired to estimate the proportion of college students who feel a sudden relief now that their statistics class is over. How many students must be sampled in order to estimate the true proportion to within 2% at the 90% confidence level? A) n = 2401 B) n = 1692 C) Cannot determine because no estimate of p or q exists in this problem D) n = 133 E) n = 189

12)

13) Suppose you want to estimate a population mean µ and that x = 145, s = 21, n = 750, and N = 4000. Find an approximate 95% confidence interval for µ. A) 145 ± 0.30 B) 145 ± 2.71 C) 145 ± 0.66 D) 145 ± 1.38

13)

14) The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, x = 19.5 and s2 = 25.

14)

15) A study was conducted to determine what proportion of all college students considered themselves as full-time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full-time students. Which of the following would represent the target parameter of interest? A) p B) µ

15)

16) The registrar's office at State University would like to determine a 95% confidence interval for the mean commute time of its students. A member of the staff randomly chooses a parking lot and surveys the first 150 students who park in the chosen lot on a given day. The confidence interval is A) not meaningful because of the lack of random sampling. B) meaningful because the sample size exceeds 30 and the Central Limit Theorem ensures normality of the sampling distribution of the sample mean. C) meaningful because the sample is representative of the population. D) not meaningful because the sampling distribution of the sample mean is not normal.

16)

Estimate the mean number of admissions per 24-hour period with a 90% confidence interval. A) 19.5 ± .563 B) 19.5 ± .129 C) 19.5 ± 1.028 D) 19.5 ± 5.141

17) A retired statistician was interested in determining the average cost of a $200,000.00 term life insurance policy for a 60-year-old male non-smoker. He randomly sampled 65 subjects (60-year-old male non-smokers) and constructed the following 95 percent confidence interval for the mean cost of the term life insurance: ($850.00, $1050.00). State the appropriate interpretation for this confidence interval. Note that all answers begin with "We are 95 percent confidence that…" A) The average term life insurance cost for sampled 65 subjects falls between $850.00 and $1050.00 B) The term life insurance cost for all 60-year-old male non-smokers' insurance policies falls between $850.00 and $1050.00 C) The average term life insurance costs for all 60-year-old male non-smokers falls between $850.00 and $1050.00 D) The term life insurance cost of the retired statistician's insurance policy falls between $850.00 and $1050.00

17)

18) A study was conducted to determine what proportion of all college students considered themselves as full-time students. A random sample of 300 college students was selected and 210 of the students responded that they considered themselves full-time students. A computer program was used to generate the following 95% confidence interval for the population proportion: (0.64814, 0.75186). Which of the following practical interpretations is correct for this confidence interval? A) We are 95% confident that the percentage of all college students who consider themselves full-time students was 0.700. B) We are 95% confident that the percentage of all college students who consider themselves full-time students falls between 0.648 and 0.752. C) We are 95% confident that the percentage of the 300 students who responded that they considered themselves full-time students falls between 0.648 and 0.752. D) We are 95% confident that the percentage of the 300 students who responded that they considered themselves full-time students was 0.700.

18)

19) Parking at a large university can be extremely difficult at times. One particular university is trying to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. Which of the following would represent the target parameter of interest? A) µ B) p

19)

20) Which statement best describes a parameter? A) A parameter is a level of confidence associated with an interval about a sample mean or proportion. B) A parameter is a numerical measure of a population that is almost always unknown and must be estimated. C) A parameter is a sample size that guarantees the error in estimation is within acceptable limits. D) A parameter is an unbiased estimate of a statistic found by experimentation or polling.

20)

21) Parking at a large university can be extremely difficult at times. One particular university is trying 21) to determine the location of a new parking garage. As part of their research, officials are interested in estimating the average parking time of students from within the various colleges on campus. A survey of 338 College of Business (COBA) students yields the following descriptive information regarding the length of time (in minutes) it took them to find a parking spot. Note that the "Lo 95%" and "Up 95%" refer to the endpoints of the desired confidence interval. Variable N Lo 95% CI Mean Up 95% CI Parking Time 338 9.1944 10.466 11.738

SD 11.885

University officials have determined that the confidence interval would be more useful if the interval were narrower. Which of the following changes in the confidence level would result in a narrower interval? A) The university could increase their confidence level. B) The university could decrease their confidence level.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 22) Calculate the percentage of the population sampled if n = 250 and N = 1000. Should the standard error in this situation be multiplied by a finite population correction factor? Explain.

22)

23) The U.S. Commission on Crime randomly selects 600 files of recently committed crimes in an area and finds 380 in which a firearm was reportedly used. Find a 90% confidence interval for p, the true fraction of crimes in the area in which some type of firearm was reportedly used.

23)

24) Let t0 be a particular value of t. Find a value of t0 such that P(t t0 or t t0 ) = .1 where df = 14.

24)

25) A random sample of 50 employees of a large company was asked the question, "Do you participate in the company's stock purchase plan?" The answers are shown below.

25)

yes no no yes no

no yes yes no yes

yes yes no yes no

no no yes yes yes

no yes yes yes yes

yes no no yes yes

yes no yes yes yes

no yes yes no yes

no yes yes yes yes

Use a 90% confidence interval to estimate the proportion of employees who participate in the company's stock purchase plan.

26) Suppose you selected a random sample of n = 7 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 90% confidence interval.

26)

27) How much money does the average professional football fan spend on food at a single football game? That question was posed to 43 randomly selected football fans. The sample results provided a sample mean and standard deviation of $18.00 and $3.20, respectively. Find and interpret a 99% confidence interval for µ.

27)

28) The following sample of 16 measurements was selected from a population that is approximately 28) normally distributed. 61 85 92 77 83 81 75 78 95 87 69 74 76 84 80 83 Construct a 90% confidence interval for the population mean.

29) Suppose (1,000, 2,100) is a 95% confidence interval for µ. To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Explain why an increase in sample size will lead to a narrower interval of the estimate of µ.

29)

30) Sales of a new line of athletic footwear are crucial to the success of a newly formed company. The company wishes to estimate the average weekly sales of the new footwear to within $350 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1575. How many weeks of data must be sampled for the company to get the information it desires?

30)

31) For n = 40 and p = .35, is the sample size large enough to construct a confidence for p?

31)

32) A random sample of n = 100 measurements was selected from a population with unknown mean µ and standard deviation . Calculate a 95% confidence interval if x = 26 and s2 = 16.

32)

33) For n = 800 and p = .99, is the sample size large enough to construct a confidence for p?

33)

34) A newspaper reports on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 505 teenagers. Estimate the proportion of all teenagers who want more family discussions about religion. Use a 95% confidence level.

34)

35) The following random sample was selected from a normal population: 9, 11, 8, 10, 14, 8. Construct a 95% confidence interval for the population mean µ.

35)

36) Suppose that 100 samples of size n = 50 are independently chosen from the same population and that each sample is used to construct its own 95% confidence interval for an unknown population mean µ. How many of the 100 confidence intervals would you expect to actually contain µ?

36)

37) Suppose you selected a random sample of n = 29 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 95% confidence interval.

37)

38) To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has recently been marketed. The FDA tests on this cigarette yielded a mean nicotine content of 26.8 milligrams and standard deviation of 2.4 milligrams for a sample of n = 81 cigarettes. Find a 95% confidence interval for µ.

38)

39) You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 95% confidence interval. You manage to obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were resold at an average price of $13,800 with a standard deviation of $600. Create a 95% confidence interval for the true mean resale value of a 5-year-old car of that model.

39)

40) A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 95% confident that the margin of error of their estimate is within 4%?

40)

41) The standard deviation of a population is estimated to be 315 units. To estimate the population mean to within 50 units with 99% reliability, what size sample should be selected?

41)

42) For n = 40 and p = .45, is the sample size large enough to construct a confidence for p?

42)

43) When is the finite population correction factor used?

43)

44) Let t0 be a particular value of t. Find a value of t0 such that P(t t0 ) = .005 where df = 9.

44)

45) Calculate the percentage of the population sampled if n = 100 and N = 10,000. Should the standard error in this situation be multiplied by a finite population correction factor? Explain.

45)

46) What is the rule of thumb for the finite population correction factor?

46)

47) A random sample of n = 144 measurements was selected from a population with unknown

47)

mean µ and standard deviation . Calculate a 90% confidence interval if x = 3.55 and s = .49.

48) Suppose you wanted to estimate a binomial proportion, p, correct to within .03 with probability 0.90. What size sample would need to be selected if p is known to be approximately 0.65?

48)

49) The following data represent the scores of a sample of 50 randomly chosen students on a standardized test. 39 71 79 85 90 a. b.

48 71 79 86 91

55 73 79 86 92

63 74 80 88 92

66 76 80 88 93

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

49)

71 79 85 89 99

Write a 95% confidence interval for the mean score of all students who took the test. Identify the target parameter and the point estimator.

50) A random sample of 80 observations produced a mean x = 35.4 and a standard deviation s = 3.1.

50)

a. Find a 90% confidence interval for the population mean µ. b. Find a 95% confidence interval for µ. c. Find a 99% confidence interval for µ. d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?

51) A computer package was used to generate the following printout for estimating the mean sale 51) price of homes in a particular neighborhood. X = sale_price SAMPLE MEAN OF X =46300 SAMPLE STANDARD DEV = 13747 SAMPLE SIZE OF X = 25 CONFIDENCE = 90 UPPER LIMIT = 51003.90 SAMPLE MEAN OF X =46300 LOWER LIMIT = 41596.10 A friend suggests that the mean sale price of homes in this neighborhood is $44,000. Comment on your friend's suggestion.

52) A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 99% confidence interval was calculated to be ($2,181,260, $5,836,180). Give a practical interpretation of the confidence interval.

52)

53) A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval was calculated to be ($2,181,260, $5,836,180). Based on the interval above, do you believe the average total compensation of CEOs in the service industry is more than $1,500,000?

53)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 54) An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). The researcher claimed that the average GPA of all male students exceeded 2.94. Using the confidence interval supplied above, how do you respond to this claim? A) We are 95% confident that the researcher is incorrect. B) We cannot make any statement regarding the average GPA of male university students at the 95% confidence level. C) We are 100% confident that the researcher is incorrect. D) We are 95% confident that the researcher is correct.

54)

55) Given the values of x, s, and n, form a 99% confidence interval for .

55)

x = 4.5, s = 2.5, n = 29 A) (3.43, 14.04)

B) (1.9, 3.59)

C) (1.37, 5.62)

D) (1.85, 3.75)

56) How much money does the average professional football fan spend on food at a single football game? That question was posed to ten randomly selected football fans. The sampled results show that the sample mean and sample standard deviation were $70.00 and $17.50, respectively. Use this information to create a 95 percent confidence interval for the population mean. 17.50 17.50 A) 70 ± 1.833 B) 70 ± 1.960 60 60 C) 70 ± 2.228

17.50 60

D) 70 ± 2.262

56)

17.50 60

Answer the question True or False. 57) If no estimate of p exists when determining the sample size for a confidence interval for a proportion, we can use .5 in the formula to get a value for n. A) True B) False Solve the problem. 58) Suppose a large labor union wishes to estimate the mean number of hours per month a union member is absent from work. The union decides to sample 351 of its members at random and monitor the working time of each of them for 1 month. At the end of the month, the total number of hours absent from work is recorded for each employee. If the mean and standard deviation of the

57)

58)

sample are x = 9.3 hours and s = 3.8 hours, find a 90% confidence interval for the true mean number of hours a union member is absent per month. Round to the nearest thousandth. A) 9.3 ± .334 B) 9.3 ± .171 C) 9.3 ± .018 D) 9.3 ± .183

59) A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 90% confident that the margin of error of their estimate is about 2%? A) 6766 B) 1029 C) 1692 D) 3383

59)

60) A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 90% confidence interval to estimate the true proportion of students who receive financial aid. A) .59 ± .057 B) .59 ± .004 C) .59 ± .002 D) .59 ± .398 61) For the given combination of

and degrees of freedom (df), find the value of

used to find the lower endpoint of a confidence interval for 2 . = 0.1, df = 14 A) 22.3621 B) 6.5706 C) 21.0642

/2 that would be

Answer the question True or False. 63) For data with two outcomes (success or failure), the binomial proportion of successes is likely to be the parameter of interest. A) True B) False Solve the problem. 64) A newspaper reported on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 536 teenagers. Estimate the proportion of all teenagers who want more family discussions about school. Use a 99% confidence level. A) .63 ± .002 B) .37 ± .002 C) .37 ± .054 D) .63 ± .054 65) A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following confidence interval: (.438, .642). Using the information above, what sample size would be necessary if we wanted to estimate the true proportion to within 3% using 95% reliability? A) 1110 B) 1061 C) 1068 D) 1025 and degrees of freedom (df), find the value of

would be used to find the upper endpoint of a confidence interval for 2 . = 0.01, df = 6 A) 18.5476 B) 0.675727 C) 0.411740

2 (1 - /2) that

62)

63)

64)

65)

66)

D) 0.872085

Answer the question True or False. 67) For quantitative data, the target parameter is most likely to be the mode of the data. A) True B) False

61)

D) 23.6848

62) Find the value of t0 such that the following statement is true: P(-t0 t t0 ) = .95 where df = 15. A) 2.947 B) 2.131 C) 1.753 D) 2.602

66) For the given combination of

60)

67)

Solve the problem. 68) A computer package was used to generate the following printout for estimating the mean sale price of 68) homes in a particular neighborhood. X = sale_price SAMPLE MEAN OF X =46,500 SAMPLE STANDARD DEV = 13,747 SAMPLE SIZE OF X = 15 CONFIDENCE = 95 UPPER LIMIT = 54,113.6 SAMPLE MEAN OF X =46,500 LOWER LIMIT = 38,886.4 Which of the following is a practical interpretation of the interval above? A) We are 95% confident that the true sale price of all homes in this neighborhood fall between $38,886.40 and $54,113.60. B) 95% of the homes in this neighborhood have sale prices that fall between $38,886.40 and $54,113.60. C) We are 95% confident that the mean sale price of all homes in this neighborhood falls between $38,886.40 and $54,113.60. D) All are correct practical interpretations of this interval.

69) A random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. Estimate the true proportion of all Americans who are in favor of gun control legislation using a 99% confidence interval. A) .5625 ± .6337 B) .5625 ± .0202 C) .4375 ± .6337 D) .4375 ± .0202

69)

70) The volumes (in ounces) of juice in eight randomly selected juice bottles are as follows: 70) 15.0 15.8 15.5 15.9 15.6 15.1 15.1 15.7 Find a 99% confidence interval for the standard deviation, , of the volumes of juice in all such bottles. Round to the nearest hundredth when necessary. A) (0.21, 0.93) B) (0.21, 0.80) C) (0.21, 0.97) D) (0.20, 0.80) 71) What type of car is more popular among college students, American or foreign? One hundred 71) fifty-nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below of a 90% confidence interval for the proportion of college students who prefer American automobiles. SAMPLE PROPORTION = .396 SAMPLE SIZE = 159 UPPER LIMIT = .460 LOWER LIMIT = .332 Based on the interval above, do you believe that 51% of all college students prefer American automobiles? A) No, and we are 100% sure of it. B) Yes, and we are 100 %sure of it. C) No, and we are 90% confident of it. D) Yes, and we are 90% confident of it.

72) What type of car is more popular among college students, American or foreign? One hundred fifty-nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below for the proportion of college students who prefer American automobiles.

72)

SAMPLE PROPORTION = .384297 SAMPLE SIZE = 159 UPPER LIMIT = .464240 LOWER LIMIT = .331153 What proportion of the sampled students prefer foreign automobiles? A) .384297 B) .331153 C) .464240

D) .615703

73) An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). What assumption is necessary for the confidence interval analysis to work properly? A) The population that we are sampling from needs to be a t-distribution with n-1 degrees of freedom. B) The Central Limit theorem guarantees that no assumptions about the population are necessary. C) The population that we are sampling from needs to be approximately normally distributed. D) The sampling distribution of the sample mean needs to be approximately normally distributed.

73)

74) You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 90% confidence interval. You manage to obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were resold at an average price of $12,610 with a standard deviation of $600. Suppose that the interval is calculated to be ($12,355.92, $12,864.08). How could we alter the sample size and the confidence coefficient in order to guarantee a decrease in the width of the interval? A) Increase the sample size but decrease the confidence coefficient. B) Keep the sample size the same but increase the confidence coefficient. C) Increase the sample size and increase the confidence coefficient. D) Decrease the sample size but increase the confidence coefficient.

74)

75) A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following 90% confidence interval: (.438, .642). Based on the interval, is the population proportion of females equal to 57%? A) No, and we are 90% sure of it. B) No, the proportion is 54%. C) Yes, and we are 90% sure of it. D) Maybe. 57% is a believable value of the population proportion based on the information above.

75)

76) How much money does the average professional football fan spend on food at a single football game? That question was posed to 10 randomly selected football fans. The sample results provided a sample mean and standard deviation of $19.00 and $2.55, respectively. Use this information to construct a 90% confidence interval for the mean. A) 19 ± 1.383(2.55/ 10) B) 19 ± 1.833(2.55/ 10) C) 19 ± 1.812(2.55/ 10) D) 19 ± 1.796(2.55/ 10)

76)

77) A computer package was used to generate the following printout for estimating the mean sale price of 77) homes in a particular neighborhood. X = sale_price SAMPLE MEAN OF X =46,500 SAMPLE STANDARD DEV = 13,747 ` SAMPLE SIZE OF X = 15 CONFIDENCE = 95 UPPER LIMIT = 54,113.6 SAMPLE MEAN OF X =46,500 LOWER LIMIT = 38,886.4 At what level of reliability is the confidence interval made? A) 95% B) 52.5% C) 5%

D) 47.5%

78) A 95% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($139,528, $154,454). Give a practical interpretation of the interval. A) We are 95% confident that the mean salary of the sampled CEOs falls in the interval $139,528 to $154,454. B) 95% of all CEOs in the electronics industry have salaries that fall between $139,528 to $154,454. C) 95% of the sampled CEOs have salaries that fell in the interval $139,528 to $154,454. D) We are 95% confident that the mean salary of all CEOs in the electronics industry falls in the interval $139,528 to $154,454. Answer the question True or False. 79) The Central Limit Theorem guarantees an approximately normal sampling distribution for the sample mean for large sample sizes, so no knowledge about the distribution of the population is necessary for the corresponding interval to be valid. A) True B) False Solve the problem. 80) A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 340 college students produced the following 95% confidence interval for the proportion of college students who prefer one of the colas: (.340, .459). What additional assumptions are necessary for the interval to be valid? A) The sample was randomly selected from an approximately normal population. B) The sample proportion equals the population proportion. C) The population proportion has an approximately normal distribution. D) No additional assumptions are necessary.

78)

79)

80)

81) You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 99% confidence interval. You manage to obtain data on 17 recently resold 5-year-old foreign sedans of the same model. These 17 cars were resold at an average price of $12,560 with a standard deviation of $700. What is the 99% confidence interval for the true mean resale value of a 5- year-old car of this model? A) 12,560 ± 2.921(700/ 16) B) 12,560 ± 2.921(700/ 17) C) 12,560 ± 2.898(700/ 17) D) 12,560 ± 2.575(700/ 17)

81)

Answer the question True or False. 82) The confidence level is the confidence coefficient expressed as a percentage. A) True B) False

82)

Solve the problem. 83) Find the value of t0 such that the following statement is true: P(-t0 t t0 ) = .99 where df = 9. A) 2.262 B) 3.250 C) 2.2821 D) 1.833

83)

84) The mean systolic blood pressure for a random sample of 28 women aged 18-24 is 115.1 mm Hg and the standard deviation is 12.7 mm Hg. Construct a 90% confidence interval for the standard deviation , of the systolic blood pressures of all women aged 18-24. Round to the nearest hundredth when necessary. A) (10.89, 15.51) B) (10.26, 16.04) C) (9.91, 17.78) D) (10.42, 16.42)

84)

85) Find the value of t0 such that the following statement is true: P(-t0 t t0 ) = .90 where df = 14. A) 1.761 B) 2.624 C) 2.145 D) 1.345

85)

86) What type of car is more popular among college students, American or foreign? One hundred fifty-nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below of a 99% confidence interval for the proportion of college students who prefer American automobiles.

86)

SAMPLE PROPORTION = .396 SAMPLE SIZE = 159 UPPER LIMIT = .496 LOWER LIMIT = .296 Which of the following is a correct practical interpretation of the interval? A) We are 99% confident that the proportion of the 159 sampled students who prefer American cars falls between .296 and .496. B) We are 99% confident that the proportion of all college students who prefer foreign cars falls between .296 and .496. C) We are 99% confident that the proportion of all college students who prefer American cars falls between .296 and .496. D) 99% of all college students prefer American cars between .296 and .496 of the time.

87) What is the confidence coefficient in a 95% confidence interval for µ? A) .05 B) .025 C) .95

D) .475

87)

88) In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a __________ interval. A) narrower B) wider C) biased D) less significant

88)

89) What type of car is more popular among college students, American or foreign? One hundred fifty-nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below for the proportion of college students who prefer American automobiles.

89)

SAMPLE PROPORTION = .396226 SAMPLE SIZE = 159 UPPER LIMIT = .46611 LOWER LIMIT = .331134 Is the sample large enough for the interval to be valid? A) No, the population of college students is not normally distributed. ^

B) Yes, since np and nq are both greater than 15. C) No, the sample size should be at 10% of the population. D) Yes, since n > 30. 90) A marketing research company is estimating which of two soft drinks college students prefer. A random sample of n college students produced the following 99% confidence interval for the proportion of college students who prefer drink A: (.373, .453). Identify the point estimate for estimating the true proportion of college students who prefer that drink. A) .373 B) .413 C) .453 D) .04

90)

91) A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 99% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What additional assumption is necessary for this confidence interval to be valid? A) The sample standard deviation is less than the degrees of freedom. B) The distribution of the sample means is approximately normal. C) The population of total compensations of CEOs in the service industry is approximately normally distributed. D) None. The Central Limit Theorem applies.

91)

92) A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following 99% confidence interval: (.438, .642). State the level of reliability used to create the confidence interval. A) 64.2% B) 99% C) between 43.8% and 64.2% D) 72%

92)

93) Fifteen SmartCars were randomly selected and the highway mileage of each was noted. The analysis yielded a mean of 47 miles per gallon and a standard deviation of 5 miles per gallon. Which of the following would represent a 90% confidence interval for the average highway mileage of all SmartCars? 5 5 5 5 A) 47 ± 1.753 B) 47 ± 1.761 C) 47 ± 1.345 D) 47 ± 1.645 15 15 15 15

93)

94) Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 68.1, 53.3, 239.8, 481.7, 111.1, 178.4, 96.2, and 223.8. What value will be used as the point estimate for the mean endowment of all private colleges in the United States? A) 207.486 B) 1452.4 C) 8 D) 181.55

94)

95) Given the values of x, s, and n, form a 99% confidence interval for 2 .

95)

x = 10.1, s = 8.5, n = 8 A) (28.5, 584.27)

B) (24.94, 511.24)

C) (3.22, 48.02)

D) (27.37, 408.18)

96) Determine the confidence level for the given confidence interval for µ. x ± 1.96

A) 97.5%

96)

B) 92.5%

C) 2.5%

D) 95%

97) A 90% confidence interval for the mean percentage of airline reservations being canceled on the day of the flight is (2.5%, 6.4%). What is the point estimator of the mean percentage of reservations that are canceled on the day of the flight? A) 1.95% B) 4.45% C) 3.9% D) 3.20%

97)

98) Which information is not shown on the screen below?

98)

A) the sample standard deviation C) the confidence level

B) the sample size D) the sample mean

99) A computer package was used to generate the following printout for estimating the mean sale price of 99) homes in a particular neighborhood. X = sale_price SAMPLE MEAN OF X =46,300 SAMPLE STANDARD DEV = 13,747 SAMPLE SIZE OF X = 15 CONFIDENCE = 95 UPPER LIMIT = 53,913.60 SAMPLE MEAN OF X =46,300 LOWER LIMIT = 38,686.40 A friend suggests that the mean sale price of homes in this neighborhood is $41,000. Comment on your friend's suggestion. A) Your friend is wrong, and you are 95% certain. B) Your friend is correct, and you are 100% certain. C) Based on this printout, all you can say is that the mean sale price might be $41,000. D) Your friend is correct, and you are 95% certain.

100) Sales of a new line of athletic footwear are crucial to the success of a company. The company wishes to estimate the average weekly sales of the new footwear to within $500 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1200. How many weeks of data must be sampled for the company to get the information it desires? A) 10 weeks B) 16 weeks C) 4 weeks D) 7794 weeks

100)

101) A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 99% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What would happen to the confidence interval if the confidence level were changed to 98%? A) It is impossible to tell until the 98% interval is constructed. B) The interval would get wider. C) The interval would get narrower. D) There would be no change in the width of the interval.

101)

Answer the question True or False. 102) Since the population standard deviation is almost always known, we use it instead of the sample standard deviation s when finding a confidence interval. A) True B) False Solve the problem. 103) Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield x = 180.975 and s = 143.042. Calculate a 90% confidence interval for the mean endowment of all private colleges in the United States. A) 180.975 ± 94.066 B) 180.975 ± 102.453 C) 180.975 ± 95.836 D) 180.975 ± 100.561

102)

103)

104) To help consumers assess the risks they are taking, the Food and Drug Administration (FDA) publishes the amount of nicotine found in all commercial brands of cigarettes. A new cigarette has recently been marketed. The FDA tests on this cigarette yielded mean nicotine content of 24.3 milligrams and standard deviation of 2.1 milligrams for a sample of n = 9 cigarettes. Construct a 99% confidence interval for the mean nicotine content of this brand of cigarette. A) 24.3 ± 2.491 B) 24.3 ± 2.275 C) 24.3 ± 2.349 D) 24.3 ± 2.413 Answer the question True or False. 105) The confidence coefficient is the relative frequency with which the interval estimator encloses the population parameter when the estimator is used repeatedly a very large number of times. A) True B) False Solve the problem. 106) A random sample of 250 students at a university finds that these students take a mean of 14.6 credit hours per quarter with a standard deviation of 1.9 credit hours. The 90% confidence interval for the mean is 14.6 ± 0.198. Interpret the interval. A) 90% of the students take between 14.402 to 14.798 credit hours per quarter. B) The probability that a student takes 14.402 to 14.798 credit hours in a quarter is 0.90. C) We are 90% confident that the average number of credit hours per quarter of the sampled students falls in the interval 14.402 to 14.798 hours. D) We are 90% confident that the average number of credit hours per quarter of students at the university falls in the interval 14.402 to 14.798 hours. 107) What is the confidence level of the following confidence interval for µ? x ± 1.645

A) 98%

104)

105)

106)

107)

B) 95%

C) 165%

D) 90%

108) A newspaper reported on the topics that teenagers most want to discuss with their parents. The findings, the results of a poll, showed that 46% would like more discussion about the family's financial situation, 37% would like to talk about school, and 30% would like to talk about religion. These and other percentages were based on a national sampling of 549 teenagers. Using 99% reliability, can we say that more than 30% of all teenagers want to discuss school with their parents? A) No, since the value .30 is not contained in the 99% confidence interval. B) Yes, since the values inside the 99% confidence interval are greater than .30. C) No, since the value .30 is not contained in the 99% confidence interval. D) Yes, since the value .30 falls inside the 99% confidence interval.

108)

109) The mean replacement time for a random sample of 12 CD players is 8.6 years with a standard deviation of 3.3 years. Construct the 99% confidence interval for the population variance, 2 .

109)

110) A previous random sample of 4000 U.S. citizens yielded 2250 who are in favor of gun control legislation. How many citizens would need to be sampled for a 90% confidence interval to estimate the true proportion within 2%? A) 1556 B) 1665 C) 1759 D) 1692

110)

Assume the data are normally distributed, and round to the nearest hundredth when necessary. A) (2.12, 6.78) B) (1.36, 13.94) C) (4.84, 39.23) D) (4.48, 46.02)

Answer the question True or False. 111) One way of reducing the width of a confidence interval is to reduce the size of the sample taken. A) True B) False Solve the problem. 112) A random sample of 250 students at a university finds that these students take a mean of 14.7 credit hours per quarter with a standard deviation of 1.9 credit hours. Estimate the mean credit hours taken by a student each quarter using a 95% confidence interval. Round to the nearest thousandth. A) 14.7 ± .236 B) 14.7 ± .171 C) 14.7 ± .011 D) 14.7 ± .015 113) Find z /2 for the given value of . = 0.02 A) 3.08

112)

113)

B) 0.18

C) 2.33

D) 2.05

114) Let t0 be a specific value of t. Find t0 such that the following statement is true: P(t t0 ) = .01 where df = 20. A) -2.539 B) 2.539

C) 2.528

114) D) -2.528

115) A marketing research company is estimating which of two soft drinks college students prefer. A random sample of 148 college students produced the following confidence interval for the proportion of college students who prefer drink A: (.344, .494). Is this a large enough sample for this analysis to work? A) Yes, since n = 148 (which is 30 or more). ^

111)

115)

B) Yes, since both np 15 and nq 15. C) No. D) It is impossible to say with the given information. 116) A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. The 95% confidence interval for p is 59 ± .07. Interpret this interval. A) We are 95% confident that between 52% and 66% of the sampled students receive some sort of financial aid. B) 95% of the students receive between 52% and 66% of their tuition in financial aid. C) We are 95% confident that 59% of the students are on some sort of financial aid. D) We are 95% confident that the true proportion of all students receiving financial aid is between .52 and .66.

116)

117) What is z /2 when

117)

A) 1.96

= 0.06?

B) 1.645

C) 2.33

D) 1.88

118) A sociologist develops a test to measure attitudes towards public transportation, and 47 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4. Construct an approximate 95% confidence interval for the mean score of all such subjects. A) 76.2 ± 0.91 B) 76.2 ± 6.87 C) 76.2 ± 6.24 D) 76.2 ± 3.12

118)

119) A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 99% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). Explain what the phrase "99% confident" means. A) 99% of the population values will fall within the interval. B) 99% of the similarly constructed intervals would contain the value of the sample mean. C) 99% of the sample means from similar samples fall within the interval. D) In repeated sampling, 99% of the intervals constructed would contain µ.

119)

120) We intend to estimate the average driving time of a group of commuters. From a previous study, we believe that the average time is 42 minutes with a standard deviation of 10 minutes. We want our 90 percent confidence interval to have a margin of error of no more than plus or minus 2 minutes. What is the smallest sample size that we should consider? A) 68 B) 136 C) 9 D) 7

120)

121) A random sample of 15 crates have a mean weight of 165.2 pounds and a standard deviation of 15.8 pounds. Construct a 95% confidence interval for the population standard deviation . Assume the population is normally distributed, and round to the nearest hundredth when necessary. A) (12.15, 23.06) B) (2.91, 6.27) C) (133.81, 620.92) D) (11.57, 24.92)

121)

Answer the question True or False. ^

122) The sampling distribution for p is approximately normal for a large sample size n, where n is ^

122)

considered large if both n p 15 and n(1 - p) 15. A) True

B) False

Solve the problem. 123) A random sample of 4000 U.S. citizens yielded 2140 who are in favor of gun control legislation. Find the point estimate for estimating the proportion of all Americans who are in favor of gun control legislation. A) .5350 B) 2140 C) .4650 D) 4000 Answer the question True or False. 124) One way of reducing the width of a confidence interval is to reduce the confidence level. A) True B) False Solve the problem. 125) A 95% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($138,395, $150,096). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width? A) Increase the sample size and decrease the confidence level. B) Increase the sample size and increase the confidence level. C) Decrease the sample size and decrease the confidence level. D) Decrease the sample size and increase the confidence level.

123)

124)

125)

126) An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). Which of the following statements is true? A) In construction of the confidence interval, a z-value was used. B) In construction of the confidence interval, a t-value with 20 degrees of freedom was used. C) In construction of the confidence interval, a z-value with 20 degrees of freedom was used. D) In construction of the confidence interval, a t-value with 19 degrees of freedom was used.

126)

127) Calculate the finite population correction factor for n = 300 and N = 1500. A) 2.000 B) .8944 C) .8000

127)

D) .4472

128) A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 2% with 95% reliability, how many students would need to be sampled? A) 2324 B) 1186 C) 47 D) 562

128)

129) A random sample of n measurements was selected from a population with unknown mean µ and known 129) standard deviation . Calculate a 95% confidence interval for µ for the given situation. Round to the nearest hundredth when necessary. n = 80, x = 65, A) 65 ± 0.49

= 20

B) 65 ± 3.68

C) 65 ± 39.2

D) 65 ± 4.38

130) The daily intakes of milk (in ounces) for ten five-year old children selected at random from one school 130) were: 15.9 10.8 22.9 16.8 29.0 17.1 20.2 14.4 21.4 29.3 Find a 99% confidence interval for the standard deviation, , of the daily milk intakes of all five-year olds at this school. Round to the nearest hundredth when necessary. A) (3.73, 12.37) B) (3.61, 12.37) C) (3.73, 13.77) D) (0.88, 3.38) 131) A survey of 865 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct an approximate 95% confidence interval for the true proportion of all voters in the state who favor approval. A) 0.47 ± 0.049 B) 0.47 ± 0.025 C) 0.47 ± 0.017 D) 0.47 ± 0.034

131)

132) Explain what the phrase 95% confident means when we interpret a 95% confidence interval for µ. A) 95% of the observations in the population fall within the bounds of the calculated interval. B) The probability that the sample mean falls in the calculated interval is 0.95. C) In repeated sampling, 95% of similarly constructed intervals contain the value of the population mean. D) 95% of similarly constructed intervals would contain the value of the sampled mean.

132)

133) The director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, x = 19.8 and s2 = 4. If

133)

the director wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 99% reliability, what is the minimum sample size she should use? A) 27 B) 11 C) 107 D) 42

Answer the question True or False. 134) When the sample size is small, confidence intervals for a population proportion are more reliable when the population proportion p is near 0 or 1. A) True B) False Solve the problem. 135) Suppose the population standard deviation is known to be x when n = 300 and N = 1500. A) 17.32 B) 3.87

= 150. Calculate the standard error of

C) 6.92

D) 7.75

134)

135)

Answer Key Testname: CHAPTER 6 1) D 2) D 3) B 4) D 5) C 6) D 7) C 8) A 9) A 10) C 11) A 12) B 13) D 14) C 15) A 16) A 17) C 18) B 19) A 20) B 21) B 22) 25%; Yes, more than 5% of the population was sampled. 23) Let p = the true fraction of crimes in the area in which some type of firearm was reportedly used. ^

^ ^ 380 = .6333 and q = 1 - p = 1 - .6333 = .3667. 600

The confidence interval for p is p ± z /2

pq . n

= 1 - .90 = .1. For confidence coefficient .90, 1 - = .90 /2 = .1/2 = .05. z /2 = z.05 = 1.645. The 90% confidence interval is: .6333 ± 1.645

.6333(.3667) = .6333 ± .032 600

24) t0 = 1.761; Use table for t.050 with 14 degrees of freedom. ^

25) p =

32 = .64; The confidence interval is .64 ± 1.645 50

(.64)(.36) 50

.64 ± .112.

26) z: 1.645 and t: 1.943; The t value is considerably bigger than the z value. = 1 - .99 = .01. 27) For confidence coefficient .99, 1 /2 = .01/2 = .005 z.005 = 2.575. The confidence interval is: x ± z /2

s 3.20 = 18.00 ± 2.575 = 18.00 ± 1.257 = ($16.74, $19.26) n 43

We are 99% confident that the average amount a fan spends on food at a single professional football game is between $16.74 and $19.26.

Answer Key Testname: CHAPTER 6 s 8.367 = 80 ± 1.753 = 80 ± 3.667 n 16

28) x = 80; s = 8.367; x ± t /2

29) An increase in the sample size reduces the sampling variation of the point estimate as it is calculated as / n. The larger the sample size, the smaller this variation which leads to narrower intervals. z /2 2 2. 30) To determine the sample size necessary to estimate µ, we use n = SE For confidence coefficient .90, 1 - = .90 = 1 - .90 = .1. /2 = .1/2 = .05. z /2 = z.05 = 1.645. 1.645 2 n= 15752 = 54.7970. Round up to n = 55. 350 ^

31) No; np = 14 < 15 s = 26 ± 1.96 32) x ± z /2 n

16 = 26 ± .784 100

33) No; nq = 8 < 15 = 1 - .95 = .05. 34) For confidence coefficient .95, 1 - = .95 /2 = .05/2 = .025. z /2 = z.025 = 1.96. The 95% confidence interval for p is: ^

p ± z /2

pq n

.30(.70) 505

.30 ± 1.96

35) x = 10; s = 2.28; x ± t /2

.30 ± .0400

s 2.28 = 10 ± 2.571 = 10 ± 2.393 n 6

36) 95% of the 100 samples, or 95, are expected to produce a confidence interval that contains µ. 37) z: 1.96 and t: 2.048; The t value is a little bigger than the z value. = 1 - .95 = .05. 38) For confidence coefficient .95, 1 - = .95 /2 = .05/2 = .025. z /2 = z.025 = 1.96. The 95% confidence interval is: x ± z /2

s = 26.8 ± 1.96 n

2.4 81

26.8 ± .523 = (26.277, 27.323)

= 1 - .95 = .05. 39) For confidence coefficient .95, 1 /2 = 0.05/2 = 0.025. With df = n - 1 = 17 - 1 = 16, t0.025 = 2.120. The 95% confidence interval is: x ± t /2

s 600 = 13,800 ± 2.120 = (13,491.49, 14,108.51) n 17

For this interval to be valid, we must assume that the population of resale values for all 5-year-old cars of this model follows an approximately normal distribution.

Answer Key Testname: CHAPTER 6

40) To determine the sample size necessary to estimate p, we use n = For confidence coefficient .95, 1 /2 = .05/2 = .025. z /2 = z.025 = 1.96.

= .95

z /2 2 SE

= 1 - .95 = .05.

Since no estimate of p exists, we use p = q = .5. 1.96 2 n= (.5)(.5) = 600.25. Round up to n = 601. .04

41) To determine the sample size necessary to estimate µ, we use n =

z /2 2

For confidence coefficient .99, 1 - = .99 = 1 - .99 = .01. /2 = .01/2 = .005. z /2 = z.005 = 2.575. 2.575 2 n= 3152 = 263.1695. Round up to n = 264. 50 ^

42) Yes; np = 18 > 15 and nq = 22 > 15 43) The finite population correction factor is used when the sample size is large relative to the size of the population. 44) t0 = 3.250; Use table for t.005 with 9 degrees of freedom. 45) 1%; No, less than 5% of the population was sampled. 46) Use the finite population correction factor when: n/N > .05. s .49 = 3.55 ± 1.645 = 3.55 ± .067 47) x ± z /2 n 144

48) To determine the sample size necessary to estimate p, we use n =

z /2 2 SE

p(1 - p).

For confidence coefficient .90, 1 - = .90 = 1 - .90 = .1. /2 = .1/2 = .05. z /2 = z.05 = 1.645. 1.645 2 n= (.65)(1 - .65) = 684.022986. Round up to n = 685. .03

49) a.

The sample mean is 79.98 and the sample standard deviation is 12.34. 12.34 The interval is 79.98 ± 1.96 79.98 ± 3.42. 50

The target parameter is the mean score of all students who took the test, and the point estimator is the sample mean 79.98. s 3.1 = 35.4 ± 1.645 = 35.4 ± .57 50) a. x ± z /2 n 80 b. x ± z /2

s 3.1 = 35.4 ± 1.96 = 35.4 ± .68 n 80

c. x ± z /2

s 3.1 = 35.4 ± 2.575 = 35.4 ± .89 n 80

d. increases 51) Your friend could be correct. $44,000 is contained in the 90% confidence interval. It cannot be ruled out as a possible value for the mean sales price.

Answer Key Testname: CHAPTER 6 52) We are 99% confident that the average total compensation of CEOs in the service industry is contained in the interval $2,181,260 to $5,836,180. 53) Since all of the values in the interval are greater than $1,500,000, it seems very likely that the mean is greater than $1,500,000, but we can't be 100% certain. 54) D 55) D 56) D 57) A 58) A 59) C 60) A 61) D 62) B 63) A 64) C 65) B 66) B 67) B 68) C 69) B 70) A 71) C 72) D 73) C 74) A 75) D 76) B 77) A 78) D 79) A 80) D 81) B 82) A 83) B 84) D 85) A 86) C 87) C 88) A 89) B 90) B 91) C 92) B 93) B 94) D 95) B 96) D 97) B 98) C 99) C 26

Answer Key Testname: CHAPTER 6 100) B 101) C 102) B 103) C 104) C 105) A 106) D 107) D 108) B 109) D 110) B 111) B 112) A 113) C 114) D 115) B 116) D 117) D 118) C 119) D 120) A 121) D 122) A 123) A 124) A 125) A 126) D 127) B 128) A 129) D 130) C 131) D 132) C 133) A 134) B 135) D

Chapter 7 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A small private college is interested in determining the percentage of its students who live off campus and drive to class. Specifically, it was desired to determine if less than 20% of their current students live off campus and drive to class. The college decided to take a random sample of 108 of their current students to use in the analysis. Is the sample size of n = 108 large enough to use this inferential procedure? A) Yes, since both np and nq are greater than or equal to 15 B) No C) Yes, since the central limit theorem works whenever proportions are used D) Yes, since n 30 2) It is desired to test H0 : µ = 50 against HA: µ 50 using

= 0.10. The population in question is

uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If µ is really equal to 45, what is the power of the test? A) 0.8959 B) 0.1041 C) 0.2456 D) 0.7544

3) A consumer product magazine recently ran a story concerning the increasing prices of digital 3) cameras. The story stated that digital camera prices dipped a couple of years ago, but now are beginning to increase in price because of added features. According to the story, the average price of all digital cameras a couple of years ago was $215.00. A random sample of cameras was recently taken and entered into a spreadsheet. It was desired to test to determine if that average price of all digital cameras is now more than $215.00. The information was entered into a spreadsheet and the following printout was obtained: One-Sample T Test Null Hypothesis: µ = 215 Alternative Hyp: µ > 215 95% Conf Interval Variable Mean SE Lower Upper T DF P Camera Price 245.23 15.620 212.740 277.720 1.94 21 0.0333 Cases Included 22 Use the p- value given above to determine which of the following conclusions is correct. A) At = 0.03, there is insufficient evidence to indicate that the mean price of all digital cameras exceeds $215.00 B) At = 0.10, there is insufficient evidence to indicate that the mean price of all digital cameras exceeds $215.00 C) At = 0.05, there is insufficient evidence to indicate that the mean price of all digital cameras exceeds $215.00 D) At = 0.01, there is sufficient evidence to indicate that the mean price of all digital cameras exceeds $215.00

4) A small private college is interested in determining the percentage of its students who live off campus and drive to class. Specifically, it was desired to determine if less than 20% of their current students live off campus and drive to class. Suppose a sample of 108 students produced a test statistic of z = -1.35. Find the p-value for the test of interest to the college. A) p = 0.9115 B) p = 0.0885 C) p = 0.1770 D) p = 0.4115

5) A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 238 students and asked each to provide the amount of time they spent traveling to campus. This variable, travel time, was then used conduct a test of hypothesis. The goal was to determine if the average travel time of all the university's students differed from 20 minutes. Suppose the sample mean and sample standard deviation were calculated to be 23.2 and 20.26 minutes, respectively. Calculate the value of the test statistic to be used in the test. A) z = 2.551 B) z = 37.59 C) z = 0.173 D) z = 2.437

6) A consumer product magazine recently ran a story concerning the increasing prices of digital cameras. The story stated that digital camera prices dipped a couple of years ago, but now are beginning to increase in price because of added features. According to the story, the average price of all digital cameras a couple of years ago was $215.00. A random sample of n = 22 cameras was recently taken and entered into a spreadsheet. It was desired to test to determine if that average price of all digital cameras is now more than $215.00. Find a rejection region appropriate for this test if we are using = 0.05. A) Reject H0 if t > 1.725 B) Reject H0 if t > 1.721

7) A consumer product magazine recently ran a story concerning the increasing prices of digital cameras. The story stated that digital camera prices dipped a couple of years ago, but now are beginning to increase in price because of added features. According to the story, the average price of all digital cameras a couple of years ago was $215.00. A random sample of n = 200 cameras was recently taken and entered into a spreadsheet. It was desired to test to determine if that average price of all digital cameras is now more than $215.00. Find the large-sample rejection region appropriate for this test if we are using = 0.05. A) Reject H0 if z > 1.645. B) Reject H0 if z < -1.96 or z > 1.96.

8) A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was 23.243 minutes and the sample standard deviation was 20.40 minutes. Find the rejection region for determining if the population standard deviation exceeds 20 minutes. Use = 0.05. A) Reject H0 if z > 1.645 B) Reject H0 if 2 > 34.7642

9) It is desired to test H0 : µ = 50 against HA: µ 50 using

C) Reject H0 if t > 1.717

D) Reject H0 if t > 2.080 or t < -2.080

C) Reject H0 if z < -1.645 or z > 1.645.

D) Reject H0 if z < -1.96.

C) Reject H0 if 2 > 67.5048

D) Reject H0 if 2 > 71.4202

= 0.10. The population in question is

uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If µ is really equal to 48, what is the probability that the hypothesis test would lead the investigator to commit a Type II error? A) 0.7567 B) 0.1006 C) 0.2433 D) 0.8994

10) The State Association of Retired Teachers has recently taken flak from some of its members regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Identify the null and alternative hypothesis that should be tested to determine if a name change is warranted. A) H0 : p = 0.6 vs. Ha : p 0.6 B) H0 : p = 0.6 vs. Ha : p < 0.6 C) H0 : p 0.6 vs. Ha : p < 0.6

10)

D) H0 : p = 0.6 vs. Ha : p > 0.6

11) A consumer product magazine recently ran a story concerning the increasing prices of digital 11) cameras. The story stated that digital camera prices dipped a couple of years ago, but now are beginning to increase in price because of added features. According to the story, the average price of all digital cameras a couple of years ago was $215.00. A random sample of cameras was recently taken and entered into a spreadsheet. It was desired to test to determine if that average price of all digital cameras is now more than $215.00. The information was entered into a spreadsheet and the following printout was obtained: One-Sample T Test Null Hypothesis: µ = 215 Alternative Hyp: µ > 215

Variable Camera Price

95% Conf Interval Mean SE Lower Upper 245.23 15.620 212.740 277.720 1.9421

T DF 0.0333

Cases Included 22 Is a sample size n = 22 large enough to utilize the central limit theorem in this inferential procedure? A) No, since either np or nq is less than 15 B) Yes, since the central limit theorem works whenever means are used C) Yes, since both np and nq are greater than or equal to 15 D) No, since n < 30

12) The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion exceeds 25%, then the lab will scale back a proposed enlargement of its facilities. Suppose 200 business students were randomly sampled and 65 have laptops. What assumptions are necessary for this test to be satisfied? A) The sample size n satisfies both np0 15 and nq0 15.

12)

13) A random sample of n observations, selected from a normal population, is used to test the null hypothesis H0 : 2 = 155. Specify the appropriate rejection region.

13)

B) The population has an approximately normal distribution. C) The sample size n satisfies n 30. D) The sample proportion is close to .5.

Ha : 2 < 155, n = 14, A) 2 < 4.66043

= .01

B) 2 < 29.1413

C) 2 < 4.10691

D) 2 < 27.6883

14) A small private college is interested in determining the percentage of its students who live off campus and drive to class. Specifically, it was desired to determine if less than 20% of their current students live off campus and drive to class. Find the large-sample rejection region for the test of interest to the college when using a level of significance of 0.01. A) Reject H0 if z < -1.28. B) Reject H0 if z < -2.33.

14)

15) A random sample of n observations, selected from a normal population, is used to test the null hypothesis H0 : 2 = 155. Specify the appropriate rejection region.

15)

C) Reject H0 if z < -2.33 or z > 2.33.

Ha : 2 > 155, n = 25, A) 2 > 33.1963

D) Reject H0 if z < -1.96.

= .10

B) 2 > 36.4151

C) 2 > 15.6587

D) 2 > 34.3816

16) The State Association of Retired Teachers has recently taken flak from some of its members 16) regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Suppose the association decided to conduct a test of hypothesis using the following null and alternative hypotheses: H0 : p = 0.6

HA: p > 0.6 Define a Type II Error in the context of this problem. A) They conclude that more than 60% of the association wants a name change when that is, in fact, true. B) They conclude that exactly 60% of the association wants a name change when, in fact, that is not true. C) They conclude that exactly 60% of the association wants a name change when that is, in fact, true. D) They conclude that more than 60% of the association wants a name change when, in fact, that is not true.

17) A bottling company produces bottles that hold 10 ounces of liquid. Periodically, the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling company randomly samples 64 bottles and finds the average amount of liquid held by the bottles is 9.9145 ounces with a standard deviation of 0.40 ounce. Suppose the p-value of this test is 0.0436. State the proper conclusion. A) At = 0.05, reject the null hypothesis. B) At = 0.10, fail to reject the null hypothesis. C) At = 0.05, accept the null hypothesis. D) At = 0.025, reject the null hypothesis.

17)

18) A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 238 students and asked each to provide the amount of time they spent traveling to campus. This variable, travel time, was then used conduct a test of hypothesis. The goal was to determine if the average travel time of all the university's students differed from 20 minutes. Suppose the large-sample test statistic was calculated to be z = 2.14. Find the p-value for this test of hypothesis. A) p = 0.9838 B) p = 0.4838 C) p = 0.0162 D) p = 0.0324

18)

19) A large university is interested in learning about the average time it takes students to drive to campus. 19) The university sampled 238 students and asked each to provide the amount of time they spent traveling to campus. This variable, travel time, was then used to create a confidence interval and to conduct a test of hypothesis, both of which are shown in the printout below. One-Sample Z Test Null Hypothesis: µ = 20 Alternative Hyp: µ > 20 95% Conf Interval Variable Mean SE Lower Upper Z P Camera Price 23.243 1.3133 20.669 25.817 2.47 0.0071 Cases Included 238 What conclusion can be made from the test of hypothesis conducted in this printout? Begin each answer with, "When testing at = 0.01…" A) …there is sufficient evidence to indicate that the average travel time of all students exceeds 20 minutes. B) …there is sufficient evidence to indicate that the average travel time of all students is equal to 20 minutes. C) …there is insufficient evidence to indicate that the average travel time of all students exceeds 20 minutes. D) …there is insufficient evidence to indicate that the average travel time of all students is equal to 20 minutes.

20) A small private college is interested in determining the percentage of its students who live off campus and 20) drive to class. Specifically, it was desired to determine if less than 20% of their current students live off campus and drive to class. A sample of 108 students was randomly selected and the following printout was obtained: Hypothesis Test - One Proportion Sample Size Successes Proportion Null Hypothesis: Alternative Hyp: Difference Standard Error -1.35 Z

108 16 0.14815 P = 0.2 P < 0.2 -0.05185 0.03418 p-value

0.0885

Based on the information contained in the printout, what conclusion would be correct when testing at = 0.05. A) Reject H0 B) Fail to reject H0 C) Accept HA D) Accept H0

21) If a hypothesis test were conducted using the null hypothesis to be rejected. A) 0.060 B) 0.040

= 0.05, to which of the following p-values would cause

C) 0.100 5

D) 0.055

21)

22) A consumer product magazine recently ran a story concerning the increasing prices of digital cameras. The story stated that digital camera prices dipped a couple of years ago, but are now beginning to increase in price because of added features. According to the story, the average price of all digital cameras a couple of years ago was $215.00. A random sample of cameras was recently taken and entered into a spreadsheet. It was desired to test to determine if that average price of all digital cameras is now more than $215.00. What null and alternative hypothesis should be tested? A) H0 : µ 215 vs. HA: µ < 215 B) H0 : µ = 215 vs. HA: µ > 215

22)

23) A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 51 students and asked each to provide the amount of time they spent traveling to campus. The sample results found that the sample mean was 23.243 minutes and the sample standard deviation was 20.40 minutes. It is desired to determine if the population standard deviation exceeds 20 minutes. Calculate the test statistic for this test of hypothesis. A) 2 = 53.06 B) 2 = 51 C) 2 = 52.02 D) 2 = 58.11

23)

C) H0 : µ = 215 vs. HA: µ 215

D) H0 : µ = 215 vs. HA: µ < 215

24) The State Association of Retired Teachers has recently taken flak from some of its members 24) regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Suppose the association decided to conduct a test of hypothesis using the following null and alternative hypotheses: H0 : p = 0.6

HA: p > 0.6 Define a Type I Error in the context of this problem. A) They conclude that more than 60% of the association wants a name change when, in fact, that is not true. B) They conclude that more than 60% of the association wants a name change when that is, in fact, true. C) They conclude that exactly 60% of the association wants a name change when, in fact, that is not true. D) They conclude that exactly 60% of the association wants a name change when that is, in fact, true.

25) A random sample of n observations, selected from a normal population, is used to test the null hypothesis H0 : 2 = 155. Specify the appropriate rejection region. Ha : 2 155, n = 10, = .05 A) 2.70039 < 2 < 19.0228

25)

B) 2 < 3.32511 or 2 > 16.9190 D) 2 < 3.24697 or 2 > 20.4831

C) 2 < 2.70039 or 2 > 19.0228

26) A large university is interested in learning about the average time it takes students to drive to campus. The university sampled 238 students and asked each to provide the amount of time they spent traveling to campus. This variable, travel time, was then used conduct a test of hypothesis. The goal was to determine if the average travel time of all the university's students differed from 20 minutes. Find the large-sample rejection region for the test of interest to the college when using a level of significance of 0.05. A) Reject H0 if z < -1.96 or z > 1.96. B) Reject H0 if z > 1.645. C) Reject H0 if z < -1.96.

D) Reject H0 if z < -1.645 or z > 1.645.

26)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 27) A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .05 to ensure proper inoculation. A random sample of 41 injections resulted in a variance of .135. Specify the rejection region for the test. Use = .10.

27)

28) An educational testing service designed an achievement test so that the range in student scores would be greater than 300 points. To determine whether the objective was achieved, the testing service gave the test to a random sample of 41 students and found that the sample mean and variance were 776 and 2517, respectively. Specify the null and alternative hypotheses for determining whether the test achieved the desired dispersion in scores. Assume that range = 6 .

28)

29) In a test of H0 : µ = 70 against Ha : µ 70, the sample data yielded the test statistic z = 2.11.

29)

Find and interpret the p-value for the test.

30) The scores on a standardized test are reported by the testing agency to have a mean of 70. Based 30) on his personal observations, a school guidance counselor believes the mean score is much higher. He collects the following scores from a sample of 50 randomly chosen students who took the test. 39 71 79 85 90

48 71 79 86 91

55 73 79 86 92

63 74 80 88 92

66 76 80 88 93

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 99

Use the data to conduct a test of hypotheses at = .05 to determine whether there is any evidence to support the counselor's suspicions.

31) A random sample of n = 15 observations is selected from a normal population to test H0 : µ = 2.89 against Ha : µ < 2.89 at = .01. Specify the rejection region.

31)

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. 32) z < -1.96 or z > 1.96 32) Solve the problem. 33) A sample of 8 measurements, randomly selected from a normally distributed population, resulted in the following summary statistics: x = 5.2, s = 1.1. Test the null hypothesis that the mean of the population is 4 against the alternative hypothesis µ 4. Use = .05.

33)

34) The scores on a standardized test are reported by the testing agency to have a mean of 75. Based 34) on his personal observations, a school guidance counselor believes the mean score is much higher. He collects the following scores from a sample of 50 randomly chosen students who took the test. 39 71 79 85 90

48 71 79 86 91

55 73 79 86 92

63 74 80 88 92

66 76 80 88 93

68 76 82 88 95

68 76 83 88 96

69 77 83 89 97

70 78 83 89 97

71 79 85 89 99

Find and interpret the p-value for the test of H0 : µ = 75 against Ha : µ > 75.

35) A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .08 to ensure proper inoculation. A random sample of 25 injections resulted in a variance of .135. Calculate the test statistic for the test of interest.

35)

36) In a test of H0 : µ = 65 against Ha : µ > 65, the sample data yielded the test statistic z = 1.38.

36)

37) The hypotheses for H0 : µ = 125.4 and Ha : µ 125.4 are tested at

37)

Find and interpret the p-value for the test.

= .10. Sketch the

appropriate rejection region.

38) State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently uses has a mean life of 980 hours. A competitor claims that its bulbs, which cost the same as the brand the university currently uses, have a mean life of more than 980 hours. The university has decided to purchase the new brand if, when tested, the evidence supports the manufacturer's claim at the .05 significance level. Suppose 82 bulbs were

38)

tested with the following results: x = 999 hours, s = 76 hours. Find the rejection region for the test of interest to the State University.

39) It has been estimated that the G-car obtains a mean of 30 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 64 G-cars and records the mileage obtained for each car over a driving course similar to that used to obtain the

39)

estimate. The following data resulted: x = 31.8 miles per gallon, s = 8 miles per gallon. Calculate the power of the test if the true value of the mean is 31 miles per gallon. Use a value of = .025.

40) A random sample of n = 18 observations is selected from a normal population to test H0: µ = 145 against Ha : µ 145 at

40)

= .10. Specify the rejection region.

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. 41) z < -2.33 or z > 2.33 41)

Solve the problem. 42) Identify the observed level of significance for the test summarized on the screen below and interpret its value.

42)

43) A revenue department is under orders to reduce the time small business owners spend filling out pension form ABC-5500. Previously the average time spent on the form was 63 hours. In order to test whether the time to fill out the form has been reduced, a sample of 72 small business owners who annually complete the form was randomly chosen and their completion times recorded. The mean completion time for the sample was 62.7 hours with a standard deviation of 20 hours. State the rejection region for the desired test at = .01.

43)

44) Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims that more than 85% of firms in the manufacturing sector still do not offer any child-care benefits. A random sample of 460 manufacturing firms is selected, and only 34 of them offer child-care benefits. Specify the rejection region that the union will use when testing at = .10.

44)

For the given rejection region, sketch the sampling distribution for z and indicate the location of the rejection region. 45) z < -1.28 45) 46) z > 2.575

46)

47) z < -1.96

47)

Solve the problem. 48) Based on the information in the screen below, what would you conclude in the test of H0 : µ 14, Ha : µ > 14. Use = .01.

48)

49) A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than .05 to ensure proper inoculation. A random sample of 49 injections was measured. Suppose the p-value for the test is p = .0024. State the proper conclusion using = .01.

49)

50) In a test of H0 : µ = 12 against Ha : µ > 12, a sample of n = 75 observations possessed mean x

50)

51) A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 23% of women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately. A random sample of 82 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in eleven. Specify the null and alternative hypotheses that the researchers wish to test.

51)

52) A sample of 6 measurements, randomly selected from a normally distributed population,

52)

= 13.1 and standard deviation s = 4.3. Find and interpret the p-value for the test.

resulted in the following summary statistics: x = 9.1, s = 1.5. Test the null hypothesis that the mean of the population is 10 against the alternative hypothesis µ < 10. Use = .05.

53) A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 15% of women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately. A random sample of 70 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 8. Calculate the test statistic used by the researchers for the corresponding test of hypothesis.

53)

54) An educational testing service designed an achievement test so that the range in student scores would be greater than 360 points. To determine whether the objective was achieved, the testing service gave the test to a random sample of 30 students and found that the sample mean and variance were 759 and 1943, respectively. Conduct the test for H0 : 2 = 3600 vs. Ha : 2 > 3600 using = .025. Assume the range is 6 .

54)

the 55) A supermarket sells rotisserie chicken at a fixed price per chicken rather than by the weight of 55) chicken. The store advertises that the average weight of their chickens is 4.6 pounds. A random sample of 30 of the store's chickens yielded the weights (in pounds) shown below. 4.4 4.7 4.6 4.4 4.5 4.3 4.6 4.5 4.6 4.9 4.6 4.8 4.3 4.4 4.7 4.5 4.2 4.3 4.1 4.0 4.5 4.6 4.2 4.4 4.7 4.8 5.0 4.2 4.1 4.5 Test whether the population mean weight of the chickens is less than 4.6 pounds. Use

= .05.

56) Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims that more than 85% of firms in the manufacturing sector still do not offer any child-care benefits. A random sample of 500 manufacturing firms is selected and asked if they offer child-care benefits. Suppose the p-value for this test was reported to be p = .1216. State the conclusion of interest to the union. Use = .10.

56)

57) A random sample of 8 observations from an approximately normal distribution is shown below. 57) 5

Find the observed level of significance for the test of H0 : µ = 5 against Ha : µ 5. Interpret the result.

58) State University uses thousands of fluorescent light bulbs each year. The brand of bulb it currently uses has a mean life of 1000 hours. A competitor claims that its bulbs, which cost the same as the brand the university currently uses, have a mean life of more than 1000 hours. The university has decided to purchase the new brand if, when tested, the evidence supports the manufacturer's claim at the .05 significance level. Suppose 64 bulbs were

58)

tested with the following results: x = 1027.5 hours, s = 80 hours. Conduct the test using = .05.

59) A random sample of 100 observations is selected from a binomial population with

59)

unknown probability of success, p. The computed value of p is equal to .56. Find the observed levels of significance in a test of H0 : p = .5 against Ha : p > .5. Interpret the result. A 60) A company reports that 80% of its employees participate in the company's stock purchase plan.60) random sample of 50 employees was asked the question, "Do you participate in the stock purchase plan?" The answers are shown below. yes no no yes no

no yes yes no yes

yes yes no yes no

no no yes yes yes

no yes yes yes yes

yes no no yes yes

yes no yes yes yes

no yes yes no yes

no yes yes yes yes

Perform the appropriate test of hypothesis to investigate your suspicion that fewer than 80% of the company's employees participate in the plan. Use = .05. the 61) A supermarket sells rotisserie chicken at a fixed price per chicken rather than by the weight of 61) chicken. The store advertises that the average weight of their chickens is 4.6 pounds. A random sample of 30 of the store's chickens yielded the weights (in pounds) shown below. 4.4 4.7 4.6 4.4 4.5 4.3 4.6 4.5 4.6 4.9 4.6 4.8 4.3 4.4 4.7 4.5 4.2 4.3 4.1 4.0 4.5 4.6 4.2 4.4 4.7 4.8 5.0 4.2 4.1 4.5 Find and interpret the p-value in a test of H0 : µ = 4.6 against Ha : µ < 4.6.

62) According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 105 bushels per acre. Fifteen farmers who belong to a cooperative plant the soybeans in soil prepared as specified. Each uses a 40-acre plot and records the mean yield per acre. The mean and variance for the sample of the 15 farms are x = 90 and s2 = 10,125. Find the rejection region used for determining if the mean yield for the soybeans is not equal to 105 bushels per acre. Use

= .05.

62)

63) Based on the information in the screen below, what would you conclude in the test of H0 : µ 14, Ha : µ > 14. Use = .01.

63)

64) In a test of H0 : µ = 250 against Ha : µ 250, a sample of n = 100 observations possessed

64)

mean x = 247.3 and standard deviation s = 11.4. Find and interpret the p-value for the test.

65) According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 531 bushels per acre. Twenty-five farmers who belong to a cooperative plant the soybeans in soil prepared as specified. Each uses a 40-acre plot and records the mean yield per acre. The mean and variance for the sample of 25 farms are x = 506 and s2 = 9720. Specify the null and alternative hypotheses

65)

66) It has been estimated that the G-car obtains a mean of 30 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 49 G-cars and records the mileage obtained for each car over a driving course similar to that used to obtain the

66)

used to determine if the mean yield for the soybeans is different than advertised.

estimate. The following data resulted: x = 31.2 miles per gallon, s = 7 miles per gallon. Calculate the value of if the true value of the mean is 32 miles per gallon. Use = .025.

67) A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 20% of women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately. A random sample of 80 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 9. Is the sample size sufficiently large to conduct this test of hypothesis? Explain.

67)

a 68) A recipe submitted to a magazine by one of its subscribers states that the mean baking time for68) cheesecake is 55 minutes. A test kitchen preparing the recipe before it is published in the magazine makes the cheesecake 10 times at different times of the day in different ovens. The following baking times (in minutes) are observed. 54

Assume that the baking times belong to a normal population. Test the null hypothesis that the mean baking time is 55 minutes against the alternative hypothesis µ > 55. Use = .05.

69) The hypotheses for H0 : µ = 65 and Ha : µ > 65 are tested at rejection region.

= .05. Sketch the appropriate

69)

70) An ink cartridge for a laser printer is advertised to print an average of 10,000 pages. A random70) sample of eight businesses that have recently bought this cartridge are asked to report the number of pages printed by a single cartridge. The results are shown. 9771 9975

9811 10,079

9885 10,145

9914 10,214

Assume that the data belong to a normal population. Test the null hypothesis that the mean number of pages is 10,000 against the alternative hypothesis µ 10,000. Use = .10.

71) A random sample of n = 12 observations is selected from a normal population to test H0: µ = 22.1 against Ha : µ > 22.1 at

71)

= .05. Specify the rejection region.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 72) Data were collected from the sale of 25 properties by a local real estate agent. The following printout concentrated on the land value variable from the sampled properties.

72)

HYPOTHESIS: MEAN X = x X = land_value SAMPLE MEAN OF X = 44,626 SAMPLE VARIANCE OF X = 273,643,254 SAMPLE SIZE OF X = 25 x = 40,183 MEAN X - x = 4443 t = 1.34293 D.F. = 24 P-VALUE = 0.1918585 P-VALUE/2 = 0.0959288 SD. ERROR = 3308.43 What assumptions are necessary for any inferences derived from this printout to be valid? A) The sampled data are approximately normal. B) The sample was selected from an approximately normal population. C) The sampling distribution of the sample mean is approximately normal. D) None. The Central Limit Theorem makes any assumptions unnecessary.

73) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time spent on a delivery does not exceed 34 minutes. The owner has randomly selected 15 customers and delivered pizzas to their homes in order to test whether the mean delivery time actually exceeds 34 minutes. Suppose the p-value for the test was found to be .0280. State the correct conclusion. A) At = .03, we fail to reject H0 . B) At = .025, we fail to reject H0 . C) At

D) At

= .05, we fail to reject H0 .

= .02, we reject H0 .

73)

Answer the question True or False. 74) A rejection region is established in each tail of the sampling distribution for a two-tailed test. A) True B) False

74)

Solve the problem. 75) Data were collected from the sale of 25 properties by a local real estate agent. The following printout concentrated on the land value variable from the sampled properties.

75)

HYPOTHESIS: MEAN X = x X = land_value SAMPLE MEAN OF X = 50,432 SAMPLE VARIANCE OF X = 273,643,254 SAMPLE SIZE OF X = 25 x = 45,989 MEAN X - x = 4443 t = 1.34293 D.F. = 24 P-VALUE = 0.1918585 P-VALUE/2 = 0.0959288 SD. ERROR = 3308.43 Suppose we are interested in testing whether the mean land value from this neighborhood differs from 45,989. Which hypotheses would you test? A) H0 : µ = 45,989 vs. Ha : µ < 45,989 B) H0 : µ 45,989 vs. Ha : µ = 45,989

C) H0 : µ = 45,989 vs. Ha : µ > 45,989

Answer the question True or False. 76) The rejection region for a two-tailed test with A) True

D) H0 : µ = 45,989 vs. Ha : µ 45,989

= .05 is -1.96 < z < 1.96. B) False

76)

For the given binomial sample size and null-hypothesized value of p0 , determine whether the sample size is large enough to use the normal approximation methodology to conduct a test of the null hypothesis H0 : p = p0 .

77) n = 800, p0 = 0.99 A) No

77)

B) Yes

Solve the problem. 78) Researchers have claimed that the average number of headaches per student during a semester of Statistics is 16. Statistics students believe the average is higher. In a sample of n = 19 students the mean is 20 headaches with a deviation of 1.9. Which of the following represent the null and alternative hypotheses necessary to test the students' belief? A) H0 : µ = 16 vs. Ha : µ < 16 B) H0 : µ = 16 vs. Ha : µ > 16 C) H0 : µ = 16 vs. Ha : µ 16

D) H0 : µ < 16 vs. Ha : µ = 16

78)

Find the rejection region for the specified hypothesis test. 79) Consider a test of H0 : µ = 4. For the following case, give the rejection region for the test in terms of the z-statistic: Ha : µ < 4,

A) z > -1.28

79)

= 0.10

B) z < 1.645

C) z < -1.28

D) z < -1.645

Solve the problem. 80) What is the probability associated with not making a Type II error? A) B) (1 - ) C) (1 - )

81) An industrial supplier has shipped a truckload of teflon lubricant cartridges to an aerospace customer. The customer has been assured that the mean weight of these cartridges is in excess of the 11 ounces printed on each cartridge. To check this claim, a sample of n = 19 cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample

80)

81)

are: x = 11.2 ounces, s = .19 ounce. To determine whether the supplier's claim is true, consider the test, H0 : µ = 11 vs. Ha : µ > 11, where µ is the true mean weight of the cartridges. Calculate the value of the test statistic. A) 1.053

B) 20.000

C) 4.588

D) 2.000

82) Suppose we wish to test H0 : µ = 33 vs. Ha : µ > 33. What will result if we conclude that the mean is greater than 33 when its true value is really 39? A) a Type I error C) a Type II error

82)

B) a correct decision D) none of the above

83) A __________ is a numerical quantity computed from the data of a sample and is used in reaching a decision on whether or not to reject the null hypothesis. A) test statistic B) critical value C) significance level D) parameter

83)

84) An industrial supplier has shipped a truckload of teflon lubricant cartridges to an aerospace customer. The customer has been assured that the mean weight of these cartridges is in excess of the 10 ounces printed on each cartridge. To check this claim, a sample of n = 10 cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample

84)

are: x = 10.11 ounces, s = .30 ounce. To determine whether the supplier's claim is true, consider the test, H0 : µ = 10 vs. Ha : µ > 10, where µ is the true mean weight of the cartridges. Find the rejection region for the test using = .01. A) t > 3.25, where t depends on 9 df C) t > 2.821, where t depends on 9 df

B) z > 2.33 D) |z| > 2.58

85) We have created a 95% confidence interval for µ with the result (8, 13). What conclusion will we make if we test H0 : µ = 15 vs. Ha : µ 15 at = .05?

85)

A) Fail to reject H0 .

B) Accept H0 rather than Ha . C) Reject H0 in favor of Ha .

D) We cannot tell what our decision will be with the information given. Answer the question True or False. 86) Type I errors and Type II errors are complementary events so that 1 - P(Type II error) = 1 - . A) True B) False 15

= P(Type I error) =

86)

Solve the problem. 87) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time spent on a delivery does not exceed 39 minutes. The owner has randomly selected 19 customers and delivered pizzas to their homes in order to test whether the mean delivery time actually exceeds 39 minutes. What assumption is necessary for this test to be valid? A) The population variance must equal the population mean. B) The sample mean delivery time must equal the population mean delivery time. C) The population of delivery times must have a normal distribution. D) None. The Central Limit Theorem makes any assumptions unnecessary. 88) A bottling company produces bottles that hold 12 ounces of liquid. Periodically, the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling company randomly samples 22 bottles and finds the average amount of liquid held by the bottles is 11.6 ounces with a standard deviation of .2 ounce. Which of the following is the set of hypotheses the company wishes to test? A) H0 : µ < 12 vs. Ha : µ = 12 B) H0 : µ = 12 vs. Ha : µ > 12 C) H0 : µ = 12 vs. Ha : µ < 12

87)

88)

D) H0 : µ = 12 vs. Ha : µ 12

89) n = 50, p0 = 0.9 A) Yes

89)

B) No

Solve the problem. 90) A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 13 miles per hour (mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. To determine whether the site meets the organization's requirements, consider the test, H0 : µ = 13 vs. Ha : µ > 13, where µ is the true mean wind speed at the site and = .05. Suppose

90)

the observed significance level (p-value) of the test is calculated to be p = 0.4413. Interpret this result. A) Since the p-value exceeds = .05, there is insufficient evidence to reject the null hypothesis. B) The probability of rejecting the null hypothesis is 0.4413. C) Since the p-value greatly exceeds = .05, there is strong evidence to reject the null hypothesis. D) We are 55.87% confident that µ = 13.

91) Suppose we wish to test H0 : µ = 39 vs. Ha : µ < 39. Which of the following possible sample results

91)

gives the most evidence to support Ha (i.e., reject H0 )?

A) x = 37, s = 3

B) x = 35, s = 2

C) x = 36, s = 5

D) x = 35, s = 5

Answer the question True or False. 92) Under the assumption that µ = µa , where µa is the alternative mean, the distribution of x is mound shaped and symmetric about µa .

A) True

B) False

92)

Solve the problem. 93) Given H0 : µ = 18, Ha : µ < 18, and p = 0.070. Do you reject or fail to reject H0 at the .05 level of

93)

significance? A) fail to reject H0

B) reject H0

C) not sufficient information to decide 94) We never conclude "Accept H0 " in a test of hypothesis. This is because: A)

= p(Type I error) is not known.

94)

B) H0 is never true.

C) We want H0 to be false.

95) It is desired to test H0 : µ = 40 against Ha: µ < 40 using

= p(Type II error) is not known.

= .10. The population in question is

95)

uniformly distributed with a standard deviation of 10. A random sample of 36 will be drawn from this population. If µ is really equal to 35, what is the probability that the hypothesis test would lead the investigator to commit a Type II error? A) .4573 B) .0854 C) .9573 D) .0427

Find the rejection region for the specified hypothesis test. 96) Consider a test of H0 : µ = 8. For the following case, give the rejection region for the test in terms of the z-statistic: Ha : µ > 8,

A) z > 1.28

96)

= 0.10

B) |z| > 1.645

C) z > 1.96

D) |z| > 1.28

Answer the question True or False. 97) A Type I error occurs when we accept a false null hypothesis. A) True B) False Solve the problem. 98) The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion differs from 35%, then the lab will modify a proposed enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is 2.6. Find the p-value for a two-tailed test of hypothesis. A) .0047 B) .4906 C) .0094 D) .4953

97)

98)

99) Consider the following printout.

99)

HYPOTHESIS: VARIANCE X = x X = gpa SAMPLE MEAN OF X = 2.4968 SAMPLE VARIANCE OF X = .25000 SAMPLE SIZE OF X = 240 HYPOTHESIZED VALUE (x) = 2.6 VARIANCE X - x = -.1032 z = -3.19754 Suppose we tested Ha : µ < 2.6. Find the appropriate rejection region if we used

A) Reject if z < -1.96. C) Reject if z < -1.645.

= .05.

B) Reject if z > 1.96 or z < -1.96. D) Reject if z > 1.645 or z < -1.645.

100) How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 42 tissues during a cold. Suppose a random sample of 10,000 people yielded the

100)

following data on the number of tissues used during a cold: x = 37, s = 24. Identify the null and alternative hypothesis for a test to determine if the mean number of tissues used during a cold is less than 42. A) H0 : µ = 42 vs. Ha : µ > 42 B) H0 : µ > 42 vs. Ha : µ 42

C) H0 : µ = 42 vs. Ha : µ 42

D) H0 : µ = 42 vs. Ha : µ < 42

101) Consider a test of H0 : µ = 70 performed with the computer. SPSS reports a two-tailed p-value of 0.0094. Make the appropriate conclusion for the given situation: Ha : µ 70, z = 2.6,

A) Fail to reject H0

101)

= 0.01

B) Reject H0

102) Consider the following printout.

102)

HYPOTHESIS: VARIANCE X = x X = gpa SAMPLE MEAN OF X = 1.9928 SAMPLE VARIANCE OF X = .16000 SAMPLE SIZE OF X = 192 HYPOTHESIZED VALUE (x) = 2.1 VARIANCE X - x = -.1072 z = -3.71352 Is this a large enough sample for this analysis to work? A) Yes, since the np > 15 and nq > 15. B) Yes, since the population of GPA scores is approximately normally distributed. C) No. D) Yes, since n = 192, which is greater than 30.

103) It is desired to test H0 : µ = 55 against Ha: µ < 55 using

= .10. The population in question is

103)

uniformly distributed with a standard deviation of 15. A random sample of 49 will be drawn from this population. If µ is really equal to 50, what is the power of this test? A) .3531 B) .1469 C) .2938 D) .8531

104) n = 75, p0 = 0.2 A) Yes

104)

B) No

Solve the problem. 105) Consider a test of H0 : µ = 90 performed with the computer. SPSS reports a two-tailed p-value of 0.2112. Make the appropriate conclusion for the given situation: Ha : µ > 90, z = 1.25,

A) Fail to reject H0

105)

= 0.10

B) Reject H0

Answer the question True or False. 106) The null hypothesis represents the status quo to the party performing the sampling experiment. A) True B) False

106)

Solve the problem. 107) How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 64 tissues during a cold. Suppose a random sample of 100 people yielded the

107)

following data on the number of tissues used during a cold: x = 49, s = 19. Using the sample information provided, set up the calculation for the test statistic for the relevant hypothesis test, but do not simplify. 49 - 64 49 - 64 49 - 64 49 - 64 A) z = B) z = C) z = D) z = 19 19 19 192 100 1002 100

108) An insurance company sets up a statistical test with a null hypothesis that the average time for processing a claim is 3 days, and an alternative hypothesis that the average time for processing a claim is greater than 3 days. After completing the statistical test, it is concluded that the average time exceeds 3 days. However, it is eventually learned that the mean process time is really 3 days. What type of error occurred in the statistical test? A) Type I error B) No error occurred in the statistical sense. C) Type III error D) Type II error

108)

109) How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 61 tissues during a cold. Suppose a random sample of 10,000 people yielded the

109)

following data on the number of tissues used during a cold: x = 56, s = 22. We want to test the alternative hypothesis Ha : µ < 61. State the correct rejection region for = .05.

A) Reject H0 if z < -1.96.

B) Reject H0 if z < -1.645.

C) Reject H0 if z > 1.96 or z < -1.96.

D) Reject H0 if z > 1.645.

110) Data were collected from the sale of 25 properties by a local real estate agent. The following printout concentrated on the land value variable from the sampled properties.

110)

HYPOTHESIS: MEAN X = x X = land_value SAMPLE MEAN OF X = 53,021 SAMPLE VARIANCE OF X = 273,643,254 SAMPLE SIZE OF X = 25 x = 48,578 MEAN X - x = 4443 t = 1.34293 D.F. = 24 P-VALUE = 0.1918585 P-VALUE/2 = 0.0959288 SD. ERROR = 3308.43 Find the p-value for testing whether the mean land value differs from $48,578. A) p = 0.1918585 B) p = 0.808142 C) p = 0.308142 D) p = 0.0959288

For the given value of and observed significance level (p-value), indicate whether the null hypothesis would be rejected. 111) = 0.05, p-value = 0.001 111) A) Fail to reject H0 B) Reject H0

Solve the problem. 112) A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 15 miles per hour (mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. Based on a sample of n = 144 wind speed recordings (taken at random intervals), the

112)

wind speed at the site averaged x = 14.6 mph, with a standard deviation of s = 3.0 mph. To determine whether the site meets the organization's requirements, consider the test, H0 : µ = 15 vs. Ha : µ > 15, where µ is the true mean wind speed at the site and = .05. Fill in the

blanks. "A Type I error in the context of this problem is to conclude that the true mean wind speed at the site _____ 15 mph when it actually _____ 15 mph." A) equals; equals B) exceeds; equals C) equals; exceeds D) exceeds; exceeds

113) I want to test H0 : p = .7 vs. Ha : p .7 using a test of hypothesis. This test would be called a(n) ____________ test. A) upper-tailed

B) two-tailed

C) one-tailed

D) lower-tailed

Answer the question True or False. 114) The null distribution is the distribution of the test statistic assuming the null hypothesis is true; it mound shaped and symmetric about the null mean µ0 . A) True

B) False

113)

114)

Solve the problem. 115) I want to test H0 : p = .3 vs. Ha : p .3 using a test of hypothesis. If I concluded that p is .3 when, in

115)

fact, the true value of p is not .3, then I have made a __________. A) correct decision B) Type I error C) Type II error D) Type I and Type II error

116) n = 700, p0 = 0.01 A) No

116)

B) Yes

Solve the problem. 117) A company claims that 9 out of 10 doctors (i.e., 90%) recommend its brand of cough syrup to their patients. To test this claim against the alternative that the actual proportion is less than 90%, a random sample of doctors was taken. Suppose the test statistic is z = -2.23. Can we conclude that H0 should be rejected at the a) = .10, b) = .05, and c) = .01 level? A) a) no; b) no; c) yes C) a) yes; b) yes; c) yes

117)

B) a) no; b) no; c) no D) a) yes; b) yes; c) no

118) Data were collected from the sale of 25 properties by a local real estate agent. The following printout concentrated on the land value variable from the sampled properties.

118)

HYPOTHESIS: MEAN X = x X = land_value SAMPLE MEAN OF X = 46,590 SAMPLE VARIANCE OF X = 273,643,254 SAMPLE SIZE OF X = 25 x = 42,147 MEAN X - x = 4443 t = 1.34293 D.F. = 24 P-VALUE = 0.1918585 P-VALUE/2 = 0.0959288 SD. ERROR = 3308.43 What is the correct conclusion when testing a greater-than alternative hypothesis at A) Reject H0 . B) Fail to reject H0 .

C) Fail to reject Ha .

= .01?

D) Accept H0 .

119) A company claims that 9 out of 10 doctors (i.e., 90%) recommend its brand of cough syrup to their patients. To test this claim against the alternative that the actual proportion is less than 90%, a random sample of 100 doctors was chosen which resulted in 86 who indicate that they recommend this cough syrup. The test statistic in this problem is approximately: A) -1.33 B) 1.33 C) -0.99 D) -0.83

119)

120) A bottling company produces bottles that hold 12 ounces of liquid. Periodically, the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling company randomly samples 19 bottles and finds the average amount of liquid held by the bottles is 11.6 ounces with a standard deviation of .3 ounce. Calculate the appropriate test statistic. A) t = -5.657 B) t = -25.333 C) t = -3.183 D) t = -5.812

120)

121) A local eat-in pizza restaurant wants to investigate the possibility of starting to deliver pizzas. The owner of the store has determined that home delivery will be successful only if the average time spent on a delivery does not exceed 35 minutes. The owner has randomly selected 25 customers and delivered pizzas to their homes. What hypotheses should the owner test to demonstrate that the pizza delivery will not be successful? A) H0 : µ = 35 vs. Ha : µ 35 B) H0 : µ = 35 vs. Ha : µ > 35

121)

C) H0 : µ < 35 vs. Ha : µ = 35

D) H0 : µ = 35 vs. Ha : µ < 35

Answer the question True or False. 122) The alternative hypothesis is accepted as true unless there is overwhelming evidence that it is false. A) True B) False Solve the problem. 123) A revenue department is under orders to reduce the time small business owners spend filling out pension form ABC-5500. Previously the average time spent on the form was 5.2 hours. In order to test whether the time to fill out the form has been reduced, a sample of 60 small business owners who annually complete the form was randomly chosen, and their completion times recorded. The mean completion time for ABC-5500 form was 4.5 hours with a standard deviation of 1.6 hours. In order to test that the time to complete the form has been reduced, state the appropriate null and alternative hypotheses. A) H0 : µ = 5.2 B) H0 : µ = 5.2 C) H0 : µ = 5.2 D) H0 : µ > 5.2 Ha : µ 5.2

124) Let

Ha : µ < 5.2

Ha : µ > 5.2

2 2 0 be a particular value of . Find the value of

A) 11.6509

B) 28.412

2 2 0 such that P( >

C) 30.1435

122)

123)

Ha : µ < 5.2

2 0 ) = .10 for n = 20.

124)

D) 27.2036

125) Consider the following printout.

125)

HYPOTHESIS: MEAN X = x X = gpa SAMPLE MEAN OF X = 3.0512 SAMPLE VARIANCE OF X = 0.230731 SAMPLE SIZE OF X = 167 HYPOTHESIZED VALUE (x) = 3 MEAN X - x = 0.0512 z = 1.3774 Suppose a two-tailed test is desired. Find the p-value for the test. A) p = 0.8324 B) p = 0.1676 C) p = 0.0838

D) p = 0.9162

126) Consider a test of H0 : µ = 55 performed with the computer. SPSS reports a two-tailed p-value of 0.0802. Make the appropriate conclusion for the given situation: Ha : µ < 55, z = -1.75,

A) Reject H0

126)

= 0.05

B) Fail to reject H0

127) Consider the following printout.

127)

HYPOTHESIS: VARIANCE X = x X = gpa SAMPLE MEAN OF X = 2.1995 SAMPLE VARIANCE OF X = .18000 SAMPLE SIZE OF X = 218 HYPOTHESIZED VALUE (x) = 2.3 VARIANCE X - x = -.1005 z = -3.49750 State the proper conclusion when testing H0 : µ = 2.3 vs. Ha : µ < 2.3 at

= .05.

A) Fail to reject H0 . B) Reject H0 . C) Accept H0 .

D) We cannot determine from the information given. 128) A significance level for a hypothesis test is given as = .01. Interpret this value. A) The probability of making a Type I error is .01. B) The probability of making a Type II error is .99. C) The smallest value of that you can use and still reject H0 is .01.

128)

129) If I specify A) True

129)

D) There is a 1% chance that the sample will be biased. to be .17, then the value of

must be .83.

B) False

130) Consider a test of H0 : µ = 40 performed with the computer. SPSS reports a two-tailed p-value of 0.0124. Make the appropriate conclusion for the given situation: Ha : µ > 40, z = -2.5,

A) Reject H0

130)

= 0.01

B) Fail to reject H0

For the given value of and observed significance level (p-value), indicate whether the null hypothesis would be rejected. 131) = 0.05, p-value = 0.25 131) A) Reject H0 B) Fail to reject H0

Answer the question True or False. 132) We do not accept H0 because we are concerned with making a Type II error. A) True

B) False

132)

133) In a test of hypothesis, the sampling distribution of the test statistic is calculated under the assumption that the alternative hypothesis is true. A) True B) False

133)

134) The rejection region refers to the values of the test statistic for which we will reject the alternative hypothesis. A) True B) False

134)

Solve the problem. 135) The owner of Get-A-Away Travel has recently surveyed a random sample of 153 customers to determine whether the mean age of the agency's customers is over 22. The appropriate hypotheses are H0 : µ = 22, Ha : µ > 22. If he concludes the mean age is over 22 when it is not, he makes a

135)

__________ error. If he concludes the mean age is not over 22 when it is, he makes a __________ error. A) Type I; Type II B) Type I; Type I C) Type II; Type I D) Type II; Type II

136) A national organization has been working with utilities throughout the nation to find sites for large wind machines that generate electricity. Wind speeds must average more than 16 miles per hour (mph) for a site to be acceptable. Recently, the organization conducted wind speed tests at a particular site. Based on a sample of n = 40 wind speed recordings (taken at random intervals), the

136)

wind speed at the site averaged x = 16.8 mph, with a standard deviation of s = 4.1 mph. To determine whether the site meets the organization's requirements, consider the test, H0 : µ = 16 vs. Ha : µ > 16, where µ is the true mean wind speed at the site and = .01. Suppose the value of the test statistic were computed to be 1.23. State the conclusion. A) At = .01, there is sufficient evidence to conclude the true mean wind speed at the site exceeds 16 mph. B) We are 99% confident that the site does not meet the organization's requirements. C) We are 99% confident that the site meets the organization's requirements. D) At = .01, there is insufficient evidence to conclude the true mean wind speed at the site exceeds 16 mph.

137) How many tissues should a package of tissues contain? Researchers have determined that a person uses an average of 51 tissues during a cold. Suppose a random sample of 10,000 people yielded the

137)

following data on the number of tissues used during a cold: x = 40, s = 21. Suppose the corresponding test statistic falls in the rejection region at = .05. What is the correct conclusion? A) At = .10, reject Ha . B) At = .05, reject H0 .

C) At

D) At

= .05, accept Ha .

= .10, reject H0 .

Answer the question True or False. 138) The value of

is the area under the bell curve for the distribution of x centered at µa for values of x

138)

that fall within the acceptance region of the distribution of x centered at µ0 .

A) True

B) False

Solve the problem. 139) Given H0 : µ = 25, Ha : µ 25, and p = 0.033. Do you reject or fail to reject H0 at the .01 level of significance? A) reject H0

B) fail to reject H0 C) not sufficient information to decide 24

139)

140) A test of hypothesis was performed to determine if the true proportion of college students who preferred 140) a particular brand of soda differs from .50. The ASP printout is supplied below. Note: All data refer to the proportion of students who preferred the brand of soda. HYPOTHESIS: PROPORTION X = x X = drink_(soda=1) SAMPLE PROPORTION OF X = .431138 SAMPLE SIZE OF X = 167 HYPOTHESIZED VALUE (x) = .5 SAMPLE PROPORTION X - x = -.068862 Z = -1.77979 P-VALUE = .0750 P-VALUE/2 = .0375 SD. ERROR = .0386912 State the proper conclusion if the test was conducted at = .10. A) There is sufficient evidence to indicate the true proportion of college students who prefer the brand of soda is less than .50. B) There is insufficient evidence to indicate the true proportion of college students who prefer the brand of soda differs from .50. C) There is insufficient evidence to indicate the true proportion of college students who prefer the brand of soda is less than .50. D) There is sufficient evidence to indicate the true proportion of college students who prefer the brand of soda differs from .50.

Find the rejection region for the specified hypothesis test. 141) Consider a test of H0 : µ = 9. For the following case, give the rejection region for the test in terms of the z-statistic: Ha : µ 9,

141)

= 0.10

A) |z| > 1.28

B) z > 1.645

C) |z| > 1.645

D) z > 1.28

Solve the problem. 142) The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion exceeds 30%, then the lab will scale back a proposed enlargement of its facilities. Suppose 250 business students were randomly sampled and 75 have laptops. Find the rejection region for the corresponding test using = .05. A) Reject H0 if z > 1.645. B) Reject H0 if z = 1.645. C) Reject H0 if z < -1.645.

142)

D) Reject H0 if z > 1.96 or z < -1.96.

Answer the question True or False. 143) The smaller the p-value in a test of hypothesis, the more significant the results are. A) True B) False Solve the problem. 144) It is desired to test H0 : µ = 12 against Ha: µ 12 using

= 0.05. The population in question is

uniformly distributed with a standard deviation of 2.0. A random sample of 100 will be drawn from this population. If µ is really equal to 11.9, what is the value of associated with this test? A) .9209 B) .0395 C) .0791 D) .4210

143)

144)

Answer Key Testname: CHAPTER 7 1) A 2) D 3) A 4) B 5) D 6) B 7) A 8) C 9) A 10) D 11) D 12) A 13) C 14) B 15) A 16) B 17) A 18) D 19) A 20) B 21) B 22) B 23) C 24) A 25) C 26) A

27) The rejection region requires = .10 in the upper tail of the X2 distribution with n - 1 = 41 - 1 = 40 df. From Table VII, Appendix B, X2 .10 = 51.805. The rejection region is X2 > 51.805. 28) To determine if the test achieved the desired dispersion, we test:

H0 : 2 = 2500 vs. Ha : 2 > 2500 29) p-value = P(z < -2.11 or z > 2.11) = 2(.5 - .4826) = .0348; The probability of a test statistic even more contradictory to the null hypothesis than the one observed is .0348. 79.98 - 70 5.72 30) H0 : µ = 70 vs. Ha : µ > 70; x = 79.98, s = 12.34; z = 12.34 / 50 Since 5.72 > 1.645, we reject the null hypothesis in favor of the alternative hypothesis. There is evidence to support the counselor's suspicions. 31) Using 14 degrees of freedom, t.01 = 2.624. The rejection region is t < -2.624.

Answer Key Testname: CHAPTER 7 32)

33) The test statistic is t =

5.2 - 4 1.1 / 8

3.09. The rejection region is t < -2.365 or t > 2.365. Since the test statistic falls in the

rejection region, we reject the null hypothesis in favor of the alternative hypothesis. We conclude that 4 is not the true population mean. 79.98 - 75 2.85. 34) x = 79.98, s = 12.34; z = 12.34 / 50 p-value = P(z >2.85) = .5 - .4978 = .0022; The probability of a test statistic even more contradictory to the null hypothesis than the one observed is .0022. (n - 1)s2 (25 - 1).135 = = 40.500. 35) The test statistic is X2 = .08 2 0

36) p-value = P(z >1.38) = .5 - .4162 = .0838; The probability of a test statistic even more contradictory to the null hypothesis than the one observed is .0838.

Answer Key Testname: CHAPTER 7 37)

38) To determine if the mean exceeds 980 hours, we test: H0 : µ = 980 vs. Ha : µ > 980 The rejection region requires

= .05 in the upper tail of the z distribution. From a z table, we find z.05 = 1.645. The

rejection region is z > 1.645. 39) Since the alternative hypothesis is Ha : µ > 30, the test is one-tailed. Thus,

= .025 is required in the upper tail of the z

distribution, and we have z.025 = 1.96. The value of x on the border between the rejection region and the acceptance region is found using x=z

+ 30

x = 1.96

8 + 30 64

= P(x < 31.96, when µa = 31) = P z <

x = 31.96

31.96 - 31 = P(z < 0.96) = .5 + .3315 = .8315 8/ 64

The power is 1 - = 1 - .8315 = .1685. 40) Using 17 degrees of freedom, t.10/2 = t.05 = 1.740. The rejection region is t < -1.740 or t > 1.740.

Answer Key Testname: CHAPTER 7 41)

42) The p-value is .010.; The probability of a test statistic even more contradictory than the one observed is .010. 43) To determine whether the mean time has been reduced, we test: H0 : µ = 63 vs. Ha : µ < 63 The rejection region requires

= .01 in the lower tail of the z distribution. From a z table, we find z.01 = 2.33. The

rejection region is z < -2.33. 44) To determine if more than 85% of the firms do not offer any child-care benefits, we test: H0 : p = .85 vs. Ha : p > .85 The rejection region requires

= .10 in the upper tail of the z distribution. The rejection region is z > z.10 = 1.28.

Answer Key Testname: CHAPTER 7 45)

46)

Answer Key Testname: CHAPTER 7 47)

48) Since the p-value, .00335, is less than .01, we reject the null hypothesis in favor of the alternative hypothesis. 49) Since = .01 > p = .0024, H0 can be rejected. There is sufficient evidence to indicate that the variance in the amount of serum injected exceeds .05. 13.1 - 12 50) The z-statistic is z = 4.3 / 75

2.22.

p-value = P(z >2.22) = .5 - .4868 = .0132; The probability of a test statistic even more contradictory to the null hypothesis than the one observed is .0132. 51) To determine if the new method is more accurate in detecting cancer than the old method, we test:

H0 : p = .23 vs. Ha : p < .23 9.1 - 10 52) The test statistic is t = 1.5 / 6

-1.47. The rejection region is t < -2.015. Since the test statistic does not fall in the

rejection region, we can not reject the null hypothesis in favor of the alternative hypothesis. We cannot conclude that the true population mean is actually less than 10. ^

53) The test statistic is z =

p - p0 p 0 q0

where p =

8 = .1143. 70

The test statistic is z =

.1143 - .15 = -.84. .15(.85) 70

Answer Key Testname: CHAPTER 7 54) We test H0 : 2 = 3600

Ha : 2 > 3600

(n -1)s2 (30 -1)1943 = = 15.652 The test statistic is X2 = 2 3600 = .025 in the upper tail of the X2 distribution with n - 1 = 30 - 1 = 29 df. So X2 .025 = 45.722. The rejection region is X2 > 45.722. The rejection region requires

Since the observed value of the test statistic does not fall in the rejection region (X2 = 15.652 45.722), H0 cannot be rejected.

There is insufficient evidence to indicate the variance is greater than 3600 at = .025. 4.48 - 4.6 -2.677; rejection region: z < -1.645; 55) H0 : µ = 4.6; Ha : µ < 4.6; x = 4.48; s = .2455; z = .2455 / 30

Since the test statistic falls within the rejection region, we reject the null hypothesis in favor of the alternative hypothesis. There is evidence that the mean weight of the chicken is less than 4.6 pounds. 56) At = .10, < p-value = .1216, so H0 cannot be rejected. There is insufficient evidence to indicate that more than 85%

of the firms do not offer any child-care benefits. 57) p = .649; The probability of a test statistic even more contradictory than the one observed is .649. 58) To determine if the mean life exceeds 1000 hours, we test: H0 : µ = 1000 vs. Ha : µ > 1000 The test statistic is z =

x - µ0

/ n

s/ n

1027.5 - 1000 = 2.75. 80/ 64

Since the test is greater than 1.645, H0 can be rejected. There is sufficient evidence to indicate the average life of the new

bulbs exceeds 1000 hours when testing at = .05. 59) The p-value is .115.; The probability of a test statistic even more contradictory than the one observed is .115. ^ 32 .64 - .8 = .64; The test statistic is z = -2.828. The rejection region is z < 1.645. Since the test statistic falls 60) p = 50 (.8)(.2) / 50

within the rejection region, we reject the null hypothesis in favor of the alternative hypothesis and conclude that fewer than 80% of the company's employees participate in the stock plan. 61) p = .0037; The probability of a test statistic even more contradictory than the one observed is .0037. 62) The rejection region requires /2 = .05/2 = .025 in both tails of the t distribution with df = n - 1 = 15 - 1 = 14. The rejection region is t > 2.145 or t < -2.145. 63) Since the p-value, .104, is greater than .01, we do not reject the null hypothesis. 247.3 - 250 -2.36. 64) The z-statistic is z = 11.4 / 100 p-value = P(z < -2.36 or z > 2.36) = 2(.5 - .4909) = .0182; The probability of a test statistic even more contradictory to the null hypothesis than the one observed is .0182. 65) To determine if the mean yield for the soybeans differs from 531 bushels per acre, we test:

H0 : µ = 531 vs. Ha : µ 531

Answer Key Testname: CHAPTER 7 66) Since the alternative hypothesis is Ha : µ > 30, the test is one-tailed. Thus,

= .025 is required in the upper tail of the z

distribution, and we have z.025 = 1.96. The value of x on the border between the rejection region and the acceptance region is found using x=z

+ 30

x = 1.96

7 + 30 49

x = 31.96

= P(x < 31.96, when µa = 32) = P z <

31.96 - 32 = P(z < -.04) = .5 - .0160 = .4840 7/ 49

67) To determine if the sample size is large enough for the test of hypothesis to work properly, we need to calculate np0 and nq0 .

np0 = 80(.2) = 16 > 15 and nq0 = 80(.8) = 64 > 15 Since both quantities are greater than 15, the sample size is large enough for the test of hypothesis to work properly. 59.4 - 55 4.29. The rejection region is t > 1.833. Since the test statistic falls in 68) x = 59.4, s = 3.24; The test statistic is t = 3.24 / 10

69)

the rejection region, we reject the null hypothesis in favor of the alternative hypothesis. We conclude that the true mean baking time is actually greater than 55 minutes.

70) x = 9974.25, s = 159.09; The test statistic is t =

9,974.25 - 10,000 159.09 / 8

-.458. The rejection region is t < 1.895 or t > 1.895.

Since the test statistic does not fall in the rejection region, we can not reject the null hypothesis in favor of the alternative hypothesis. We cannot conclude that the mean number of pages printed per cartridge is different than 10,000. 71) Using 11 degrees of freedom, t.05 = 1.796. The rejection region is t > 1.796.

72) B 73) B 74) A 75) D 76) B 77) A 78) B

Answer Key Testname: CHAPTER 7 79) C 80) C 81) C 82) B 83) A 84) C 85) C 86) B 87) C 88) C 89) B 90) A 91) B 92) A 93) A 94) D 95) D 96) A 97) B 98) C 99) C 100) D 101) B 102) D 103) D 104) A 105) A 106) A 107) C 108) A 109) B 110) A 111) B 112) B 113) B 114) A 115) C 116) A 117) D 118) B 119) A 120) D 121) B 122) B 123) B 124) D 125) B 126) A 127) B 128) A 34

Answer Key Testname: CHAPTER 7 129) B 130) B 131) B 132) A 133) B 134) B 135) A 136) D 137) B 138) A 139) B 140) D 141) C 142) A 143) A 144) A

Chapter 8 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) An inventor has developed a new spray coating that is designed to improve the wear of bicycle 1) tires. To test the new coating, the inventor randomly selects one of the two tires on each of 50 bicycles to be coated with the new spray. The bicycle is then driven for 100 miles and the amount of the depth of the tread left on the two bicycle tires is measured (in millimeters). It is desired to determine whether the new spray coating improves the wear of the bicycle tires. The data and summary information is shown below: Bicycle Coated Tire (C) Non-Coated Tire (N) 1 1.452 0.785 2 1.634 0.844 50

1.211

Mean Std. Dev. Sample Size

0.954 Coated 1.38 0.12 50

Non-Coated Difference 0.85 0.53 0.11 0.06 50 50

Use the summary data to construct a 90% confidence interval for the difference between the means. A) 0.53 ± 0.01663 B) 0.53 ± 0.03787 C) 0.53 ± 0.01396 D) 0.53 ± 0.04512

2) A marketing study was conducted to compare the mean age of male and female purchasers of a certain2) product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the mean age of all female purchasers exceeds the mean age of all male purchasers. The sample data is shown here: Female: n = 10, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 10, sample mean = 39.80, sample standard deviation = 10.040 Which of the following assumptions must be true in order for the pooled test of hypothesis to be valid? I. Both the male and female populations of ages must possess approximately normal probability distributions. II. Both the male and female populations of ages must possess population variances that are equal. III. Both samples of ages must have been randomly and independently selected from their respective populations. A) II only B) I, II, and III C) I only D) III only

3) University administrators are trying to decide where to build a new parking garage on campus. The state legislature has budgeted just enough money for one parking structure on campus. The administrators have determined that the parking garage will be built either by the college of engineering or by the college of business. To help make the final decision, the university has randomly and independently asked students from each of the two colleges to estimate how long they usually take to find a parking spot on campus (in minutes). Suppose that the sample sizes selected by the university for the two samples were both n e = n b = 15. What critical value should be

used by the university in the calculations for the 95% confidence interval for µe - µb? Assume that the university used the pooled estimate of the population variances in the calculation of the confidence interval. A) t = 1.701 B) t = 2.048 C) t = 2.042 D) z = 1.96 E) z = 1.645

4) A researcher is investigating which of two newly developed automobile engine oils is better at prolonging the life of an engine. Since there are a variety of automobile engines, 20 different engine types were randomly selected and were tested using each of the two engine oils. The number of hours of continuous use before engine breakdown was recorded for each engine oil. Based on the information provided, what type of analysis will yield the most useful information? A) Independent samples comparison of population means. B) Independent samples comparison of population proportions. C) Matched pairs comparison of population means. D) Matched pairs comparison of population proportions.

5) A marketing study was conducted to compare the mean age of male and female purchasers of a certain5) product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the mean age of all female purchasers exceeds the mean age of all male purchasers. The sample data is shown here: Female: n = 20, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 20, sample mean = 39.80, sample standard deviation = 10.040 Suppose the test statistic was calculated to be the value, t = 2.83. Use the rejection region to state the correct conclusion when testing at alpha = 0.05. A) We accept H0 . B) We fail to reject H0 . C) We reject H0 .

6) When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes of the male and female responses. Suppose 45 of the females and 60 of the males responded that they were able to give blood. Find the test statistic that would be used if it is desired to test to determine if a difference exists between the proportion of the females and males who responds to the call to donate blood. A) z = 1.645 B) z = 2.01 C) z = 1.93 D) z = 1.96

7) A certain manufacturer is interested in evaluating two alternative manufacturing plans consisting 7) of different machine layouts. Because of union rules, hours of operation vary greatly for this particular manufacturer from one day to the next. Twenty-eight random working days were selected and each plan was monitored and the number of items produced each day was recorded. Some of the collected data is shown below: DAY PLAN 1 OUTPUT PLAN 2 OUTPUT 1 1234 units 1311 units 2 1355 units 1366 units 3 1300 units 1289 units What assumptions are necessary for the above test to be valid? A) None of these listed, since the Central Limit Theorem can be applied. B) The population of paired differences must be approximately normally distributed. C) The population variances must be approximately equal. D) Both populations must be approximately normally distributed.

8) A marketing study was conducted to compare the mean age of male and female purchasers of a certain8) product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the mean age of all female purchasers exceeds the mean age of all male purchasers. The sample data is shown here: Female: n = 10, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 10, sample mean = 39.80, sample standard deviation = 10.040 Find the rejection region to state the correct conclusion when testing at alpha = 0.01. A) Reject H0 if t > 2.528 B) Reject H0 if t > 1.330

C) Reject H0 if t > 2.552

D) Reject H0 if t > 2.878

9) Suppose you want to estimate the difference between two population proportions correct to within 0.03 with probability 0.90. If prior information suggests that p1 0.4 and p2 0.8, and you want to

10) Suppose you want to estimate the difference between two population means correct to within 2.5 with probability 0.95. If prior information suggests that the population variances are both equal to the value 20, and you want to select independent random samples of equal size from the populations, how large should the sample sizes be? A) n 1 = n 2 = 18 B) n 1 = n 2 = 25 C) n 1 = n 2 = 62 D) n 1 = n 2 = 44

10)

select independent random samples of equal size from the populations, how large should the sample sizes be? A) n 1 = n 2 = 1708 B) n 1 = n 2 = 1203 C) n 1 = n 2 = 1925 D) n 1 = n 2 = 963

11) A certain manufacturer is interested in evaluating two alternative manufacturing plans consisting 11) of different machine layouts. Because of union rules, hours of operation vary greatly for this particular manufacturer from one day to the next. Twenty-eight random working days were selected and each plan was monitored and the number of items produced each day was recorded. Some of the collected data is shown below: DAY PLAN 1 OUTPUT PLAN 2 OUTPUT 1 1234 units 1311 units 2 1355 units 1366 units 3 1300 units 1289 units What type of analysis will best allow the manufacturer to determine which plan is more effective? A) An independent samples comparison of population proportions. B) A test of a single population proportion. C) An independent samples comparison of population means. D) A paired difference comparison of population means.

12) A marketing study was conducted to compare the mean age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. What type of analysis should be used to compare the mean age of male and female purchasers? A) A paired difference comparison of population means. B) An independent samples comparison of population proportions. C) An independent samples comparison of population means. D) A test of a single population mean.

12)

13) The owners of an industrial plant want to determine which of two types of fuel (gas or electricity) will produce more useful energy at a lower cost. The cost is measured by plant investment per delivered quad ($ invested /quadrillion BTUs). The smaller this number, the less the industrial plant pays for delivered energy. Suppose we wish to determine if there is a difference in the average investment/quad between using electricity and using gas. Our null and alternative hypotheses would be: A) H0 : (µe - µg ) = 0 vs. Ha: (µe - µg ) = 0 B) H0 : (µe - µg ) = 0 vs. Ha: (µe - µg ) < 0

13)

C) H0 : (µe - µg ) = 0 vs. Ha: (µe - µg ) 0

D) H0 : (µe - µg ) = 0 vs. Ha: (µe - µg ) > 0

14) In a controlled laboratory environment, a random sample of 10 adults and a random sample of 10 children 14) were tested by a psychologist to determine the room temperature that each person finds most comfortable. The data are summarized below:

Adults (1) Children (2)

Sample Mean 77.5° F 74.5°F

Sample Variance 4.5 2.5

Suppose that the psychologist decides to construct a 99% confidence interval for the difference in mean comfortable room temperatures instead of proceeding with a test of hypothesis. The 99% confidence interval turns out to be (-2.9, 3.1). Select the correct statement. A) It can be concluded at the 99% confidence level that the true mean comfortable room temperature is between -2.9 and 3.1. B) It can be concluded at the 99% confidence level that the true mean room temperature for adults exceeds that for children. C) It cannot be concluded at the 99% confidence level that there is actually a difference between the true mean comfortable room temperatures for the two groups. D) It can be concluded at the 99% confidence level that the true mean comfortable room temperature for children exceeds that for adults.

15) When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes between the male and female responses. What type of analysis should be used? A) A test of a single population proportion. B) A paired difference comparison of population means. C) An independent samples comparison of population proportions. D) An independent samples comparison of population means.

15)

16) Salary data were collected from CEOs in the consumer products industry and CEOs in the 16) telecommunication industry. The data were analyzed using a software package in order to compare mean salaries of CEOs in the two industries. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM SALARY X = Consumer Products Y = Telecommunications SAMPLE MEAN OF X = 1761 SAMPLE VARIANCE OF X = 3.97555E6 SAMPLE SIZE OF X = 21 SAMPLE MEAN OF Y = 1093.5 SAMPLE VARIANCE OF Y =103255 SAMPLE SIZE OF Y = 21 MEAN X - MEAN Y = test statistic = D. F. = P-VALUE = P-VALUE/2 = SD. ERROR =

667.5 1.47809 40 0.147626 0.0738131 451.597

What of the following assumptions is necessary to perform the test described above? A) The means of the two populations of salaries are equal. B) The population of salaries for each of the two industries has an approximately normal distribution. C) The standard deviations of the two populations of salaries are both large. D) None. The Central Limit Theorem takes care of all assumptions

17) In a controlled laboratory environment, a random sample of 10 adults and a random sample of 10 children 17) were tested by a psychologist to determine the room temperature that each person finds most comfortable. The data are summarized below:

Adults (1) Children (2)

Sample Mean 77.5° F 74.5°F

Sample Variance 4.5 2.5

If the psychologist wished to test the hypothesis that children prefer warmer room temperatures than adults, which set of hypotheses would he use? A) H0 : (µ1 - µ2) = 0 vs. H0 : (µ1 - µ2 ) = 0 B) H0 : (µ1 - µ2) = 0 vs. H0 : (µ1 - µ2 ) > 0

C) H0 : (µ1 - µ2) = 0 vs. H0 : (µ1 - µ2 ) < 0

D) H0 : (µ1 - µ2) = 3 vs. H0 : (µ1 - µ2 )

18) Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low-tech industry with CEOs in the consumer products industry.

18)

HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM RETURN industry 1 (low tech) (NUMBER = 15) industry 3 (consumer products) (NUMBER = 15) ___________________________________________________ X = industry1 Y = industry3 SAMPLE MEAN OF X = 157.286 SAMPLE VARIANCE OF X =1563.45 SAMPLE SIZE OF X = 14 SAMPLE MEAN OF Y = 217.583 SAMPLE VARIANCE OF Y =1601.54 SAMPLE SIZE OF Y = 12 MEAN X - MEAN Y = -60.2976 t = -4.23468 P-VALUE = 0.000290753 P-VALUE/2 = 0.000145377 SD. ERROR = 14.239 Using the printout, which of the following assumptions is not necessary for the test to be valid? A) The samples were randomly and independently selected. B) The population means are equal. C) The population variances are equal. D) Both populations have approximately normal distributions.

19) Given v1 = 15 and v2 = 20, find P(F > 1.84). A) 0.01

19)

B) 0.05

C) 0.025

D) 0.10

a 20) A marketing study was conducted to compare the variation in the age of male and female purchasers of20) certain product. Random and independent samples were selected for both male and female purchasers of the product. The sample data is shown here: Female: n = 31, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 21, sample mean = 39.80, sample standard deviation = 10.040 Calculate the test statistics that should be used to determine if the variation in the female ages exceeds the variation in the male ages. A) F = 1.597 B) F = 1.732 C) F = 1.264 D) F = 1.316

21) A researcher is investigating which of two newly developed automobile engine oils is better at prolonging the life of an engine. Since there are a variety of automobile engines, 20 different engine types were randomly selected and were tested using each of the two engine oils. The number of hours of continuous use before engine breakdown was recorded for each engine oil. Suppose the following 95% confidence interval for µA - µB was calculated: (100, 2500). Which of the following

21)

inferences is correct? A) We are 95% confident that an engine using oil A has a higher mean number of hours of continuous use before breakdown than does an engine using oil B. B) We are 95% confident that an engine using oil B has a higher mean number of hours of continuous use before breakdown than does an engine using oil A. C) We are 95% confident that the mean number of hours of continuous use of an engine using oil A is between 100 and 2500 hours. D) We are 95% confident that no significant differences exists in the mean number of hours of continuous use before breakdown of engines using oils A and B.

22) An inventor has developed a new spray coating that is designed to improve the wear of bicycle 22) tires. To test the new coating, the inventor randomly selects one of the two tires on each of 50 bicycles to be coated with the new spray. The bicycle is then driven for 100 miles and the amount of the depth of the tread left on the two bicycle tires is measured (in millimeters). It is desired to determine whether the new spray coating improves the wear of the bicycle tires. The data and summary information is shown below: Bicycle Coated Tire (C) Non-Coated Tire (N) 1 1.452 0.785 2 1.634 0.844 50

1.211

Mean Std. Dev. Sample Size

0.954 Coated 1.38 0.12 50

Non-Coated Difference 0.85 0.53 0.11 0.06 50 50

Identify the correct null and alternative hypothesis for testing whether the new spray coating improves the mean wear of the bicycle tires (which would result in a larger amount of tread left on the tire). A) H0 : µd = 0 vs. Ha : µd > 0

B) H0 : µd = 0 vs. Ha : µd 0

C) H0 : µd = 0 vs. Ha : µd < 0

23) We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table:

Supermarket 1 2 3 4 5 6 7 8

23)

Price Brand 1 Brand 2 Difference $2.25 $2.30 $-0.05 2.47 2.45 0.02 -0.06 2.38 2.44 2.27 2.29 -0.02 -0.10 2.15 2.25 2.25 2.25 0.00 -0.06 2.36 2.42 2.37 2.40 -0.03 x1 = 2.3125 x2 = 2.3500 d = -0.0375 s1 = 0.1007 s2 = 0.0859 sd = 0.0381

If the problem above represented a paired difference, what assumptions are needed for a confidence interval for the mean difference to be valid? A) The samples were independently selected from each population. B) The population variances are equal. C) The population of paired differences has an approximately normal distribution. D) All of the above are needed.

24) The owners of an industrial plant want to determine which of two types of fuel (gas or electricity) will produce more useful energy at a lower cost. The cost is measured by plant investment per delivered quad ($ invested /quadrillion BTUs). The smaller this number, the less the industrial plant pays for delivered energy. Random samples of 11 similar plants using electricity and 16 similar plants using gas were taken, and the plant investment/quad was calculated for each. In an analysis of the difference of means of the two samples, the owners were able to reject H0 in the test

24)

H0 : (µE - µG) = 0 vs. Ha: (µE - µG) > 0. What is our best interpretation of the result? A) The mean investment/quad for electricity is greater than the mean investment/quad for gas. B) The mean investment/quad for electricity is less than the mean investment/quad for gas. C) The mean investment/quad for electricity is not different from the mean investment/quad for gas. D) The mean investment/quad for electricity is different from the mean investment/quad for gas.

a 25) A marketing study was conducted to compare the variation in the age of male and female purchasers of25) certain product. Random and independent samples were selected for both male and female purchasers of the product. The sample data is shown here: Female: n = 31, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 21, sample mean = 39.80, sample standard deviation = 10.040 Identify the rejection region to that should be used to determine if the variation in the female ages exceeds the variation in the male ages when testing at = 0.05. A) F > 1.93 B) F > 2.12 C) F > 2.04 D) F > 1.74

26) When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes of the male and female responses. Find the rejection region that would be used if it is desired to test to determine if a difference exists between the proportion of the females and males who responds to the call to donate blood. Use = 0.10. A) Reject H0 if z < -1.645 or z > 1.645. B) Reject H0 if z < -1.96 or z > 1.96.

26)

27) We sampled 100 men and 100 women and asked: "Do you think the environment is a major concern?" Of those sampled, 67 women and 53 men responded that they believed it is. For the confidence interval procedure to work properly, what additional assumptions must be satisfied? A) Both populations have approximate normal distributions. B) The population variances are equal. C) Both samples were randomly and independently selected from their respective populations. D) All of the above are necessary.

27)

28) University administrators are trying to decide where to build a new parking garage on campus. The state legislature has budgeted just enough money for one parking structure on campus. The administrators have determined that the parking garage will be built either by the college of engineering or by the college of business. To help make the final decision, the university has randomly and independently asked students from each of the two colleges to estimate how long they usually take to find a parking spot on campus (in minutes). Based on their sample, the following 95% confidence interval (for µe - µb) was created - (4.20, 10.20). What conclusion can the

28)

C) Reject H0 if z < -1.96.

D) Reject H0 if z > 1.645.

university make about the population mean parking times based on this confidence interval? A) They are 95% confident that the mean parking time of all business students equals the mean parking time of all engineering students. B) They are 95% confident that the mean parking time of all business students exceeds the mean parking time of all engineering students. C) They are 95% confident that the mean parking time of all business students is less than the mean parking time of all engineering students.

29) A marketing study was conducted to compare the mean age of male and female purchasers of a certain29) product. Random and independent samples were selected for both male and female purchasers of the product. It was desired to test to determine if the mean age of all female purchasers exceeds the mean age of all male purchasers. The sample data is shown here: Female: n = 20, sample mean = 50.30, sample standard deviation = 13.215 Male: n = 20, sample mean = 39.80, sample standard deviation = 10.040 Use the pooled estimate of the population standard deviation to calculate the value of the test statistic to use in this test of hypothesis. A) t = 3.17 B) t = 2.83 C) t = 2.65 D) t = 2.17

30) Consider the following set of salary data:

Sample Size Mean Standard Deviation

30)

Men (1) Women (2) 100 80 $12,850 $13,000 $345 $500

What assumptions are necessary to perform a test for the difference in population means? A) The two samples were independently selected from the populations of men and women. B) The population variances of salaries for men and women are equal. C) Both of the target populations have approximately normal distributions. D) All of the above are necessary.

31) An inventor has developed a new spray coating that is designed to improve the wear of bicycle 31) tires. To test the new coating, the inventor randomly selects one of the two tires on each of 50 bicycles to be coated with the new spray. The bicycle is then driven for 100 miles and the amount of the depth of the tread left on the two bicycle tires is measured (in millimeters). It is desired to determine whether the new spray coating improves the wear of the bicycle tires. The data and summary information is shown below: Bicycle Coated Tire (C) Non-Coated Tire (N) 1 1.452 0.785 2 1.634 0.844 50

1.211

Mean Std. Dev. Sample Size

0.954 Coated 1.38 0.12 50

Non-Coated Difference 0.85 0.53 0.11 0.06 50 50

Use the summary data to calculate the test statistic to determine if the new spray coating improves the mean wear of the bicycle tires. A) z = 55.26 B) z = 34.25 C) z = 62.46 D) z = 23.02

32) Which supermarket has the lowest prices in town? All claim to be cheaper, but an independent agency 32) recently was asked to investigate this question. The agency randomly selected 100 items common to each of two supermarkets (labeled A and B) and recorded the prices charged by each supermarket. The summary results are provided below: x A = 2.09 sA = 0.22

x B = 1.99 d = .10 sB = 0.19 sd = .03

Assuming a matched pairs design, which of the following assumptions is necessary for a confidence interval for the mean difference to be valid? A) The samples are randomly and independently selected. B) None of these assumptions are necessary. C) The population of paired differences has an approximate normal distribution. D) The population variances must be equal.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 33) In order to compare the means of two populations, independent random samples of 225 observations are selected from each population with the following results. Sample 1

Sample 2

x 1 = 478

x 2 = 481

s1 = 14.2

s2 = 11.2

Test the null hypothesis H0 : (µ1 - µ2 ) = 0 against the alternative hypothesis Ha : (µ1 - µ2 ) using

33)

= .10. Give the significance level, and interpret the result.

34) A paired difference experiment has 75 pairs of observations. What is the rejection region for testing Ha: µd > 0? Use = .01.

34)

35) A paired difference experiment yielded the following results.

35)

nd = 50,

d = 967,

d 2 = 19,201

Test H0 : µd = 20 against Ha : µd 20 , where µd = µ1 - µ2 , using

= .05.

36) Determine whether the sample sizes are large enough to conclude that the sampling distributions 36) are approximately normal. ^

n1 = 45, n2 = 52, p 1 = .3, p 2 = .6

37) A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table. Let µ1 be the true mean weight of

individuals before starting the diet and let µ2 be the true mean weight of individuals after 3 weeks on the diet. Person 1 2 3 4 5

Weight Before Diet 162 207 200 209 216

Weight After Diet 155 202 197 203 212

Summary information is as follows: d = 5, sd = 1.58. Test to determine if the diet is effective at reducing weight. Use

= .10.

37)

38) A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table. Let µ1 be the true mean weight of

38)

individuals before starting the diet and let µ2 be the true mean weight of individuals after 3 weeks on the diet. Person 1 2 3 4 5

Weight Before Diet 159 204 197 206 213

Weight After Diet 152 199 194 200 209

Summary information is as follows: d = 5, sd = 1.58. Calculate a 90% confidence interval for the difference between the mean weights before and after the diet is used.

39) Independent random samples selected from two normal populations produced the following 39) sample means and standard deviations. Sample 1 n1 = 14

Sample 2 n2 = 11

x 1 = 7.1

x 2 = 8.4

s1 = 2.3

s2 = 2.9

Find and interpret the 95% confidence interval for (µ1 - µ2 ).

40) Independent random samples, each containing 500 observations were selected from two binomial populations. The samples from populations 1 and 2 produced 210 and 320 successes, respectively. Test H0 : (p 1 - p 2 ) = 0 against Ha: (p 1 - p 2 ) < 0. Use = .05.

40)

41) A government housing agency is comparing home ownership rates among several immigrant groups. In a sample of 235 families who emigrated to the U.S. from Eastern Europe five years ago, 165 now own homes. In a sample of 195 families who emigrated to the U.S. from Pacific islands five years ago, 125 now own homes. Write a 95% confidence interval for the difference in home ownership rates between the two groups. Based on the confidence interval, can you conclude that there is a significant difference in home ownership rates in the two groups of immigrants?

41)

42) One indication of how strong the real estate market is performing is the proportion of properties that sell in less than 30 days after being listed. Of the condominiums in a Florida beach community that sold in the first six months of 2006, 75 of the 115 sampled had been on the market less than 30 days. For the first six months of 2007, 25 of the 85 sampled had been on the market less than 30 days. Test the hypothesis that the proportion of condominiums that sold within 30 days decreased from 2006 to 2007. Use = .01.

42)

43) Suppose it desired to compare two physical education training programs for preadolescent 43) girls. A total of 42 girls are randomly selected, with 21 assigned to each program. After three 6-week periods on the program, each girl is given a fitness test that yields a score between 0 and 100. The means and variances of the scores for the two groups are shown in the table.

Program 1 Program 2

n 21 21

x 78.3 75.1

s2 201.9 259.5

Test to determine if the variances of the two programs differ. Use

= .05.

44) The data for a random sample of five paired observations are shown below. Pair Observation 1 1 3 2 4 3 3 4 2 5 5 a.

44)

Observation 2 5 4 4 5 6

Calculate the difference between each pair of observations by subtracting observation 2 from

observation 1. Use the differences to calculate d and sd. b. c.

Calculate the means x 1 and x 2 of each column of observations. Show that d = x 1 - x2 . Form a 90% confidence interval for µD.

45) Assume that

1 2 = 2 2 = 2 . Calculate the pooled estimator of 2 for s1 2 = .88,

45)

s2 2 = 1.01, n1 = 10, and n2 = 12.

1 2 = 2 2 = 2 . Calculate the pooled estimator of 2 for s1 2 = 50, s2 2 = 57, and n1 = n2 = 18.

46) Assume that

46)

47) The screen below shows the 95% confidence interval for (µ1 - µ2).

47)

What does the interval suggest about the relationship between µ1 and µ2?

48) Independent random samples, each containing 1,000 observations were selected from two binomial populations. The samples from populations 1 and 2 produced 475 and 550 successes, respectively. Test H0 : (p 1 - p 2 ) = 0 against Ha : (p 1 - p 2 ) 0. Use = .01.

48)

49) Independent random samples selected from two normal populations produced the following 49) sample means and standard deviations. Sample 1 n1 = 14

Sample 2 n2 = 11

x 1 = 7.1

x 2 = 8.4

s1 = 2.3

s2 = 2.9

Conduct the test H0 : (µ1 - µ2 ) = 0 against. Ha: (µ1 - µ2 )

0, Use

= .05.

50) In an exit poll, 42 of 75 men sampled supported a ballot initiative to raise the local sales tax to build a new football stadium. In the same poll, 41 of 85 women sampled supported the initiative. Find and interpret the p-value for the test of hypothesis that the proportions of men and women who support the initiative are different.

50)

51) In order to compare the means of two populations, independent random samples of 144 observations are selected from each population with the following results.

51)

Sample 1

Sample 2

x 1 = 7,123

x 2 = 6,957

s1 = 175

s2 = 225

Use a 95% confidence interval to estimate the difference between the population means (µ1 - µ2 ). Interpret the confidence interval.

52) Determine whether the sample sizes are large enough to conclude that the sampling distributions 52) are approximately normal. ^

n1 = 48, n2 = 55, p 1 = .4, p 2 = .7

53) A new type of band has been developed for children who have to wear braces. The new bands are designed to be more comfortable, look better, and provide more rapid progress in realigning teeth. An experiment was conducted to compare the mean wearing time necessary to correct a specific type of misalignment between the old braces and the new bands. One hundred children were randomly assigned, 50 to each group. A summary of the data is shown in the table.

x s

Old Braces

New Bands

410 days 41 days

380 days 57 days

How many patients would need to be sampled to estimate the difference in means to within days with probability 99%?

53)

54) Independent random samples from normal populations produced the results shown below.

54)

Sample 1: 5.8, 5.1, 3.9, 4.5, 5.4 Sample 2: 4.4, 6.1, 5.2, 5.7 a. b. c.

Calculate the pooled estimator of 2 . Test µ1 < µ2 using = .10.

Find a 90% confidence interval for (µ1 - µ2 ).

55) A paired difference experiment produced the following results.

55)

nd = 40, x 1 = 18.4, x2 = 19.7, d = -1.3, sd2 = 5 Perform the appropriate test to determine whether there is sufficient evidence to conclude that µ1 < µ2 using = .10.

56) Construct a 90% confidence interval for (p 1 - p 2 ) when n1 = 400, n2 = 550, p 1 = .42, and p 2

56)

57) The screens below show the results of a test of H0 : (µ1 - µ2 ) = 0 against Ha: (µ1 - µ2 ) 0

57)

= .63.

Comment on the validity of the results.

58) The data for a random sample of six paired observations are shown below. Pair Observation 1 1 1 2 2 3 3 4 4 5 5 6 6

58)

Observation 2 3 4 5 6 7 8

Calculate the difference between each pair of observations by subtracting observation 2 from observation 1. Use the differences to calculate sd 2 . b.

Calculate the standard deviations s1 2 and s2 2 of each column of observations. Then find

pooled estimate of the variance sp 2 . c.

Comparing sd 2 and sp 2 , explain the benefit of a paired difference experiment.

59) An experiment has been conducted at a university to compare the mean number of study hours expended per week by student athletes with the mean number of hours expended by non athletes. A random sample of 55 athletes produced a mean equal to 20.6 hours studied per week and a standard deviation equal to 5.9 hours. A second random sample of 200 non athletes produced a mean equal to 23.5 hours per week and a standard deviation equal to 4.5 hours. How many students would need to be sampled in order to estimate the difference in means to within 2 hours with probability 95%?

59)

60) A paired difference experiment has 15 pairs of observations. What is the rejection region for testing Ha: µd > 0? Use = .05.

60)

61) Two surgical procedures are widely used to treat a certain type of cancer. To compare the success rates of the two procedures, random samples of surgical patients were obtained and the numbers of patients who showed no recurrence of the disease after a 1-year period were recorded. The data are shown in the table.

61)

n 100 100

Procedure A Procedure B

Number of Successes 85 98

How large a sample would be necessary in order to estimate the difference in the true success rates to within .10 with 95% reliability?

62) Independent random samples of 100 observations each are chosen from two normal populations 62) with the following means and standard deviations. Population 1 µ1 = 15

Population 2 µ2 = 13

1=3

2=2

Find the mean and standard deviation of the sampling distribution of (x 1 - x 2 ).

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 63) A paired difference experiment yielded n d pairs of observations. For the given case, what is the rejection region for testing H0 : µd = 9 against Ha: µd 9? n d = 26,

= 0.01

A) t > 2.779

B) t > 2.787

C) t > 2.485

D) t > 2.787

63)

64) The FDA is comparing the mean caffeine contents of two brands of cola. Independent random 64) samples of 6-oz. cans of each brand were selected and the caffeine content of each can determined. The study provided the following summary information.

Sample size Mean Variance

Brand A 15 18 1.2

Brand B 10 20 1.5

How many cans of each soda would need to be sampled in order to estimate the difference in the mean caffeine content to within .10 with 90% reliability? A) n1 = n2 = 104 B) n1 = n2 = 1038 C) n1 = n2 = 74 D) n1 = n2 = 731

65) Identify the rejection region that should be used to test H0 : 1 2 = 22 against Ha: 1 2 > 2 2 for v 1 = 10, v 2 = 29, and = .10. A) F > 1.83

B) F > 2.53

C) F > 3.00

65)

D) F > 2.18

66) Data was collected from CEOs of companies within both the low-tech industry and the consumer 66) products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low tech industry with CEOs in the consumer products industry. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM RETURN industry 1 (low tech) (NUMBER = 15) industry 3 (consumer products) (NUMBER = 15) _____________________________________________________________ X = industry1 Y = industry3 SAMPLE MEAN OF X = 157.286 SAMPLE VARIANCE OF X =1563.45 SAMPLE SIZE OF X = 14 SAMPLE MEAN OF Y = 217.583 SAMPLE VARIANCE OF Y =1601.54 SAMPLE SIZE OF Y = 12 MEAN X - MEAN Y = -60.2976 t = -4.23468 P-VALUE = 0.000290753 P-VALUE/2 = 0.000145377 SD. ERROR = 14.239 If we conclude that the mean return-to-pay ratios of the consumer products and low tech CEOs are equal when, in fact, a difference really does exist between the means, we would be making a __________. A) Type III error B) Type II error C) Type I error D) correct decision

67) Find F.05 where v 1 = 8 and v 2 = 11. A) 2.30

67)

B) 3.66

C) 2.95

D) 4.74

68) A paired difference experiment yielded n d pairs of observations. For the given case, what is the

68)

rejection region for testing H0 : µd = 15 against Ha: µd < 15? n d = 21,

= 0.05

A) t < -1.725

B) t < 1.725

C) t < -1.721

D) t < 2.086

69) A confidence interval for (µ1 - µ2 ) is (5, 8). Which of the following inferences is correct? A) µ1 > µ2

69)

B) no significant difference between means

C) µ1 < µ2

D) µ1 = µ2

Answer the question True or False. 70) The sample mean difference d is equal to the difference of the sample means x1 - x 2 . A) True

70)

B) False

Solve the problem. 71) Which supermarket has the lowest prices in town? All claim to be cheaper, but an independent agency 71) recently was asked to investigate this question. The agency randomly selected 100 items common to each of two supermarkets (labeled A and B) and recorded the prices charged by each supermarket. The summary results are provided below: x A = 2.09 sA = 0.22

x B = 1.99 d = .10 sB = 0.19 sd = .03

Assuming the data represent a matched pairs design, calculate the confidence interval for comparing mean prices using a 95% confidence level. A) .10 ± .1255 B) .10 ± .056975 C) .10 ± .00588 D) .10 ± .004935

72) A confidence interval for (µ1 - µ2 ) is (-5, 8). Which of the following inferences is correct? A) µ1 < µ2

B) µ1 > µ2 D) µ1 = µ2

C) no significant difference between means

73) Calculate the degrees of freedom associated with a small-sample test of hypothesis for (µ1 - µ2 ), assuming 1 2 = 2 2 and n1 = n2 = 12. A) 25 B) 11

C) 23

74) Given v 1 = 9 and v 2 = 5, find P(F > 6.68). A) 0.025

73)

D) 12 74)

B) 0.05

C) 0.10

75) Which of the following represents the difference in two population means? A) µ1 - µ2 B) p 1 - p 2 C) p 1 + p 2 76) Find F.10 where v 1 = 20 and v 2 = 40. A) 1.99

72)

D) 0.01

D) µ1 + µ2

75)

76)

B) 1.84

C) 1.71

D) 1.61

77) Independent random samples were selected from each of two normally distributed populations, n1 = 7 from population 1 and n2 = 9 from population 2. The data are shown below. Population 1:

2.5 3.1 2.3 1.8 4.2 3.5 3.9

Population 2:

2.9 1.7 4.6 3.5 3.7 2.8 4.6 3.4 1.9

Find the test statistic for the test of H0 : 1 2 = 2 2 against Ha : 12 A) 1.36 B) .735 C) .857

77)

22 .

D) 1.17

Answer the question True or False. 78) In order for the results of a paired difference experiment to be unbiased, the experimental units in each pair must be chosen independently of one another. A) True B) False

78)

Solve the problem. 79) Data were collected from CEOs in the consumer products industry and CEOs in the telecommunication79) industry. The data were analyzed using a software package in order to compare mean salaries of CEOs in the two industries. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM SALARY X = Consumer Products Y = Telecommunications SAMPLE MEAN OF X = 1761 SAMPLE VARIANCE OF X = 3.97555E6 SAMPLE SIZE OF X = 21 SAMPLE MEAN OF Y = 1093.5 SAMPLE VARIANCE OF Y =103255 SAMPLE SIZE OF Y = 21 MEAN X - MEAN Y = test statistic = D. F. = P-VALUE = P-VALUE/2 = SD. ERROR =

667.5 1.47809 40 0.147626 0.0738131 451.597

Find the p-value for testing a two-tailed alternative hypothesis. A) 0.295252 B) 0.0738131 C) 0.9261869

D) 0.147626

80) Data was collected from CEOs of companies within both the low-tech industry and the consumer products industry. The following printout compares the mean return-to-pay ratios between CEOs in the low-tech industry and CEOs in the consumer products industry.

80)

HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM RETURN industry 1 (low tech) (NUMBER = 15) industry 3 (consumer products) (NUMBER = 15) ___________________________________________________ X = industry1 Y = industry3 SAMPLE MEAN OF X = 157.286 SAMPLE VARIANCE OF X =1563.45 SAMPLE SIZE OF X = 14 SAMPLE MEAN OF Y = 217.583 SAMPLE VARIANCE OF Y =1601.54 SAMPLE SIZE OF Y = 12 MEAN X - MEAN Y = -60.2976 t = -4.23468 P-VALUE = 0.000290753 P-VALUE/2 = 0.000145377 SD. ERROR = 14.239 Using the printout above, find the test statistic necessary for testing whether the mean return-to-pay ratio of low tech CEO's exceeds the return-to-pay ratio of consumer product CEOs. A) 14.239 B) .000145377 C) -60.2976 D) -4.23468

81) We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table:

Supermarket 1 2 3 4 5 6 7 8

Price Brand 1 Brand 2 $2.25 $2.30 2.47 2.45 2.38 2.44 2.27 2.29 2.15 2.25 2.25 2.25 2.36 2.42 2.37 2.40 x1 = 2.3125 x2 = 2.3500

s1 = 0.1007

s2 = 0.0859

Difference $-0.05 0.02 -0.06 -0.02 -0.10 0.00 -0.06 -0.03 d = -0.0375 sd = 0.0381

Find a 98% confidence interval for the difference in mean price of brand 1 and brand 2. A) 0.0375 ± 0.1393 B) 0.0375 ± 0.0404 C) 0.0375 ± 0.0471 D) 0.0375 ± 0.0347

81)

82) A consumer protection agency is comparing the work of two electrical contractors. The agency plans to inspect residences in which each of these contractors has done the wiring in order to estimate the difference in the proportions of residences that are electrically deficient. Suppose the proportions of residences with deficient work are expected to be about .2 for both contractors. How many homes should be sampled in order to estimate the difference in proportions using a 95% confidence interval of width .1? A) n1 = n2 = 492 B) n1 = n2 = 246 C) n1 = n2 = 615 D) n1 = n 2 = 984

82)

83) The FDA is comparing the mean caffeine contents of two brands of cola. Independent random 83) samples of 6-oz. cans of each brand were selected and the caffeine content of each can determined. The study provided the following summary information.

Sample size Mean Variance

Brand A 15 18 1.2

Brand B 10 20 1.5

How many cans of each soda would need to be sampled in order to estimate the difference in the mean caffeine content to within .5 with 95% reliability? A) n1 = n2 = 18 B) n1 = n2 = 21 C) n1 = n2 = 42 D) n1 = n2 = 57

84) Calculate the degrees of freedom associated with a small-sample test of hypothesis for (µ1 - µ2 ), assuming 1 2 A) 31

2 2 and n1 = n2 = 20.

B) 30

C) 15

D) 33

85) Consider the following set of salary data:

Sample Size Mean Standard Deviation

85)

Men (1) Women (2) 100 80 $12,850 $13,000 $345 $500

Calculate the appropriate test statistic for a test about µ1 - µ2.

A) z = -2.81

84)

B) z = -2.28

C) z = -2.45

86) Which of the following represents the ratio of variances? 2 µ1 1 A) B) C) p 1 - p 2 2 µ2 2

D) z = -3.02 86) D)

p1 p2

Answer the question True or False. 87) The sample standard deviation of differences sd is equal to the difference of the sample standard deviations s1 - s2 .

A) True

B) False

87)

Solve the problem. 88) A cola manufacturer invited consumers to take a blind taste test. Consumers were asked to decide which 88) of two sodas they preferred. The manufacturer was also interested in what factors played a role in taste preferences. Below is a printout comparing the taste preferences of men and women. HYPOTHESIS: PROP. X = PROP. Y SAMPLES SELECTED FROM soda(brand1,brand2) males females

(sex=0, males) (sex=1, females)

(NUMBER = 115) (NUMBER = 56) X = males Y = females

SAMPLE PROPORTION OF X = 0.422018 SAMPLE SIZE OF X =109 SAMPLE PROPORTION OF Y = 0.25 SAMPLE SIZE OF Y = 52 PROPORTION X - PROPORTION Y = 0.172018 Z = 2.11825 Suppose the manufacturer wanted to test to determine if the males preferred its brand more than the females. Using the test statistic given, compute the appropriate p-value for the test. A) .0340 B) .4681 C) .0170 D) .2119

89) Data were collected from CEOs in the consumer products industry and the CEOs in the telecommunication 89) industry. The data were analyzed using a software package in order to compare mean salaries of CEOs in the two industries. HYPOTHESIS: MEAN X = MEAN Y SAMPLES SELECTED FROM SALARY X = Consumer Products Y = Telecommunications SAMPLE MEAN OF X = 1761 SAMPLE VARIANCE OF X = 3.97555E6 SAMPLE SIZE OF X = 21 SAMPLE MEAN OF Y = 1093.5 SAMPLE VARIANCE OF Y =103255 SAMPLE SIZE OF Y = 21 MEAN X - MEAN Y = test statistic = D. F. = P-VALUE = P-VALUE/2 = SD. ERROR =

667.5 1.47809 40 0.147626 0.0738131 451.597

Using = .05, give the rejection region for a two-tailed test. A) Reject H0 if t > 1.684 or t < -1.684. B) Reject H0 if t > 2.021.

C) Reject H0 if t > 1.684.

D) Reject H0 if t > 2.021 or t < -2.021.

90) Calculate the degrees of freedom associated with a small-sample test of hypothesis for (µ1 - µ2 ), assuming 1 2 A) 23

2 2 and n1 = 13, n2 = 12, s1 = 1.3, s2 = 1.5.

B) 25

C) 11

D) 12

91) Consider the following set of salary data:

Sample Size Mean Standard Deviation

90)

91)

Men (1) Women (2) 100 80 $12,850 $13,000 $345 $500

Suppose the test statistic turned out to be z = -1.20 (not the correct value). Find a two-tailed p-value for this test statistic. A) .1151 B) .2302 C) .6151 D) .3849

92) Given v 1 = 30 and v 2 = 60, find P(F < 1.68). A) 0.10

92)

B) 0.95

C) 0.90

D) 0.05

93) A paired difference experiment yielded n d pairs of observations. For the given case, what is the

93)

rejection region for testing H0 : µd = 9 against Ha: µd > 9? n d = 11,

= 0.05

A) t > 1.812

B) t < 1.812

Specify the appropriate rejection region for testing H0 :

94) Ha :

2 1 <

2 2;

C) t < 2.228 2 1 =

D) t > 1.796

2 2 in the given situation.

= 0.01, n 1 = 9, n 2 = 21

A) F > 4.00

94)

B) F > 5.36

C) F > 5.36

D) F > 3.56

Solve the problem. 95) Consider the following set of salary data:

Sample Size Mean Standard Deviation

95)

Men (1) Women (2) 100 80 $12,850 $13,000 $345 $500

To determine if women have a higher mean salary than men, we would test: A) H0 : µ1 - µ2 = 0 vs. Ha : µ1 - µ2 > 0 B) H0 : µ1 - µ2 = 0 vs. Ha : µ1 - µ2 < 0

C) H0 : µ1 - µ2 = 0 vs. Ha : µ1 - µ2 = 0

D) H0 : µ1 - µ2 = 0 vs. Ha : µ1 - µ2 0

96) Identify the rejection region that should be used to test H0 : 1 2 = 22 against Ha: 1 2 v 1 = 5, v 2 = 8, and = .05. A) F > 6.63

B) F > 6.76

C) F > 4.82

2 2 for

D) F > 3.69

97) Calculate the degrees of freedom associated with a small-sample test of hypothesis for (µ1 - µ2 ), assuming 1 2 = 2 2 and n1 = n2 = 16. A) 31 B) 15

C) 30

96)

97)

D) 33

Answer the question True or False.

98) Using paired differences removes sources of variation that tend to inflate 2. A) True B) False

98)

Solve the problem. 99) In a controlled laboratory environment, a random sample of 10 adults and a random sample of 10 children 99) were tested by a psychologist to determine the room temperature that each person finds most comfortable. The data are summarized below:

Adults (1) Children (2)

Sample Mean 77.5° F 74.5°F

Sample Variance 4.5 2.5

Find the standard error of the estimate for the difference in mean comfortable room temperatures between adults and children. A) 1.6279 B) 0.7000 C) 0.1871 D) 0.8367

Specify the appropriate rejection region for testing H0 :

100) Ha :

2 1

2 2;

2 1 =

2 2 in the given situation.

= 0.05, n 1 = 16, n 2 = 7

100)

2 2 Assume that s 2 > s 1 .

A) F > 5.27 101) Ha :

2 1 >

2 2;

A) F > 2.41

B) F > 5.27

C) F > 3.41

D) F > 2.79

= 0.01, n 1 = 13, n 2 = 31

101)

B) F > 2.84

C) F > 2.84

D) F > 3.70

Solve the problem. 102) Which of the following represents the difference in two population proportions? A) p 1 + p 2

B) µ1 - µ2

C) p 1 - p 2

1 2

2 2

102)

Answer Key Testname: CHAPTER 8 1) C 2) B 3) B 4) C 5) C 6) C 7) B 8) C 9) B 10) B 11) D 12) C 13) C 14) C 15) C 16) B 17) C 18) B 19) D 20) B 21) A 22) A 23) C 24) A 25) C 26) A 27) C 28) C 29) B 30) A 31) C 32) B 33) The test statistic is z =

478 - 481 14.12 11.22 + 225 225

-2.50

34) The rejection region is z > 2.575. 19,201 - 9672 / 50 10.188, so sd 3.19. 35) sd 2 = 49 The test statistic is z =

19.34 - 20 3.19/ 50

-.1.46

The rejection region is z < -1.96 or z > 1.96. Since the test statistic does not fall in the rejection region, we have insufficient evidence to conclude that the mean difference is not 20. ^

36) Since n1p 1 = 45(.3) = 13.5 < 15, the sample size is not large enough.

Answer Key Testname: CHAPTER 8 37) To determine if the diet is effective at reducing weight, we test: H0 : µD = 0 Ha : µD > 0 The test statistic is t =

xd - 0 sd / n

The rejection region requires rejection region is t > 1.533.

5-0 = 7.07. 1.58 5 = .10 in the upper tail of the t distribution with df = n - 1 = 5 - 1 = 4. t.10 = 1.533. The

Since the observed value of the test statistic falls in the rejection region (t = 7.07 > 1.533), H0 is rejected. There is sufficient

evidence to indicate that the diet is effective at reducing weight when testing at sd . 38) The matched pairs confidence interval for µd is x d ± t /2 n Confidence coefficient .90

= .10.

= 1 - .90 = .10. /2 = .10/2 = .05. t.05 = 2.132

with n - 1 = 5 - 1 = 4 df. The 90% confidence interval is: 5 ± 2.132

1.58 5

5 ± 1.51

(3.49, 6.51)

39) The confidence interval is (-3.449, .849). There is no significant difference between the means. ^ ^ (.42 - .64) - 0 -7.15. 40) p 1 = .42 and p 2 = .64; The test statistic is z = .42(.58) .64(.36) + 500 500 The rejection region is z < 1.645. Since the test statistic falls in the rejection region, we reject the null hypothesis in favor of the alternative hypothesis that (p1 - p 2 ) < 0. ^

41) p 1 .70 and p 2 = .64; The confidence interval is (.70 - .64) ± 1.96

.70(.30) .64(.36) + 235 195

.06 ± .089.

Since the confidence interval includes 0, we cannot conclude that there is a difference in home ownership rates. ^ ^ (.65 - .29) - 0 5.43. 42) p 1 .65 and p 2 = .29; The test statistic is z = .65(.35) .29(.71) + 115 85

The rejection region is z > 2.33. Since the test statistic falls in the rejection region, we reject the null hypothesis in favor of the alternative hypothesis that (p 1 - p2 ) > 0. We conclude that the proportion of condominiums that sold within 30 days was greater in the first half of 2006 than in the first half of 2007.

Answer Key Testname: CHAPTER 8 43) To determine if the variances of the two programs differ, we test: H0 :

2 1 =

2 2

Ha :

2 1

2 2

The test statistic is F =

2 s2 2 s1

259.5 = 1.285. 201.9

This test requires /2 = .05/2 = .025 in the upper tail of the F distribution with v 1 = n2 - 1 = 20 and v 2 = n2 - 1 = 20 df. From Table X, Appendix A, F.025 = 2.46. The rejection region is F > 2.46.

Since the observed value of the test statistic does not fall in the rejection region (F = 1.285 2.46), H0 cannot be rejected. There is insufficient evidence to indicate the variances of the two programs differ when testing at

44) a. b. c.

= .05.

The differences are 2, 0, 1, 3, and 1; d = -1.4; sd = 1.14 x 1 = 3.4, x 2 = 4.8, x 1 - x 2 = 3.4 - 4.8 = - 1.4 = d 1.14 -1.4 ± 2.132 -1.4 ± 1.09 5

45) sp 2 =

(n1 - 1)s1 2 + (n2 - 1) s22 (10 - 1).88 + (12 - 1)1.01 = = .9515 (n1 - 1) + (n2 -1) (10 - 1) + (12 -1)

46) sp 2 =

(n1 - 1)s1 2 + (n2 - 1) s22 (18 - 1)50 + (18 - 1)57 = = 53.5 (n1 - 1) + (n2 -1) (18 - 1) + (18 -1)

47) µ1 > µ2 ^

48) p 1 = .475 and p 2 = .550; The test statistic is z =

(.475 - .550) - 0 .475(.525) .550(.45) + 1,000 1,000

-3.36.

The rejection region is z < 2.575 or z > 2.575. Since the test statistic falls in the rejection region, we reject the null hypothesis in favor of the alternative hypothesis that (p 1 - p 2 ) 0.

49) The observed level of significance is .223, which is not less than .05. There is no significant difference between the means. ^ ^ (.56 - .48) - 0 1.01 50) p 1 .56 and p 2 = .48; The test statistic is z = .56(.44) .48(.52) + 75 85 The p-value is p = 2(.5 - .3438) = .3124. The probability of observing a value of z more contradictory to the null hypothesis is .3124. 1752 2252 166 ± 46.56 + 51) (7,123 - 6,957) ± 1.96 144 144 ^

52) Since n1p 1 = 48(.4) = 19.2 > 15, n1 q1 = 48(.6) = 28.8 > 15, n2 p 2 = 55(.7) = 38.5 > 15, and n2 q2 = 55(.3) = 16.5 > 15, the sample sizes are large enough.

Answer Key Testname: CHAPTER 8

z /2 2

53) To determine the sample size necessary, we use n1 = n2 = Using

2 1 +

2 2

(ME)2

= .01, /2 = .01/2 = .005. Thus z.005 = 2.575.

n1 = n2 =

(2.575)2 (412 + 572 ) = 48.36 262

Round up to n1 = n2 = 49.

54) x 1 = 4.94, s1 = .75, n1 = 5, x 2 = 5.35, s2 = .73, n2 = 4 (5 - 1).752 + (4 - 1).732 (5 - 1) + (4 -1)

sp 2 =

The test statistic is t =

.550

4.94 - 5.35 1 1 .550 + 5 4

-.82. There are 7 degrees of freedom, so the rejection region for

= .10 is t <

1.415. Since the test statistic does not fall within the rejection region, we have insufficient evidence to conclude that µ1 < µ2 . c.

(4.94 - 5.35) ± 1.895

55) The test statistic is z =

.550

1 1 + 5 4

-1.3 - 0 5 / 40

.41 ± .943.

-3.677.

The rejection region is z < -1.28.

Since the test statistic does fall in the rejection region, we have sufficient evidence to conclude that µ1 < µ2. .42(.58) .63(.37) + -.21 ± .053. 56) (.42 - .63) ± 1.645 400 550

57) The results are not valid because the sample sizes are too small to use a z-test. 58) a. The differences are all -2, so sd 2 = 0. 5(3.5) + 5(3.5) = 3.5 10

s1 2 = s2 2 = 3.5; sp 2 =

For the paired difference experiment, the variance is much smaller. z /2 2

59) To determine the sample size necessary, we use n1 = n2 = Using confidence coefficient .95 = 1 Thus z.025 = 1.96. n1 = n2 =

2 1 +

2 2

(ME)2

= 1 - .95 = .05. /2 = .05/2 = .025.

(1.96)2 (5.92 + 4.52 ) = 52.88 22

Round up to n1 = n2 = 53.

60) The rejection region is t > 1.761. 30

Answer Key Testname: CHAPTER 8

61) To determine the sample size necessary, we use n1 = n2 = Confidence coefficient .95 = 1 n1 = n2 =

(z /2)2 (p 1 q1 + p 2 q2 )

= 1 - .95 = .05. /2 = .05/2 = .025. z.025 = 1.96.

(1.96)2 [(0.85)(0.15) + (0.98)(0.02)] = 56.509936 .102

Round up to n1 = n2 = 57.

62) The mean is µ1 - µ2 = 15 - 13 = 2. The standard deviation is

(ME)2

12 22 + = n1 n2

32 22 + 100 100

.361.

63) B 64) D 65) A 66) B 67) C 68) A 69) A 70) A 71) C 72) C 73) C 74) A 75) A 76) D 77) A 78) B 79) D 80) D 81) B 82) A 83) C 84) B 85) B 86) A 87) B 88) C 89) D 90) A 91) B 92) B 93) A 94) B 95) B 96) C 97) C 98) A 99) D 31

Answer Key Testname: CHAPTER 8 100) C 101) B 102) C

Chapter 9 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Which method generally produces wider confidence intervals? A) Tukey B) Bonferroni C) ANOVA

D) Scheffé

2) 307 diamonds were sampled and randomly sorted into three groups of diamonds. These diamonds were randomly assigned to one of the three organizations, or groups (HRD, GIA, or IGI), that certify the appraisal of diamonds. A study was conducted to determine if the average size of diamonds reported by these three certification groups differ. A completely randomized design was used and the resulting ANOVA table is shown below.

One-Way AOV for CARAT by CERT Source DF SS MS F CERT 2 8.3265 4.16326 ??? Error 305 15.2604 0.05003 Total 307 23.5869 Find the F-value that is missing in the ANOVA table. A) 0.5000 B) 83.215 C) 0.0120

D) 0.5242

3) 307 diamonds were sampled and randomly sorted into three groups of diamonds. These diamonds were randomly assigned to one of the three organizations, or groups (HRD, GIA, or IGI), that certify the appraisal of diamonds. A study was conducted to determine if the average size of diamonds reported by these three certification groups differ. A completely randomized design was used and the Bonferroni multiple comparison results are shown below. Bonferroni All-Pairwise Comparisons Test of CARAT by CERT CERT HRD GIA IGI

Mean 0.8129 0.6723 0.3665

Bonferroni Groups A B C

Alpha 0.05 Give the population mean(s) which are in the statistically largest group. A) µIGI B) µHRD C) µGIA

D) µHRD & µGIA

4) In a study to determine the least amount of time necessary to clean an SUV while maintaining a high quality standard, the owner of a chain of car washes designed an experiment where 20 employees were divided into four groups, each with five members. Each member of each group was assigned an SUV to clean within a certain time limit. The time limits for the groups were 20 minutes, 25 minutes, 30 minutes, and 35 minutes. After the time limits for each group had expired, the owner inspected each SUV and rated the quality of the cleaning job on a scale of 1 to 10. What are the factor levels for this study? A) the time limits: 20 min, 25 min, 30 min, 35 min B) the number of employees in each group: 5 C) the number of groups: 4 D) the quality ratings: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

5) A study was conducted to test the effectiveness of supermarket sales strategies. At one supermarket, the price level (regular, reduced price, and at cost to supermarket) and display level (normal display space, normal display space plus end-of-aisle display, and twice the normal display space) were tested to determine if they had any effect on the weekly sales of a particular supermarket product. Each of the combinations of price level and display level were put in place for a randomly selected week and the weekly sales of the product was recorded. Each combination was used three times over the course of the experiment. The results of the study are shown here:

Identify the response variable used in this experiment. A) The nine combinations of price level and display level used by the supermarket. B) The weekly sales collected for each of the weeks. C) The three price levels used by the supermarket. D) The three display levels used by the supermarket.

6) A scientist is hoping to compare the mean levels of DDT toxin found in three species of fish in a local river. He randomly samples 50 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. Ideally he will be able to rank the species based on the mean level of toxin found in each of the three species. Identify the response variable in this study. A) The number of fish B) The amount of DDT in a fish C) The three fish species D) The scientist

7) Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The average distance of the five longest hits using each bat is shown in the table below. The goal is to determine if the average hit distance differs for the four brands of baseball bats.

Batter Brand 1 Brand 2 Brand 3 Brand 4 1 307 315 300 275 2 310 317 305 285 3 335 335 330 302 4 325 328 320 300 5 300 305 295 270 6 345 350 340 310 7 312 315 308 300 8 298 302 295 288 The ANOVA table output is shown here: Source Batter Brand Error Total

DF 7 3 21 31

SS MS F P 6227.4 946.77 39.70 0.0000 4117.6 1372.54 57.55 0.0000 500.9 23.85 11245.9

Based on the p-value for this test, make the proper conclusion about the treatments in this experiment. A) There is insufficient evidence (at = 0.01) to indicate differences among the mean distances for the four brands of baseball bats. B) There is sufficient evidence (at = 0.01) to indicate differences among the mean distances for the eight batters. C) There is sufficient evidence (at = 0.01) to indicate differences among the mean distances for the four brands of baseball bats. D) There is insufficient evidence (at = 0.01) to indicate differences among the mean distances for the eight batters.

8) Which of the following is not one of the multiple comparison method options available to compare treatment means? A) The Einstein Method B) The Bonferroni Method C) The Tukey Method D) The Scheffe Method

9) 307 diamonds were sampled and randomly sorted into three groups of diamonds. These diamonds were randomly assigned to one of the three organizations, or groups (HRD, GIA, or IGI), that certify the appraisal of diamonds. A study was conducted to determine if the average size of diamonds reported by these three certification groups differ. A completely randomized design was used and the resulting ANOVA table is shown below. One-Way AOV for CARAT by CERT Source DF SS MS CERT 2 8.3265 4.16326 Error 305 15.2604 0.05003 Total 307 23.5869

F 83.21

P 0.0000

Specify the null hypothesis for a test to compare the mean size of a diamond for the three certification groups (HRD, GIA, and IGI). A) H0 : µHRD = µGIA = µIGI, where µj = mean carat weight for certification group i

B) At least two of the population mean carat weights differ for the three certification groups. C) H0 : µ = 0, where µ = mean carat weight.

D) H0 : µHRD = µGIA = µIGI = 0, where µj = mean carat weight for certification group i

10) An appliance manufacturer is interested in determining whether the brand of laundry detergent 10) used affects the average amount of dirt removed from standard household laundry loads. An experiment is set up in which 10 laundry loads are randomly assigned to each of four laundry detergents-Brands A, B, C, and D (a total of 40 loads in the experiment). The amount of dirt removed, y, (measured in milligrams) for each load is recorded and subjected to an ANOVA analysis, including a follow-up Tukey analysis. Which of the following inferences concerning the Tukey results below is incorrect? Brands D C B A

A) µC < µD

Sample Means 186 174 148 133

B) µB < µD

C) µA < µC

D) µD > µA

11) 307 diamonds were sampled and randomly sorted into three groups of diamonds. These diamonds were randomly assigned to one of the three organizations, or groups (HRD, GIA, or IGI), that certify the appraisal of diamonds. A study was conducted to determine if the average size of diamonds reported by these three certification groups differ. A completely randomized design was used and the resulting ANOVA table is shown below. One-Way AOV for CARAT by CERT Source DF SS MS CERT 2 8.3265 4.16326 Error 305 15.2604 0.05003 Total 307 23.5869

F 83.21

11)

P 0.0000

Give a practical conclusion for the test in the words of the problem. Use = 0.10 to make your conclusion. A) There is insufficient evidence to indicate that differences exist among the mean carat weights for the three certification groups. B) There is sufficient evidence to indicate that differences exist among the mean carat weights for the three certification groups. C) There is sufficient evidence to indicate that the mean carat weight for the HRD group equals the mean carat weight for the IGI group. D) There is sufficient evidence to indicate that the mean carat weight for the GIA group is lower than the other two groups.

12) Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The average distance of the five longest hits using each bat is shown in the table below. The goal is to determine if the average hit distance differs for the four brands of baseball bats. Batter Brand 1 Brand 2 Brand 3 Brand 4 1 307 315 300 275 2 310 317 305 285 3 335 335 330 302 4 325 328 320 300 5 300 305 295 270 6 345 350 340 310 7 312 315 308 300 8 298 302 295 288 The ANOVA table output is shown here: Source Batter Brand Error Total

DF 7 3 21 31

SS MS F P 6227.4 946.77 39.70 0.0000 4117.6 1372.54 57.55 0.0000 500.9 23.85 11245.9

Identify the test statistic that should be used for testing whether the average distance hit for the four brands of baseball bats differ. A) 0.0000 B) 39.7 C) 23.85 D) 57.55

12)

13) Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The average distance of the five longest hits using each bat is shown in the table below. The goal is to determine if the average hit distance differs for the four brands of baseball bats.

13)

Batter Brand 1 Brand 2 Brand 3 Brand 4 1 307 315 300 275 2 310 317 305 285 3 335 335 330 302 4 325 328 320 300 5 300 305 295 270 6 345 350 340 310 7 312 315 308 300 8 298 302 295 288 Identify the response variable in this experiment. A) A batter C) The brand of bat

B) The brand of baseball D) The average distance hit

14) Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The average distance of the five longest hits using each bat is shown in the table below. The goal is to determine if the average hit distance differs for the four brands of baseball bats. Batter Brand 1 Brand 2 Brand 3 Brand 4 1 307 315 300 275 2 310 317 305 285 3 335 335 330 302 4 325 328 320 300 5 300 305 295 270 6 345 350 340 310 7 312 315 308 300 8 298 302 295 288 How should the data be analyzed? A) Randomized block design with eight treatments and four blocks B) Completely randomized design with four treatments C) 4 × 8 factorial design D) Randomized block design with four treatments and eight blocks

14)

15) An economist is investigating the impact of today's economy on workers in the manufacturing 15) industry who have been laid off. A sample of 50 workers was randomly selected from all workers in manufacturing that have been laid off in the past year. The following variables were measured for each laid off worker: length of time jobless (number of weeks) and tax status (single, married, or married/head of household). The data for the 50 workers were entered into the computer and analyzed to determine if the mean number of weeks jobless differed for the three tax status groups. The Tukey multiple comparison printout is shown below: Tukey HSD All-Pairwise Comparisons Test of JOBLESS by STATUS STATUS Married Single Mar/Head

Mean 50.375 48.000 33.789

Tukey Groups A A B

Alpha 0.1

Critical Q Value 2.975

Give the population mean(s) which are in the statistically smallest group. A) µMarried B) µMar/Head

C) µSingle

D) µMarried & µSingle

16) A scientist is hoping to compare the mean levels of DDT toxin found in three species of fish in a local river. He randomly samples 50 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. Ideally he will be able to rank the species based on the mean level of toxin found in each of the three species. Identify the treatments for this study. A) The amount of DDT in a fish B) The scientist C) The three fish species D) The 50 fish

16)

17) Which of the following is not a condition required for a valid ANOVA F-test for a completely randomized experiment? A) The sampled populations all have distributions that are approximately normal. B) The sample chosen from each of the populations is sufficiently large. C) The samples are chosen from each population in an independent manner. D) The variances of all the sampled populations are equal.

17)

18) Which procedure was specifically developed for pairwise comparisons when the sample sizes of the treatments are equal? A) ANOVA B) Scheffé C) Bonferroni D) Tukey

18)

19) A study was conducted to test the effectiveness of supermarket sales strategies. At one supermarket, the price level (regular, reduced price, and at cost to supermarket) and display level (normal display space, normal display space plus end-of-aisle display, and twice the normal display space) were tested to determine if they had any effect on the weekly sales of a particular supermarket product. Each of the combinations of price level and display level were put in place for a randomly selected week and the weekly sales of the product was recorded. Each combination was used three times over the course of the experiment. The results of the study are shown here:

The ANOVA table is shown below: Source Display Price Display*Price Error Total

DF 2 2 4 18 26

SS MS F P 1691393 845696 1709.37 0.0000 3089054 154427 3121.89 0.0000 510705 127676 258.07 0.0000 8905 495 5300057

Which of the following tests should be conducted first? A) A test of the Price Main Effect. B) A test of the Display Main Effect. C) A test of the interaction between Price and Display. D) A test of the Weekly Sales Main Effect.

19)

20) A study was conducted to test the effectiveness of supermarket sales strategies. At one supermarket, the price level (regular, reduced price, and at cost to supermarket) and display level (normal display space, normal display space plus end-of-aisle display, and twice the normal display space) were tested to determine if they had any effect on the weekly sales of a particular supermarket product. Each of the combinations of price level and display level were put in place for a randomly selected week and the weekly sales of the product was recorded. Each combination was used three times over the course of the experiment. The results of the study are shown here:

20)

Identify the treatments used in this experiment. A) The weekly sales collected for each of the weeks. B) The three display levels used by the supermarket. C) The nine combinations of price level and display level used by the supermarket. D) The three price levels used by the supermarket.

21) A scientist is hoping to compare the mean levels of DDT toxin found in three species of fish in a local river. He randomly samples 50 of each species to use in the analysis. For each fish, he measures the amount of DDT toxin present. Ideally he will be able to rank the species based on the mean level of toxin found in each of the three species. How many factors are present in this study? A) 1 B) 3 C) 6 D) 50

21)

22) A study was conducted to test the effectiveness of supermarket sales strategies. At one supermarket, the price level (regular, reduced price, and at cost to supermarket) and display level (normal display space, normal display space plus end-of-aisle display, and twice the normal display space) were tested to determine if they had any effect on the weekly sales of a particular supermarket product. Each of the combinations of price level and display level were put in place for a randomly selected week and the weekly sales of the product was recorded. Each combination was used three times over the course of the experiment. The results of the study are shown here:

The ANOVA table is shown below: Source Display Price Display*Price Error Total

DF 2 2 4 18 26

SS MS 1691393 845696 3089054 1544527 510705 127676 8905 495 5300057

Find the test statistic for determining whether the interaction between Price and Display is significant. A) 257.93 B) 1709.37 C) 3121.89 D) 495

22)

23) Four brands of baseball bats were tested to determine which bat allowed hitters to hit a baseball farthest. Eight different batters were thrown 25 pitches while hitting with each of the four bats (assigned in random order). The average distance of the five longest hits using each bat is shown in the table below. The goal is to determine if the average hit distance differs for the four brands of baseball bats.

23)

Batter Brand 1 Brand 2 Brand 3 Brand 4 1 307 315 300 275 2 310 317 305 285 3 335 335 330 302 4 325 328 320 300 5 300 305 295 270 6 345 350 340 310 7 312 315 308 300 8 298 302 295 288 A partial ANOVA table is shown below. Source Batter Brand Error Total

DF 7 3 21 31

MS 946.77

500.9 11245.9

Find the F-value in the table above for testing whether the average distance hit for the four brands of baseball bats differ. A) 39.7 B) 2.8 C) 23.9 D) 57.6

24) Find the critical value F0 for a one-tailed test using

= 0.05, with 8 numerator degrees of freedom

and 15 denominator degrees of freedom A) 3.22 B) 2.64

C) 4.10

D) 3.20

25) In a study to determine the least amount of time necessary to clean an SUV while maintaining a high quality standard, the owner of a chain of car washes designed an experiment where 20 employees were divided into four groups, each with five members. Each member of each group was assigned an SUV to clean within a certain time limit. The time limits for the groups were 20 minutes, 25 minutes, 30 minutes, and 35 minutes. After the time limits for each group had expired, the owner inspected each SUV and rated the quality of the cleaning job on a scale of 1 to 10. What are the possible values of the response variable? A) the number of groups: 4 B) the number of employees in each group: 5 C) the quality ratings: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 D) the time limits: 20 min, 25 min, 30 min, 35 min

24)

25)

26) A study was conducted to test the effectiveness of supermarket sales strategies. At one supermarket, the price level (regular, reduced price, and at cost to supermarket) and display level (normal display space, normal display space plus end-of-aisle display, and twice the normal display space) were tested to determine if they had any effect on the weekly sales of a particular supermarket product. Each of the combinations of price level and display level were put in place for a randomly selected week and the weekly sales of the product was recorded. Each combination was used three times over the course of the experiment. The results of the study are shown here:

26)

The ANOVA table is shown below: Source Display Price Display*Price Error Total

DF 2 2 4 18 26

SS MS F P 1691393 845696 1709.37 0.0000 3089054 154427 3121.89 0.0000 510705 127676 258.07 0.0000 8905 495 5300057

Based on the results found in the ANOVA table, should the Main Effects tests for either Display or Price be conducted? A) It depends on whether the main effects tests will be significant or not. B) Yes. The interaction of Display and Price indicates that the Main Effects should be tested. C) Yes. The main effects tests are both significant and should be tested. D) No. The interaction of Display and Price indicates that the Main Effects should not be tested.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 27) Complete the ANOVA table. Source Treatments Error Total

df 3 8

27) SS 857.1 372.8

28) Complete the ANOVA table. Source A B AB ERROR Total

28)

df 3 1

SS 331.10

3756.40

MS 534.70

140.40

29) In a completely randomized design experiment, 10 experimental units were randomly chosen for 29) each of three treatment groups and a quantity was measured for each unit within each group. In the first steps of testing whether the means of the three groups are the same, the sum of squares for treatments was calculated to be 3,110 and the sum of squares for error was calculated to be 27,000. Complete the ANOVA table. Source Treatments Error Total

30) The __________ in a designed experiment to compare k means is the probability of making at least one Type I error in a series of inferences about the population means, based on (1 ) 100% confidence intervals.

30)

31) An experiment was conducted using a randomized block design. The data from the experiment 31) are displayed in the following table. TREATMENT BLOCK 1 2 3 1 14 19 12 2 13 22 12 3 16 18 13 Fill in the missing entries for an ANOVA table. SOURCE df SS MS F ________________________________________________ Treatments 86.22 Blocks Error ________________________________________________ Total 100.22

32) The results of a Tukey multiple comparison are summarized below. Treatment B C A a. b. c.

32)

Sample Mean 35.4 31.5 20.7

How many pairwise comparisons of the three treatments are there? Which treatments are significantly different from each other? Which treatments are not significantly different from each other?

33) A company that employs a large number of salespeople is interested in learning which of the 33) salespeople sell the most: those strictly on commission, those with a fixed salary, or those with a reduced fixed salary plus a commission. The previous month's records for a sample of salespeople are inspected and the amount of sales (in dollars) is recorded for each, as shown in the table. Commissioned $507 $450 $465 $483 $466 $410

Fixed Salary $492 $376 $437 $432 $444

ANALYSIS OF VARIANCE SOURCE FACTOR ERROR TOTAL

DF 2 12 14

Commission Plus Salary $425 $492 $470 $506

SS 4195 7945 12140

MS 2097.7 662.1

F 3.17

Test to determine if a difference exists in the mean sale amounts among the three compensation systems. Test using = .025.

34) A beverage distributor wanted to determine the combination of advertising agency (two levels) and advertising medium (three levels) that would produce the largest increase in sales per advertising dollar. Each of the advertising agencies prepared ads as required for each of the media-- newspaper, radio, and television. Twelve small towns of roughly the same size were selected for the experiment, and two each were randomly assigned to receive an advertisement prepared and transmitted by each of the six agency-medium combinations. The dollar increases in sales per advertising dollar, based on a 1-month sales period, are shown in the table.

34)

Advertising Medium Newspaper Radio Television Agency 1 12.7, 15.3 20.1, 17.4 12.7, 16.2 Agency 2 22.4, 18.9 24.3, 28.8 9.4, 12.5 The SPSS analysis is shown below. _____________________________________________________________________ ***ANALYSIS OF VARIANCE*** SALES BY AGENCY MEDIUM

Source of Variation

Sum of Squares

Mean Square

Signif of F

Main Effects AGENCY MEDIUM

238.299 39.967 198.332

3 1 2

79.433 39.967 99.166

13.934 7.011 17.395

.004 .038 .003

77.345

38.672

6.784

.029

315.644 5 34.205 6 349.849 11

63.129 5.701 31.804

11.074

.005

AGENCY*MEDIUM Explained Residual Total

(Note: SPSS uses "Explained" instead of "Treatment" in the factorial analysis. Also, SPSS uses "Residual" instead of "Error.") Would you test the main effects factors, agency and medium, in this example? Explain why or why not.

35) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket. The results of the Bonferroni analysis are summarized below. ___________________ ________________ Supermarket A B C Mean Price 1.66 1.80 1.94 Fully interpret the Bonferroni analysis.

35)

36) A partially completed ANOVA table for a completely randomized design is shown here. Source Time Error Total

df 11 13

SS 25.2

36)

86.4

a. Complete the ANOVA table. b. How many treatments are involved in the experiment? c. Do the data provide sufficient evidence to indicate a difference among the population means? Test using = .05.

37) Psychologists have found that people are generally reluctant to transmit bad news to their peers. This phenomenon has been named the "MUM effect." To investigate the cause of the MUM effect, undergraduates at a university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Twenty-five subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions. Then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) Describe the experiment, including the response variable, factors, factor levels, replications, and treatments.

37)

38) A market research firm is interested in the possible success of new flavors of ice cream. A study was conducted with three different flavors peach, almond, and coconut. Three participants were given a sample of each ice cream, in random order, and asked to rate the flavors on a 100-point scale. The results are given in the table below.

38)

FLAVOR PARTICIPANT Peach Almond Coconut 1 55 63 53 2 60 78 58 3 63 69 54 a.

What is the purpose of blocking on participants in this study?

Construct an ANOVA summary table using the information given.

Is there sufficient evidence of a difference in the mean ratings for the three flavors? Use = 0.05.

39) Psychologists have found that people are generally reluctant to transmit bad news to their 39) peers. This phenomenon has been named the "MUM effect." To investigate the cause of the MUM effect, 40 undergraduates at a university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions, then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) The data was subject to an analysis of variance, with the following results: Source df SS MS F PR > F ___________________________________________________________________ Subject visibility 1 1325.16 1325.16 4.09 0.50 -1316.81 0.046 Test taker success 1 1380.24 1380.24 Visibility x success 1 3385.80 3385.80 10.45 .002 Error 36 11,664.00 324.00 ___________________________________________________________________ Total 39 17,755.20 Is there evidence to indicate that subject visibility and test taker success interact? Use

40) FACTOR A

Level 1 2

FACTOR B 1 4.2, 4.0 5.6, 5.8

2 5.0, 5.2 5.0, 5.4

40)

3 6.1, 6.3 8.8, 9.0

Calculate the mean response for each treatment

b. The MINITAB ANOVA printout is shown here. Test for interaction at the significance. Analysis of variance for response. Source df SS MS F __________________________________________________________ A 1 0.53777 0.53777 0.11851 B 2 5.02708 2.51334 0.55391 AB 2 13.493346.74667 1.48678 Error 6 27.226674.53778 __________________________________________________________ Total 11 46.28486 c.

= .01.

Does the result warrant tests of the two factor mean effects?

= 0.05 level of

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 41) An article in a journal examined the attitudes of three groups of professionals on the condition of the environment, and quantified their responses on a seven-point scale (1 = no problem to 7 = disaster very likely). The mean scores for the groups are: A: 4.60 , B: 4.28 , and C: 4.19 . Using an experimentalwise error rate of = .05, Tukey's minimum significance for comparing means is 0.29. Use this information to conduct a multiple comparisons of the means. A) The lowest mean differs significantly from the other two, but there is no significant difference in the other two means. B) There is no significant difference in any of the means. C) All means are significantly different. D) The highest mean differs significantly from the other two, but there is no significant difference in the other two means.

41)

42) A multiple-comparison procedure for comparing four treatment means produced the confidence 42) intervals shown below. For each pair of means, indicate which mean is larger or indicate that there is no significant difference. (µ1 - µ2): (8, 20) (µ1 - µ3): (-8, 4)

(µ1 - µ4): (9, 21) (µ2 - µ3): (-19, -13) (µ2 - µ4): (-5, 7) (µ3 - µ4): (13, 21)

A) no significant difference between µ1 and µ2; no significant difference between µ1 and µ3 ; µ1 > µ4 ; µ2 < µ3 ; no significant difference between µ2 and µ4 ; µ3 > µ4 B) µ1 > µ2 ; no significant difference between µ1 and µ3 ; µ1 > µ4 ; µ2 < µ3 ; no significant difference between µ2 and µ4 ; µ3 > µ4

C) µ1 > µ2 ; µ1 < µ3 ; µ1 > µ4 ; µ2 < µ3 ; µ2 < µ4 ; µ3 > µ4

D) µ1 < µ2 ; no significant difference between µ1 and µ3 ; µ1 < µ4 ; µ2 > µ3 ; no significant difference between µ2 and µ4 ; µ3 < µ4

43) Suppose a company makes 6 different frozen dinners, and tests their ability to attract customers. They test the frozen dinners in 19 different stores in order to account for any extraneous sources of variation. The company records the number of customers who purchase each product at each store. What assumptions are necessary for the validity of the F statistic for comparing the response means of the 6 frozen dinners? A) The means of the observations corresponding to all the block-treatment combinations are equal, and the variances of all the probability distributions are equal. B) The probability distributions of observations corresponding to all the block-treatment combinations are normal, and the sampling distributions of the variances of all the block-treatment combinations are normally distributed. C) The probability distributions of observations corresponding to all the block-treatment combinations are normal, and the variances of all the probability distributions are equal. D) None. The Central Limit Theorem eliminates the need for any assumptions.

43)

44) A multiple-comparison procedure for comparing four treatment means produced the confidence 44) intervals shown below. Rank the means from smallest to largest. Use solid lines to connect those means which are not significantly different. (µ1 - µ2): (9, 19) (µ1 - µ3): (-7, 3)

(µ1 - µ4): (12, 18) (µ2 - µ3): (-21, -11) (µ2 - µ4): (-6, 8) (µ3 - µ4): (12, 22)

A) 4 2 1 3

B) 4 2 1 3

C) 2 4 1 3

D) 4 1 2 3

45) A certain HMO is attempting to show the benefits of managed care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that certification level is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 19 physicians from each of the three certification levels Board certified (C); Uncertified, board eligible (E); and Uncertified, board ineligible (I) and recorded the total per-member, per-month charges for each (a total of 57 physicians). Identify the dependent (response) variable for this study. A) the three certifications groups C, E, and I B) the 57 physicians C) the HMO D) the total per-member, per-month charge

45)

46) Given that the mean square for treatments (MST) for an ANOVA F-test is 5,000 and the mean square for error (MSE) is 3,750, find the value of the test statistic F. A) .750 B) 1.33 C) .800 D) 1.25

46)

47) An appliance manufacturer is interested in determining whether the brand of laundry detergent used affects the average amount of dirt removed from standard household laundry loads. An experiment is set up in which 28 laundry loads are randomly assigned to each of four laundry detergents-- Brands A, B, C, and D. (A total of 40 loads in the experiment.) A manufacturer of Brand A claims that the design of the experiment is flawed. According to the manufacturer, Brand A is better in cold water than in hot water. If all 112 loads in the above experiment were run in hot water, the results will be biased against Brand A. Explain how to redesign the experiment so that the main effects of both brand and water temperature (hot or cold) on amount of dirt removed, and their possible interaction, can be investigated. A) Use one detergent brand (Brand A). Put 56 loads in hot water and 56 loads in cold water, and compare the results. B) Randomly select two brands (say, A and B) and wash 28 loads in with each brand in cold water. Use hot water in all loads washed by the remaining two brands (say, C and D). C) Consider all eight combinations of brand and temperature (e.g., A-hot, A-cold, B-hot, B-cold, etc.). Randomly assign 14 loads to each of the eight combinations. D) For each of the 112 loads, randomly select one of the detergent brands and randomly select hot or cold water.

47)

48) An experiment was conducted to compare the mean iron content in iron ore pieces determined by 48) three different methods: (1) mechanical, (2) manual, and (3) laser. Five 1-meter long pieces of iron ore were removed from a conveyor belt, and the iron content of each piece was determined using each of the three methods. The data are shown below. How should the data be analyzed? Piece Mechanical Manual Laser 1 61.58 62.61 61.55 2 66.68 67.18 67.70 3 61.56 61.62 62.90 4 59.58 59.61 59.45 5 54.44 53.46 55.40

A) randomized block design with five treatments and three blocks B) completely randomized design with three treatments C) randomized block design with three treatments and five blocks D) 3 × 5 factorial design 49) The intensity of a factor is called __________. A) the treatment C) the experimental unit

B) a factor level D) the design

49)

50) A certain HMO is attempting to show the benefits of managed health care to an insurance 50) company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that primary specialty is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 20 HMO physicians from each of four primary specialties General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP) and recorded the total per-member, per-month charges for each. In order to compare the mean charges for the four specialty groups, the data were be subjected to a one-way analysis of variance. The results of the Tukey analysis are summarized below. Group IM GP FP PED

Sample Mean 55.9 41.4 39.20 21.20

Which primary specialties have significantly lower mean charges than Internal Medicine (IM)? A) PED and FP B) PED C) PED, FP, and GP D) none

51) An industrial psychologist is investigating the effects of work environment on employee attitudes. A group of 36 recently hired sales trainees were randomly assigned to one of 9 different "home rooms" - four trainees per room. Each room is identical except for wall color, with 9 different colors used. The psychologist wants to know whether room color has an effect on attitude, and, if so, wants to compare the mean attitudes of the trainees assigned to the 9 room colors. At the end of the training program, the attitude of each trainee was measured on a 100-pt. scale (the lower the score, the poorer the attitude). How many treatments are in this study? A) 4 B) 36 C) 9 D) 100

51)

52) Consider a completely randomized design with five treatments. How many pairwise comparisons of treatments are made in a Bonferroni analysis? A) 20 B) 5! = 120 C) 5 D) 10

52)

53) Psychologists have found that people are generally reluctant to transmit bad news to their peers. 53) This phenomenon has been named the "MUM effect." To investigate the cause of the MUM effect, 40 undergraduates at a certain university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions, then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) The data was subject to an analysis of variance, with the following results: Source df SS MS F PR > F ___________________________________________________________________ Subject visibility 1 1325.16 1325.16 4.09 0.50 Test taker success 1 1380.24 1380.24 4.26 0.046 Visibility x success 1 3385.80 3385.80 10.45 .002 Error 36 11,664.00 324.00 ___________________________________________________________________ Total 39 17,755.20 Which conclusion can you draw from the analysis? Use = .01. A) At = .01, there is no evidence of interaction between subject visibility and test taker success. B) At = .01, neither subject visibility nor test taker success are important predictors of latency to feedback. C) At = .01, there is sufficient evidence to indicate that subject visibility and test taker success interact. D) At = .01, the model is not useful for predicting latency to feedback.

54) __________ is a particular combination of levels of the factors involved in a study. A) The sampling design B) A treatment C) An analysis of variance D) The factor level

54)

55) An industrial psychologist is investigating the effects of work environment on employee attitudes. A group of 20 recently hired sales trainees were randomly assigned to one of four different "home rooms" five trainees per room. Each room is identical except for wall color. The four colors used were light green, light blue, gray, and red. The psychologist wants to know whether room color has an effect on attitude, and, if so, wants to compare the mean attitudes of the trainees assigned to the four room colors. At the end of the training program, the attitude of each trainee was measured on a 60-pt. scale (the lower the score, the poorer the attitude). The data was subjected to a one-way analysis of variance.

55)

ONE-WAY ANOVA FOR ATTITUDE BY COLOR SOURCE DF SS MS F P BETWEEN 3 1678.15 559.3833 59.03782 0.0000 WITHIN 16 151.6 9.475 TOTAL 19 1829.75 SAMPLE GROUP COLOR MEAN SIZE Blue 60.900 5 Green 60.700 5 Gray 41.900 5 Red 43.100 5

STD DEV 4.3589 3.9623 1.5811 0.8367

Give the null hypothesis for the ANOVA F-test shown on the printout. A) H0 : µ1 = µ2 = µ3 = µ4 = µ5 , where the µi represent attitude means for the ith person in each

room B) H0 : µgreen = µblue = µgray = µred, where the µ's represent mean attitudes for the four rooms

C) H0 : p green = pblue = p gray = p red, where the p's represent the proportion with the

corresponding attitude D) H0 : x 1 = x 2 = x 3 = x 4 , where the x's represent the room colors

Answer the question True or False. 56) The randomized block design is an extension of the matched pairs comparison of µ1 and µ2 . A) True

56)

B) False

Solve the problem. 57) Given that the sum of squares for treatments (SST) for an ANOVA F-test is 9,000 and there are four total treatments, find the mean square for treatments (MST). A) 1,800 B) 3,000 C) 1,500 D) 2,250

57)

58) Given that the sum of squares for error (SSE) for an ANOVA F-test is 12,000 and there are 40 total experimental units with eight total treatments, find the mean square for error (MSE). A) 308 B) 300 C) 375 D) 400

58)

59) A city monitors ozone levels daily over a 9 year period in order to relate the ozone levels to the seasons. Determine whether the study is observational or designed. A) observational B) designed

59)

60) A certain HMO is attempting to show the benefits of managed care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that certification level is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 20 physicians from each of the three certification levels Board certified (C); Uncertified, board eligible (E); and Uncertified, board ineligible (I) and recorded the total per member per month charges for each (a total of 60 physicians). In order to compare the mean charges for the three groups, the data were subjected to an analysis of variance. The results of the ANOVA are summarized in the following table. Take = 0.01

60)

Source df SS MS F Value Prob > F Treatments 2 2562.228 1281.114 20.73 .0001 Error 57 3522.6 61.8 Total 59 6084.828 _____________________________________________________ Interpret the p-value of the ANOVA F-test. A) The variances of the total per number per month charges for the three groups of physicians differ at = .01. B) The model is not statistically useful (at = .01) for prediction purposes. C) The means of the total per member per month charges for the three groups of physicians differ at = .01. D) The means of the total per member per month charges for the three groups of physicians are equal at = .01.

61) The goal of an experiment is to investigate the factors that affect visitor travel time in a complex, multilevel building on campus. Specifically, we want to determine whether the effect of directional aid (wall signs or map) on travel time depends on starting room location (interior or exterior). Three visitors were assigned to each of the combinations of directional aid and starting room location, and the travel times of each (in seconds) to reach the goal destination room were recorded. DIRECTIONAL STARTING ROOM AID Interior Exterior Wall signs 149 235 108 337 235 140 Map

153 113 247

239 331 142

Explain how to properly analyze these data. A) ANOVA F-test for a completely randomized design with four treatments B) Chi-square test for a 2 x 2 factorial design C) ANOVA F-test for a randomized block design with two treatments D) ANOVA F-test for interaction in a 2 x 2 factorial design with 3 replications

61)

62) Four different leadership styles used by Big-Six accountants were investigated. As part of a 62) designed study, 15 accountants were randomly selected from each of the four leadership style groups (a total of 60 accountants). Each accountant was asked to rate the degree to which their subordinates performed substandard field work on a 10-point scale called the "substandard work scale". The objective is to compare the mean substandard work scales of the four leadership styles. The data on substandard work scales for all 60 observations were subjected to an analysis of variance. ONE-WAY ANOVA FOR SUBSTAND BY STYLE SOURCE BETWEEN WITHIN TOTAL

DF 3 56 59

SS MS F P 2356.16 785.387 6.050 0.0012 7269.70 129.816 9625.86

Interpret the results of the ANOVA F-test shown on the printout for = 0.05. A) At = .05, there is no evidence of interaction. B) At =.05, there is insufficient evidence of differences among the substandard work scale means for the four leadership styles. C) At =.05, nothing can be said. D) At = .05, there is sufficient evidence of differences among the substandard work scale means for the four leadership styles.

63) A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that both primary specialty and whether the physician is a foreign or USA medical school graduate are an important factors in measuring the cost-effectiveness of physicians. To investigate this, the president obtained independent random samples of 40 HMO physicians, half foreign graduates and half USA graduates, from each of four primary specialties General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP) and recorded the total per-member, per-month charges for each. Thus, information on charges were obtained for a total of n = 160 doctors. The ANOVA results are summarized in the following table. Source Specialty Medschool Interaction Error Total

df 3 1 3 152 159

SS 22855 105 890 18950 42800

MS 7618 105 297 125

F Value 60.94 0.84 2.38

63)

Prob > F .0001 .6744 .1348

Assuming no interaction, is there evidence of a difference between the mean charges of USA and foreign medical school graduates? Use = 0.1. A) No, because the test for the interaction is not significant at = 0.1, the test for the main effect for medical school is not valid. B) No, the test for the main effect for medical school is not significant at = 0.1. C) Yes, the test for the main effect for medical school is significant at = 0.1. D) It is impossible to make conclusions about the main effect of medical school based on the given information

64) A multiple-comparison procedure for comparing four treatment means produced the confidence 64) intervals shown below. For each pair of means, indicate which mean is larger or indicate that there is no significant difference. (µA - µB): (18, 34) (µA - µC): (7, 23)

(µA - µD): (6, 18) (µB - µC): (-21, -1)

(µB - µD): (-23, -5) (µC - µD): (-11, 5)

A) µA > µB; µA > µC; µA > µD; µB < µC; µB < µD; µC < µD

B) µA < µB; µA < µC; µA < µD; µB > µC; µB > µD; no significant difference between µC and µD

C) µA > µB; µA > µC; µA > µD; µB < µC; µB < µD; no significant difference between µC and µD

D) no significant difference between µA and µB; µA < µC; µA < µD; µB > µC; µB > µD; no significant difference between µC and µD

65) The variable measured in the study is called __________. A) the response variable B) the treatment C) the factor level D) a sampling unit

65)

Answer the question True or False. 66) When a variable is identified as reducing variation in the response variable, but no additional knowledge concerning the variable is desired, it should be used as the blocking factor in the randomized block design. A) True B) False Solve the problem. 67) Use the appropriate table to find the following F value: F0.025, v 1 = 4, v 2 = 14 A) 4.15

B) 8.66

C) 3.80

66)

67) D) 3.89

68) A counselor obtains SAT averages for incoming freshmen each year for a period covering 10 years, with the objective of determining the relationship between the SAT score and the year the test was given. The averages are then subjected to analysis for the purpose of drawing a conclusion regarding a trend. Determine whether the study is observational or designed. A) observational B) designed

68)

69) An advertising firm conducts 9 different campaigns, each in 13 different cities, to promote a certain product, and tracks the product sales attributable to each campaign in each city. Determine whether the study is observational or designed. A) observational B) designed

69)

70) Find the following: P(F 2.71), for v 1 = 5, v 2 = 20

70)

A) 0.05

B) 0.95

C) 0.975

D) 0.025

71) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket.

71)

The results of a Bonferroni analysis are summarized below. ___________ Supermarket A B C Mean Price 1.665 1.919 1.925 Interpret the Bonferroni analysis results. A) C has a significantly larger mean price than either of the other two supermarkets. B) A has a significantly smaller mean price than either of the other two supermarkets. C) A has a significantly larger mean price than either of the other two supermarkets. D) B and C have significantly different mean prices.

72) Psychologists have found that people are generally reluctant to transmit bad news to their peers. This phenomenon has been named the "MUM effect." To investigate the cause of the MUM effect, undergraduates at a university participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. (Unknown to the subject, the test taker was a bogus student who was working with the researchers.) The experimenters manipulated two factors, subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of test taker was either top 20% or bottom 20%. Twenty-five subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions. Then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) What type of experimental design was employed in this study? A) completely randomized design with four treatments B) 2 x 2 factorial design with 25 replications C) randomized block design with four treatments and 25 blocks D) 4 x 20 factorial design with no replications

72)

73) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket.

73)

Item A 1) paper towels 1.26 2) cereal 2.79 3) floor cleaner 6.08 | | | 59) shaving cream 1.09 60) canned green beans 0.57

B 1.46 3.24 5.96 | 0.99 0.72

C 1.41 2.99 6.97 | 1.05 0.49

Identify the blocks for this experiment. A) the 60 × 3 = 180 prices C) the three supermarkets

B) the 60 grocery items D) the day on which the data were collected

74) Consider a completely randomized design with k treatments. Assume all pairwise comparisons of treatment means are to be made using a multiple comparisons procedure. Determine the total number of treatment means to be compared for the value k = 12. A) 24 B) 78 C) 66 D) 12

74)

75) A multiple-comparison procedure for comparing four treatment means produced the confidence 75) intervals shown below. Rank the means from smallest to largest. Use solid lines to connect those means which are not significantly different. (µA - µB): (20, 32) (µA - µC): (7, 23)

(µA - µD): (6, 18) (µB - µC): (-20, -2)

(µB - µD): (-21, -7) (µC - µD): (-10, 1)

A) B C D A

B) C D B A

C) B C D A

76) Find the critical value F0 for a one-tailed test using A) 2.19

= 0.05, d.f.N = 6, and d.f.D = 16.

B) 2.74

C) 2.66

77) Find the following: P(F > 2.52), for v 1 = 3, v 2 = 14 A) 0.85

D) B C A D 76)

D) 3.94 77)

B) 0.90

C) 0.05

D) 0.1

78) Suppose an experiment utilizing a random block design has 4 treatments and 10 blocks for a total of 40 observations. Assume that the total Sum of Squares for the response is SS(Total) = 300. If the Sum of Squares for Treatments (SST) is 40% of SS(Total), and the Sum of Squares for Blocks (SSB) is 10% of SS (Total), find the F values for this experiment. A) treatments: F = 10.4; blocks: F = 0.87 B) treatments: F = 16.67; blocks: F = 5.56 C) treatments: F = 7.20; blocks: F = 0.60 D) treatments: F = 5.40; blocks: F = 0.54

78)

79) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket.

79)

Item A 1) paper towels 1.20 2) cereal 2.77 3) floor cleaner 6.01 | | | 59) shaving cream 0.90 60) canned green beans 0.44

B 1.40 3.22 5.89 | 0.80 0.59

C 1.35 2.97 6.90 | 0.86 0.36

Identify the dependent (response) variable for this experiment. A) the supermarkets B) the grocery items C) the prices of the grocery items D) the mean prices of the grocery items at each supermarket

80) In an experiment with 10 treatments, how many pairs of means can be compared? A) 90 B) 100 C) 20 D) 45

80)

81) A certain HMO is attempting to show the benefits of managed care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that certification level is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 23 physicians from each of the three certification levels Board certified (C); Uncertified, board eligible (E); and Uncertified, board ineligible (I) and recorded the total per member per month charges for each (a total of 69 physicians). How many factors are present in this study? A) 69 B) 23 C) 1 D) 3

81)

82) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket.

82)

The results of the ANOVA are summarized in the following table. Source Supermkt Item Error Corrected Total

df 2 59 118 179

Anova SS 2.6412678 215.5949311 3.9725322 222.2087311

Mean Square 1.3206399 3.6541514 0.0336655

F Value 39.23 108.54

Pr > F 0.0001 0.0001

Based on the p-value of the test, make the proper conclusion. A) There is insufficient evidence (at = .01) to indicate differences among the mean prices of grocery items at the three supermarkets. B) No conclusions can be drawn from the given information. C) There is sufficient evidence (at = .01) to indicate differences among the mean prices of grocery items at the three supermarkets. D) There is sufficient evidence (at = .01) to indicate that the mean prices of grocery items at the three supermarkets are identical.

83) A certain HMO is attempting to show the benefits of managed care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that certification level is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 22 physicians from each of the three certification levels Board certified (C); Uncertified, board eligible (E); and Uncertified, board ineligible (I) and recorded the total per-member, per-month charges for each (a total of 66 physicians). In order to compare the mean charges for the three groups, the data will be subjected to an analysis of variance. Give the degrees of freedom appropriate for conducting the ANOVA F-test. A) numerator df = 64, denominator df = 3 B) numerator df = 3, denominator df = 63 C) numerator df = 2, denominator df = 63 D) numerator df = 64, denominator df = 2

83)

84) The variables, quantitative or qualitative, whose effect on a response variable is of interest are called __________. A) the factor level B) factors C) the treatments D) the experimental units

84)

85) A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that both primary specialty and whether the physician is a foreign or USA medical school graduate are an important factors in measuring the cost-effectiveness of physicians. To investigate this, the president obtained independent random samples of 40 HMO physicians, half foreign graduates and half USA graduates, from each of four primary specialties General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP) and recorded the total per-member, per-month charges for each. Thus, information on charges were obtained for a total of n = 160 doctors. The ANOVA results are summarized in the following table. Source Specialty Medschool Interaction Error Total

df 3 1 3 152 159

SS 22855 105 890 18950 42800

MS 7618 105 297 125

F Value 60.94 0.84 2.38

85)

Prob > F .0001 .6744 .1348

Interpret the test for interaction shown in the ANOVA table. Use = 0.05. A) There is sufficient evidence at the = 0.05 level to say that primary specialty and medical school interact. B) There is sufficient evidence at the = 0.05 level to say that primary specialty and medical school do not interact. C) There is insufficient evidence at the = 0.05 level to say that primary specialty and medical school interact. D) It is impossible to make conclusions about primary specialty and medical school interaction based on the given information.

86) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket. Item A 1) paper towels 1.27 2) cereal 2.83 3) floor cleaner 5.97 | | | 59) shaving cream 1.00 60) canned green beans 0.48

B C 1.47 1.42 3.28 3.03 5.85 6.86 | | 0.90 0.96 0.63 0.40

Identify the treatments for this experiment. A) the day on which the data were collected C) the three supermarkets

B) the 60 × 3 = 180 prices D) the 60 grocery items

86)

87) A certain HMO is attempting to show the benefits of managed care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that certification level is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 27 physicians from each of the three certification levels Board certified (C); Uncertified, board eligible (E); and Uncertified, board ineligible (I) and recorded the total per-member, per-month charges for each (a total of 27 physicians). In order to compare the mean charges for the three groups, the data will be subjected to an analysis of variance. Write the null hypothesis tested by the ANOVA. A) H0 : µC = µE = µI B) H0 : 1 = 2 = 3 = 0

87)

88) A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that both primary specialty and whether the physician is a foreign or USA medical school graduate are an important factors in measuring the cost-effectiveness of physicians. To investigate this, the president obtained independent random samples of 40 HMO physicians, half foreign graduates and half USA graduates, from each of four primary specialties General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP) and recorded the total per-member, per-month charges for each. Thus, information on charges were obtained for a total of n = 160 doctors. The sample mean charges for each of the eight categories are shown in the table.

88)

C) H0 : p 1 = p 2 = p 3

D) H0 : µC = µE = µI = 0

Primary Specialty Foreign Grad USA Grad GP 39.90 42.30 IM 52.80 50.50 PED 23.40 25.20 FP 33.30 37.10 What type of design was used for this experiment? A) completely randomized design with eight treatments B) completely randomized design with two treatments C) 4 x 2 factorial design with 20 replications D) 2 x 2 factorial design with 160 replications

89) Define the statistical term "treatments." A) combinations of factor-levels employed in a designed study B) objects on which the responses are measured C) correlations among the factors used in an analysis of variance D) assumptions that are satisfied exactly

89)

90) A local consumer reporter wants to compare the average costs of grocery items purchased at three different supermarkets, A, B, and C. Prices (in dollars) were recorded for a sample of 60 randomly selected grocery items at each of the three supermarkets. In order to reduce item-to-item variation, the prices were recorded for each item on the same day at each supermarket.

90)

The results of the ANOVA test are summarized in the following table. Source Supermkt Item Error Corrected Total

df 2 59 118 179

Anova SS 2.6412678 215.5949311 3.9725322 222.2087311

Mean Square 1.3206399 3.6541514 0.0336655

F Value 39.23 108.54

Pr > F 0.0001 0.0001

What is the value of the test statistic for determining whether the three supermarkets have the same average prices? A) 39.23 B) 0.0001 C) 1.3206 D) 108.54

91) A certain HMO is attempting to show the benefits of managed health care to an insurance company. The HMO believes that certain types of doctors are more cost-effective than others. One theory is that primary specialty is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtained independent random samples of 25 HMO physicians from each of four primary specialties-- General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physician (FP)-- and recorded the total per-member, per-month charges for each. Identify the treatments for this group. A) the HMO B) the 100 physicians C) the four specialty groups GP, IM, PED, and FP D) the total per-member, per-month charges

91)

Answer Key Testname: CHAPTER 9 1) D 2) B 3) B 4) A 5) B 6) B 7) C 8) A 9) A 10) A 11) B 12) D 13) D 14) D 15) B 16) C 17) B 18) D 19) C 20) C 21) A 22) A 23) D 24) B 25) C 26) D 27) Source Treatments Error Total

df 3 8 11

SS 857.1 372.8 1229.9

MS 285.7 46.6

F 6.13

df SS MS F 28) Source _____________________________________________________________________ A 3 1604.1 534.70 6.11 B

331.10

3.78

421.20

140.40

1.60

ERROR 16 1400.00 87.50 _____________________________________________________________________ Total 23 3756.40

29)

Source Treatments Error Total

df 2 27 29

SS 3,110 27,000 30,110

MS 1,555 1,000

F 1.555

Answer Key Testname: CHAPTER 9 30) experimentwise error rate df SS MS F 31) SOURCE _________________________________________________ Treatments 2 86.22 43.11 13.15 Blocks 2 0.889 0.444 0.136 Error 4 13.11 3.28 _________________________________________________ Total 8 100.22 32) a. 3 b. A and C; A and B c. B and C 33) To determine if a difference exists in the mean sale amounts among the three compensation systems, we test: H0 : µ1 = µ2 = µ3 vs. Ha : At least two means differ. The test statistic is F = 3.17. The rejection region requires = .025 in the upper tail of the F distribution with v 1 = p - 1 = 3 - 1 = 2 df and v2 = n - p = 15 - 3 = 12 df. So F.025 = 5.10, and the rejection region is F > 5.10. Since the

observed value of the test statistic does not fall in the rejection region (F = 3.17 5.10), H0 cannot be rejected. There is insufficient evidence to indicate a difference in the mean sale amounts among the three compensation systems when testing at = .025. 34) The main effect factors, agency and medium, would not be tested since the interaction of these factors is a significant factor. 35) Supermarkets connected by a line cannot be determined to have significantly different mean prices. From the Bonferroni Summary, we can conclude only that µA < µC.

36) a.

b. c.

S T E T

df 2 11 13

SS 25.2 61.2 86.4

MS 12.6 5.56

F 2.26

3 No; F = 2.26 is less than F.05 = 3.98 with df = 2 and 11.

37) The data were collected using a 2 x 2 factorial design with 25 replications. The two factors in the experiment are subject visibility (levels: visible and not visible) and test taker success (levels: top 20% and bottom 20%). The treatments are the 2 x 2 = 4 combinations of the factor levels: Visible, Top 20% Visible, Bottom 20% Not Visible, Top 20% Not Visible, Bottom 20% The response variable is the latency to feedback times.

Answer Key Testname: CHAPTER 9 38) a.

Blocking on participants controls possible participant-to-participant variation in rating the ice cream flavors.

b. SOURCE df SS MS F ____________________________________________________ Flavors 2 357.56 178.78 12.98 Participants 2 105.56 52.78 3.83 Error 4 55.111 13.78 _____________________________________________________ Total 8 518.23 c.

Yes. Since F = 12.98 > F.05 = 6.94 (df1 = 2, df2 = 4), we reject the null hypothesis of equal means. There is sufficient

statistical evidence that the three flavors of ice cream have different mean ratings. 39) To determine if subject visibility and test taker success interact, we test:

H0 : Subject visibility and test taker success do not interact. Ha : Subject visibility and test taker success do interact. The test statistic is F = 10.45. The p-value for this test is p = .002. Since evidence to indicate that subject visibility and test taker success interact. FACTOR B 40) a. Level 1 2 3 FACTOR A 1 4.1 5.1 6.2 2 5.7 5.2 8.9 b.

= .01 > p = .002, H0 is rejected. There is sufficient

Since F = 1.48678 < F.05 = 5.14 (df1 = 2, df2 = 6), we do not reject the null hypothesis of no interaction. No, there is not

significant evidence of interaction. c. 41) D 42) B 43) C 44) B 45) D 46) B 47) C 48) C 49) B 50) B 51) C 52) D 53) C 54) B 55) B 56) A 57) B 58) C 59) A 60) C

Yes, since the null hypothesis of no interaction was not rejected, we should test the main effects.

Answer Key Testname: CHAPTER 9 61) D 62) D 63) B 64) C 65) A 66) A 67) D 68) A 69) B 70) B 71) B 72) B 73) B 74) C 75) A 76) B 77) D 78) C 79) C 80) D 81) C 82) C 83) C 84) B 85) C 86) C 87) A 88) C 89) A 90) A 91) C

Chapter 10 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Economists at USF are researching the problem of absenteeism at U.S. firms. A random sample of 1) 100 U.S. organizations was selected to participate in a 1-year study. As part of the study, the economists had collected data on the following two variables for each company: shiftwork available (Yes or No), and union-management relationship (Good or Poor). As part of their analyses, the economists wanted to determine whether or not a company makes shiftwork available depends on the relationship between union and management. The collected data are shown below: Relation Shiftwork Good Bad No 11 22 Yes 25 42 Total 36 64

Total 33 67 100

Use the chi-square distribution to determine the rejection region for this test when testing at 0.05. A) Reject H0 if 2 > 3.84146 B) Reject H0 if 2 > 5.99147

C) Reject H0 if 2 > 7.81473

D) Reject H0 if 2 > 5.02389

2) A drug company developed a honey-based liquid medicine designed to calm a child's cough at 2) night. To test the drug, 105 children who were ill with an upper respiratory tract infection were randomly selected to participate in a clinical trial. The children were randomly divided into three groups - one group was given a dosage of the honey drug, the second was given a dosage of liquid DM (an over-the-counter cough medicine), and the third (control group) received a liquid placebo (no dosage at all). After administering the medicine to their coughing child, parents rated their children's cough diagnosis as either better or worse. The results are shown in the table below: Diagnosis Treatment Better Worse Total Control 4 33 37 DM 12 21 33 Honey 24 11 35 Total 40 65 105 In order to determine whether the treatment group is independent of the coughing diagnosis, a two-way chi-square test was conducted. Use the chi-square distribution to determine the rejection region for this test when testing at = 0.025. A) Reject H0 if 2 > 7.81473 B) Reject H0 if 2 > 5.99147

C) Reject H0 if 2 > 9.34840

D) Reject H0 if 2 > 7.37776

3) An adverse drug effect (ADE) is an unintended injury caused by prescribed medication. The table summarizes the proximal cause of 95 ADEs that resulted from a dosing error at a Boston hospital.

WRONG DOSAGE USE NUMBER OF ADEs (1) Lack of knowledge of drug 29 (2) Rule violation 17 (3) Faulty dose checking 13 (4) Slips 9 (5) Other 27 TOTAL 95 In order to determine whether the true percentages of ADEs in the five "cause" categories differ, a chi-square analysis was conducted. Use the chi-square distribution to determine the rejection region when testing at = 0.025. A) Reject H0 if 2 > 14.4494 B) Reject H0 if 2 > 11.1433

C) Reject H0 if 2 > 9.48773

D) Reject H0 if 2 > 12.8325

4) A drug company developed a honey-based liquid medicine designed to calm a child's cough at 4) night. To test the drug, 105 children who were ill with an upper respiratory tract infection were randomly selected to participate in a clinical trial. The children were randomly divided into three groups - one group was given a dosage of the honey drug, the second was given a dosage of liquid DM (an over-the-counter cough medicine), and the third (control group) received a liquid placebo (no dosage at all). After administering the medicine to their coughing child, parents rated their children's cough diagnosis as either better or worse. The results are shown in the table below: Diagnosis Treatment Better Worse Total Control 4 33 37 DM 12 21 33 Honey 24 11 35 Total 40 65 105 In order to determine whether the treatment group is independent of the coughing diagnosis, a two-way chi-square test was conducted. Suppose the p-value for the test was calculated to be p = 0.0016. What is the appropriate conclusion to make when testing at = 0.05? A) There is insufficient evidence to indicate the treatment group is independent of the coughing diagnosis. B) There is sufficient evidence to indicate the treatment group is dependent on the coughing diagnosis. C) There is insufficient evidence to indicate the treatment group is dependent on the coughing diagnosis. D) There is sufficient evidence to indicate the treatment group is independent of the coughing diagnosis.

5) A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. U.S. 40

Europe 35

Japan 25

In order to determine whether the true proportions in each response category differ, a one-way chi-square analysis should be conducted. As part of that analysis, a 95% confidence interval for the multinomial probability associated with the "Europe" response was desired. Which of the following confidence intervals should be used? A) (0.257, 0.443) B) (0.265, 0.440) C) (0.227, 0.473) D) (0.271, 0.428)

6) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

NUMBER 60 110 80 50

Specify the null hypothesis for testing whether the true proportions of Internet users in each response category are equal. A) H0 : Internet users and Government scanning are independent

B) H0 : p1 = 60, p2 = 110, p3 = 80, p4 = 50, where pi represents the proportion of Internet users in

one of the four response categories C) H0 : p1 = p2 = p3 = p4 = 0.25, where pi represents the proportion of Internet users in one of

the four response categories D) H0 : µ1 = µ2 = µ3 = µ4 , where µi represents the average number of Internet users in one of the found response categories

7) A business professor conducted a campus survey to estimate demand among all students for a protein 7) supplement for smoothies and other nutritional drinks. Each of 113 students, randomly selected from all students on campus, provided the following information: (1) How health conscious are you? (Very, Moderately, Slightly, Not very) (2) Do you prefer protein supplements in your smoothies? (Yes, No) As part of his analysis, the professor claims that whether or not the student prefers a protein supplement in smoothies is independent of health consciousness level (Very, Moderate, Slightly, or Not very). Use the chi-square distribution to determine the rejection region for this test when testing at = 0.05. A) Reject H0 if 2 > 9.48773 B) Reject H0 if 2 > 5.99147

C) Reject H0 if 2 > 7.81473

D) Reject H0 if 2 > 9.34840

8) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

NUMBER 60 110 80 50

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. As part of that analysis, a 90% confidence interval for the multinomial probability associated with the "Disagree Somewhat" response was desired. Which of the following confidence intervals should be used? A) (0.216, 0.317) B) (0.201, 0.332) C) (0.206, 0.327) D) (0.225, 0.309)

9) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

NUMBER 60 110 80 50

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. When calculating the test statistic, what values for the expected counts should be used in the calculation? A) E1 = 60, E2 = 110, E3 = 80, E4 = 50 B) E1 = 75, E2 = 75, E3 = 75, E4 = 75

C) E1 = 0.25, E2 = 0.25, E3 = 0.25, E4 = 0.25

D) E1 = 300, E2 = 300, E3 = 300, E4 = 300

10) A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. U.S. 40

Europe 35

Japan 25

In order to determine whether the true proportions in each response category differ, a one-way chi-square analysis should be conducted. Use the chi-square distribution to determine the rejection region when testing at = 0.025. A) Reject H0 if 2 > 11.1433 B) Reject H0 if 2 > 7.37776

C) Reject H0 if 2 > 9.34840

D) Reject H0 if 2 > 9.21034

10)

11) Economists at USF are researching the problem of absenteeism at U.S. firms. A random sample of 11) 100 U.S. organizations was selected to participate in a 1-year study. As part of the study, the economists had collected data on the following two variables for each company: shiftwork available (Yes or No), and union-management relationship (Good or Poor). As part of their analyses, the economists wanted to determine whether or not a company makes shiftwork available depends on the relationship between union and management. The collected data are shown below: Relation Shiftwork Good Bad No 11 22 Yes 25 42 Total 36 64

Total 33 67 100

In order to determine whether the shiftwork responses depend on the relationship responses, a two-way chi-square analysis should be conducted. Calculate the value of the test statistic for the desired analysis. A) 2 = 0.70 B) 2 = 0.07 C) 2 = 0.15 D) 2 = 11.88

12) A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. U.S. 40

Europe 35

Japan 25

In order to determine whether the true proportions in each response category differ, a one-way chi-square analysis should be conducted. When calculating the test statistic, what values for the expected counts should be used in the calculation? A) E1 = 0.45, E2 = 0.46, E3 = 0.09 B) E1 = 100, E2 = 100, E3 = 100

C) E1 = 46, E2 = 44, E3 = 9

D) E1 = 33.33, E2 = 33.33, E3 = 33.33

12)

13) Economists at USF are researching the problem of absenteeism at U.S. firms. A random sample of 13) 100 U.S. organizations was selected to participate in a 1-year study. As part of the study, the economists had collected data on the following two variables for each company: shiftwork available (Yes or No), and union-management relationship (Good or Poor). As part of their analyses, the economists wanted to determine whether or not a company makes shiftwork available depends on the relationship between union and management. The collected data are shown below: Relation Shiftwork Good Bad No 11 22 Yes 25 42 Total 36 64

Total 33 67 100

In order to determine whether the shiftwork responses depend on the relationship responses, a two-way chi-square analysis should be conducted. When calculating the test statistic, what values for the expected counts should be used in the calculation? A) E11 = 36, E12 = 33, E21 = 64, E22 = 67

B) E11 = 0.065, E12 = 0.036, E21 = 0.032, E22 = 0.018

C) E11 = 11.88, E12 = 21.12, E21 = 24.12, E22 = 42.88 D) E11 = 11, E12 = 22, E21 = 25, E22 = 42

14) A business professor conducted a campus survey to estimate demand among all students for a protein 14) supplement for smoothies and other nutritional drinks. Each of 113 students, randomly selected from all students on campus, provided the following information: (1) How health conscious are you? (Very, Moderately, Slightly, Not very) (2) Do you prefer protein supplements in your smoothies? (Yes, No) As part of his analysis, the professor claims that whether or not the student prefers a protein supplement in smoothies is independent of health consciousness level (Very, Moderate, Slightly, or Not very). Identify the appropriate alternative hypothesis that the professor should use in the test of hypothesis he desires. A) HA: Preference and Health Consciousness level are dependent variables.

B) HA: Preference and Health Consciousness level are mutually exclusive variables.

C) HA: Preference and Health Consciousness level are independent variables. D) HA: There is interaction between the Preference and Health Consciousness variables.

15) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

15)

NUMBER 60 110 80 50

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. As part of that analysis, a 95% confidence interval for the multinomial probability associated with the "Agree Strongly" response was desired. Which of the following confidence intervals should be used? A) (0.141, 0.259) B) (0.145, 0.255) C) (0.162, 0.238) D) (0.155, 0.245)

16) A drug company developed a honey-based liquid medicine designed to calm a child's cough at 16) night. To test the drug, 105 children who were ill with an upper respiratory tract infection were randomly selected to participate in a clinical trial. The children were randomly divided into three groups - one group was given a dosage of the honey drug, the second was given a dosage of liquid DM (an over-the-counter cough medicine), and the third (control group) received a liquid placebo (no dosage at all). After administering the medicine to their coughing child, parents rated their children's cough diagnosis as either better or worse. The results are shown in the table below: Diagnosis Treatment Better Worse Total Control 4 33 37 DM 12 21 33 Honey 24 11 35 Total 40 65 105 In order to determine whether the treatment group is independent of the coughing diagnosis, a two-way chi-square test was conducted. Calculate the value of the test statistic for the desired analysis. A) 2 = 25.51 B) 2 = 15.79 C) 2 = 28.54 D) 2 = 9.72

17) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

17)

NUMBER 60 110 80 50

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. Use the chi-square distribution to determine the rejection region when testing at = .05. A) Reject H0 if 2 > 9.48773 B) Reject H0 if 2 > 0.351846

C) Reject H0 if 2 > 7.81473

D) Reject H0 if 2 > 0.710721

18) A survey of entrepreneurs focused on their job characteristics, work habits, social activities, leisure time, etc. One question put to each entrepreneur was, "What make of car (U.S., Europe, or Japan) do you drive?" The responses (number in each category) for a sample of 100 entrepreneurs are summarized below. The goal of the analysis is to determine if the proportions of entrepreneurs who drive American, European, and Japanese cars differ. U.S. 40

Europe 35

Japan 25

In order to determine whether the true proportions in each response category differ, a one-way chi-square analysis should be conducted. Suppose the p-value for the test was calculated to be p = 0.1738. What is the appropriate conclusion to make when testing at = 0.10? A) There is sufficient evidence to indicate the proportion of entrepreneurs driving Japanese cars is less than the proportion driving U.S. cars. B) There is sufficient evidence to indicate the proportion of entrepreneurs driving the three makes of car are equal. C) There is insufficient evidence to indicate the proportion of entrepreneurs driving the three makes of car differ. D) There is sufficient evidence to indicate the proportion of entrepreneurs driving the three makes of car differ.

18)

19) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

19)

NUMBER 60 110 80 50

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. Calculate the value of the test statistic for the desired analysis. A) 2 = 75 B) 2 = 28.0 C) 2 = 0.25 D) 2 = 22.54

20) Inc. Technology reported the results of consumer survey in which 300 Internet users indicated their level of agreement with the following statement: "The government needs to be able to scan Internet messages and user communications to prevent fraud and other crimes." The possible responses were "agree strongly", "agree somewhat", "disagree somewhat", and "disagree strongly". The number of Internet users in each category is summarized in the table. RESPONSE Agree Strongly Agree Somewhat Disagree Somewhat Disagree Strongly

NUMBER 60 110 80 50

In order to determine whether the true proportions of Internet users in each response category differ, a one-way chi-square analysis should be conducted. Which of the following statements is not necessary for the analysis to be valid? A) The 300 internet users sampled are independent from one another. B) The expected cell counts all must be 30 or more. C) The probabilities for the four response outcomes remain the same from one internet user to the next.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) Consider the accompanying contingency table.

Row 1 2

1 13 19

21)

Column 2 3 15 16 26 11

a. Convert the values in row 1 to percentages by calculating the percentage of each column total falling in row 1. b. Create a bar graph with row 1 percentage on the vertical axis and column number on the horizontal axis. c. What pattern do you expect to see if the rows and columns are not independent? Is this pattern present in your graph?

20)

22) A multinomial experiment with k = 3 cells and n =30 has been conducted and the results are shown in the table.

Cell 1 2

22)

Explain why the sample size is not large enough to test whether p 1 = .55, p 2 = .40, and p 3 = .05.

23) A multinomial experiment with k = 4 cells and n = 300 produced the data shown in the following table.

Cell 1 2 69 65

3 80

23)

4 86

Do these data provide sufficient evidence to contradict the null hypothesis that p 1 = .20, p 2 = .20, p 3 = .30, and p 4 = .30? Test using = .05.

24) A new coffeehouse wishes to see whether customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Test the claim that the probabilities show no preference. Use = 0.01.

24)

Brand 1 2 3 4 5 Customers 55 18 30 65 32

25) The null hypothesis for a test of data resulting from a multinomial experiment is given as p1 = p 2 = p 3 = p 4 = .25. What is the alternative hypothesis for the test?

25)

26) Describe probabilities of the k outcomes of the multinomial experiment trials.

26)

27) A random sample of 160 car accidents are selected and categorized by the age of the driver determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is 18% for the under 26 group, 39% for the 26-45 group, 31% for the 46-65 group, and 12% for the group over 65. Test the claim that all ages have crash rates proportional to their number of drivers. Use = 0.05.

27)

Age Under 26 26 - 45 46 - 65 Over 65 Drivers 66 39 25 30

28) Many track runners believe that they have a better chance of winning if they start in the 28) inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The table displays the starting positions for the winners of 240 competitions. Test the claim that the probability of winning is the same regardless of starting position. Use = 0.05. The results are based on 240 wins. Starting Position Number of Wins

1 2 3 4 5 6 50 45 44 33 32 36

29) What is categorical data?

29)

30) A professor chose a random sample of 50 recent graduates of an MBA program and recorded the 30) gender of each graduate (M or F) and whether the graduate chose to complete his or her degree requirements by completing a research project (RP) or by taking comprehensive exams (CE). The results are shown below. M, CE F, CE F, CE F, RP M, RP F, CE F, CE M, CE F, RP M, RP a.

M, RP F, RP, M, RP M, RP F, CE M, RP F, RP, M, RP M, RP F, CE

F, RP M, CE F, RP F, CE F, RP F, RP M, CE F, RP M, CE F, RP

M, CE M, RP F, CE M, CE M, CE M, CE M, RP F, CE M, CE M, CE

M, CE F, RP M, RP M, CE M, CE M, CE F, RP M, RP M, CE F, CE

Create a contingency table for the data. Perform a 2 -test to determine if there is any evidence that gender and choice of

b. research project or comprehensive exams are not independent. Use

= 0.05.

31) A sports researcher is interested in determining if there is a relationship between the number of home team and visiting team wins and different sports. A random sample of 526 games is selected and the results are given below. Test the claim that the number of home team and visiting team wins is independent of the sport. Use = 0.01.

31)

Football Basketball Soccer Baseball 39 156 25 83 31 98 19 75

Home team wins Visiting team wins

32) A multinomial experiment with k = 3 cells and n =100 has been conducted and the results are 32) shown in the table.

Cell 1 2

Construct a 99% confidence interval for the multinomial probability associated with cell 2.

33) Test the null hypothesis of independence of the two classifications, A and B, of the 3 × 3 contingency table shown below. Test using = .005.

A1 A2 A3

B B2

33)

34) The data below show the age and favorite type of music of 779 randomly selected people. Test the claim that age and preferred music type are independent. Use = 0.05.

34)

Age Country Rock Pop Classical 15 - 21 21 45 90 33 21 - 30 68 55 42 48 30 - 40 65 47 31 57 40 - 50 60 39 25 53

35) A teacher finds that final grades in the statistics department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following grades were recorded. Determine if the grade distribution for the department is different than expected. Use = 0.01.

35)

Grade A B C D F Number 36 42 60 8 14

36) What are characteristics of the trials in a multinomial experiment?

36)

37) The contingency table below shows the results of a random sample of 200 state representatives37) that was conducted to see whether their opinions on a bill are related to their party affiliations. Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Test the claim of independence. Use

= .05.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 38) Use the appropriate table to find the following probability: P( 2 > 7.815) for df = 3. A) 0.950 B) 0.900 C) 0.050 D) 0.100 Answer the question True or False. 39) In a test of independence, it is safe to conclude that the events are independent when the value of 2 is very small. A) True

B) False

38)

39)

Solve the problem. 40) The contingency table below shows the results of a random sample of 200 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliations. Use = 0.05.

40)

Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the rejection region used to test the claim of independence. A) 2 > 7.779 B) 2 > 13.277 C) 2 > 9.488

D) 2 > 11.143

41) The contingency table below shows the results of a random sample of 400 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliations. Assuming the row and column classifications are independent, find an estimate for the expected cell count E22. Opinion Party Approve Disapprove No Opinion Republican 84 40 28 Democrat 100 48 36 Independent 20 32 12 A) 93.84 B) 55.2

C) 34.96

41)

D) 45.6

42) The contingency table below shows the results of a random sample of 200 state representatives that was42) conducted to see whether their opinions on a bill are related to their party affiliation. Opinion Party Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the chi-square test statistic 2 used to test the claim of independence. A) 11.765 B) 7.662 C) 8.030

D) 9.483

Answer the question True or False. 43) The 2 -test for independence is a useful tool for establishing a causal relationship between two factors. A) True

B) False

43)

Solve the problem. 44) A sports researcher is interested in determining if there is a relationship between the number of home team and visiting team wins and different sports. A random sample of 526 games is selected and the results are given below. Assuming the row and column classifications are independent, find an estimate for the expected cell count of cell E22. Football Basketball Soccer Baseball Home team wins 38 153 26 86 Visiting team wins 29 95 21 78 A) 27.1 B) 19.9 C) 142.9

D) 105.1

45) A teacher finds that final grades in the statistics department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following grades were recorded. Calculate the chi-square test statistic 2 used to determine if the grade distribution for the department is different than expected. Use

C) 6.87

D) 4.82

46) Many track runners believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The table displays the starting positions for the winners of 240 competitions. Find the rejection region used to test the claim that the probability of winning is the same regardless of starting position. Use = 0.05. The results are based on 240 wins. 1 2 3 4 5 6 32 45 36 33 44 50 B) 2 > 9.236

C) 2 > 12.833

Age Under 26 26 - 45 46 - 65 Over 65 Drivers 66 39 25 30 2 2 > 7.815 > 9.348 A) B)

C) 2 > 6.251

3 101

4 40

Previous studies in this area have shown that p 1 = p 2 = p 3 = p 4 = .25. Construct a 95% confidence interval for the multinomial probability associated with cell 2. A) (0.507, 0.588) B) (0.071, 0.129) C) (0.505, 0.590)

47)

D) 2 > 11.143

48) A multinomial experiment with k = 4 cells and n = 400 produced the data shown in the following table.

46)

D) 2 > 11.070

47) A random sample of 160 car accidents are selected and categorized by the age of the driver determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is 18% for the under 26 group, 39% for the 26-45 group, 31% for the 45-65 group, and 12% for the group over 65. Find the rejection region used to test the claim that all ages have crash rates proportional to their number of drivers. Use = 0.05.

Cell 1 2 40 219

45)

= 0.01.

Grade A B C D F Number 42 36 60 8 14 A) 3.41 B) 5.25

Starting Position Number of Wins A) 2 > 15.086

44)

D) (0.499, 0.596)

48)

49) A coffeehouse wishes to see if customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Find the rejection region used to test the claim that the probabilities show no preference. Use = 0.01. Brand 1 2 3 4 5 Customers 30 32 55 18 65 A) 2 > 14.860 B) 2 > 13.277

C) 2 > 9.488

49)

D) 2 > 11.143

50) A coffeehouse wishes to see if customers have any preference among 5 different brands of coffee. A sample of 200 customers provided the data below. Calculate the chi-square test statistic 2 used to

50)

test the claim that the probabilities show no preference. Brand 1 2 3 4 5 Customers 30 55 65 32 18 A) 45.91 B) 55.63

C) 48.91

D) 37.45

Answer the question True or False. 51) When using any procedure to perform a hypothesis test, the user should always be certain that the experiment satisfies the assumptions given with the procedure. A) True B) False

51)

Solve the problem. 52) Use the appropriate table to find the following chi-square value: A) 9.348

B) 5.024

C) 0.051

2 .025 for df = 2.

52)

D) 7.378

53) Many track runners believe that they have a better chance of winning if they start in the inside lane that is closest to the field. For the data below, the lane closest to the field is Lane 1, the next lane is Lane 2, and so on until the outermost lane, Lane 6. The table displays the starting positions for the winners of 240 competitions. Calculate the chi-square test statistic 2 used to test the claim that the

53)

probability of winning is the same regardless of starting position.. Starting Position Number of Wins A) 15.541

1 2 3 4 5 6 45 32 36 44 33 50 B) 6.750

C) 12.592

D) 9.326

54) A sports researcher is interested in determining if there is a relationship between the number of home team and visiting team wins and different sports. A random sample of 526 games is selected and the results are given below. Calculate the chi-square test statistic 2 used to test the claim that the number of home team and visiting team wins is independent of the sport. Use

Home team wins Visiting team wins A) 5.391

Football Basketball Soccer Baseball 39 156 25 83 31 98 19 75 B) 2.919 C) 4.192

= 0.01.

D) 3.290

54)

55) A random sample of 160 car accidents are selected and categorized by the age of the driver 55) determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is 18% for the under 26 group, 39% for the 26-45 group, 31% for the 45-65 group, and 12% for the group over 65. Calculate the chi-square test statistic 2 used to test the claim that all ages have crash rates proportional to their driving rates. Age Under 26 26 - 45 46 - 65 Over 65 Drivers 66 39 25 30 A) 85.123 B) 95.431

C) 75.101

D) 101.324

56) A teacher finds that final grades in the statistics department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following grades were recorded. Find the rejection region used to determine if the grade distribution for the department is different than expected. Use = 0.01. Grade A B C D F Number 42 36 60 14 8 A) 2 > 7.779 B) 2 > 13.277

C) 2 > 9.488

D) 2 > 11.143

57) A sports researcher is interested in determining if there is a relationship between the number of home team and visiting team wins and different sports. A random sample of 526 games is selected and the results are given below. Find the rejection region used to test the claim that the number of home team and visiting team wins is independent of the sport. Use = 0.01. Football Basketball Soccer Baseball Home team wins 39 156 25 83 Visiting team wins 31 98 19 75 2 2 > 12.838 > 7.815 A) B) C) 2 > 11.345

B) 2 > 10.645

C) 2 > 12.017

57)

D) 2 > 9.348

58) Find the rejection region for a one-dimensional chi-square test of a null hypothesis concerning p 1 , p 2 , . . . p k if k = 6 and = .10. A) 2 > 9.236

56)

58)

D) 2 > 1.61

Answer the question True or False.

59) The sampling distribution for 2 works well when expected counts are very small. A) True B) False

Solve the problem. 60) Find the rejection region for a test of independence of two classifications where the contingency table contains r = 4 rows and c = 5 columns and = .05. A) 2 > 31.410 B) 2 > 34.170 C) 2 > 43.773 D) 2 > 21.026

59)

60)

Answer Key Testname: CHAPTER 10 1) A 2) D 3) B 4) B 5) A 6) C 7) C 8) D 9) B 10) B 11) C 12) D 13) C 14) A 15) D 16) A 17) C 18) C 19) B 20) B 21) a.

40.6%, 36.6%, 59.3%

c. One bar would be much taller or much shorter than the others. Yes, the bar for column 3 is much taller than the other two bars. 22) E(n3 ) = .05(30) = 1.5 < 5, so the sample size is too small.

23) Since 2 = 3.056 <

2 .05 = 7.815 (df = 3), we do not reject the null hypothesis. There is not sufficient evidence that the

cell proportions differ from those given in the null hypothesis. 24) The rejection region is 2 > 13.277; chi-square test statistic is 2 37.45; reject H0 ; There is sufficient evidence to reject the claim that customers show no preference for the brands. 25) The alternative hypothesis is that at least two of the proportions differ from .25. 26) The probabilities of the k outcomes remain the same from trial to trial and sum to 1. 27) The rejection region is 2 > 7.815; chi-square test statistic is 2 75.101; reject H0 ; There is sufficient evidence to reject the claim that the crash rates are proportional to the number of drivers.

Answer Key Testname: CHAPTER 10 28) The rejection region is 2 >11.070; chi-square test statistic is 2 6.750; fail to reject H0 ; There is not sufficient evidence to reject the claim of equal probabilities of winning in the six lanes. 29) data that represent the counts for each category of a multinomial experiment RP CE Total 30) a. M 12 16 28 F 12 10 22 Total 24 26 50

b. The rejection region is 2 > 3.84146; the test statistic is 2 .6743; we cannot reject thee null hypothesis. There is no evidence that gender and choice of research project or comprehensive exams are not independent. 31) The rejection region is 2 > 11.345; chi-square test statistic is 2 3.290; fail to reject H0 ; There is not sufficient

evidence to reject the claim of independence. ^

32) p 2 = .32; The confidence interval is .32 ± 2.575 33) Since 2 = 42.8567 >

.32(.68) 100

.32 ± .12.

2 .005 = 14.860, we reject the null hypothesis. There is sufficient evidence to reject the claim that A

and B are independent.

34) The rejection region is 2 > 16.919; chi-square test statistic is 2 91.097; reject H0 ; There is sufficient evidence to reject the claim of independence. 35) The rejection region is 2 > 13.277; chi-square test statistic is 2 5.25; fail to reject H0 ; There is not sufficient evidence

to reject the claim that the grade distribution matches the expected one. 36) The trials are independent; there are k possible outcomes to each trial; the experiment consists of n identical trials. 37) The rejection region is 2 > 9.488; chi-square test statistic is 2 8.030; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence. 38) C 39) B 40) C 41) B 42) C 43) B 44) D 45) B 46) D 47) A 48) D 49) B 50) D 51) A 52) D 53) B 54) D 55) C 56) B 57) C 58) A 59) B 60) D

Chapter 11 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000’s) of graduates of the program based on the amount of tuition (in $1000’s) charged by the program. We are told that the coefficient of correlation was calculated to be r = 0.7763. Use this information to calculate the test statistic that would be used to determine if a positive linear relationship exists between the two variables. A) t = 1.475 B) t = 0.6027 C) t = 10.52 D) t = 1.760 2) If a least squares line were determined for the data set in each scattergram, which would have the smallest variance? A) B)

3) A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000’s) of 3) graduates of the program based on the amount of tuition (in $1000’s) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Size

Coefficient 18.1849 1.47494

Std Error 10.3336 0.14017

T 1.76 10.52

R-Squared 0.6027 Resid. Mean Square (MSE) Adjusted R-Squared 0.5972 Standard Deviation

P 0.0826 0.0000 532.986 23.0865

In addition, we are told that the coefficient of correlation was calculated to be r = 0.7763. Interpret this result. A) There is a very weak positive linear relationship between the amount of tuition charged and the average starting salary variables. B) There is a fairly strong negative linear relationship between the amount of tuition charged and the average starting salary variables. C) There is a fairly strong positive linear relationship between the amount of tuition charged and the average starting salary variables. D) There is almost no linear relationship between the amount of tuition charged and the average starting salary variables.

4) Consider the data set shown below. Find the estimate of the slope of the least squares regression line. 4) y 0 x -2

A) 1.5

3 0

2 2

3 4

8 10 11 6 8 10

B) 0.9003

C) 0.94643

D) 1.49045

5) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below:

Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

Interpret the coefficient of determination for the regression model. A) There is sufficient evidence to indicate that the size of the diamond is a useful predictor of the price of a diamond when testing at alpha = 0.05. B) We expect most of the sampled diamond prices to fall within $2235.12 of their least squares predicted values. C) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $1117.56. D) We can explain 89.25% of the variation in the sampled diamond prices around their mean using the size of the diamond in a linear model.

6) Locate the values of SSE, s2 , and s on the printout below.

Model Summary Model 1

R .859

Model 1 Regression Residual Total

R Square .737

Adjusted R Square .689

ANOVA Sum of Squares df 4512.024 1 1678.115 12 6190.139 13

Std. Error of the Estimate 11.826

Mean Square 4512.024 139.843

A) SSE = 4512.024; s2 = 139.843; s = 11.826 C) SSE = 1678.115; s2 = 139.843; s = 11.826

F 32.265

Sig. .001

B) SSE = 4512.024; s2 = 4512.024; s = 32.265 D) SSE = 6190.139; s2 = 4512.024; s = 32.265

7) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below:

Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

Which of the following conclusions is correct when testing to determine if the size of the diamond is a useful positive linear predictor of the price of a diamond? A) The sample size is too small to make any conclusions regarding the regression line. B) There is insufficient evidence to indicate that the size of the diamond is a useful positive linear predictor of the price of a diamond when testing at = 0.05. C) There is sufficient evidence to indicate that the size of the diamond is a useful positive linear predictor of the price of a diamond when testing at = 0.05. D) There is insufficient evidence to indicate that the price of the diamond is a useful positive linear predictor of the size of a diamond when testing at = 0.05.

8) Is there a relationship between the raises administrators at County University receive and their performance on the job? A faculty group wants to determine whether job rating (x) is a useful linear predictor of raise (y). Consequently, the group considered the linear regression model E(y) = 0 + 1 x. The faculty group obtained the following prediction equation: ^

y = 14,000 - 2,000x Which of the following statements about the model E(y) = 0 + 1 x is correct? A) The model hypothesizes that knowing an administrator's rating (x) will determine exactly the administrator's raise (y). B) The model hypothesizes that, on average, administrators make more money than professors. C) The model hypothesizes that the raises for the administrators fall in a perfect straight line. D) The model hypothesizes a line of means; as rating (x) increases, the mean raise E(y) moves up or down along a straight line.

9) The dean of the Business School at a small Florida college wishes to determine whether the grade-point average (GPA) of a graduating student can be used to predict the graduate's starting salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting salaries. Records for 23 of last year's Business School graduates are selected at random, and data on GPA (x) and starting salary (y, in $thousands) for each graduate were used to fit the model

E(y) = 0 + 1 x. The value of the test statistic for testing 1 is 17.169. Select the proper conclusion. A) There is insufficient evidence (at = .05) to conclude that GPA is positively linearly related to starting salary. B) There is insufficient evidence (at = .10) to conclude that GPA is a useful linear predictor of starting salary. C) At any reasonable , there is no relationship between GPA and starting salary. D) There is sufficient evidence (at = .05) to conclude that GPA is positively linearly related to starting salary.

10) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below: Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

Interpret the standard deviation of the regression model. A) We expect most of the sampled diamond prices to fall within $2235.12 of their least squares predicted values. B) We can explain 89.25% of the variation in the sampled diamond prices around their mean using the size of the diamond in a linear model. C) We expect most of the sampled diamond prices to fall within $1117.56 of their least squares predicted values. D) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $1117.56.

10)

11) A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000’s) of 11) graduates of the program based on the amount of tuition (in $1000’s) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Size

Coefficient 18.1849 1.47494

Std Error 10.3336 0.14017

T 1.76 10.52

R-Squared 0.6027 Resid. Mean Square (MSE) Adjusted R-Squared 0.5972 Standard Deviation

P 0.0826 0.0000 532.986 23.0865

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the tuition charged by the MBA program was $75,000. The results are shown here: 95% confidence interval for E(Y): ($123,390, $134,220) 95% prediction interval for Y: ($82,476, $175,130) Which of the following interpretations is correct if you want to use the model to estimate E(Y) for all MBA programs? A) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $123,390 and $134,220. B) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $82,476 and $175,130. C) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $123,390 and $134,220. D) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $82,476 and $175,130.

12) Consider the data set shown below. Find the estimate of the y-intercept of the least squares regression 12) line. y 0 x -2

3 0

2 2

3 4

A) 0.9003

8 10 11 6 8 10

B) 1.49045

C) 1.5

D) 0.94643

13) Consider the data set shown below. Find the coefficient of correlation for between the variables x and y.13) y 0 x -2

A) 0.9003

3 0

2 2

3 4

8 10 11 6 8 10

B) 0.9383

C) 0.8804

D) 0.9489

14) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 14) in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the linear regression model: E(y) = 0 + 1 x, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following result was obtained: ^

y = 74.80 + 19.72x Which of the following statements concerning the deterministic model, E(y) = 0 + 1 x is true? A) In theory, if the appraised values y and number of rooms x for the entire population of houses in East Meadow were obtained and the (x, y) data points plotted, the points would fall in a straight line. B) In theory, a plot of the mean appraised value E(y) against the number of rooms x for the entire population of houses in east Meadow would result in a straight line. C) All of the above statements are true. ^

D) A plot of the predicted appraised values y against the number of rooms x for the sample of houses in East Meadow would not result in a straight line. 15) A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000’s) of 15) graduates of the program based on the amount of tuition (in $1000’s) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Size

Coefficient 18.1849 1.47494

Std Error 10.3336 0.14017

T 1.76 10.52

R-Squared 0.6027 Resid. Mean Square (MSE) Adjusted R-Squared 0.5972 Standard Deviation

P 0.0826 0.0000 532.986 23.0865

Interpret the estimated slope of the regression line. A) For every $1000 increase in the average starting salary, we estimate that the tuition charged by the MBA program will increase by $1474.94. B) For every $1474.94 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $18,184.90. C) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will decrease by $1474.94. D) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $1474.94.

16) What is the relationship between diamond price and carat size? 307 diamonds were sampled 16) (ranging in size from 0.18 to 1.1 carats) and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below: Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

Interpret the estimated y-intercept of the regression line. A) When a diamond is 11598.9 carats in size, we estimate the price of the diamond to be $2298.36. B) When a diamond is 0 carats in size, we estimate the price of the diamond to be $2298.36. C) When a diamond is 0 carats in size, we estimate the price of the diamond to be $11,598.90. D) No practical interpretation of the y-intercept exists since a diamond of 0 carats cannot exist and falls outside the range of the carat sizes sampled.

17) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below: Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

Interpret the estimated slope of the regression line. A) For every 2298.36-carat decrease in the size of a diamond, we estimate that the price of the diamond will increase by $11,598.90. B) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $11,598.90. C) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will decrease by $2298.36. D) For every $1 decrease in the price of the diamond, we estimate that the size of the diamond will increase by 11,598.9 carats.

17)

18) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below:

18)

Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the carat size of the diamond was 1 carat. The results are shown here: 95% confidence interval for E(Y): ($9091.60, $9509.40) 95% prediction interval for Y: ($7091.50, $11,510.00) Which of the following interpretations is correct if you want to use the model to determine the price of a single 1-carat diamond? A) We are 95% confident that the average price of all 1-carat diamonds will fall between $9091.60 and $9509.40. B) We are 95% confident that the average price of all 1-carat diamonds will fall between $7091.50 and $11,510.00. C) We are 95% confident that the price of a 1-carat diamond will fall between $7091.50 and $11,510.00. D) We are 95% confident that the price of a 1-carat diamond will fall between $9091.60 and $9509.40.

19) A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000’s) of 19) graduates of the program based on the amount of tuition (in $1000’s) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Size

Coefficient 18.1849 1.47494

Std Error 10.3336 0.14017

T 1.76 10.52

R-Squared 0.6027 Resid. Mean Square (MSE) Adjusted R-Squared 0.5972 Standard Deviation

P 0.0826 0.0000 532.986 23.0865

Fill in the blank. At = 0.05, there is _________________ between the amount of tuition charged by an MBA program and the average starting salary of graduates of the program. A) …sufficient evidence of a positive linear relationship… B) …insufficient evidence of a positive linear relationship… C) …sufficient evidence of a negative linear relationship…

20) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below:

20)

Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the carat size of the diamond was 1 carat. The results are shown here: 95% confidence interval for E(Y): ($9091.60, $9509.40) 95% prediction interval for Y: ($7091.50, $11,510.00) Which of the following interpretations is correct if you want to use the model to estimate E(Y) for all 1-carat diamonds? A) We are 95% confident that the price of a 1-carat diamond will fall between $7091.50 and $11,510.00. B) We are 95% confident that the price of a 1-carat diamond will fall between $9091.60 and $9509.40. C) We are 95% confident that the average price of all 1-carat diamonds will fall between $9091.60 and $9509.40. D) We are 95% confident that the average price of all 1-carat diamonds will fall between $7091.50 and $11,510.00.

21) Consider the data set shown below. Find the coefficient of determination for the simple linear regression 21) model. y 0 x -2

A) 0.9003

3 0

2 2

3 4

8 10 11 6 8 10

B) 0.9489

C) 0.9383

D) 0.8804

22) An academic advisor wants to predict the typical starting salary of a graduate at a top business school 22) using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT was created from a set of 25 data points. Which of the following is not an assumption required for the simple linear regression analysis to be valid? A) The errors of predicting SALARY have a variance that is constant for any given value of GMAT. B) SALARY is independent of GMAT. C) The errors of predicting SALARY have a mean of 0. D) The errors of predicting SALARY are normally distributed.

23) What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between y = diamond price (in dollars) and x = size of the diamond (in carats). The simple linear regression for the analysis is shown below:

23)

Least Squares Linear Regression of PRICE Predictor Variables Constant Size

Coefficient -2298.36 11598.9

R-Squared Adjusted R-Squared

Std Error 158.531 230.111

0.8925 0.8922

T P -14.50 0.0000 50.41 0.0000

Resid. Mean Square (MSE) Standard Deviation

1248950 1117.56

Which of the following assumptions is not stated correctly? A) The mean of the probability distribution of is 0. B) The values of associated with any two observations are dependent on one another. C) The probability distribution of is normal. D) The variance of the probability distribution of is constant for all settings of the independent variable.

24) A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000’s) of 24) graduates of the program based on the amount of tuition (in $1000’s) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Size

Coefficient 18.1849 1.47494

Std Error 10.3336 0.14017

T 1.76 10.52

R-Squared 0.6027 Resid. Mean Square (MSE) Adjusted R-Squared 0.5972 Standard Deviation

P 0.0826 0.0000 532.986 23.0865

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the tuition charged by the MBA program was $75,000. The results are shown here: 95% confidence interval for E(Y): ($123,390, $134,220) 95% prediction interval for Y: ($82,476, $175,130) Which of the following interpretations is correct if you want to use the model to predict Y for a single MBA programs? A) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $123,390 and $134,220. B) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $123,390 and $134,220. C) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $82,476 and $175,130. D) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $82,476 and $175,130.

25) Consider the data set shown below. Find the standard deviation of the least squares regression line. y 0 x -2 A) 1.5

3 0

2 2

3 4

8 10 11 6 8 10 B) 1.49045

C) 0.9003

25)

D) 0.94643

26) Consider the data set shown below. Find the 95% confidence interval for the slope of the regression line. 26) y 0 x -2

3 0

2 2

3 4

8 10 11 6 8 10

A) 0.94643 ± 0.28377 C) 0.94643 ± 0.36203

B) 0.94643 ± 0.33306 D) 0.94643 ± 0.27603

(Situation P) Below are the results of a survey of America's best graduate and professional schools. The top 25 business schools, as determined by reputation, student selectivity, placement success, and graduation rate, are listed in the table. For each school, three variables were measured: (1) GMAT score for the typical incoming student; (2) student acceptance rate (percentage accepted of all students who applied); and (3) starting salary of the typical graduating student.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

The academic advisor wants to predict the typical starting salary of a graduate at a top business school using GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using the 25 data points in the table are shown below. ----------------------------------------------------------------------^ 0 = -92040 1 = 228 s = 3213 r2 = .66 r = .81 df = 23 t = 6.67 -----------------------------------------------------------------------

27) For the situation above, which of the following is not an assumption required for the simple linear regression analysis to be valid? A) The errors of predicting SALARY have a mean of 0. B) The errors of predicting SALARY have a variance that is constant for any given value of GMAT. C) The errors of predicting SALARY are normally distributed. D) SALARY is independent of GMAT.

27)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 28) To investigate the relationship between yield of potatoes, y, and level of fertilizer 28) application, x, a researcher divides a field into eight plots of equal size and applies differing amounts of fertilizer to each. The yield of potatoes (in pounds) and the fertilizer application (in pounds) are recorded for each plot. The data are as follows: x y

1 25

1.5 31

2 27

2.5 28

3 36

3.5 35

4 32

4.5 34

Summary statistics yield SSxx = 10.5, SSyy = 112, SSxy = 25, x = 2.75, and y = 31. Find the least squares prediction equation.

29) A realtor collected the following data for a random sample of ten homes that recently sold in her 29) area. House A B C D E F G H I J

Asking Price $114,500 $149,900 $154,700 $159,900 $160,000 $165,900 $169,700 $171,900 $175,000 $289,900

Days on Market 29 16 59 42 72 45 12 39 81 121

a. Find a 90% confidence interval for the mean number of days on the market for all houses listed at $150,000. b. Suppose a house has just been listed at $150,000. Find a 90% prediction interval for the number of days the house will be on the market before it sells.

30) The data for n = 24 points were subjected to a simple linear regression with the results: ^

1 = 0.88 and s ^1 = 0.13.

a. Test whether the two variables, x and y, are positively linearly related. Use b. Construct and interpret a 90% confidence interval for 1 .

= .05.

31) Suppose you fit a least squares line to 20 data points and the calculated value of SSE is 0.484. a. Find s2 , the estimator of 2 .

b. What is the largest deviation you might expect between any one of the 20 points and the least squares line?

30)

31)

32) In team-teaching, two or more teachers lead a class. An researcher tested the use of team-teaching in mathematics education. Two of the variables measured on each sample of 164 mathematics teachers were years of teaching experience x and reported success rate y (measured as a percentage) of team-teaching mathematics classes.

32)

a. The researcher hypothesized that mathematics teachers with more years of experience will report higher perceived success rates in team-taught classes. State this hypothesis in terms of the parameter of a linear model relating x to y. b. The correlation coefficient for the sample data was reported as r = -0.32. Interpret this result. c. Does the value of r support the hypothesis? Test using = .05.

33) Consider the following pairs of observations: x y

2 1.3

3 1.6

33)

5 2.1

5 2.2

6 2.7

Find and interpret the value of the coefficient of determination.

34) Is the number of games won by a major league baseball team in a season related to the team's 34) batting average? Data from 14 teams were collected and the summary statistics yield: y = 1,134,

x = 3.642,

y2 = 93,110,

x2 = .948622, and

xy = 295.54

Find the least squares prediction equation for predicting the number of games won, y, using a straight-line relationship with the team's batting average, x.

35) Plot the line y = 3x. Then give the slope and y-intercept of the line.

35)

36) A breeder of Thoroughbred horses wishes to model the relationship between the gestation 36) period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse

1 2 3 4

Gestation period x (days) 416 279 298 307

Life Length y (years) 24 25.5 20 21.5

Horse

5 6 7

Gestation period x (days) 356 403 265

Life Length y (years) 22 23.5 21

Summary statistics yield SSxx = 21,752, SSxy = 236.5, SSyy = 22, x = 332, and y = 22.5. Test to determine if a linear relationship exists between the gestation period and the length of life of a horse. Use = .05 and use s = 1.97 as an estimate of .

37) Suppose you fit a least squares line to 22 data points and the calculated value of SSE is .678. a. Find s2 , the estimator of 2 .

37)

b. Find s, the estimator of . c. What is the largest deviation you might expect between any one of the 22 points and the least squares line?

38) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding 38) by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and and the age of the warthog (in days) are listed below: Number of Grunts 90 68 39 44 63 40 62 17 20

Age (days) 125 141 155 160 167 174 183 189 195

a. Write the equation of a straight-line model relating number of grunts (y) to age (x). b. Give the least squares prediction equation. c.

Give a practical interpretation of the value of 0 , if possible.

d. Give a practical interpretation of the value of

39) Construct a 90% confidence interval for

1 , if possible.

1 when 1 = 49, s = 4, SSxx = 55, and n = 15.

39)

40) A breeder of Thoroughbred horses wishes to model the relationship between the gestation 40) period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse

1 2 3 4

Gestation period x (days) 416 279 298 307

Life Length y (years) 24 25.5 20 21.5

Horse

5 6 7

Gestation period x (days) 356 403 265

Life Length y (years) 22 23.5 21

Summary statistics yield SSxx = 21,752, SSxy = 236.5, SSyy = 22, x = 332, and y = 22.5. Calculate SSE, s2 , and s.

41) Consider the following pairs of measurements: x y a. b. c. d.

1 3

3 6

4 8

41) 6 12

7 13

Construct a scattergram for the data. What does the scattergram suggest about the relationship between x and y? Find the least squares estimates of 0 and 1 .

Plot the least squares line on your scattergram. Does the line appear to fit the data well?

42) Consider the following pairs of observations: x y

2 1.3

3 1.6

42)

5 2.1

5 2.2

6 2.7

a. Construct a scattergram for the data. Does the scattergram suggest that y is positively linearly related to x? b. Find the slope of the least squares line for the data and test whether the data provide sufficient evidence that y is positively linearly related to x. Use = .05.

43) Plot the line y = 1.5 + .5x. Then give the slope and y-intercept of the line.

43)

44) Consider the following pairs of measurements:

44)

x y

5 6.2

8 3.4

3 7.5

4 8.1

9 3.2

a. Construct a scattergram for the data. b. Use the method of least squares to model the relationship between x and y. c. Calculate SSE, s2 , and s. ^

d. What percentage of the observed y-values fall within 2s of the values of y predicted by the least squares model?

45) To investigate the relationship between yield of potatoes, y, and level of fertilizer 45) application, x, a researcher divides a field into eight plots of equal size and applies differing amounts of fertilizer to each. The yield of potatoes (in pounds) and the fertilizer application (in pounds) are recorded for each plot. The data are as follows: x y

1 25

1.5 31

2 27

2.5 28

3 36

3.5 35

4 32

4.5 34

Summary statistics yield SSxx = 10.5, SSyy = 112, SSxy = 25, and SSE = 52.476. Calculate the coefficient of correlation.

46) To investigate the relationship between yield of potatoes, y, and level of fertilizer 46) application, x, a researcher divides a field into eight plots of equal size and applies differing amounts of fertilizer to each. The yield of potatoes (in pounds) and the fertilizer application (in pounds) are recorded for each plot. The data are as follows: x y

1 25

1.5 31

2 27

2.5 28

3 36

3.5 35

4 32

4.5 34

Summary statistics yield SSxx = 10.5, SSyy = 112, SSxy = 25, and SSE = 52.476. Calculate the coefficient of determination.

47) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding 47) by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and and the age of the warthog (in days) are listed below: Number of Grunts 92 70 41 46 65 42 64 19 22

Age (days) 127 143 157 162 169 176 185 191 197

Find and interpret the value of r.

48) A company keeps extensive records on its new salespeople on the premise that sales should increase with experience. A random sample of seven new salespeople produced the data on experience and sales shown in the table. Months on Job 2 4 8 12 1 5 9

Monthly Sales y ($ thousands) 2.4 7.0 11.3 15.0 .8 3.7 12.0

Summary statistics yield SSxx = 94.8571, SSxy = 124.7571, SSyy = 176.5171, x = 5.8571, and y = 7.4571. Calculate a 90% confidence interval for E(y) when x = 5 months. Assume ^

s = 1.577 and the prediction equation is y = -.25 + 1.315x.

48)

49) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding 49) by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and and the age of the warthog (in days) are listed below: Number of Grunts 83 61 32 37 56 33 55 10 13

Age (days) 118 134 148 153 160 167 176 182 188

Find and interpret the value of r2.

50) Is the number of games won by a major league baseball team in a season related to the team's 50) batting average? Data from 14 teams were collected and the summary statistics yield: y = 1,134,

x = 3.642,

y2 = 93,110,

x 2 = .948622, and

xy = 295.54

Assume 1 = 455.27. Estimate and interpret the estimate of .

51) a.

Totals b. c.

51)

Complete the table. xi

2 5 3 8

3 2 4 0

xi =

xi 2

yi =

xi2 =

x iy i

xiy i =

Find SSxy, SSxx, 1 , x, y, and 0 . Write the equation of the least squares line.

52) Calculate SSE and s2 for n = 25,

y2 = 950,

y = 65, SSxy = 3000, and 1 = .2.

53) Plot the line y = 4 - 2x. Then give the slope and y-intercept of the line.

52) 53)

54) In a comprehensive road test for new car models, one variable measured is the time it takes the54) car to accelerate from 0 to 60 miles per hour. To model acceleration time, a regression analysis is conducted on a random sample of 129 new cars. TIME60: y = Elapsed time (in seconds) from 0 mph to 60 mph MAX: x = Maximum speed attained (miles per hour) The simple linear model E(y) = 0 + 1 x was fit to the data. Computer printouts for the analysis are given below: NWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME60 PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT'S T P CONSTANT 18.7171 0.63708 29.38 0.0000 MAX 0.00491 0.0000 -0.08365 -17.05 R-SQUARED ADJUSTED R-SQUARED SOURCE REGRESSION RESIDUAL TOTAL

DF 1 127 128

0.6960 0.6937

SS 374.285 163.443 537.728

RESID. MEAN SQUARE (MSE) STANDARD DEVIATION MS 374.285 1.28695

F 290.83

CASES INCLUDED 129 MISSING CASES 0 ^

Find and interpret the estimate 1 in the printout above.

P 0.0000

1.28695 1.13444

55) Operations managers often use work sampling to estimate how much time workers spend 55) on each operation. Work sampling which involves observing workers at random points in time was applied to the staff of the catalog sales department of a clothing manufacturer. The department applied regression to data collected for 40 randomly selected working days. The simple linear model E(y) = 0 + 1 x was fit to the data. The printouts for the analysis are given below: TIME:

y = Time spent (in hours) taking telephone orders during the day

ORDERS:

x = Number of telephone orders received during the day

UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT'S T P CONSTANT 10.1639 1.77844 5.72 0.0000 ORDERS 0.05836 0.00586 9.96 0.0000 R-SQUARED ADJUSTED R-SQUARED SOURCE REGRESSION RESIDUAL TOTAL

DF 1 38 39

0.7229 0.7156

SS 1151.55 441.464 1593.01

RESID. MEAN SQUARE (MSE) STANDARD DEVIATION MS 1151.55 11.6175

F 99.12

11.6175 3.40844

P 0.0000

CASES INCLUDED 40 MISSING CASES 0 Conduct a test of hypothesis to determine if time spent (in hours) taking telephone orders during the day and the number of telephone orders received during the day are positively linearly related. Use = .01.

56) Consider the following pairs of observations: x y a. b. c. d.

2 1

0 3

56)

3 4

3 6

5 7

Construct a scattergram for the data. Find the least squares line, and plot it on your scattergram. Find a 99% confidence interval for the mean value of y when x = 1. Find a 99% prediction interval for a new value of y when x = 1.

by 57) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding 57) a lake in the 15 minute period following the addition of food. The data showing the number of grunts and the age of the warthog (in days) are listed below: Number of Grunts 104 82 53 58 77 54 76 31 36 a. b.

Age (days) 132 148 162 167 174 181 190 196 202

Find SSE, s2, and s. Interpret the value of s.

58) A company keeps extensive records on its new salespeople on the premise that sales should increase with experience. A random sample of seven new salespeople produced the data on experience and sales shown in the table. Months on Job 2 4 8 12 1 5 9

Monthly Sales y ($ thousands) 2.4 7.0 11.3 15.0 .8 3.7 12.0

Summary statistics yield SSxx = 94.8571, SSxy = 124.7571, SSyy = 176.5171, x = 5.8571, and y = 7.4571. Using SSE = 12.435, find and interpret the coefficient of determination.

58)

59) A company keeps extensive records on its new salespeople on the premise that sales should increase with experience. A random sample of seven new salespeople produced the data on experience and sales shown in the table. Months on Job 2 4 8 12 1 5 9

59)

Monthly Sales y ($ thousands) 2.4 7.0 11.3 15.0 .8 3.7 12.0

Summary statistics yield SSxx = 94.8571, SSxy = 124.7571, SSyy = 176.5171, x = 5.8571, and y = 7.4571. State the assumptions necessary for predicting the monthly sales based on the linear relationship with the months on the job.

60) The equation for a (deterministic) straight line is y = 0 + 1 x. If the line passes through the points (5, 8) and (7, 9), find the values of 0 and 1 , respectively.

60)

61) State the four basic assumptions about the general form of the probability distribution of the random error .

61)

62) A realtor collected the following data for a random sample of ten homes that recently sold in her 62) area. House A B C D E F G H I J

Asking Price $114,500 $149,900 $154,700 $159,900 $160,000 $165,900 $169,700 $171,900 $175,000 $289,900

Days on Market 29 16 59 42 72 45 12 39 81 121

a. Construct a scattergram for the data. b. Find the least squares line for the data and plot the line on your scattergram. c. Test whether the number of days on the market, y, is positively linearly related to the asking price, x. Use = .05.

63) In team-teaching, two or more teachers lead a class. A researcher tested the use of team-teaching in mathematics education. Two of the variables measured on each teacher in a sample of 180 mathematics teachers were years of teaching experience x and reported success rate y (measured as a percentage) of team-teaching mathematics classes. The correlation coefficient for the sample data was reported as r = -0.33. Interpret this result.

63)

64) Construct a 95% confidence interval for

1 when 1 = 49, s = 4, SSxx = 55, and n = 15.

64)

65) A breeder of Thoroughbred horses wishes to model the relationship between the gestation 65) period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse

Gestation period x (days) 416 279 298 307

1 2 3 4

Life Length y (years) 24 25.5 20 21.5

Horse

5 6 7

Gestation period x (days) 356 403 265

Life Length y (years) 22 23.5 21

Summary statistics yield SSxx = 21,752, SSxy = 236.5, SSyy = 22, x = 332, and y = 22.5. Find a 95% prediction interval for the length of life of a horse that had a gestation period of 300 days. Use s = 2 as an estimate of

and use y = 18.89 + .01087x.

66) Is the number of games won by a major league baseball team in a season related to the team's 66) batting average? Data from 14 teams were collected and the summary statistics yield: y = 1,134,

x = 3.642,

y2 = 93,110,

x 2 = .948622, and

xy = 295.54

Assume 1 = 455.27 and = 9.18. Conduct a test of hypothesis to determine if a positive linear relationship exists between team batting average and number of wins. Use = .05. ^

67) Calculate SSE and s2 for n = 30, SSyy = 100, SSxy = 60, and 1 = .8.

67)

68) Consider the following pairs of observations:

68)

x y

2 1.3

3 1.6

5 2.1

5 2.2

Find and interpret the value of the coefficient of correlation.

6 2.7

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 69) In a comprehensive road test on new car models, one variable measured is the time it takes the car to 69) accelerate from 0 to 60 miles per hour. To model acceleration time, a regression analysis is conducted on a random sample of 129 new cars. TIME60: MAX

y = Elapsed time (in seconds) from 0 mph to 60 mph x = Maximum speed attained (miles per hour)

The simple linear model E(y) = 0 + 1 x was fit to the data. Computer printouts for the analysis are given below: NWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME60 PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT'S T P CONSTANT 18.7171 0.63708 29.38 0.0000 MAX 0.00491 0.0000 -0.08365 -17.05 R-SQUARED ADJUSTED R-SQUARED SOURCE REGRESSION RESIDUAL TOTAL

DF 1 127 128

0.6960 0.6937

SS 374.285 163.443 537.728

RESID. MEAN SQUARE (MSE) STANDARD DEVIATION MS 374.285 1.28695

F 290.83

1.28695 1.13444

P 0.0000

CASES INCLUDED 129 MISSING CASES 0 Approximately what percentage of the sample variation in acceleration time can be explained by the simple linear model? A) -17% B) 70% C) 8% D) 0%

Answer the question True or False. 70) The Method of Least Squares specifies that the regression line has an average error of 0 and has an SSE that is minimized. A) True B) False

70)

Solve the problem. 71) Is there a relationship between the raises administrators at State University receive and their performance 71) on the job? A faculty group wants to determine whether job rating (x) is a useful linear predictor of raise (y). Consequently, the group considered the straight-line regression model E(y) = 0 + 1 x. Using the method of least squares, the faculty group obtained the following prediction equation: ^

y = 14,000 - 2,000x Interpret the estimated slope of the line. A) For a $1 increase in an administrator's raise, we estimate the administrator's rating to decrease 2,000 points. B) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to decrease $2,000. C) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to increase $2,000. D) For an administrator with a rating of 1.0, we estimate his/her raise to be $2,000.

72) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 72) in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(y) = 0 + 1 x, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 73 houses in East Meadow, the following results were obtained: ^

y = 73.80 + 19.72x ^

What are the properties of the least squares line, y = 73.80 + 19.72x? A) It is normal, mean 0, constant variance, and independent. B) All 73 of the sample y-values fall on the line. C) It will always be a statistically useful predictor of y. D) Average error of prediction is 0, and SSE is minimum.

73) A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect drums in the future. The manufacturer collected a random sample of 35 boilers and measured the following two variables: MANHRS: PRESSURE:

y = Number of man-hours required to erect the drum x = Boiler design pressure (pounds per square inch, i.e., psi)

The simple linear model E(y) = 1 + 1 x was fit to the data. A printout for the analysis appears below: UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF MANHRS PREDICTOR VARIABLES CONSTANT PRESSURE

COEFFICIENT 1.88059 0.00321

R-SQUARED ADJUSTED R-SQUARED SOURCE REGRESSION RESIDUAL TOTAL

DF 1 34 35

0.4342 0.4176

SS 111.008 144.656 255.665

STD ERROR 0.58380 0.00163

STUDENT'S T 3.22 2.17

P 0.0028 0.0300

RESID. MEAN SQUARE (MSE) STANDARD DEVIATION

4.25460 2.06267

MS 111.008 4.25160

F 5.19

P 0.0300

Give a practical interpretation of the coefficient of determination, r2 . A) About 2.06% of the sample variation in number of man-hours can be explained by the simple linear model. B) About 43% of the sample variation in number of man-hours can be explained by the simple linear model. C) Approximately 95% of the actual man-hours required to build a drum will fall within 43 hours of their predicted values. D) We are 43% confident that the design pressure will be a useful predictor of number of man-hours required to build a steam drum.

73)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

74) For the situation above, write the equation of the least squares line. A) GMAT = 228 - 92040 (SALARY) B) SALARY = 228 + 92040 (GMAT) C) GMAT = -92040 + 228 (SALARY) D) SALARY = -92040 + 228 (GMAT)

74)

Solve the problem. 75) In a comprehensive road test on new car models, one variable measured is the time it takes a car to 75) accelerate from 0 to 60 miles per hour. To model acceleration time, a regression analysis is conducted on a random sample of 129 new cars. TIME60: MAX:

y = Elapsed time (in seconds) from 0 mph to 60 mph x = Maximum speed attained (miles per hour)

DF 1 127 128

0.6960 0.6937

SS 374.285 163.443 537.728

RESID. MEAN SQUARE (MSE) STANDARD DEVIATION MS 374.285 1.28695

F 290.83

1.28695 1.13444

P 0.0000

CASES INCLUDED 129 MISSING CASES 0 Fill in the blank: "At =.05, there is ________________ between maximum speed and acceleration time." A) insufficient evidence of a linear relationship B) sufficient evidence of a positive linear relationship C) insufficient evidence of a negative linear relationship D) sufficient evidence of a negative linear relationship

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

76) A 95% prediction interval for SALARY when GMAT = 600 is ($37,915, $51,948). Interpret this interval for the situation above. A) We are 95% confident that the SALARY of a top business school graduate will fall between $37,915 and $51,984. B) We are 95% confident that the SALARY of a top business school graduate with a GMAT of 600 will fall between $37,915 and $51,984. C) We are 95% confident that the increase in SALARY for a 600-point increase in GMAT will fall between $37,915 and $51,984. D) We are 95% confident that the mean SALARY of all top business school graduates with GMATs of 600 will fall between $37,915 and $51,984.

76)

Answer the question True or False. 77) The least squares model provides very good estimates of y for values of x far outside the range of x values contained in the sample. A) True B) False 78) A high value of the correlation coefficient r implies that a causal relationship exists between x and y. A) True B) False

77)

78)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

79) For the situation above, give a practical interpretation of 1 = 228. A) The value has no practical interpretation since a GMAT of 228 is nonsensical and outside the range of the sample data. B) We estimate SALARY to increase $228 for every 1-point increase in GMAT. C) We estimate GMAT to increase 228 points for every $1 increase in SALARY. D) We expect to predict SALARY to within 2(228) = $456 of its true value using GMAT in a straight-line model.

79)

Solve the problem. 80) The dean of the Business School at a small Florida college wishes to determine whether the grade-point average (GPA) of a graduating student can be used to predict the graduate's starting salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting salaries. Records for 23 of last year's Business School graduates are selected at random, and data on GPA (x) and starting salary (y, in $thousands) for each graduate were used to fit the model

80)

E(y) = 0 + 1 x. The results of the simple linear regression are provided below. ^

y = 4.25 + 2.75x,

SSxy = 5.15, SSxx = 1.87 SSyy = 15.17, SSE = 1.0075 Range of the x-values: 2.23 - 3.85 Range of the y-values: 9.3 - 15.6 Suppose a 95% prediction interval for y when x = 3.00 is (16, 21). Interpret the interval. A) We are 95% confident that the mean starting salary of all Business School graduates with GPAs of 3.00 will fall between $16,000 and $21,000. B) We are 95% confident that the starting salary of a Business School graduate will increase between $16,000 and $21,000 for every 3-point increase in GPA. C) We are 95% confident that the starting salary of a Business School graduate with a GPA of 3.00 will fall between $16,000 and $21,000. D) We are 95% confident that the starting salary of a Business School graduate will fall between $16,000 and $21,000.

81) An academic advisor wants to predict the typical starting salary of a graduate at a top business school 81) using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. ^

0 = -92040

1 = 228 s = 3213 r2 = .66 r = .81 df = 23 t = 6.67

Give a practical interpretation of r = .81. A) 81% of the sample variation in SALARY can be explained by using GMAT in a straight-line model. B) We estimate SALARY to increase 81% for every 1-point increase in GMAT. C) We can predict SALARY correctly 81% of the time using GMAT in a straight-line model. D) There appears to be a positive correlation between SALARY and GMAT.

82) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 82) in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(y) = 0 + 1 x, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were obtained: ^

y = 74.80 + 19.84x Give a practical interpretation of the estimate of the slope of the least squares line. A) For each additional dollar of appraised value, we estimate the number of rooms in the house to increase by 19.84. B) For each additional room in the house, we estimate the appraised value to increase $19,840. C) For each additional room in the house, we estimate the appraised value to increase $74,800. D) For a house with 0 rooms, we estimate the appraised value to be $74,800.

Answer the question True or False. 83) A low value of the correlation coefficient r implies that x and y are unrelated. A) True B) False

83)

Graph the line that passes through the given points. 84) (5, -2) and (-6, 5)

84)

Answer the question True or False. 85) The probabilistic model allows the E(y) values to fall around the regression line while the actual values of y must fall on the line. A) True B) False

85)

Solve the problem. 86) An academic advisor wants to predict the typical starting salary of a graduate at a top business school 86) using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. ^

0 = -92040

1 = 228 s = 3213 r2 = .66 r = .81 df = 23 t = 6.67

Give a practical interpretation of r2 = .66. A) We can predict SALARY correctly 66% of the time using GMAT in a straight-line model. B) We expect to predict SALARY to within 2 .66 of its true value using GMAT in a straight-line model. C) We estimate SALARY to increase $.66 for every 1-point increase in GMAT. D) 66% of the sample variation in SALARY can be explained by using GMAT in a straight-line model.

Graph the line that passes through the given points. 87) (2, -6) and (-1, 3)

87)

Solve the problem. 88) A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect drums in the future. The manufacturer collected a random sample of 35 boilers and measured the following two variables: MANHRS: PRESSURE:

y = Number of man-hours required to erect the drum x 1 = Boiler design pressure (pounds per square inch, i.e., psi)

The simple linear model E(y) = 0 + 1 x was fit to the data. A printout for the analysis appears below: UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF MANHRS PREDICTOR VARIABLES CONSTANT PRESSURE

COEFFICIENT 1.88059 0.00321

R-SQUARED ADJUSTED R-SQUARED SOURCE REGRESSION RESIDUAL TOTAL

DF 1 34 35

0.4342 0.4176

SS 111.008 144.656 255.665

STD ERROR 0.58380 0.00163

STUDENT'S T 3.22 2.17

P 0.0028 0.0300

RESID. MEAN SQUARE (MSE) STANDARD DEVIATION

4.25460 2.06267

MS 111.008 4.25160

F 5.19

P 0.0300

Fill in the blank. At =.01, there is ____________ between man-hours and pressure. A) sufficient evidence of a positive linear relationship B) sufficient evidence of a linear relationship C) insufficient evidence of a positive linear relationship D) sufficient evidence of a negative linear relationship

88)

Graph the line that passes through the given points. 89) (0, 7) and (7, 0)

89)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

90) For the situation above, give a practical interpretation of s = 3213. A) Our predicted value of SALARY will equal 2(3213) = $6,426 for any value of GMAT. B) We estimate SALARY to increase $3,213 for every 1-point increase in GMAT. C) We expect the predicted SALARY to deviate from actual SALARY by at least 2(3213) = $6,426 using GMAT in a straight-line model. D) We expect to predict SALARY to within 2(3213) = $6,426 of its true value using GMAT in a straight-line model. Answer the question True or False. 91) The coefficient of correlation is a useful measure of the linear relationship between two variables. A) True B) False

90)

91)

Solve the problem. 92) Consider the following model y = 0 + 1 x + , where y is the daily rate of return of a stock, and x is the daily rate of return of the stock market as a whole, measured by the daily rate of return of Standard & Poor's (S&P) 500 Composite Index. Using a random sample of n = 12 days from 1980, the least squares lines shown in the table below were obtained for four firms. The estimated ^

standard error of 1 is shown to the right of each least squares prediction equation. Firm

Estimated Market Model

Company A Company B Company C Company D

y = .0010 + 1.40x y = .0005 - 1.21x y = .0010 + 1.62x y = .0013 + .76x

Estimated Standard Error of 1 .03 .06 1.34 .15

For which of the three stocks, Companies B, C, or D, is there evidence (at = .05) of a positive linear relationship between y and x? A) Company D only B) Company C only C) Companies B and D only D) Companies B and C only

92)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

93) Set up the null and alternative hypotheses for testing whether a positive linear relationship exists between SALARY and GMAT in the situation above. A) H0 : 1 > 0 vs. Ha : 1 < 0 B) H0 : 1 = 0 vs. Ha : 1 0 C) H0 : 1 = 228 vs. Ha : 1 > 228

93)

D) H0 : 1 = 0 vs. Ha : 1 > 0

Answer the question True or False. 94) Probabilistic models are commonly used to estimate both the mean value of y and a new individual value of y for a particular value of x. A) True B) False

94)

Solve the problem. 95) A large national bank charges local companies for using their services. A bank official reported the 95) results of a regression analysis designed to predict the bank's charges (y), measured in dollars per month, for services rendered to local companies. One independent variable used to predict service charge to a company is the company's sales revenue (x), measured in $ million. Data for 21 companies who use the bank's services were used to fit the model E(y) = 0 + 1 x. Suppose a 95% confidence interval for 1 is (15, 25). Interpret the interval. A) We are 95% confident that service charge (y) will increase between $15 and $25 for every $1 million increase in sales revenue (x). B) We are 95% confident that the mean service charge will fall between $15 and $25 per month. C) We are 95% confident that service charge (y) will decrease between $15 and $25 for every $1 million increase in sales revenue (x). D) We are 95% confident that sales revenue (x) will increase between $15 and $25 million for every $1 increase in service charge (y).

Graph the line that passes through the given points. 96) (-8, -8) and (4, 4)

96)

Solve the problem. 97) Is there a relationship between the raises administrators at State University receive and their performance 97) on the job? A faculty group wants to determine whether job rating (x) is a useful linear predictor of raise (y). Consequently, the group considered the straight-line regression model E(y) = 0 + 1 x. Using the method of least squares, the faculty group obtained the following prediction equation: ^

y = 14,000 - 2,000x Interpret the estimated y-intercept of the line. A) The base administrator raise at State University is $14,000. B) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to increase $14,000. C) There is no practical interpretation, since rating of 0 is nonsensical and outside the range of the sample data. D) For an administrator who receives a rating of zero, we estimate his or her raise to be $14,000.

98) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 98) in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(y) = 0 + 1 x, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. What set of hypotheses would you test to determine whether appraised value is positively linearly related to number of rooms? A) H0 : 1 = 0 vs. Ha : 1 0 B) H0 : 1 = 0 vs. Ha : 1 > 0

C) H0 : 1 = 0 vs. Ha : 1 < 0

D) H0 : 1 < 0 vs. Ha : 1 > 0

99) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 99) in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(y) = 0 + 1 x, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were obtained: ^

y = 74.80 + 19.72x

R = 0.539

R 2 = 0.290

s = 58.031

Give a practical interpretation of the estimate of , the standard deviation of the random error term in the model. A) We expect 95% of the observed appraised values to lie on the least squares line. B) We expect to predict the appraised value of an East Meadow house to within about $29,000 of its true value. C) About 29% of the total variation in the sample of y-values can be explained by the linear relationship between appraised value and number of rooms. D) We expect to predict the appraised value of an East Meadow house to within about $58,000 of its true value.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

100) For the situation above, give a practical interpretation of r = .81. A) We estimate SALARY to increase 81% for every 1-point increase in GMAT. B) We can predict SALARY correctly 81% of the time using GMAT in a straight-line model. C) 81% of the sample variation in SALARY can be explained by using GMAT in a straight-line model. D) There appears to be a positive correlation between SALARY and GMAT.

100)

101) For the situation above, give a practical interpretation of r2 = .66. A) We expect to predict SALARY to within 2( .66) of its true value using GMAT in a straight-line model. B) We can predict SALARY correctly 66% of the time using GMAT in a straight-line model. C) We estimate SALARY to increase $.66 for every 1-point increase in GMAT. D) 66% of the sample variation in SALARY can be explained by using GMAT in a straight-line model. Solve the problem. 102) The dean of the Business School at a small Florida college wishes to determine whether the grade-point average (GPA) of a graduating student can be used to predict the graduate's starting salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting salaries. Records for 23 of last year's Business School graduates are selected at random, and data on GPA (x) and starting salary (y, in $thousands) for each graduate were used to fit the model

101)

102)

E(y) = 0 + 1 x The results of the simple linear regression are provided below. ^

y = 4.25 + 2.75x,

SSxy = 5.15, SSxx = 1.87 SSyy = 15.17, SSE = 1.0075

Compute an estimate of , the standard deviation of the random error term. A) 1.0075 B) 0.219 C) .048

D) .689

103) A large national bank charges local companies for using their services. A bank official reported the 103) results of a regression analysis designed to predict the bank's charges (y), measured in dollars per month, for services rendered to local companies. One independent variable used to predict service charge to a company is the company's sales revenue (x), measured in $ million. Data for 21 companies who use the bank's services were used to fit the model E(y) = 0 + 1 x. The results of the simple linear regression are provided below. ^

y = 2,700 + 20x, s = 65, 2-tailed p-value = .064 (for testing 1 ) Interpret the p-value for testing whether 1 exceeds 0. A) For every $1 million increase in sales revenue (x), we expect a service charge (y) to increase $.064. B) There is sufficient evidence (at = .05) to conclude that service charge (y) is positively linearly related to sales revenue (x) . C) Sales revenue (x) is a poor predictor of service charge (y). D) There is insufficient evidence (at = .05) to conclude that service charge (y) is positively linearly related to sales revenue (x).

104) An academic advisor wants to predict the typical starting salary of a graduate at a top business school 104) using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. ^

0 = -92040

1 = 228 s = 3213 df = 23 t = 6.67

Set up the null and alternative hypotheses for testing whether a linear relationship exists between SALARY and GMAT.

A) H0 : 1 = 0 vs. Ha : 1 0

B) H0 : 1 > 0 vs. Ha : 1 < 0

C) H0 : 1 = 0 vs. Ha : 1 > 0

D) H0 : 1 = 228 vs. Ha : 1 > 228

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

105) For the situation above, write the equation of the probabilistic model of interest. A) Salary = 0 + 1 (GMAT) + B) GMAT = 0 + 1 (SALARY) + C) GMAT = 0 + 1 (SALARY)

105)

D) Salary = 0 + 1 (GMAT) ^

106) For the situation above, give a practical interpretation of 0 = -92040. A) We estimate the base SALARY of graduates of a top business school to be -$92,040. B) We expect to predict SALARY to within 2(92040) = $184,080 of its true value using GMAT in a straight-line model. C) The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the sample data. D) We estimate SALARY to decrease $92,040 for every 1-point increase in GMAT.

106)

Solve the problem. 107) Consider the following model y = 0 + 1 x + , where y is the daily rate of return of a stock, and x is the daily rate of return of the stock market as a whole, measured by the daily rate of return of Standard & Poor's (S&P) 500 Composite Index. Using a random sample of n = 12 days from 2007, the least squares lines shown in the table below were obtained for four firms. The estimated

107)

standard error of 1 is shown to the right of each least squares prediction equation. Firm

Estimated Market Model

Company A Company B Company C Company D

y = .0010 + 1.40x y = .0005 - 1.21x y = .0010 + 1.62x y = .0013 + .76x

Estimated Standard Error of 1 .03 .06 1.34 .15

Calculate the test statistic for determining whether the market model is useful for predicting daily rate of return of Company A's stock. A) 1.40 B) 161.6 C) 1.40 ± .067 D) 46.7

108) A large national bank charges local companies for using its services. A bank official reported the results of a regression analysis designed to predict the bank's charges (y), measured in dollars per month, for services rendered to local companies. One independent variable used to predict the service charge to a company is the company's sales revenue (x), measured in $ million. Data for 21 companies who use the bank's services were used to fit the model E(y) = 0 + 1 x. The results of the simple linear regression are provided below. ^

y = 2,700 + 20x Interpret the estimate of 0 , the y-intercept of the line. A) About 95% of the observed service charges fall within $2,700 of the least squares line. B) All companies will be charged at least $2,700 by the bank. C) There is no practical interpretation since a sales revenue of $0 is a nonsensical value. D) For every $1 million increase in sales revenue, we expect a service charge to increase $2,700.

108)

109) An academic advisor wants to predict the typical starting salary of a graduate at a top business school 109) using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. ^

0 = -92040

1 = 228 s = 3213 r2 = .66 r = .81 df = 23 t = 6.67

A 95% prediction interval for SALARY when GMAT = 600 is approximately ($37,915, $51,984). Interpret this interval. A) We are 95% confident that the SALARY of a top business school graduate will fall between $37,915 and $51,984. B) We are 95% confident that the SALARY of a top business school graduate with a GMAT of 600 will fall between $37,915 and $51,984. C) We are 95% confident that the mean SALARY of all top business school graduates with GMATs of 600 will fall between $37,915 and $51,984. D) We are 95% confident that the increase in SALARY for a 600-point increase in GMAT will fall between $37,915 and $51,984.

110) The dean of the Business School at a small Florida college wishes to determine whether the grade-point average (GPA) of a graduating student can be used to predict the graduate's starting salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting salaries. Records for 23 of last year's Business School graduates are selected at random, and data on GPA (x) and starting salary (y, in $thousands) for each graduate were used to fit the model E(y) = 0 + 1 x The results of the simple linear regression are provided below. ^

y = 4.25 + 2.75x,

SSxy = 5.15, SSxx = 1.87 SSyy = 15.17, SSE = 1.0075

Calculate the value of r2 , the coefficient of determination. A) 0.339 B) 0.872 C) 0.661

D) 0.934

110)

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

School Harvard Stanford Penn Northwestern MIT Chicago Duke Dartmouth Virginia Michigan Columbia Cornell CMU UNC Cal-Berkeley UCLA Texas Indiana NYU Purdue USC Pittsburgh Georgetown Maryland Rochester

GMAT 644 665 644 640 650 632 630 649 630 620 635 648 630 625 634 640 612 600 610 595 610 605 617 593 605

Acc. Rate 15.0% 10.2 19.4 22.6 21.3 30.0 18.2 13.4 23.0 32.4 37.1 14.9 31.2 15.4 24.7 20.7 28.1 29.0 35.0 26.8 31.9 33.0 31.7 28.1 35.9

Salary $63,000 60,000 55,000 54,000 57,000 55,269 53,300 52,000 55,269 53.300 52,000 50,700 52,050 50,800 50,000 51,494 43,985 44,119 53,161 43,500 49,080 43,500 45,156 42,925 44,499

111) For the situation above, give a practical interpretation of t = 6.67. A) There is evidence (at = .05) of at least a positive linear relationship between SALARY and GMAT. B) There is evidence (at = .05) to indicate that 1 = 0. C) We estimate SALARY to increase $6.67 for every 1-point increase in GMAT. D) Only 6.67% of the sample variation in SALARY can be explained by using GMAT in a straight-line model.

111)

Graph the line that passes through the given points. 112) (-2, 0) and (6, -1)

112)

Solve the problem. 113) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 113) in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: E(y) = 0 + 1 x, where y = appraised value of the house (in thousands of dollars) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were obtained: ^

y = 74.80 + 19.72x Give a practical interpretation of the estimate of the y-intercept of the least squares line. A) For each additional room in the house, we estimate the appraised value to increase $19,720. B) There is no practical interpretation, since a house with 0 rooms is nonsensical. C) For each additional room in the house, we estimate the appraised value to increase $74,800. D) We estimate the base appraised value for any house to be $74,800.

Answer Key Testname: CHAPTER 11 1) C 2) D 3) A 4) C 5) D 6) C 7) C 8) D 9) D 10) A 11) A 12) C 13) D 14) B 15) D 16) D 17) B 18) C 19) A 20) C 21) A 22) B 23) B 24) C 25) B 26) C 27) D

SSxy 25 1 = SS = 10.5 xx

28) ^

2.3810

0 = y - 1 x = 31 - 2.3810(2.75) = 24.4523 ^

The least squares prediction equation is y = 24.4523 + 2.3810x

29) a.

The regression line is y = -43.94 + .0005583x, so for x = 150,000 we have ^

y = -43.94 + .0005583(150,000) = 39.805. For df = 8, t.05 = 1.860.

s = 22.58236; x = 171,140; SSxx = 1.8329 x 1010

The interval is 39.805 ± 14.813. b. The interval is 39.805 ± 44.539. 30) a. t = 6.769 > t.05 = 1.717, reject H0; we concluded that x and y are positively linearly related.

b. (0.657, 1.103); We can be 90% confident that the true slope is between 0.657 and 1.103. 31) a. s2 = 0.027; b. 0.329

32) a. H0 : 1 > 0. b. There is a weak negative correlation between years of teaching experience and success in team-teaching mathematics classes. c. No; t = -4.299 .040741 .9657; 33) SSyy = 1.188; SSE = .040741; r2 = 1 1.188 96.57% of the sample variation in y values can be attributed to the linear relationship between x and y. 56

Answer Key Testname: CHAPTER 11

34) SSxx =

x2 -

SSxy =

xy -

y n x n

n x

= .948622 -

(3.642)2 = .00118171 14

= 295.54 -

(3.642)(1,134) = .538 14

1,134 = 81 14

3.642 = .26014 14

SSxy .538 = = 455.27 SSxx .00118171

0 = y - 1x = 81 - 455.27(.26014) = -37.434

35)

The least squares equation is y = -37.434 + 455.27x.

0 slope: 3; y-intercept: 0

Answer Key Testname: CHAPTER 11

36)

SSxy 236.5 = = .01087 SSxx 21,752

We test: H0 : 1 = 0 Ha : 1 0 ^

The test statistic is t =

1- 0 .01087 - 0 = = .814 s/ SSxx 1.97/ 21,752

The rejection region requires /2 = .05/2 = .025 in both tails of the t distribution with n - 2 = 7 - 2 = 5 df. From a t table, t.025 = 2.571. The rejection region is t > 2.571 or t < -2.571. Since the observed value of the test statistic does not fall in the rejection region (t = .814 2.571), H0 cannot be rejected. There

is insufficient evidence to indicate that the gestation period and the length of life of a horse are linearly related at = .05. SSE .678 .0339 = 37) a. s2 = n - 2 22 - 2 b. s = s2 .0339 .1841 c. 2s = 2(.1841) = .3682 38) a. E(y) = 0 + 1 x ^

b. y = 0 + 1x = 184.8 - .8195x c. We would expect approximately 172 grunts after feeding a warthog that was just born. However, since the value 0 is outside the range of the original data set, this estimate is highly unreliable. d. For each additional day, we estimate the number of grunts will decrease by .8195. ^ 4 = 49 ± .96 39) 1 ± t.05 s ^ = 49 ± 1.771 1 55 ^

40) 1 =

SSxy 236.5 = = .01087 SSxx 21,752 ^

SSE = SSyy - 1 SSxy = 22 - .01087(236.5) = 19.4286 SSE 19.4286 = = 3.8857 s2 = n- 2 7 -2

s2 =

3.8857 = 1.971

Answer Key Testname: CHAPTER 11 41) a.

b. y increases as x increases in an approximately linear pattern. c. 1 1.7368; 0 1.1053 d. The line appears to fit the data well. 42) a.

The scattergram does suggest that y is positively linearly related to x. b. SSxy = 45.1 ^

3.52 1 = 10.8

(21)(9.9) (21)2 = 3.52; SSxx = 99 = 10.8; 5 5

.3259; s = .1165

The test statistic is t =

.3259 .1165/ 10.8

9.19.

Based on 3 degrees of freedom, the rejection region is t > 2.353. Since the test statistic falls in the rejection region, we reject the null hypothesis and conclude that y is positively linearly related to x.

Answer Key Testname: CHAPTER 11 43)

0 slope: .5; y-intercept: 1.5 44) a.

b. c. d.

45) r = 46) r2 =

y = 10.6187 = .8515x SSE = 1.35677, s2 = .4523, and s = .6725 100% SSxy

SSxx · SSyy

25 = .729 (10.5)(112)

SSyy - SSE 112 - 52.4762 = = .53146 SSyy 112

47) r =.792; There is a positive linear correlation between age and number of grunts.

Answer Key Testname: CHAPTER 11

48) For x = 5, y = -.25 + 1.315(5) = 6.325 The confidence interval is of the form: ^ 1 (x - x)2 + y ± t /2s n SSxx Confidence coefficient .90 = 1 -

= 1 - .90 = .10. /2 = .10/2 = .05. From a t table, t.05 = 2.015 with n - 2 = 7 - 2 = 5 df.

The confidence interval is:

1 (5 - 5.8571)2 + 7 94.8571

6.325 ± 2.015(1.577)

6.325 ± 1.233

(5.092, 7.558)

49) r2 = .627; 62.7% of the variation in number of grunts can be explained by using age in a linear model. ^

50) SSE = SSyy - 1 SSxy SSxy =

xy -

SSyy =

y2 -

n y n

= 295.54 -

(3.642)(1,134) = .538 14

= 93,110 -

(1,134)2 = 1,256 14

SSE = 1,256 - 455.27(.538) = 1,011.06 SSE 1011.06 s2 = = = 84.26 n- 2 14 - 2 s=

s2 =

84.26 = 9.179

We expect most of the sample number of games won, y, to fall within 2s 2(9.179) values. 51) a. xi yi x iy i xi 2 2 5 3 8 xi = 18

Totals

SSxy = 28 -12.5 21

3 2 4 0 yi = 9

4 25 9 64 xi 2 = 102

(18)(9) 182 = -12.5; SSxx = 102 = 21; 4 4 18 9 = 4.5; y = = 2.25; 4 4

0 = 2.25 + .5952(4.5) = 4.9284

-.5952; x =

y = 4.9284 + .5952x

6 10 12 0 xi yi = 28

18.358 of their least squares predicted

Answer Key Testname: CHAPTER 11

52) SSyy = 950 -

^ 652 = 781, SSE = SSyy - 1SSxy = 781 - .2(3000) = 181; 25

SSE 181 s2 = = n - 2 25 - 2

7.87

53)

0 slope: -2; y-intercept: 4 ^

1 = -.08365. For every 1 mile per hour increase in the maximum speed attained, we estimate the elapsed 0-to-60 acceleration time to decrease by .08365 second. 55) To determine if time spent taking telephone orders during the day is positively linearly related with the number of telephone orders received during the day, we test:

54)

H0 : 1 = 0 Ha : 1 > 0 The test statistic is given on the printout as t = 9.96. The p-value for the desired test is p = .0000/2 = .0000 (divided in half because a one-tailed test is desired.) Since

= .01 > p-value 0, H0 is rejected. There is sufficient evidence to indicate that the time spent taking telephone orders during the day is positively linearly related with the number of telephone orders received during the day.

Answer Key Testname: CHAPTER 11 56) a.

y = 2.7273 + .8182x

For x = 1, y = 2.7273 + .8182(1) = 3.5455. For 3 degrees of freedom, t.005 = 5.841.

s = .8528; x = 1.8; SSxx = 30.8 The interval is 3.5455 ± 5.841(.8528)

1 (1 - 1.8)2 + 5 30.8

3.5455 ± 2.3405.

The interval is 3.5455 ± 5.841(.8528)

1 (1 - 1.8)2 + 5 30.8

3.5455 ± 5.5037.

57) a.

SSE = 1650.36; s2 = 235.77; s = 15.35 ^

We expect most of the observed y values to lie within 30.70 units of their respective least squares predicted values, y. SSyy - SSE 176.5171 - 12.435 = = .9296 58) r2 = SSyy 176.5171 92.96% of the variation in the sample monthly sales values can be explained by using months on the job in a linear model. 59) The assumptions necessary are: 1. The random errors are normally distributed. 2. The random errors are independent of one another. 3. The mean of the random errors is 0. 4. The variance of the random errors, 2 , is constant for all levels of the independent variable x.

11 1 , 1= 60) 0 = 2 2

61) Assumption 1: The mean of the probability distribution of is 0. Assumption 2: The variance of the probability distribution of is constant for all settings of the independent variable, x. Assumption 3: The probability distribution of is normal. Assumption 4: The values of associated with any two observed values of y are independent.

Answer Key Testname: CHAPTER 11 62) a.

SSxy = 10,232,660; SSxx 1.833 x 1010;

1 =.0005583; x = 171,140;

y = 51.6; 0 = 51.6 - .0005583(171,140) = -43.94 ^

y = 43.94 + .0005583x c.

The test statistic is t =

.0005583 22.58/ 1.833 x 1010

3.35.

Based on 8 degrees of freedom, the rejection region is t > 1.860. Since the test statistic falls in the rejection region, we reject the null hypothesis and conclude that the number of days on the market, y, is positively linearly related to the asking price, x. 63) There is a weak negative correlation between years of teaching experience and success in team-teaching mathematics classes. ^ 4 = 49 ± 1.17 64) 1 ± t.025 s ^ = 49 ± 2.160 1 55

65) The prediction interval is of the form: ^ 1 (x - x)2 y ± t /2s 1 + + n SSxx ^

y = 18.89 + .01087(300) = 22.151 Confidence coefficient .95 = 1 -

= 1 - .95 = .05.

/2 = .05/2 = .025. From a t table, t.025 = 2.571 with

n - 2 = 7 - 2 = 5 df. The 95% prediction interval is: 1 (300 - 332)2 22.151 ± 2.571(2) 1 + + 22.151 ± 5.609 7 21,752

(16.542, 27.760)

Answer Key Testname: CHAPTER 11 66) We test:

H0 : 1 = 0 Ha : 1 > 0 ^

The test statistic is t =

SSxx =

x2 -

1- 0 . s/ SSxx

2 = .948622 -

(3.642)2 = .00118 14

455.27 - 0 = 1.704 9.18/ .00118

The rejection region requires = .05 in the upper tail of the t distribution with df = n - 2 = 14 - 2 = 12. From a t table, t.05 = 1.782. The rejection region is t > 1.782. Since the observed value of the test statistic does not fall in the rejection region (t = 1.704 1.782), H0 cannot be rejected. There is insufficient evidence to indicate that team wins is positively linearly related with team batting average. ^ SSE 52 1.857 = 67) SSE = SSyy - 1SSxy = 100 - .8(60) = 52; s2 = n - 2 30 - 2

68) SSxy = 45.1 -

(21)(9.9) (21)2 = 3.52; SSxx = 99 = 10.8; 5 5

SSyy = 20.79 -

(9.9)2 = 1.188; r = 5

3.52 10.8 · 1.188

.9827;

There is a strong linear relationship between x and y. 69) B 70) A 71) B 72) D 73) B 74) D 75) D 76) B 77) B 78) B 79) B 80) C 81) D 82) B 83) B 84) A 85) B 86) D 87) D 88) C 89) A 90) D

Answer Key Testname: CHAPTER 11 91) A 92) A 93) D 94) A 95) A 96) A 97) D 98) B 99) D 100) D 101) D 102) B 103) B 104) A 105) A 106) C 107) D 108) C 109) B 110) D 111) A 112) B 113) B

Chapter 12 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) It is desired to build a regression model to predict y = the sales price of a single family home, based 1) on the x1 = size of the house and x2 = the neighborhood the home is located in. The goal is to compare the prices of homes that are located in two different neighborhoods. The following model is proposed: E(y) = 0 + 1 x1 + 2 x2 A regression model was fit and the following residual plot was observed.

Which of the following assumptions appears violated based on this plot? A) The variance of the errors is constant B) The mean of the errors is zero C) The errors are normally distributed D) The errors are independent

2) In regression, it is desired to predict the dependent variable based on values of other related independent variables. Occasionally, there are relationships that exist between the independent variables. Which of the following multiple regression pitfalls does this example describe? A) Multicollinearity B) Stepwise Regression C) Extrapolation D) Estimability

3) It is dangerous to predict outside the range of the data collected in a regression analysis. For instance, we shouldn't predict the price of a 5000 square foot home if all our sample homes were smaller than 4500 square feet. Which of the following multiple regression pitfalls does this example describe? A) Extrapolation B) Multicollinearity C) Stepwise Regression D) Estimability

4) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 4) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Gmat Tuition

Coefficient Std Error T 51.6573 -203.402 -3.94 0.39412 0.09039 4.36 0.92012 0.17875 5.15

P VIF 0.0002 0.0 0.0000 2.0 0.0000 2.0

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the tuition charged by the MBA program was $75,000 and the GMAT score was 675. The results are shown here: 95% confidence interval for E(Y): ($126,610, $136,640) 95% prediction interval for Y: ($90,113, $173,160) Which of the following interpretations is correct if you want to use the model to estimate Y for a single MBA program? A) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition and has an average GMAT score of 675 will fall between $90,113 and $173,16,30. B) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition and have an average GMAT score of 675 will fall between $126,610 and $136,640. C) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition and has an average GMAT score of 675 will fall between $126,610 and $136,640. D) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition and have an average GMAT score of 675 will fall between $90,113 and $173,16,30.

5) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 5) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Tuition GMAT TxG

Coefficient Std Error T 165.406 4.16 -687.851 -11.3197 -5.15 2.19724 -0.96727 -3.79 0.25535 0.01850 0.00331 5.58

R-Squared Adjusted R-Squared Source Regression Residual Total

DF 3 71

0.7816 0.7723

P 0.0001 0.0000 0.0003 0.0000

Resid. Mean Square (MSE) Standard Deviation

SS MS 76523.8 25510.9 84.68 21388.8 301.3 74 97921.7

F 0.0000

301.251 17.3566 P

Cases Included 75 Missing Cases 0 One of the t-test test statistics is shown on the printout to be the value t = 5.58. Interpret this value. A) There is insufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. B) There is insufficient evidence, at = 0.05, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs. C) There is sufficient evidence, at = 0.05, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs. D) There is sufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.

6) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 6) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Tuition TxT

Coefficient Std Error T 169.910 26.5350 6.40 -3.37373 -4.16 0.81171 0.03563 0.00590 6.03

R-Squared Adjusted R-Squared Source Regression Residual Total

DF 2 72

0.7361 0.7288

P 0.0000 0.0001 0.0000

Resid. Mean Square (MSE) Standard Deviation

SS MS F 72081.8 36040.9 100.42 0.0000 25839.8 358.9 74 97921.7

358.887 18.9443 P

Cases Included 75 Missing Cases 0 The global-f test statistic is shown on the printout to be the value F = 100.42. Interpret this value. A) There is insufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. B) There is sufficient evidence, at = 0.05, to indicate that there is a curvilinear relationship between average starting salary of graduates of MBA programs and the tuition of the MBA program. C) There is sufficient evidence, at = 0.05, to indicate that there is a linear relationship between average starting salary of graduates of MBA programs and the tuition of the MBA program. D) There is sufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.

7) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 7) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Gmat Tuition

Coefficient Std Error T 51.6573 -203.402 -3.94 0.39412 0.09039 4.36 0.92012 0.17875 5.15

R-Squared Adjusted R-Squared

0.6857 0.6769

P VIF 0.0002 0.0 0.0000 2.0 0.0000 2.0

Resid. Mean Square (MSE) Standard Deviation

427.511 20.6763

Identify the test statistic that should be used to test to determine if the amount of tuition charged by a program is a useful predictor of the average starting salary of the graduates of the program. A) t = -3.94 B) t = 4.36 C) t = 20.67 D) t = 5.15

8) Consider the interaction model E(y) = 7 + 3x 1 - 4x2 + 5x 1x 2 . Find the slope of the line relating E(y) and x 1 when x 2 = 2.

A) 13

B) 16

C) 10

D) 1

9) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 9) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Gmat Tuition

Coefficient Std Error T 51.6573 -203.402 -3.94 0.39412 0.09039 4.36 0.92012 0.17875 5.15

P VIF 0.0002 0.0 0.0000 2.0 0.0000 2.0

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the tuition charged by the MBA program was $75,000 and the GMAT score was 675. The results are shown here: 95% confidence interval for E(Y): ($126,610, $136,640) 95% prediction interval for Y: ($90,113, $173,160) Which of the following interpretations is correct if you want to use the model to estimate E(Y) for all MBA programs? A) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition and has an average GMAT score of 675 will fall between $126,610 and $136,640. B) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition and have an average GMAT score of 675 will fall between $126,610 and $136,640. C) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition and have an average GMAT score of 675 will fall between $90,113 and $173,16,30. D) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition and has an average GMAT score of 675 will fall between $90,113 and $173,16,30.

10) During its manufacture, a product is subjected to four different tests in sequential order. An efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score (y), as a function of Test1 score (x 1 ), Test 2 score (x 2 ), and Test3 score (x 3 ). [Note: All test

10)

scores range from 200 to 800, with higher scores indicative of a higher quality product.] Consider the model: E(y) = 1 + 1 x1 + 2 x 2 + 3 x 3 The first-order model was fit to the data for each of 12 units sampled from the production line. The results are summarized in the printout. _____________________________________________________________________ SOURCE

F VALUE

PROB > F

MODEL ERROR TOTAL

3 8 12

151417 22231 173648

50472 2779

18.16

.0075

ROOT MSE DEP MEAN

52.72 645.8

R-SQUARE ADJ R-SQ

0.872 0.824

VARIABLE

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X1(TEST1) X2(TEST2) X3(TEST3)

11.98 0.2745 0.3762 0.3265

80.50 0.1111 0.0986 0.0808

0.15 2.47 3.82 4.04

0.885 0.039 0.005 0.004

Suppose the 95% confidence interval for 3 is (.15, .47). Which of the following statements is incorrect? A) At = .05, there is insufficient evidence to reject H0 : 3 = 0 in favor of Ha : 3 0.

B) We are 95% confident that the estimated slope for the Test4-Test3 line falls between .15 and .47 holding Test1 and Test2 fixed. C) We are 95% confident that the increase in Test4 score for every 1-point increase in Test3 score falls between .15 and .47, holding Test1 and Test2 fixed. D) We are 95% confident that the Test3 is a useful linear predictor of Test4 score, holding Test1 and Test2 fixed.

11) Consider the model y = 0 + 1 x 1 + 2x 2 + 3 x3 + where x 1 is a quantitative variable and x 2 and x 3 are dummy variables describing a qualitative variable at three levels using the coding scheme 1 0

x2 =

if level 2 otherwise

x3 =

1 0

if level 3 otherwise ^

The resulting least squares prediction equation is y = 16.3 + 2.3x 1 + 3.5x 2 + 18x 3 . What is the response line (equation) for E(y) when x 2 = 0 and x 3 = 1? ^

A) y = 16.3 + 2.3x 1

B) y = 18.1 + 2.3x 1

C) y = 18.6 + 2.3x 1

D) y = 16.3 + 4.1x 1

11)

12) Which equation represents a complete second-order model for two quantitative independent variables? A) E(y) = 0 + 1 x 1 x 2 +

12)

2 2 2x 1 + 3x 2

2 2 B) E(y) = 0 + 1 x 1 + 2 x 2 +

2 3 x 1 x2 +

C) E(y) = 0 + 1 x 1 +

2x 2 +

2 2 3x 1 + 4x 2

D) E(y) = 0 + 1 x 1 +

2x 2 +

3x 1 x 2 +

2 2 2 4x1 x 2 + 5 x 1 x 2

2 2 4x 1 + 5x 2

13) Retail price data for n = 60 hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive:

13)

y = Retail PRICE (measured in dollars) x 1 = Microprocessor SPEED (measured in megahertz)

(Values in sample range from 10 to 40) x 2 = CHIP size (measured in computer processing units) (Values in sample range from 286 to 486)

A first-order regression model was fit to the data. Part of the printout follows: Parameter Estimates

VARIABLE DF

PARAMETERSTANDARD T FOR 0: ESTIMATE ERROR PARAMETER = 0PROB > |T|

INTERCEPT 1 SPEED 1 CHIP 1

-373.526392 1258.1243396 -0.297 104.838940 22.36298195 4.688 3.571850 3.89422935 0.917

0.7676 0.0001 0.3629

Identify and interpret the estimate for the SPEED -coefficient,

1 = 105; For every 1-megahertz increase in SPEED, we estimate PRICE (y) to increase $105, holding CHIP fixed.

1 = 105; For every $1 increase in PRICE, we estimate SPEED to increase 105 megahertz, holding CHIP fixed.

1 = 3.57; For every $1 increase in PRICE, we estimate SPPED to increase by about 4 megahertz, holding CHIP fixed.

1 = 3.57; For every 1-megahertz increase in SPEED, we estimate PRICE to increase $3,57, holding CHIP fixed.

14) Consider the interaction model E(y) = 3.6+ 1.2x 1 + 2.4x2 + .2x 1 x 2 . Determine the change in E(y) when x 1 is changed from 6 to 7 and x 2 is held fixed at 3.

A) 1.8

B) 11.4

C) 4.2

D) 10.8

14)

15) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 15) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Tuition GMAT TxG

Coefficient Std Error T 165.406 4.16 -687.851 -11.3197 -5.15 2.19724 -0.96727 -3.79 0.25535 0.01850 0.00331 5.58

R-Squared Adjusted R-Squared Source Regression Residual Total

DF 3 71

0.7816 0.7723

P 0.0001 0.0000 0.0003 0.0000

Resid. Mean Square (MSE) Standard Deviation

SS MS 76523.8 25510.9 84.68 21388.8 301.3 74 97921.7

F 0.0000

301.251 17.3566 P

Cases Included 75 Missing Cases 0 The global-f test statistic is shown on the printout to be the value F = 84.68. Interpret this value. A) There is insufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. B) There is sufficient evidence, at = 0.05, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs. C) There is sufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. D) There is insufficient evidence, at = 0.05, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs.

16) It is desired to build a regression model to predict y = the sales price of a single family home, based on the x1 = size of the house and x2 = the neighborhood the home is located in. The goal is to compare the prices of homes that are located in two different neighborhoods. The following complete 2nd-order model is proposed: E(y) = 0 + 1 x1 + 2 x1 2 + 3 x2 + 4 x1 x2 + 5 x1 2 x2 .

What hypothesis should be tested to determine if the quadratic terms are necessary to predict the sales price of a home? A) H0 : 1 = 2 = 3 = 0 B) H0 : 2 = 5 = 0

C) H0 : 1 = 2 = 3 = 4 = 5 = 0

D) H0 : 1 = 3 = 4 = 0

16)

17) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 17) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Gmat Tuition Source Regression Residual Total

Coefficient Std Error T 51.6573 -203.402 -3.94 0.39412 0.09039 4.36 0.92012 0.17875 5.15 DF 2 72

SS MS 67140.9 33570.5 78.53 30780.8 427.5 74 97921.7

P VIF 0.0002 0.0 0.0000 2.0 0.0000 2.0 F 0.0000

Interpret the p-value for the global f-test shown on the printout. A) At = 0.05, there is sufficient evidence to indicate that the average GMAT score of the MBA program's students is useful for predicting the average starting salary of the graduates of an MBA program. B) At = 0.05, there is insufficient evidence to indicate that the average GMAT score of the MBA program's students is useful for predicting the average starting salary of the graduates of an MBA program. C) At = 0.05, there is sufficient evidence to indicate that something in the regression model is useful for predicting the average starting salary of the graduates of an MBA program. D) At = 0.05, there is insufficient evidence to indicate that something in the regression model is useful for predicting the average starting salary of the graduates of an MBA program.

18) A public health researcher wants to use regression to predict the sun safety knowledge of 18) pre-school children. The researcher randomly sampled 35 preschoolers, assigned them to one of two groups, and then measured the following three variables: SUNSCORE:y = Score on sun-safety comprehension test READING: x1 = Reading comprehension score GROUP:

x2 = 1 if child received a Be Sun Safe demonstration, 0 if not

A regression model was fit and the following residual plot was observed.

Which of the following assumptions appears violated based on this plot? A) The mean of the errors is zero B) The errors are independent C) The variance of the errors is constant D) The errors are normally distributed

19) There are four independent variables, x 1 , x 2 , x3 , and x 4 , that might be useful in predicting a

response y. A total of n = 40 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to useful. The computer fits all possible one-variable models of the form E(y) = 0 + 1 x i , i = 1, 2, 3, 4. The information in the table is provided from the computer printout. Variable X1 X2 X3 X4

2.4 -0.2 3.6 0.8

s 0.52 0.03 2.11 0.44

Which independent variable is declared the best one-variable predictor of y? A) x 1 B) x 2 C) x 3

D) x 4

19)

20) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 20) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Tuition TxT

Coefficient Std Error T 169.910 26.5350 6.40 -3.37373 -4.16 0.81171 0.03563 0.00590 6.03

R-Squared Adjusted R-Squared Source Regression Residual Total

DF 2 72

0.7361 0.7288

P 0.0000 0.0001 0.0000

Resid. Mean Square (MSE) Standard Deviation

SS MS F 72081.8 36040.9 100.42 0.0000 25839.8 358.9 74 97921.7

358.887 18.9443 P

Cases Included 75 Missing Cases 0 One of the t-test test statistics is shown on the printout to be the value t = 6.03. Interpret this value. A) There is sufficient evidence, at = 0.05, to indicate that there is a linear relationship between average starting salary of graduates of MBA programs and the tuition of the MBA program. B) There is sufficient evidence, at = 0.05, to indicate that there is a curvilinear relationship between average starting salary of graduates of MBA programs and the tuition of the MBA program. C) There is insufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. D) There is sufficient evidence, at = 0.05, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs.

21) It is desired to build a regression model to predict y = the sales price of a single family home, based on the neighborhood the home is located in. The goal is to compare the prices of homes that are located in four different neighborhoods. Which regression model should be built? A) E(y) = 0 + 1 x1 + 2 x2 + 3 x3 + 4 x4 , where x1 - x4 are qualitative variables that describe the four neighborhoods. B) E(y) = 0 + 1 x1, where x1 is a qualitative variable that describes the four neighborhoods.

C) E(y) = 0 + 1 x1 + 2 x2 + 3 x3, where x1 - x3 are qualitative variables that describe the four neighborhoods. D) E(y) = 0 + 1 x1 + 2 x1 2 , where x1 is a qualitative variable that describes the four neighborhoods.

21)

22) A public health researcher wants to use regression to predict the sun safety knowledge of 22) pre-school children. The researcher randomly sampled 35 preschoolers, assigned them to one of two groups, and then measured the following three variables: SUNSCORE:y = Score on sun-safety comprehension test READING: x1 = Reading comprehension score GROUP:

x2 = 1 if child received a Be Sun Safe demonstration, 0 if not

The following two models were hypothesized: Model 1: E(y) = 0 + 1 x1 + 2 x1 2 + 3 x2 + 4x1 x2 + 5 x1 2 x2 Model 2: E(y) = 0 + 1 x1 + 3 x2 + 4 x1 x2 A partial f-test was conducted to compare the two models and the resulting p-value was found to be 0.0023. Fill in the blank. The results lead us to conclude that there is (at = 0.05). A) sufficient evidence of interaction between sun-safety score and reading score. B) sufficient evidence of a quadratic relationship between sun-safety score to reading score. C) sufficient evidence of a statistically useful model for sun-safety score. D) insufficient evidence of quadratic relationship between sun-safety score to reading score.

23) A graphing calculator was used to fit the model E(y) = 0 + 1 x + 2 x 2 to a set of data. The resulting screen is shown below.

Which number on the screen represents the estimator of

A) 5.5

2? C) .9286

23)

B) 11

D) .9405

24) Which of the following is not a possible indicator of multicollinearity? A) non-significant t-tests for individual parameters when the F-test for overall model adequacy is significant B) significant correlations between pairs of independent variables C) signs opposite from what is expected in the estimated parameters D) non-random patterns in the plot of the residuals versus the fitted values

24)

25) A study of the top MBA programs attempted to predict y = the average starting salary (in $1000's) of graduates of the program based on x = the amount of tuition (in $1000's) charged by the program. After first considering a simple linear model, it was decided that a quadratic model should be proposed. Which of the following models proposes a 2nd-order quadratic relationship between x and y? A) E(y) = 0 + 1 x1 + 2 x1 2 B) E(y) = 0 + 1 x1 + 2 x1 2 + 3 x1 3

25)

C) E(y) = 0 + 1 x1 + 2 x2 + 3 x1x2

D) E(y) = 0 + 1 x1

26) Consider the model

26)

y = 0 + 1 x 1 + 2x 1 2 + 3x 2 + 4 x3 + 5 x 1x 2 + 6 x 1x 3 + 7 x1 2 x 2 +

8 x 12 x 3 +

where x 1 is a quantitative variable and x 2 and x 3 are dummy variables describing a qualitative variable at three levels using the coding scheme 1 0

x2 =

if level 2 otherwise

x3 =

1 0

if level 3 otherwise

The resulting least squares prediction equation is ^ y = 8.8 - 1.1x 1 + 3.2x 1 2 + 1.6x 2 - 4.4x 3 + .02x 1 x 2 + 1.3x 1 x 3 + .01x 1 2 x 2 - .06x 1 2 x 3

What is the equation of the response curve for E(y) when x2 = 0 and x 3 = 0? ^

B) y = 8.8 - 1.3x 1 + 3.2x1 2

D) y = 8.8 - 1.6x 2 - 4.4x3

A) y = 8.8 - .22x 1 + 3.15x 1 2

C) y = 8.8 - 1.1x 1 + 3.2x1 2

27) It is desired to build a regression model to predict y = the sales price of a single family home, based on the x1 = size of the house and x2 = the neighborhood the home is located in. The goal is to

27)

compare the prices of homes that are located in two different neighborhoods. A complete 2nd-order model is proposed. Which regression model proposes the complete 2nd-order model? A) E(y) = 0 + 1 x1 + 2 x1 2 + 3 x2 + 4 x1 x2 + 3 x1 2 x2

B) E(y) = 0 + 1 x1 + 2 x2 + 3 x1x2 C) E(y) = 0 + 1 x1 + 2 x2 D) E(y) = 0 + 1 x1 + 2 x1 2 + 3 x2 + 4 x2 2

28) Consider the second-order model

28)

^ 2 2 y = -3.24 + 1.12x 1 + 2.57x 2 - 3.22x 1 x 2 + 5.78 x 1 = 4.69 x 2 ^

If x 2 is held fixed at x 2 = 3, describe the relationship between y and x 1 . ^

A) The relationship between y and x1 is quadratic with downward concavity. ^

B) The relationship between y and x1 is linear with positive slope. ^

C) The relationship between y and x1 is quadratic with upward concavity. ^

D) The relationship between y and x1 is linear with negative slope.

29) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 29) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Gmat Tuition

Coefficient Std Error T 51.6573 -203.402 -3.94 0.39412 0.09039 4.36 0.92012 0.17875 5.15

R-Squared Adjusted R-Squared

0.6857 0.6769

P VIF 0.0002 0.0 0.0000 2.0 0.0000 2.0

Resid. Mean Square (MSE) Standard Deviation

427.511 20.6763

Interpret the coefficient for the tuition variable shown on the printout. A) For every $1000 increase in the average starting salary, we estimate that the tuition charged by the MBA program will increase by $920.12. B) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will decrease by $203,402, holding the GMAT score constant. C) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $394.12, holding the GMAT score constant D) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $920.12, holding the GMAT score constant

Answer the question True or False. 30) The confidence interval for the mean E(y) is narrower that the prediction interval for y. A) True B) False

30)

Solve the problem. 31) A study of the top MBA programs attempted to predict the average starting salary (in $1000's) of 31) graduates of the program based on the amount of tuition (in $1000's) charged by the program and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: Least Squares Linear Regression of Salary Predictor Variables Constant Gmat Tuition

Coefficient Std Error T 51.6573 -203.402 -3.94 0.39412 0.09039 4.36 0.92012 0.17875 5.15

R-Squared Adjusted R-Squared Source Regression Residual Total

DF 2 72

0.6857 0.6769

P VIF 0.0002 0.0 0.0000 2.0 0.0000 2.0

Resid. Mean Square (MSE) Standard Deviation

SS MS 67140.9 33570.5 78.53 30780.8 427.5 74 97921.7

427.511 20.6763

F 0.0000

Interpret the coefficient of determination value shown in the printout. A) We expect most of the average starting salaries to fall within $41,353 of their least squares predicted values. B) At = 0.05, there is insufficient evidence to indicate that something in the regression model is useful for predicting the average starting salary of the graduates of an MBA program. C) We can explain 68.57% of the variation in the average starting salaries around their mean using the model that includes the average GMAT score and the tuition for the MBA program. D) We expect most of the average starting salaries to fall within $20,676 of their least squares predicted values.

32) What relationship between x and y is suggested by the scattergram?

A) a quadratic relationship with upward concavity B) a linear relationship with positive slope C) a linear relationship with negative slope D) a quadratic relationship with downward concavity

32)

33) Consider the model y = 0 + 1 x 1 + 2x 2 + 3 x3 + where x 1 is a quantitative variable and x 2 and x 3 are dummy variables describing a qualitative variable at three levels using the coding scheme 1 0

x2 =

if level 2 otherwise

x3 =

1 0

33)

if level 3 otherwise ^

The resulting least squares prediction equation is y = 36.7 + 1.3x 1 + 5.4x 2 + 3.2x 3 . What is the least squares regression equation associated with level 2? ^

A) y = 39.9 + 5.4x 2

B) y = 38.0 + 5.4x 2

C) y = 39.9 + 1.3x 1

D) y = 42.1 + 1.3x 1

34) We decide to conduct a multiple regression analysis to predict the attendance at a major league baseball game. We use the size of the stadium as a quantitative independent variable and the type of game as a qualitative variable (with two levels - day game or night game). We hypothesize the following model: E(y) = 0 + 1 x1 + 2 x2 + 3 x3 Where x1 = size of the stadium

x2 = 1 if a day game, 0 if a night game

A plot of the y-x1 relationship would show:

A) Two non-parallel curves C) Two non-parallel lines

B) Two parallel lines D) Two parallel curves

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 35) A college admissions officer proposes to use regression to model a student's college GPA at graduation in terms of the following two variables:

35)

x 1 = high school GPA x 2 = SAT score

34)

36) Consider the data given in the table below. X 1 2 2 3 3 4 4 4 5 5 6

36)

Y 7 6 5 5 4 4 3 2 4 5 6

Plot the data on a scattergram. Does a second-order model seem to be a good fit for the data? Explain.

37) A statistics professor gave three quizzes leading up to the first test in his class. The quiz grades37) and test grade for each of eight students are given in the table.

The professor would like to use the data to find a first-order model that he might use to predict a student's grade on the first test using that student's grades on the first three quizzes. a. b. c.

Identify the dependent and independent variables for the model. What is the least squares prediction equation? Find the SSE and the estimator of 2 for the model.

38) Consider the partial printout for an interaction regression analysis of the relationship between a dependent variable y and two independent variables x 1 and x 2 .

a. b. c. d.

Write the prediction equation for the interaction model. Test the overall utility of the interaction model using the global F-test at Test the hypothesis (at = .05) that x 1 and x 2 interact positively.

38)

= .05.

Estimate the change in y for each additional 1-unit increase in x 1 when x2 = 6.

39) A statistics professor gave three quizzes leading up to the first test in his class. The quiz grades39) and test grade for each of eight students are given in the table.

The professor fit a first-order model to the data that he intends to use to predict a student’s grade on the first test using that student’s grades on the first three quizzes. Test the null hypothesis H0 : 1 = 2 = 3 = 0 against the alternative hypothesis Ha : at least one i 0. Use = .05. Interpret the result.

40) A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model E(y) = 0 + 1 x + 2 x 2 where y = Demand (in thousands) and x = Retail price per carat (dollars). This model was fit to data collected for a sample of 12 rare gems. A portion of the printout is given below: SOURCE

PR > F

Model Error TOTAL

2 9 11

115145 1388 116533

57573 154

373

.0001

Root MSE

12.42

R-Square

VARIABLES INTERPCEP X X ·X

.988

PARAMETER ESTIMATES STD. ERROR 286.42 -.31 .000067

T for HO: PARAMETER = 0

PR > |T|

29.64 -5.14 .95

.0001 .0006 .3647

9.66 .06 .00007

Does the quadratic term contribute useful information for predicting the demand for the gem? Use = .10.

40)

41) The concessions manager at a beachside park recorded the high temperature, the number of 41) people at the park, and the number of bottles of water sold for each of 12 consecutive Saturdays. The data are shown below.

Fit the model E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 1 x 2 to the data, letting y represent the number of bottles of water sold, x 1 the temperature, and x 2 the number of people at the

park. b. Identify at least two indicators of multicollinearity in the model. c. Comment on the usefulness of the model to predict the number of bottles of water sold on a Saturday when the high temperature is 103°F and there are 3500 people at the park.

42) Consider the partial printout below.

Is there evidence (at

42)

= .05) that x 1 and x2 interact? Explain.

43) The printout below shows part of the least squares regression analysis for the model E(y) = 0 + 1 x 1 + 2 x 2 fit to a set of data. The model attempts to predict a score on the final

43)

exam in a statistics course based on the scores on the first two tests in the class.

Is there evidence of multicollinearity in the printout? Explain.

44) In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked (y) per day by the clerical staff depends on the number of pieces of mail processed per day (x 1 ) and the number of checks cashed per day (x 2 ). Data collected for n = 20 working days were used to fit the model: E(y) = 0 + 1 x 1 + 2 x 2 A printout for the analysis follows: _____________________________________________________________________ Analysis of Variance SOURCE

F VALUE

PROB > F

MODEL ERROR C TOTAL

2 17 19

7089.06512 4541.72142 11630.78654

3544.53256 267.16008

13.267

0.0003

ROOT MSE DEP MEAN C.V.

16.34503 93.92682 17.40188

R-SQUARE ADJ R-SQ

0.6095 0.5636

Parameter Estimates

VARIABLE

INTERCEPT X1 X2

1 1 1

PARAMETER STANDARD T FOR 0: ESTIMATE ERROR PARAMETER = 0 PROB > |T| 114.420972 -0.007102 0.037290

18.68485744 0.00171375 0.02043937

6.124 -4.144 1.824

0.0001 0.0007 0.0857

44)

OBS 1

Actual Value

Predict Value Residual

7781 644

74.707

83.175

Lower 95% CL Upper 95% CL Predict Predict 47.224

-8.468

119.126

Test to determine if there is a positive linear relationship between the number of man-hours worked, y, and the number of checks cashed per day, x 2 . Use = .05.

45) The first-order model below was fit to a set of data.

45)

E(y) = 0 + 1x 1 + 2 x2 Explain how to determine if the constant variance assumption is satisfied.

46) Retail price data for n = 60 hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive: y = Retail PRICE (measured in dollars) x 1 = Microprocessor SPEED (measured in megahertz)

(Values in sample range from 10 to 40) x 2 = CHIP size (measured in computer processing units) (Values in sample range from 286 to 486)

A first-order regression model was fit to the data. Part of the printout follows: _____________________________________________________________________ Analysis of Variance SOURCE DF SS MS F VALUE PROB > F MODEL ERROR C TOTAL

2 57 59

ROOT MSE DEP MEAN C.V.

34593103.008 51840202.926 86432305.933

17296051.504 909477.24431

19.018

953.66516 3197.96667 29.82099

R-SQUARE ADJ R-SQ

0.4002 0.3792

0.0001

Test to determine if the model is adequate for predicting the price of a computer. Use

= .01.

46)

47) The model E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 3 + 4 x 4 was used to relate E(y) to a single qualitative variable, where x1 =

1 0

if level 2 if not

x2 =

1 0

if level 3 if not

x3 =

1 0

if level 4 if not

x4 =

1 0

if level 5 if not

47)

This model was fit to n = 40 data points and the following result was obtained: ^

y = 14.5 + 3x1 - 4x 2 + 10x 3 + 8x4 a. Use the least squares prediction equation to find the estimate of E(y) for each level of the qualitative variable. b. Specify the null and alternative hypothesis you would use to test whether E(y) is the same for all levels of the independent variable.

48) A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model

48)

E(y) = 0 + 1 x + 2 x 2 where y = Demand (in thousands) and x = Retail price per carat (dollars). This model was fit to data collected for a sample of 12 rare gems. A portion of the printout is given below: SOURCE

PR > F

Model Error TOTAL

2 9 11

115145 1388 116533

57573 154

373

.0001

Root MSE

12.42

R-Square

VARIABLES INTERPCEP X X ·X

.988

PARAMETER ESTIMATES STD. ERROR 286.42 -.31 .000067

T for HO: PARAMETER = 0

PR > |T|

29.64 -5.14 .95

.0001 .0006 .3647

9.66 .06 .00007

Is there sufficient evidence to indicate the model is useful for predicting the demand for the gem? Use = .01.

49) A college admissions officer proposes to use regression to model a student's college GPA at graduation in terms of the following two variables:

49)

x 1 = high school GPA x 2 = SAT score

The admissions officer believes the relationship between college GPA and high school GPA is linear and the relationship between SAT score and college GPA is linear. She also believes that the relationship between college GPA and high school GPA depends on the student's SAT score. She proposes the regression model: E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 1 x 2 Explain how to determine if the relationship between college GPA and SAT score depends on the high school GPA.

50) Retail price data for n = 60 hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive: y = Retail PRICE (measured in dollars) x 1 = Microprocessor SPEED (measured in megahertz)

(Values in sample range from 10 to 40) x 2 = CHIP size (measured in computer processing units) (Values in sample range from 286 to 486)

A first-order regression model. was fit to the data. Part of the printout follows: Parameter Estimates

VARIABLE DF

PARAMETERSTANDARD T FOR 0: ESTIMATE ERROR PARAMETER = 0PROB > |T|

INTERCEPT 1 SPEED 1 CHIP 1

-373.526392 1258.1243396 -0.297 104.838940 22.36298195 4.688 3.571850 3.89422935 0.917

Identify and interpret the estimate of 2 .

0.7676 0.0001 0.3629

50)

51) The table below shows data for n = 20 observations.

a. b. c. d. e.

51)

Use a first-order regression model to find a least squares prediction equation for the model. Find a 95% confidence interval for the coefficient of x 1 in your model. Interpret the result. Find a 95% confidence interval for the coefficient of x 2 in your model. Interpret the result. Find R2 and Ra2 and interpret these values.

Test the null hypothesis H0 : 1 = 2 = 0 against the alternative hypothesis Ha : at least one i 0. Use = .05. Interpret the result.

52) The table shows the profit y (in thousands of dollars) that a company made during a month when the price of its product was x dollars per unit. Profit, y 12 17 20 21 24 26 27 23 21 20 15 11 10 5

52)

Price, x 1.20 1.25 1.29 1.30 1.35 1.39 1.40 1.45 1.49 1.50 1.55 1.59 1.60 1.65

Fit the model y = 0 + 1 x + 2 x2 + to the data and give the least squares prediction equation. b. Plot the fitted equation on a scattergram of the data. c. Is there sufficient evidence of downward curvature in the relationship between profit and price? Use = .05.

53) Twenty colleges each recommended one of its graduating seniors for a prestigious graduate 53) fellowship. The process to determine which student will receive the fellowship includes several interviews. The gender of each student and his or her score on the first interview are shown below. Student 1 2 3 4 5 6 7 8 9 10

Gender Male Female Female Female Male Female Female Male Male Female

Score 18 17 19 16 12 15 18 16 18 20

Student 11 12 13 14 15 16 17 18 19 20

Gender Female Male Male Female Female Male Female Male Female Female

Score 17 16 16 19 16 15 12 14 16 18

a. Suppose you want to use gender to model the score on the interview y. Create the appropriate number of dummy variables for gender and write the model. b. Fit the model to the data. c. Give the null hypothesis for testing whether gender is a useful predictor of the score y. d. Conduct the test and give the appropriate conclusion. Use = .05.

54) Operations managers often use work sampling to estimate how much time workers spend 54) on each operation. Work sampling which involves observing workers at random points in time was applied to the staff of the catalog sales department of a clothing manufacturer. The department applied regression to the following data collected for 40 consecutive working days: TIME:

y = Time spent (in hours) taking telephone orders during the day

ORDERS:

x 1 = Number of telephone orders received during the day

WEEK:

x 2 = 1 weekday, 0 if Saturday or Sunday

Consider the complete 2nd-order model: E(y) = 0 + 1 x 1 + 2 (x1 )2 + 3 x 2 + 4 x 1 x 2 + 5 (x 1 )2 x 2 Explain how to conduct a test to determine if a quadratic relationship between total order time and the number of orders taken is necessary in the regression model above. Specify the null and alternative hypotheses that are to be tested.

55) The complete second-order model E(y) = 0 + 1 x 1 +

2 2 x 2 + 3 x 1 x2 + 4 x 1 +

2 5x 2

55)

was fit to n = 25 data points. The printout is shown below.

a. Write the complete second-order model for the data. b. Is there sufficient evidence to indicate that at least one of the parameters 1 , 2 , 3 , 4 , and 5 is nonzero? Test using = .05. c.

Test H0 : 3 = 0 against Ha : 3 0. Use d. Test H0 : 4 = 0 against Ha : 4 0. Use

= .01. = .01.

56) The model E(y) = 0 + 1 x was fit to a set of data, and the following plot of residuals against x values was obtained.

56)

Interpret the residual plot.

57) As part of a study at a large university, data were collected on n = 224 freshmen computer science (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): x 1 = average high school grade in mathematics (HSM) x 2 = average high school grade in science (HSS)

x 3 = average high school grade in English (HSE) x 4 = SAT mathematics score (SATM) x 5 = SAT verbal score (SATV)

2 A first-order model was fit to data with R a = .193. 2 Interpret the value of the adjusted coefficient of determination R a .

57)

58) The model E(y) = 0 + 1 x was fit to a set of data, and the following plot of residuals against x values was obtained.

58)

Interpret the residual plot.

59) A fast food chain test marketing a new sandwich chose 18 of its stores in one major metropolitan 59) area. Nine of the stores were in malls and nine were free standing. The sandwich was offered at three different introductory prices. The table shows the number of new sandwiches sold at each location for each location type and price combination.

a. Write a model for the mean number of sandwiches sold, E(y), assuming that the relationship between E(y) and price, x 1 , is first-order.

b. Fit the model to the data. c. Write the prediction equations for mall and free-standing stores. d. Do the data provide sufficient evidence that the change in number of sandwiches sold with respect to price is different for mall and free-standing stores? Use = .01.

60) Consider the data given in the table below. X 1 2 2 3 4 4 5 5 6

60)

Y 4 6 5 7 7 6 4 5 3

a. Plot the data on a scattergram. Does a quadratic model seem to be a good fit for the data? Explain. b. Use the method of least squares to find a quadratic prediction equation. c. Graph the prediction equation on your scattergram.

61) The model E(y) = 0 + 1 x was fit to a set of data, and the following plot of residuals against x values was obtained.

Interpret the residual plot.

61)

62) The staff of a test kitchen is attempting to determine the baking time, y, of a roast, i.e., the time it takes the internal temperature of the roast to reach 165°F, using two variables, the temperature setting of the oven, x 1 , and the weight of the roast, x 2 , in pounds. The data for

62)

24 roasts are shown below.

a. Fit a complete second-order model to the data. b. Do the data provide sufficient evidence to indicate that the second-order terms contribute information for the prediction of y? State the null and alternative hypotheses and the test statistic. Use = .05.

63) Why is the random error term added to a multiple regression model?

63)

64) In Hawaii, proceedings are under way to enable private citizens to own the property that their64) homes are built on. In prior years, only estates were permitted to own land, and homeowners leased the land from the estate. In order to comply with the new law, a large Hawaiian estate wants to use regression analysis to estimate the fair market value of the land. The following variables are proposed: y = Sale price of property ($ thousands) x 2 = 1 if property near Cove, 0 if not Write a regression model relating the sale price of a property to the qualitative variable x. Interpret all the s in the model.

65) As part of a study at a large university, data were collected on n = 224 freshmen computer science (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): x 1 = average high school grade in mathematics (HSM) x 2 = average high school grade in science (HSS)

x 3 = average high school grade in English (HSE) x 4 = SAT mathematics score (SATM) x 5 = SAT verbal score (SATV)

A first-order model was fit to the data with the following results: _____________________________________________________________________ SOURCE DF SS MS F VALUE PROB > F MODEL ERROR TOTAL

5 218 223

ROOT MSE DEP MEAN

28.64 106.82 135.46 0.700 4.635

5.73 0.49

11.69

R-SQUARE ADJ R-SQ

.0001

0.211 0.193

VARIABLE

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X1 (HSM) X2 (HSS) X3 (HSE) X4 (SATM) X5 (SATV)

2.327 0.146 0.036 0.055 0.00094 -0.00041

0.039 0.037 0.038 0.040 0.00068 0.0059

5.817 3.718 0.950 1.397 1.376 -0.689

0.0001 0.0003 0.3432 0.1637 0.1702 0.4915

Test to determine if the model is adequate for predicting GPA. Use

= .01.

65)

66) The concessions manager at a beachside park recorded the high temperature, the number of 66) people at the park, and the number of bottles of water sold for each of 12 consecutive Saturdays. The data are shown below.

Fit the model E(y) = 0 + 1 x 1 + 2 x 2 to the data, letting y represent the number of bottles of water sold, x 1 the temperature, and x 2 the number of people at the park.

b. Find the 95% confidence interval for the mean number of bottles of water sold when the temperature is 84°F and there are 2700 people at the park. c. Find the 95% prediction interval for the number of bottles of water sold when the temperature is 84°F and there are 2700 people at the park.

67) In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked (y) per day by the clerical staff depends on the number of pieces of mail processed per day (x 1 ) and the number of checks cashed per

67)

day (x 2 ). Data collected for n = 20 working days were used to fit the model: E(y) = 0 + 1 x 1 + 2 x 2

A partial printout for the analysis follows: ___________________________________________________________________________ Analysis of Variance SOURCE

F VALUE

PROB > F

MODEL ERROR C TOTAL

2 17 19

7089.06512 4541.72142 11630.78654

3544.53256 267.16008

13.267

0.0003

ROOT MSE DEP MEAN C.V.

16.34503 93.92682 17.40188

R-SQUARE ADJ R-SQ

0.6095 0.5636

Test to determine if the model is adequate for predicting the number of man-hours worked. Use = .025.

68) Retail price data for n = 60 hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive:

68)

y = Retail PRICE (measured in dollars) x1 = Microprocessor SPEED (measured in megahertz)

(Values in sample range from 10 to 40) x2 = CHIP size (measured in computer processing units) (Values in sample range from 286 to 486)

A first-order regression model was fit to the data. Part of the printout follows:

OBS SPEED CHIP 1

386

Dep Var Predict Std Err Lower 95% Upper 95% PRICE Value Predict Predict Predict Residual 5099.0 4464.9 260.768

3942.7

4987.1

634.1

Interpret the 95% prediction interval for y when x 1 = 33 and x 2 = 386.

69) The model E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 3 was used to relate E(y) to a single qualitative variable. How many levels does the qualitative variable have?

69)

70) The printout shows the results of a first-order regression analysis relating the sales price y of a product to the time in hours x 1 and the cost of raw materials x 2 needed to make the

70)

product.

a. b. c.

What is the least squares prediction equation? Identify the SSE from the printout. Find the estimator of 2 for the model.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 71) The model E(y) = 0 + 1 x 1 + 2 x 2 was fit to a set of data.

71)

A partial printout for the analysis follows:

OBS

Actual Predict Value Value Residual

7781

644

74.707

83.175

Lower 95% CL Upper 95% CL Predict Predict 47.224

-8.468

119.126

Interpret the value of the residual when x 1 = 7,781 and x 2 = 644. ^

A) The predicted y is 8.468 less than the observed value of y. ^

B) The predicted y exceeds the observed value of y by 8.468. C) Since the residual is not 0, the model is not useful for predicting y. D) Since the residual is negative, there is evidence of a negative linear relationship between y and at least one of the two independent variables. Answer the question True or False. 72) In an interaction model, the relationship between E(y) and x 1 is linear for each fixed value of x 2 but

72)

the slopes of the lines relating E(y) and x 1 may be different for two different fixed values of x 2 .

A) True

B) False

73) A nested model F-test can only be used to determine whether second-order terms should be included in the model. A) True B) False

73)

74) For any given model fit to a data set, the sum of the residuals is 0. A) True B) False

74)

Solve the problem. 75) Operations managers often use work sampling to estimate how much time workers spend on each 75) operation. Work sampling which involves observing workers at random points in time was applied to the staff of the catalog sales department of a clothing manufacturer. The department applied regression to the following data collected for 40 consecutive working days: TIME:

y = Time spent (in hours) taking telephone orders during the day

ORDERS:

x 1 = Number of telephone orders received during the day

WEEK:

x 2 = 1 weekday, 0 if Saturday or Sunday

Consider the following 2 models: Model 1: E(y) = 0 + 1 x 1 + 2 (x1 )2 + 3 x 2 + 4 x 1 x2 + 5 (x 1 )2 x 2 Model 2: E(y) = 0 + 1 x 1 + 3 x 2 What strategy should you employ to decide which of the two models, the higher-order model or the simple linear model, is better? A) Compare R2 values; the model with the larger R2 will always be the better model.

B) Always choose the more parsimonious of the two models, i.e., the model with the fewest number of -coefficients. C) Compare the two models with a nested model F-test, i.e., test the null hypothesis, H0 : 2 = 4 = 5 = 0.

D) Compare the two models with a t-test, i.e., test the null hypothesis, H0 : 1 = 0.

Answer the question True or False. 76) The method of fitting first-order models is the same as that of fitting the simple straight-line model, i.e. the method of least squares. A) True B) False

76)

77) When testing the utility of the quadratic model E(y) = 0 + 1x + 2 x 2, the most important tests involve the null hypotheses H0 : 0 = 0 and H0 : 1 = 0.

77)

78) When using the model E(y) = 0 + 1 x for one qualitative independent variable with a 0 1 coding convention, 1 represents the difference between the mean responses for the level assigned the

78)

A) True

B) False

value 1 and the base level. A) True

B) False

79) A first-order model may include terms for both quantitative and qualitative independent variables. A) True B) False

79)

80) A regression residual is the difference between an observed y value and its corresponding predicted value. A) True B) False

80)

81) Residual analysis can be used to check for violations of the assumptions that the distribution of the random error component is normally distributed with mean 0. A) True B) False

81)

82) In stepwise regression, the probability of making one or more Type I or Type II errors is quite small. A) True B) False

82)

Solve the problem. 83) During its manufacture, a product is subjected to four different tests in sequential order. An 83) efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score (y), as a function of Test1 score (x 1 ), Test 2 score (x 2), and Test3 score (x 3 ). [Note: All test scores range from 200 to 800, with higher scores indicative of a higher quality product.] Consider the model: E(y) = 1 + 1 x 1 + 2 x 2 + 3 x 3 The first-order model was fit to the data for each of 12 units sampled from the production line. A 95% prediction interval for Test4 score of a product with Test1 = 590, Test2 = 750, and Test3 = 710 is (583, 793). Interpret this result. A) We are 95% confident that a product's Test4 score will fall between 583 and 793 points when the first three scores are 590, 750, and 710, respectively. B) We are 95% confident that a product's Test4 score increases by an amount between 583 and 793 points for every 1 point increase in Test1 score, holding Test 2 and Test 3 score constant. C) We are 95% confident that the mean Test4 score of all manufactured products falls between 583 and 793 points. D) Since 0 is outside the interval, there is evidence of a linear relationship between Test4 score and any of the other test scores.

84) In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked (y) per day by the clerical staff depends on the number of pieces of mail processed per day (x 1 ) and the number of checks cashed per day (x 2 ). Data collected for n = 20

84)

working days were used to fit the model:

E(y) = 0 + 1 x 1 + 2x 2 A partial printout for the analysis follows: Parameter Estimates

VARIABLE

INTERCEPT X1 X2

1 1 1

PARAMETER STANDARD T FOR 0: ESTIMATE ERROR PARAMETER = 0 PROB > |T| 114.420972 -0.007102 0.037290

18.6848744 0.00171375 0.02043937

6.124 -4.144 1.824

Calculate a 95% confidence interval for 1 . A) -.007 ± .0007 B) -4.144 ± .0007

C) -.007 ± .0017

0.0001 0.0007 0.0857

D) -.007 ± .0036

Answer the question True or False. 85) We expect all or almost all of the residuals to fall within 2 standard deviations of 0. A) True B) False 86) When modeling E(y) with a single qualitative independent variable, the number of 0 1 dummy variables in the model is equal to the number of levels of the qualitative variable. A) True B) False

85)

86)

Solve the problem. 87) Retail price data for n = 60 hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive:

87)

y = Retail PRICE (measured in dollars) x 1 = Microprocessor SPEED (measured in megahertz)

(Values in sample range from 10 to 40) x 2 = CHIP size (measured in computer processing units) (Values in sample range from 286 to 486)

a first-order regression model was fit to the data. Part of the printout follows: Dep Var Predict Std Err Lower 95% Upper 95% OBS SPEED CHIP PRICE Value Predict Predict Predict Residual 1

286

5099.0 4464.9 260.768

3942.7

4987.1

634.1

Interpret the interval given in the printout. A) We are 95% confident that the price of a single hard drive falls between $3,943 and $4,987. B) We are 95% confident that the average price of all hard drives falls between $3,943 and $4,987. C) We are 95% confident that the average price of all hard drives with 33 megahertz speed and 386 CPU falls between $3,943 and $4,987. D) We are 95% confident that the price of a single hard drive with 33 megahertz speed and 386 CPU falls between $3,943 and $4,987.

Answer the question True or False. 88) The value of R 2 is only useful when the number of data points is substantially larger than the number of A) True

parameters in the model.

88)

B) False

Solve the problem. 89) A collector of grandfather clocks believes that the price received for the clocks at an auction 89) increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model E(y) = 0 + 1 x + 2 x 2 This model was fit to data collected for a sample of 32 clocks sold at auction; the resulting estimate of was -.31. Interpret this estimate of 1 . A) 1 is a shift parameter that has no practical interpretation.

B) We estimate the auction price will be -$.31 when there are no bidders at the auction. C) We estimate the auction price will increase $.31 for each additional bidder at the auction. D) We estimate the auction price will decrease $.31 for each additional bidder at the auction.

90) In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked (y) per day by the clerical staff depends on the number of pieces of mail processed per day (x 1 ) and the number of checks cashed per day (x 2 ). Data collected for n = 20

90)

working days were used to fit the model:

E(y) = 0 + 1 x 1 + 2 x 2 A partial printout for the analysis follows:

OBS

Actual Predict Value Value Residual

7781

644

74.707

83.175

Lower 95% CL Upper 95% CL Predict Predict 47.224

-8.468

119.126

Interpret the 95% prediction interval for y shown on the printout. A) We expect to predict number of man-hours worked per day to within an amount between 47.224 and 119.126 of the true value. B) We are 95% confident that the number of man-hours worked per day falls between 47.224 and 119.126. C) We are 95% confident that the mean number of man-hours worked per day falls between 47.224 and 119.126 for all days in which 7,781 pieces of mail are processed and 644 checks are cashed. D) We are 95% confident that between 47.224 and 119.126 man-hours will be worked during a single day in which 7,781 pieces of mail are processed and 644 checks are cashed.

91) As part of a study at a large university, data were collected on n = 224 freshmen computer science 91) (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): x 1 = average high school grade in mathematics (HSM) x 2 = average high school grade in science (HSS)

x 3 = average high school grade in English (HSE) x 4 = SAT mathematics score (SATM) x 5 = SAT verbal score (SATV)

A first-order model was fit to data. Give the null hypothesis for testing the overall adequacy of the model. A) H0 : 0 = 1 = 2 = 3 = 4 = 5 = 0

B) H0 : 1 = 2 = 3 = 4 = 5 = 0 C) H0 : 0 + 1 x1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 5 = 0 D) H0 : 1 = 0

92) Which residual plot would you examine to determine whether the assumption of constant error variance is satisfied for a model with two independent variables x 1 and x2 ?

92)

A) Plot the residuals against the independent variable x 2 . B) Plot the residuals against observed y values. C) Plot the residuals against the independent variable x 1 . ^

D) Plot the residuals against predicted values, y. Answer the question True or False. 93) It is safe to conduct t-tests on the individual parameters in a first-order linear model in order to determine which independent variables are useful for predicting y and which are not. A) True B) False 94) A first-order model does not contain any higher-order terms. A) True B) False

93)

94)

Solve the problem. 95) During its manufacture, a product is subjected to four different tests in sequential order. An 95) efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score (y), as a function of Test1 score (x 1 ), Test 2 score (x 2), and Test3 score (x 3 ). [Note: All test scores range from 200 to 800, with higher scores indicative of a higher quality product.] Consider the model: E(y) = 1 + 1 x 1 + 2 x 2 + 3 x 3 The global F statistic is used to test the null hypothesis, H0 : 1 = 2 = hypothesis in words. A) The first three test scores are reliable predictors of Test4 score. B) The model is not statistically useful for predicting Test4 score. C) The model is statistically useful for predicting Test4 score. D) The first three test scores are poor predictors of Test4 score.

3 = 0. Describe this

Answer the question True or False. 96) The number of levels of observed x-values must be equal to the order of the polynomial in x that you want to fit. A) True B) False

96)

97) The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions. A) True B) False

97)

98) Once interaction has been established between x 1 and x 2 , the first-order terms for x 1 and x 2 may

98)

be deleted from the regression model leaving the higher-order term containing the product of x 1 and x 2 .

A) True

B) False

99) The independent variables x 1 and x 2 interact when the effect on E(y) of a change in x 1 depends on x2 .

A) True

99)

B) False

Solve the problem. 100) A collector of grandfather clocks believes that the price received for the clocks at an auction 100) increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model E(y) = 0 + 1 x + 2 x 2 This model was fit to data collected for a sample of 32 clocks sold at auction; a portion of the printout follows:

VARIABLES

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X X ·X

286.42 .31 -.000067

9.66 .06 .00007

29.64 5.14 -0.95

.0001 .0016 .3600

Give the p-value for testing H0 : 2 = 0 against Ha : 2 0. A) .18 B) .05 C) .36

D) .0016

101) As part of a study at a large university, data were collected on n = 224 freshmen computer science 101) (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): x 1 = average high school grade in mathematics (HSM) x 2 = average high school grade in science (HSS)

x 3 = average high school grade in English (HSE) x 4 = SAT mathematics score (SATM) x 5 = SAT verbal score (SATV)

A first-order model was fit to data. A 95% confidence interval for 1 is (.06, .22). Interpret this result. A) We are 95% confident that a CS freshman's HS math grade increases by an amount between .06 and .22 for every 1-point increase in GPA, holding x 2 - x5 constant.

B) We are 95% confident that the mean GPA of all CS freshmen after three semesters falls between .06 and .22. C) We are 95% confident that a CS freshman's GPA increases by an amount between .06 and .22 for every 1-point increase in average HS math grade, holding x 2 - x 5 constant.

D) 95% of the GPAs fall within .06 to .22 of their true values.

Answer the question True or False. 102) One of three surfaces is produced by a complete second-order model with two quantitative independent variables: a paraboloid that opens upward, a paraboloid that opens downward, or a saddle-shaped surface. A) True B) False

102)

103) The complete second-order model with two quantitative independent variables does not allow for interaction between the two independent variables. A) True B) False

103)

104) In situations where two competing models have essentially the same predictive power (as determined by an F-test), it is standard procedure to use the model with the greater number of parameters. A) True B) False

104)

Solve the problem. 105) An elections officer wants to model voter turnout (y) in a precinct as a function of the type of precinct. 105) Consider the model relating mean voter turnout, E(y), to precinct type: E(y) = 0 + 1 x 1 + 2 x 2 , where

x 1 = 1 if urban, 0 if not

x 2 = 1 if suburban, 0 if not (Base level = rural)

The p-value for the test H0 : 1 = 2 = 0 is .14. Interpret the result. A) Do not reject H0 at = .10; there is no evidence of a difference between the mean voter

turnouts for urban, suburban, and rural precincts. B) Reject H0 at = .10; the model is useful for predicting voter turnout.

C) Reject the model since it only explains 14% of the variation. D) Reject H0 at = .01; there is evidence of a difference between the mean voter turnouts for urban, suburban, and rural precincts.

Answer the question True or False. 106) For a multiple regression model, we assume that the mean of the probability distribution of the random error is 0. A) True B) False Solve the problem. 107) An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state. Write a model for mean voter turnout, E(y), as a function of type of election. A) E(y) = 0 + 1 x + 2 x 2 , where x = voter turnout

B) E(y) = 0 + 1 x, where x = voter turnout C) E(y) = 0 + 1 x, where x = 1 if national, 0 if state D) E(y) = 0 + 1 x 1 + 2 x 2 , where x 1 = 1 if national, 0 if not and x 2 = 1 if state, 0 if not

106)

107)

Answer the question True or False.

108) In the quadratic model E(y) = 0 + 1 x + 2 x 2 , a negative value of concavity. A) True B) False

1 indicates downward

108)

Solve the problem. 109) A collector of grandfather clocks believes that the price received for the clocks at an auction 109) increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model E(y) = 0 + 1 x + 2 x 2 This model was fit to data collected for a sample of 32 clocks sold at auction. Suppose the p-value for the test of H0 : 2 = 0 vs. Ha : 2 > 0 is .02. What is the proper conclusion? A) There is evidence (at = .05) of upward curvature in the relationship between auction price (y) and number of bidders (x). B) There is evidence (at = .05) of downward curvature in the relationship between auction price (y) and number of bidders (x). C) Reject H0 at = .05; the model is not useful for predicting auction price (y).

D) There is no evidence (at = .05) of upward curvature in the relationship between auction price (y) and number of bidders (x).

110) A collector of grandfather clocks believes that the price received for the clocks at an auction 110) increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model E(y) = 0 + 1 x + 2 x 2 This model was fit to data collected for a sample of 32 clocks sold at auction; a portion of the printout follows:

VARIABLES

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X X ·X

286.42 -.31 .000067

9.66 .06 .00007

29.64 -5.14 .95

.0001 .0016 .3600

Find the p-value for testing H0 : 2 = 0 against Ha : 2 > 0. A) .36 B) .0016 C) .05

D) .18

111) A collector of grandfather clocks believes that the price received for the clocks at an auction 111) increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model E(y) = 0 + 1 x + 2 x 2 This model was fit to data collected for a sample of 32 clocks sold at auction; a portion of the printout follows: _____________________________________________________________________ SOURCE DF SS MS F VALUE PROB > F MODEL ERROR TOTAL

2 29 31

4277160 514034 4791194 ROOT MSE DEP MEAN

2138579 17725

120

133 1327

R-SQUARE ADJ R-SQ

.0005

893 .885

VARIABLES

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X X ·X

286.42 .31 -.000067

9.66 .06 .00007

29.64 5.14 -0.95

.0001 .0016 .3600

An outlier for the model is a clock with a residual that _____ in absolute value. (Fill in the blank.) A) exceeds 133 B) is less than 266 C) exceeds .893 D) exceeds 399

Answer the question True or False. 112) A term that contains the value of a quantitative variable raised to the second power is called a higher-order term. A) True B) False

112)

Solve the problem. 113) An elections officer wants to model voter turnout (y) in a precinct as a function of the type of precinct. 113) Consider the model relating mean voter turnout, E(y), to precinct type: E(y) = 0 + 1 x 1 + 2 x 2 , where

x 1 = 1 if urban, 0 if not

x 2 = 1 if suburban, 0 if not (Base level = rural)

Interpret the value of 2 . A) the mean voter turnout for suburban precincts B) the rate of increase in voter turnout (y) for suburban precincts, i.e., the slope of the y-x 2 line

C) the difference between the mean voter turnout for suburban and urban precincts D) the difference between the mean voter turnout for suburban and rural precincts

Answer the question True or False. 114) One advantage to writing a single model that includes all levels of a qualitative variable rather a separate model for each level is that we obtain a pooled estimate of 2 . A) True

114)

B) False

115) In the first-order model E(y) = 0 + 1x 1 + 2 x2 + 3 x 3, 2 represents the slope of the line relating y to x 2 when 1 and 3 are both held fixed. A) True

115)

B) False

Solve the problem. 116) A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model

116)

E(y) = 0 + 1 x + 2 x 2 where y = Demand (in thousands) and x = Retail price per carat (dollars). This model was fit to data collected for a sample of 12 rare gems. If the experts are correct in their assumptions about the relationship between price and demand, which of the following should be true? A) 2 < 0 B) 1 > 0 C) 2 > 0 D) 1 < 0

Answer the question True or False. 117) The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model. A) True B) False

117)

118) The stepwise regression procedure may not be used when the inclusion of one or more dummy variables is under consideration. A) True B) False

118)

119) In the presence of multicollinearity, you should avoid making inferences about the parameters based on the t-tests. A) True B) False

119)

Solve the problem. 120) A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand decreases as the price of the gem increases. However, experts hypothesize that when the gem is valued at very high prices, the demand increases with price due to the status the owners believe they gain by obtaining the gem. Thus, the model proposed to best explain the demand for the gem by its price is the quadratic model

120)

E(y) = 0 + 1 x + 2 x 2 where y = Demand (in thousands) and x = Retail price per carat (dollars). This model was fit to data collected for a sample of 12 rare gems.

VARIABLES INTERPCEP X X ·X

PARAMETER ESTIMATES STD. ERROR 286.42 -.31 .000067

T for HO: PARAMETER = 0

PR > |T|

29.64 -5.14 .95

.0001 .0006 .3647

9.66 .06 .00007

Does there appear to be upward curvature in the response curve relating y (demand) to x (retail price)? A) No, since the p-value for the test is greater than .10. B) Yes, since the p-value for the test is less than .01. C) Yes, since the value of 2 is positive.

D) No, since the value of 2 is near 0.

121) As part of a study at a large university, data were collected on n = 224 freshmen computer science 121) (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): x 1 = average high school grade in mathematics (HSM) x 2 = average high school grade in science (HSS)

x 3 = average high school grade in English (HSE) x 4 = SAT mathematics score (SATM) x 5 = SAT verbal score (SATV)

A first-order model was fit to data with R 2 = 0.211. What is the correct interpretation of R2 , the coefficient of determination for the model? A) We are 79% confident that the model is useful for predicting y. B) Approximately 21% of the sample variation in GPAs can be explained by the first-order model. C) Approximately 79% of the sample variation in GPAs can be explained by the first-order model. D) We expect to predict GPA to within approximately .21 of its true value.

122) An elections officer wants to model voter turnout (y) in a precinct as a function of the type of precinct. 122) Consider the model relating mean voter turnout, E(y), to precinct type: E(y) = 0 + 1 x 1 + 2 x 2 , where

x 1 = 1 if urban, 0 if not

x 2 = 1 if suburban, 0 if not (Base level = rural)

Interpret the value of 0 . A) the y-intercept of the line B) the difference between the mean voter turnout for urban and rural precincts C) the mean voter turnout for urban precincts D) the mean voter turnout for rural precincts

123) As part of a study at a large university, data were collected on n = 224 freshmen computer science 123) (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): x 1 = average high school grade in mathematics (HSM) x 2 = average high school grade in science (HSS)

x 3 = average high school grade in English (HSE) x 4 = SAT mathematics score (SATM) x 5 = SAT verbal score (SATV)

A first-order model was fit to data with the following results: _____________________________________________________________________ SOURCE DF SS MS F VALUE PROB > F MODEL 5 28.64 5.73 11.69 .0001 ERROR 218 106.82 0.49 TOTAL 223 135.46 ROOT MSE 0.700 DEP MEAN4.635

R-SQUARE 0.211 ADJ R-SQ 0.193

VARIABLE

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X1 (HSM) X2 (HSS) X3 (HSE) X4 (SATM) X5 (SATV)

2.327 0.146 0.036 0.055 0.00094 -0.00041

0.039 0.037 0.038 0.040 0.00068 0.00059

5.817 3.718 0.950 1.397 1.376 -0.689

0.0001 0.0003 0.3432 0.1637 0.1702 0.4915

Interpret the value under the column heading PROB > F. A) Over 99% of the variation in GPAs can be explained by the model. B) Accept H0 (at = .01); at least one of the -coefficients in the first-order model is equal to 0.

C) There is insufficient evidence (at = .01) to conclude that the first-order model is statistically useful for predicting GPA. D) There is sufficient evidence (at = .01) to conclude that the first-order model is statistically useful for predicting GPA.

Answer the question True or False. 124) If when using the model E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 1 x 2 we determine that interaction between x 1 and x 2 is not significant, we can drop the x 1 x 2 term from the model and use the simpler model E(y) = 0 + 1 x 1 + 2 x 2 . A) True

124)

B) False

125) The stepwise regression model should not be used as the final model for predicting y. A) True B) False

125)

126) A qualitative variable whose outcomes are assigned numerical values is called a coded variable. A) True B) False

126)

Solve the problem. 127) Suppose that the following model was fit to a set of data.

127)

E(y) = 0 + 1x 1 + 2 x 2 ^

The corresponding plot if residuals against predicted values y is shown. Interpret the plot.

A) It appears that a quadratic model would be a better fit. B) It appears that the data contain an outlier. C) It appears that the variance of is not constant. D) The residuals appear to be randomly scattered so that no model modifications are necessary. Answer the question True or False. 128) Stepwise regression is used to determine which variables, from a large group of variables, are useful in predicting the value of a dependent variable. A) True B) False

128)

Solve the problem. 129) During its manufacture, a product is subjected to four different tests in sequential order. An efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score (y), as a function of Test1 score (x 1 ), Test 2 score (x 2 ), and Test3 score (x 3 ). [Note: All test

129)

scores range from 200 to 800, with higher scores indicative of a higher quality product.] Consider the model: E(y) = 1 + 1 x 1 + 2 x 2 + 3 x 3 The first-order model was fit to the data for each of 12 units sampled from the production line. The results are summarized in the printout. _____________________________________________________________________ SOURCE

F VALUE

PROB > F

MODEL ERROR TOTAL

3 8 12

151417 22231 173648

50472 2779

18.16

.0075

ROOT MSE DEP MEAN

52.72 645.8

R-SQUARE ADJ R-SQ

0.872 0.824

VARIABLE

PARAMETER ESTIMATE

STANDARD ERROR

T FOR 0: PARAMETER = 0

PROB > |T|

INTERCEPT X1(TEST1) X2(TEST2) X3(TEST3)

11.98 0.2745 0.3762 0.3265

80.50 0.1111 0.0986 0.0808

0.15 2.47 3.82 4.04

0.885 0.039 0.005 0.004

Compute a 95% confidence interval for 3 . A) .33 ± .08 B) .33 ± 4.04

C) .33 ± 105

D) .33 ± .19

Answer the question True or False. ^

130) In the presence of multicollinearity, the predicted values of y are actually quite good for values of x far outside the range of the sampled values of x. A) True B) False

130)

131) Probabilistic models that include more than one dependent variable are called multiple regression models. A) True B) False

131)

Answer Key Testname: CHAPTER 12 1) B 2) A 3) A 4) A 5) C 6) D 7) D 8) A 9) B 10) A 11) B 12) D 13) A 14) A 15) C 16) B 17) C 18) C 19) A 20) B 21) C 22) B 23) C 24) D 25) A 26) C 27) A 28) C 29) D 30) A 31) C 32) D 33) D 34) C 35) E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 1 x 2

Answer Key Testname: CHAPTER 12 36) A second-order (quadratic) model seems to be a good fit because as x increases the y values initially decrease and then increase.

37) a. b.

The dependent variable is “test grade.” There are three independent variables: “quiz 1,” “quiz 2,” and “quiz 3.” ^

y = 15.8687 + 2.6796x 1 + 1.9780x 2 + 3.6967x 3

SSE = 317.1874; The estimator of 2 is 79.2968.

38) a.

y = 16.7220 - 3.0373x 1 - 1.0465x 2 + 4.0717x 1 x 2

We test the null hypothesis H0 : 1 = 2 = 3 = 0. The F-statistic is 9394, and the associated level of significance is 2.11 -11 × 10 . Since the level of significance is less than , we reject the null hypothesis and conclude the model is useful for estimating or predicting values of y. c. We test the null hypothesis H0 : 3 = 0 against Ha : 3 > 0. The t-statistic is 9.17, and two-tailed p-value is 9.48 × 10-5 . The upper-tailed p-value is 4.74 ×10-5 . Since this value is less than , we reject the null hypothesis and conclude that x 1 and x 2 interact positively.

d. 31.39 39) The p-value for the test is p = .4271. Since p > , we do not reject the null hypothesis. There is insufficient evidence to conclude that the quiz grades are a useful predictor of the test grade. 40) To determine if the quadratic term is useful for predicting the demand for the gem, we test: H0 : 2 = 0 Ha : 2 0 The test statistic is t = .95. The p-value is p = .3647. Since

= .10 < p = .3647, H0 cannot be rejected. There is insufficient evidence to indicate the quadratic term is useful for

predicting the demand for the gem.

Answer Key Testname: CHAPTER 12

41) a. y = 3449.598 - 41.306x 1 - 1.627x 2 + .02183x 1 x 2

b. First indicator: The F-test for overall model adequacy shows much greater significance than any of the t-tests for the individual parameters. Second indicator: The signs of two of the parameters are negative when we expect them to be positive. c. Since the high temperature of 103°F and 3500 people at the park do not fall within the ranges of observed values for the corresponding variables, the model should not be used to make a prediction for these values. 42) No, the p-value for the coefficient of x 1 x 2 is p = .21 which exceeds the value of .

43) Yes, there is evidence of multicollinearity. The F-test for overall model adequacy shows much greater significance than either of the t-tests for the individual parameters. 44) To determine if number of checks cashed per day is a positive linear predictor of number of man-hours worked, we test: H0 : 2 = 0 Ha : 2 > 0 The test statistic is t = 1.824 The p-value is p = .0857/2 = .04285 At

= .05, > p and H0 is rejected. There is sufficient evidence to indicate that the number of checks cashed per day is a positive linear predictor of the number of man-hours worked at = .05. ^

45) In order to check the variance assumption, both y and = y - y must be calculated for each of the data values. A ^

scatterplot of the residuals, , versus the predicted sale prices, y, should be constructed. If the plot reveals a random scattering of points, then the assumption of equal variances is satisfied. 46) To determine if the model is useful for predicting the retail price of a computer, we test: H0 : 1 = 2 = 0 Ha : At least one i 0 The test statistic is F = 19.018. The p-value is p = .0001. At

= .01,

> p-value and H0 is rejected. There is sufficient evidence to indicate the model is useful for predicting the

retail price of a computer at = .01. 47) a. level 1: 14.5; level 2: 14.5 + 3 = 17.5; level 3: 14.5 - 4 = 10.5; level 4: 14.5 + 10 = 24.5; level 5: 14.5 + 8 = 22.5 b. H0 : 1 = 2 = 3 = 4 = 0

Ha : At least one of the parameters 1 , 2 , 3 , and 4 differs from 0

Answer Key Testname: CHAPTER 12 48) To determine if the model is useful for predicting the demand for the gem, we test: H0 : 1 = 2 = 0 Ha : At least one i 0 The test statistic is F = 373. The p-value is p = .0001. Since

= .01 > p = .0001, H0 is rejected. There is sufficient evidence to indicate the model is useful for predicting the demand for the gem. 49) To determine if the relationship between college GPA and SAT score depends on high school GPA, we test H0 : 3 = 0

vs Ha : 3 0. A t-test on 3 would provide the necessary information.

50)

2 = 3.57. For every one computer processing unit increase in chip size, we estimate the retail price to increase by $3.57, holding microprocessor speed constant.

51) a. b.

y = 7.6253 + 2.4190x 1 + .5632x 2

The 95% confidence interval for 1 is (1.33, 3.51). We are 95% confident that y increases between 1.33 and 3.51 for each 1-unit increase in x 1 , holding x 2 fixed. c.

The 95% confidence interval for 2 is (-.42, 1.54). Since the interval includes 0, we cannot conclude that a linear relationship between y and x 2 exists, while holding x 1 fixed. d.

R2 = .8745 and Ra2 = .8597; About 87.45% of the variation in y values can be attributed to the relationship between

y and x 1 and x 2 . About 85.97% of the variation in y-values can be attributed to this relationship, after adjusting for sample size and number of independent variables.

e. The p-value for the test is p = 2.18 × 108 . Since p < , we reject the null hypothesis in favor of the alternative hypothesis. There is sufficient evidence to conclude that the model is statistically useful for estimating or predicting y. ^ 52) a. y = -616.64 + 919.10x - 329.43x 2

Test H0 : 2 = 0 against Ha : 2 < 0. The t-value for the test is -15.79 and the p-value is 3.32 × 109 , which is less than = .05. Thus, we may reject the null hypothesis and conclude that 2 is negative. There is sufficient evidence of downward curvature in the relationship between profit and price. c.

Answer Key Testname: CHAPTER 12 53) a. b. c.

E(y) = 0 + 1 x, where x = {1 if male, 0 if not} ^

y = 16.9167 - 1.2917x H0 : 1 = 0

d. The p-value is 0.1929, which is greater than , so we do not reject the null hypothesis. There is not sufficient evidence to conclude that the model is useful. 54) To determine if the quadratic component of the model is useful, we need to conduct a nested model F-test. The test would compare the current model (i.e., the full model) to the one that does not contain the quadratic terms (i.e., the reduced model). We test the following hypotheses: H0 : 2 = 5 = 0 Ha : At least one of 2 and 5 differs from 0

55) a.

y = -.2023 + .5796x 1 + .5030x 2 + 1.9761x 1 x 2 - .0268x1 2 + .0129x 2 2

The F-value for the test of H0 : 1 = 2 = 3 = 4 = 5 = 0 is 56,487.98 and the p-value is 6.13 × 1039, which is less than = .05. Thus, we may reject the null hypothesis and conclude that at least one of 1 , 2 , 3 , 4 , and 5 is nonzero. b. c.

The t-value for the test of H0: 3 = 0 is 89.78 and the p-value is 1.93 × 10-26, which is less than may reject the null hypothesis and conclude that 3 is not zero.

= .01. Thus, we

= .01. Thus, we may

The t-value for the test of H0: 4 = 0 is -1.06 and the p-value is .3033, which is greater than not reject the null hypothesis. There is insufficient evidence to conclude that 4 is not zero.

56) The curvilinear trend suggests that the model might benefit from an x 2 term. 57) 19.3% of the total variation of the sampled student GPAs can be explained by the least squares regression model after adjusting for sample size and number of independent variables in the model. 58) The random pattern about the 0 line indicates that the linear model is a good fit. 59) a. E(y) = 0 + 1 x 1 + 2 x 2 + 3 x 1 x 2 where x 1 = price and x2 = {1 if free standing, 0 if mall} ^

y = 491.446 - 95.450x 1 - 37.520x 2 + 19.189x 1 x2

mall: y = 491.446 - 95.450x 1 ; free standing: y = 453.926 - 76.261x 1

Test H0 : 3 = 0 against Ha : 3 0. The test value is t = .5195, and the p-value is p = .6115. Since the p-value is greater than .01, there is insufficient evidence of interaction between the two variables. We cannot conclude that the change in number of sandwiches sold with respect to price is different for mall and free-standing stores.

Answer Key Testname: CHAPTER 12 60) a. A quadratic model seems to be a good fit because as x increases the y values initially increase and then decrease.

^ b. y = 1.0303 + 3.3968x - .5218x 2 c. See part a. 61) It appears that the data may contain an outlier. ^

y = 14.2298 - .0690x 1 + .3511x 2 + .001252x 1 x 2 + .000085x 1 2 - .03334x2 2

62) a. b.

H0 : 3 = 4 = 5 = 0 Ha : At least one of the parameters 3 , 4 , and 5 differs from 0 (.1435 - .0689)/3 The test statistic is F = = 6.49. The rejection region based on v 1 = 3 and v 2 = 18 is F > 3.16. Since the test .003829 statistic falls in the rejection region, we reject H0 and conclude that at least one of the second-order terms contributes

information for the prediction of y. 63) The random error term is added to make the mode probabilistic rather than deterministic. 1 if property near Cove 64) E(y) = 0 + 1 x, where x = 0 if not

0 = mean sale price of properties not near a cove 1 = difference in mean sale prices of properties near a cove and properties not near a cove

65) To determine if the model is useful for predicting y, we test: H0 : 1 = 2 = 3 = 4 = 5 = 0 Ha : At least one i 0 The test statistic is F = 11.69. The p-value is p = .0001. At

= .01,

GPA.

> p so H0 is rejected. There is sufficient evidence to indicate the model is adequate for predicting student

66) a. y = -1125.433 + 16.8969x 1 + .097421x 2 b. (499.9, 614.0) c. (369.0, 744.9)

Answer Key Testname: CHAPTER 12 67) To determine if the model is useful for predicting y, we test: H0 : 1 = 2 = 0 Ha : At least one i 0 The test statistic is F = 13.267. The p-value is p = .0003. At

= .025, > p-value, so H0 is rejected. There is sufficient evidence to indicate the model is useful for predicting the number of man-hours worked at = .025. 68) We are 95% confident that a 386 CPU computer with 33 megahertz speed will have a retail price between $3,942.70 and $4,987.10. 69) The number of levels of the qualitative variable is one greater than the number of dummy variables, so there are 4 levels of the qualitative variable.

70) a. b. c. 71) B 72) A 73) B 74) A 75) C 76) A 77) B 78) A 79) B 80) A 81) A 82) B 83) A 84) D 85) B 86) B 87) D 88) A 89) A 90) D 91) B 92) D 93) B 94) A 95) B 96) B 97) B 98) B 99) A 100) C 101) C

y = -26.4843 - 2.1687x1 + 8.1422x 2

SSE = 2.8096

The estimator of 2 is 1.4048.

Answer Key Testname: CHAPTER 12 102) A 103) B 104) B 105) A 106) A 107) C 108) B 109) A 110) D 111) D 112) A 113) D 114) A 115) B 116) C 117) A 118) B 119) A 120) A 121) B 122) D 123) D 124) A 125) A 126) A 127) C 128) A 129) D 130) B 131) B

Chapter 13 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The capability index for a process centered on the desired mean is (USL - LSL) (USL - LSL) A) Cp = B) Cp = 3 6 C) Cp =

6 (USL - LSL)

D) Cp =

3 (USL - LSL)

2) _______ are boundary points that define the acceptable values for an output variable. A) Control bounds B) Tolerance limits C) Specification limits D) Capability limits

3) The upper and lower control limits are usually a distance of _______ from the centerline. A) 2 standard deviations B) 3.5 standard deviations C) 3 standard deviations D) 1 standard deviation

4) The process of monitoring and eliminating variation in order to keep a process in a state of control or to bring a process into control is called A) a control chart. B) statistical process control. C) random behavior. D) a process distribution.

5) A process is considered capable if A) Cp 1. B) Cp < 1.

C) Cp > 1.

D) Cp 1.

6) The primary goal of quality-improvement activities is A) to increase the mean. B) to increase variation. C) to reduce variation. D) to change the process.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. (Situation C) Ten samples of n = 5 were collected to construct an x-chart. The sample means and ranges for the 10 samples are shown below. Sample Mean 1 20.2 2 22.4 3 21.2 4 18.2 5 23.2

Range 2.7 1.8 1.5 1.2 2.4

Sample Mean 6 20.4 7 15.9 8 22.3 9 20.7 10 21.1

Range 1.9 1.0 2.1 1.6 1.8

7) Calculate the upper and lower control limits for the x-chart.

Solve the problem. 8) The table below shows the data from samples of size n = 5 randomly chosen from the outputs 8) of a process on 20 different days. Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a.

4.5 3.6 5.1 4.9 4.1 3.7 5.6 4.8 6.1 2.4 3.7 4.5 5.2 3.4 4.8 4.1 4.6 5.1 3.7 5.4

2.1 3.5 6.2 5.4 3.8 4.6 4.7 5.1 4.6 4.8 5.3 3.4 2.7 5.5 3.6 4.6 3.7 4.2 4.8 3.9

Data 5.4 6.1 2.4 3.5 3.8 4.8 4.1 4.3 4.7 5.6 2.9 3.4 5.1 4.6 4.8 4.9 3.5 5.6 4.9 4.2

2.7 4.9 3.7 5.4 4.6 4.2 5.1 4.6 3.8 4.1 4.7 5.2 6.2 2.4 4.4 4.9 3.7 4.1 2.3 5.4

4.5 4.2 5.1 3.7 4.9 5.2 3.1 2.7 4.2 4.8 6.3 4.5 2.9 3.4 4.6 4.3 5.2 5.7 4.5 4.3

Calculate x.

b. Calculate R. c. Find d2 and A2 . d. Construct an x-chart. e. Is the process out of control? Explain. (Situation B) A manufacturing company makes hemostats for hospital emergency rooms. The company is interested in implementing statistical process control procedures in its production operation. The production manager believes that the proportion of defective hemostats generated by the process is about 3%. The company collected one sample of 300 consecutively manufactured hemostats each day for 20 days. The data are shown below. Sample 1 2 3 4 5 6 7 8 9 10

Sample Size 300 300 300 300 300 300 300 300 300 300

Defectives 8 6 11 15 12 11 9 6 5 4

Sample 11 12 13 14 15 16 17 18 19 20

Sample Size 300 300 300 300 300 300 300 300 300 300

9) Find the upper and lower control limits for the p-chart.

Defectives 12 11 14 8 7 3 9 11 10 6

Solve the problem. 10) Does the following control chart represent a process that is in control or out of control? If it is out 10) of control, explain how you arrived at this conclusion.

(Situation E) A machine at K-Company fills boxes with bran flake cereal. The target weight for the filled boxes is 24 ounces. The company would like to use control charts to monitor the performance of the machine. The company decides to sample and weigh 10 consecutive boxes of cereal at randomly selected times over a two-week period. Twenty measurement times are selected and the following information is recorded. Time 1 2 3 4 5 6 7 8 9 10

Mean (oz) 23.8 24.5 23.9 24.2 23.7 23.5 24.2 24.4 24.1 24.2

Range (oz) 1.05 0.85 1.12 0.95 1.22 1.42 1.02 1.10 0.75 0.60

Time 11 12 13 14 15 16 17 18 19 20

Mean (oz) Range (oz) 24.5 1.21 24.7 0.65 24.0 0.55 25.5 3.21 24.2 1.25 24.4 1.35 24.5 0.98 25.0 1.30 24.1 0.88 24.2 1.01

11) Calculate the centerline for constructing the x-chart.

11)

12) Calculate the centerline of the R-chart.

12)

(Situation D) A walk-in freezer thermostat at a restaurant is set at 5°F. Because of the perishability of the food in the freezer, the restaurant manager has decided to begin monitoring the temperature inside the freezer. The managers used a precision thermometer to take sample temperature readings at five randomly chosen times per day for 10 days. The data are presented below Day 1 2 3 4 5 6 7 8 9 10

5.22 4.40 5.11 5.65 4.68 5.01 5.20 4.30 5.45 5.06

Temperature (°F) 5.29 5.11 4.95 4.41 4.63 6.03 5.43 4.90 4.55 4.24 5.09 4.82 5.92 4.71 4.67 5.26 6.10 5.20 4.99 5.15 5.96 4.91 5.03 4.97 5.62 6.11 5.13 5.13 4.95 5.59

4.78 4.83 5.23 5.50 4.75 5.25 5.35 4.80 4.90 5.80

13) Create the R-chart and interpret it.

13)

Mean (oz) 23.8 24.5 23.9 24.2 23.7 23.5 24.2 24.4 24.1 24.2

Range (oz) 1.05 0.85 1.12 0.95 1.22 1.42 1.02 1.10 0.75 0.60

Time 11 12 13 14 15 16 17 18 19 20

Mean (oz) Range (oz) 24.5 1.21 24.7 0.65 24.0 0.55 25.5 3.21 24.2 1.25 24.4 1.35 24.5 0.98 25.0 1.30 24.1 0.88 24.2 1.01

14) Calculate the upper and lower control limits for the x-chart.

14)

15) Create the x-chart and interpret it.

15)

Solve the problem. 16) Does the following control chart represent a process that is in control or out of control? If it is out 16) of control, explain how you arrived at this conclusion.

Mean (oz) 23.8 24.5 23.9 24.2 23.7 23.5 24.2 24.4 24.1 24.2

Range (oz) 1.05 0.85 1.12 0.95 1.22 1.42 1.02 1.10 0.75 0.60

Time 11 12 13 14 15 16 17 18 19 20

Mean (oz) Range (oz) 24.5 1.21 24.7 0.65 24.0 0.55 25.5 3.21 24.2 1.25 24.4 1.35 24.5 0.98 25.0 1.30 24.1 0.88 24.2 1.01

17) Find the upper and lower control limits for the R-chart.

17)

(Situation C) Ten samples of n = 5 were collected to construct an x-chart. The sample means and ranges for the 10 samples are shown below. Sample Mean 1 20.2 2 22.4 3 21.2 4 18.2 5 23.2

Range 2.7 1.8 1.5 1.2 2.4

Sample Mean 6 20.4 7 15.9 8 22.3 9 20.7 10 21.1

Range 1.9 1.0 2.1 1.6 1.8

18) Calculate the centerline for constructing the x-chart.

18)

Solve the problem. 19) Does the following control chart represent a process that is in control or out of control? If it is out 19) of control, explain how you arrived at this conclusion.

20) Does the following control chart represent a process that is in control or out of control? If it is out 20) of control, explain how you arrived at this conclusion.

21) Does the following control chart represent a process that is in control or out of control? If it is out 21) of control, explain how you arrived at this conclusion.

Temperature (°F) 5.29 5.11 4.95 4.41 4.63 6.03 5.43 4.90 4.55 4.24 5.09 4.82 5.92 4.71 4.67 5.26 6.10 5.20 4.99 5.15 5.96 4.91 5.03 4.97 5.62 6.11 5.13 5.13 4.95 5.59

5.22 4.40 5.11 5.65 4.68 5.01 5.20 4.30 5.45 5.06

4.78 4.83 5.23 5.50 4.75 5.25 5.35 4.80 4.90 5.80

22) Calculate the centerline of the R-chart.

22)

Solve the problem. 23) The table below shows the data from samples of size n = 5 randomly chosen from the outputs 23) of a process on 20 different days. Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a.

4.5 3.6 5.1 4.9 4.1 3.7 5.6 4.8 6.1 2.4 3.7 4.5 5.2 3.4 4.8 4.1 4.6 5.1 3.7 5.4

2.1 3.5 6.2 5.4 3.8 4.6 4.7 5.1 4.6 4.8 5.3 3.4 2.7 5.5 3.6 4.6 3.7 4.2 4.8 3.9

Data 5.4 6.1 2.4 3.5 3.8 4.8 4.1 4.3 4.7 5.6 2.9 3.4 5.1 4.6 4.8 4.9 3.5 5.6 4.9 4.2

2.7 4.9 3.7 5.4 4.6 4.2 5.1 4.6 3.8 4.1 4.7 5.2 6.2 2.4 4.4 4.9 3.7 4.1 2.3 5.4

4.5 4.2 5.1 3.7 4.9 5.2 3.1 2.7 4.2 4.8 6.3 4.5 2.9 3.4 4.6 4.3 5.2 5.7 4.5 4.3

Find D3 and D4 .

b. Construct an R-chart. c. Is the process out of control? Explain.

24) Does the following control chart represent a process that is in control or out of control? If it is out 24) of control, explain how you arrived at this conclusion.

25) Does the following control chart represent a process that is in control or out of control? If it is out 25) of control, explain how you arrived at this conclusion.

26) Does the following control chart represent a process that is in control or out of control? If it is out 26) of control, explain how you arrived at this conclusion.

(Situation B) A manufacturing company makes hemostats for hospital emergency rooms. The company is interested in implementing statistical process control procedures in its production operation. The production manager believes that the proportion of defective hemostats generated by the process is about 3%. The company collected one sample of 300 consecutively manufactured hemostats each day for 20 days. The data are shown below. Sample 1 2 3 4 5 6 7 8 9 10

Sample Size 300 300 300 300 300 300 300 300 300 300

Defectives 8 6 11 15 12 11 9 6 5 4

Sample 11 12 13 14 15 16 17 18 19 20

Sample Size 300 300 300 300 300 300 300 300 300 300

Defectives 12 11 14 8 7 3 9 11 10 6

27) Construct the p-chart and interpret it.

27)

Solve the problem. 28) The table below shows the data from samples of size n = 5 randomly chosen from the outputs of a process on 20 different days. Assume the specification limits are USL = 2.1 and LSL = 5.7. Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

4.5 3.6 5.1 4.9 4.1 3.7 5.6 4.8 6.1 2.4 3.7 4.5 5.2 3.4 4.8 4.1 4.6 5.1 3.7 5.4

2.1 3.5 6.2 5.4 3.8 4.6 4.7 5.1 4.6 4.8 5.3 3.4 2.7 5.5 3.6 4.6 3.7 4.2 4.8 3.9

Data 5.4 6.1 2.4 3.5 3.8 4.8 4.1 4.3 4.7 5.6 2.9 3.4 5.1 4.6 4.8 4.9 3.5 5.6 4.9 4.2

2.7 4.9 3.7 5.4 4.6 4.2 5.1 4.6 3.8 4.1 4.7 5.2 6.2 2.4 4.4 4.9 3.7 4.1 2.3 5.4

4.5 4.2 5.1 3.7 4.9 5.2 3.1 2.7 4.2 4.8 6.3 4.5 2.9 3.4 4.6 4.3 5.2 5.7 4.5 4.3

a. Assuming the process is under control, construct a capability analysis diagram for the process. b. Find the percentage of data items that fall outside the specification limits. c. Is the process capable? Support your answer with a numerical measure of capability.

28)

5.22 4.40 5.11 5.65 4.68 5.01 5.20 4.30 5.45 5.06

Temperature (°F) 5.29 5.11 4.95 4.41 4.63 6.03 5.43 4.90 4.55 4.24 5.09 4.82 5.92 4.71 4.67 5.26 6.10 5.20 4.99 5.15 5.96 4.91 5.03 4.97 5.62 6.11 5.13 5.13 4.95 5.59

4.78 4.83 5.23 5.50 4.75 5.25 5.35 4.80 4.90 5.80

29) Calculate the upper and lower control limits for the R-chart.

29)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the statement is true or false. 30) In constructing a p-chart, it is advisable to use a much smaller sample size than is typically used for x- and R-charts. A) True

30)

B) False

31) Control charts are used to help us differentiate between process variation due to common causes and special causes. A) True B) False

31)

32) When n 6, the R-chart contains only one control limit, the lower control limit. A) True B) False

32)

33) An unbiased estimator for constant. A) True

33)

can be found by dividing the mean of the ranges, R by an appropriate

B) False

(Situation F) Ten samples of n = 5 were collected to construct an R-chart. The sample means and ranges for the 10 samples are shown below. Sample Mean 1 20.2 2 22.4 3 21.2 4 18.2 5 23.2

Range 2.7 1.8 1.5 1.2 2.4

Sample Mean 6 20.4 7 15.9 8 22.3 9 20.7 10 21.1

Range 1.9 1.0 2.1 1.6 1.8

34) Calculate the upper and lower control limits for the R-chart. A) Upper = 3.6553, Lower = 0.4014 B) Upper = 4.3485, Lower = 0 C) Upper = 3.8052, Lower = 0 D) Upper = 3.1986, Lower = 0.4014

34)

Determine whether the statement is true or false. 35) With rare exceptions, all items produced by a process are identical. A) True B) False Solve the problem. 36) Find the value of Cp when USL = 1.0155, LSL = 1.0065, s = 0.002. A) 1.5

B) 0.75

C) 0.375

Determine whether the statement is true or false. 37) A process adds value to the inputs of the process. A) True

35)

36) D) 4.5

37)

B) False

38) The upper and lower control limits are positioned so that when the process is in control the probability of an individual value of the output variable falling outside the control limits is very large. A) True B) False

38)

39) Control charts may only be used for quantitative quality variables. A) True B) False

39)

Solve the problem. 40) A process is in control and has a normally distributed output distribution with mean of 1000 and a standard deviation of 100. The upper and lower specification limits for the process are 1060 and 940, respectively. Assuming no changes in the behavior of the process, what percentage of the output will be unacceptable? A) 45.14% B) 38.49% C) 61.51% D) 54.86% 41) Estimate the process spread when s = 0.0032. A) 0.0096 B) 0.0192

C) 0.0064

D) 0.0384

Determine whether the statement is true or false. 42) The quality of a product is indicated by the extent to which it satisfies the needs and preferences of its manufacturer. A) True B) False 43) If a capability analysis study indicates that an in-control process is not capable, it is usually off-centeredness, rather than variation, that is the culprit. A) True B) False Solve the problem. 44) A process is in control and has a normally distributed output distribution with mean of 1000 and a standard deviation of 100. The upper and lower specification limits for the process are 1060 and 940, respectively. Find the Cp value of the process. Is the system capable? A) Cp = 3.333; the system is not capable.

40)

41)

42)

43)

44)

B) Cp = 0.2; the system is not capable.

C) Cp = 3.333; the system is capable.

D) Cp = 0.2; the system is capable.

Determine whether the statement is true or false. 45) The p-chart is typically used to monitor the proportion of units that conform to specification. A) True B) False 11

45)

(Situation A) To construct a p-chart for a manufacturing process, 20 samples of size 100 were selected. The results are shown below: Sample 1 2 3 4 5 6 7 8 9 10

Sample Size 100 100 100 100 100 100 100 100 100 100

Defectives 10 8 9 6 7 3 8 6 11 10

Sample 11 12 13 14 15 16 17 18 19 20

Sample Size 100 100 100 100 100 100 100 100 100 100

Defectives 8 8 5 9 7 9 11 4 6 8

46) Calculate the centerline used in constructing a p-chart. A) .0706 B) .0266 C) .0532

D) .0765

Determine whether the statement is true or false. 47) Performance, reliability, and durability are some of the factors used to evaluate quality. A) True

46)

47)

B) False

48) A system receives inputs from its environment, transforms those inputs to outputs, and delivers them to its environment. A) True B) False

48)

49) Control charts are the tool of choice for continuously monitoring processes. A) True B) False

49)

Solve the problem. 50) Find the process spread when = 3.6. A) 10.8 B) 7.2

C) 21.6

D) 43.2

Determine whether the statement is true or false. 51) A process may be in control but still not be capable of producing output that is acceptable to customers. A) True B) False Solve the problem. 52) An in-control, centered process that follows a normal distribution has a Cp of 3.0. How many standard deviations away from the process mean is the upper specification limit? A) 6 B) 9 C) 18 D) 3

50)

51)

52)

(Situation A) To construct a p-chart for a manufacturing process, 20 samples of size 100 were selected. The results are shown below: Sample 1 2 3 4 5 6 7 8 9 10

Sample Size 100 100 100 100 100 100 100 100 100 100

Defectives 10 8 9 6 7 3 8 6 11 10

Sample 11 12 13 14 15 16 17 18 19 20

Sample Size 100 100 100 100 100 100 100 100 100 100

Defectives 8 8 5 9 7 9 11 4 6 8

53) Calculate the upper and lower control limits for the p-chart. A) p ± .1783 B) p ± .0532 C) p ± .0266 Solve the problem. 54) Find the specification spread when USL = 5.05 and LSL = 3.78. A) 2 B) 1.27 C) 1.37

D) p ± .0797

D) 2.27

Determine whether the statement is true or false. 55) The diagnosis phase of statistical process control is concerned with tracking down causes of variation. A) True B) False Solve the problem. 56) Find the process spread when = 13. A) 39 B) 26

C) 156

D) 78

Determine whether the statement is true or false. 57) Most processes are naturally in a state of statistical control. A) True B) False

standard deviations away from the process mean is the upper specification limit?

B) 4

C) 2

55)

56)

58)

2 3

Determine whether the statement is true or false. 59) Capability analysis is used to determine when process variation is unacceptably high. A) True B) False 60) People, machines, and raw materials can all contribute to the variability in the output of a system. A) True B) False

54)

57)

Solve the problem. 58) An in-control, centered process that follows a normal distribution has a Cp = 0.6667. How many A) 3

53)

59)

60)

61) The variation in the output of processes that are out of control can be entirely attributed to random behavior. A) True B) False

61)

62) If all points fall between the control limits, then we may safely conclude that the process is in statistical control. A) True B) False

62)

63) The x-chart is typically used in conjunction with an R-chart. A) True B) False

63)

64) Special causes of variation can often be diagnosed and eliminated by workers or their immediate supervisors. A) True B) False

64)

65) Control charts are useful for evaluating the past performance of a process, for monitoring its current performance, and for predicting future performance. A) True B) False

65)

Solve the problem. 66) Find the value of Cp when USL = 1.0155, LSL = 1.0065, s = 0.0015. A) 6

B) 2

C) 0.1667

66) D) 1

5.22 4.40 5.11 5.65 4.68 5.01 5.20 4.30 5.45 5.06

Temperature (°F) 5.29 5.11 4.95 4.41 4.63 6.03 5.43 4.90 4.55 4.24 5.09 4.82 5.92 4.71 4.67 5.26 6.10 5.20 4.99 5.15 5.96 4.91 5.03 4.97 5.62 6.11 5.13 5.13 4.95 5.59

4.78 4.83 5.23 5.50 4.75 5.25 5.35 4.80 4.90 5.80

67) Calculate the centerline for constructing the x-chart. A) 5.0700 B) 5.1224

C) 5.3062

D) 10.245

Determine whether the statement is true or false. 68) The R-chart is used to detect changes in process variation. A) True B) False 69) If quality is designed into products and process management is used in their production, mass inspection of finished products will not be necessary. A) True B) False

67)

68)

69)

(Situation F) Ten samples of n = 5 were collected to construct an R-chart. The sample means and ranges for the 10 samples are shown below. Sample Mean 1 20.2 2 22.4 3 21.2 4 18.2 5 23.2

Range 2.7 1.8 1.5 1.2 2.4

Sample Mean 6 20.4 7 15.9 8 22.3 9 20.7 10 21.1

Range 1.9 1.0 2.1 1.6 1.8

70) Calculate the centerline of the R-chart. A) 1.0386 B) 4.114

C) 2.057

D) 1.80

Determine whether the statement is true or false. 71) Control limits and specification limits are essentially the same thing. A) True B) False

71)

72) The distribution that describes the output variable of a process may change over time. A) True B) False Solve the problem. 73) Estimate the process spread when s = 107.08. A) 214.16 B) 1284.96

C) 321.24

D) 642.48

Determine whether the statement is true or false. 74) Conformance refers to the extent to which a good or service can be adapted for use in new situations. A) True B) False Solve the problem. 75) Find the value of Cp when USL = 1.0155, LSL = 1.0065, s = 0.0005. A) 0.3333

B) 18

C) 3

70)

72)

73)

74)

75) D) 0.0556

Sample Size 300 300 300 300 300 300 300 300 300 300

Defectives 8 6 11 15 12 11 9 6 5 4

Sample 11 12 13 14 15 16 17 18 19 20

Sample Size 300 300 300 300 300 300 300 300 300 300

Defectives 12 11 14 8 7 3 9 11 10 6

76) Calculate the centerline used in constructing a p-chart. A) .0245 B) .0593 C) .0297

D) .0317

76)

5.22 4.40 5.11 5.65 4.68 5.01 5.20 4.30 5.45 5.06

Temperature (°F) 5.29 5.11 4.95 4.41 4.63 6.03 5.43 4.90 4.55 4.24 5.09 4.82 5.92 4.71 4.67 5.26 6.10 5.20 4.99 5.15 5.96 4.91 5.03 4.97 5.62 6.11 5.13 5.13 4.95 5.59

4.78 4.83 5.23 5.50 4.75 5.25 5.35 4.80 4.90 5.80

77) Calculate the upper and lower control limits for the x-chart. A) upper: 5.89 B) upper: 5.45 C) upper: 5.63 lower: 4.35 lower: 4.80 lower: 4.61

D) upper: 5.73 lower: 4.51

Determine whether the statement is true or false. 78) The p-chart is based on the assumption that the number of defective units in each sample is a binomial random variable. A) True B) False 79) If the R-chart indicates that the process variation is in control, then it makes sense to construct and interpret the x chart. A) True

B) False

77)

78)

79)

80) If we are unable to conclude that the process is out of control, it is better to behave as if the process were under control than to tamper with the process. A) True B) False

80)

81) A business that operates out-of-control processes risks losing its customers and threatens its own survival. A) True B) False

81)

82) When constructing an x-chart, x is used as the estimator of µ. A) True B) False

82)

83) When using a cause-and-effect diagram in process diagnosis, you begin by specifying the cause of interest and then move forward to identify potential effects of this cause. A) True B) False

83)

84) The centerline and control limits of a p-chart should be developed using samples that were collected during a period in which the process was in control. A) True B) False

84)

85) Samples should be chosen in such a way that a change in the process mean occurs within a sample rather than between samples. A) True B) False

85)

86) Statistical process control consists of monitoring process variation, diagnosing causes of variation, and eliminating those causes. A) True B) False

86)

87) The elimination of common causes of variation is typically the responsibility of workers, not management. A) True B) False

87)

88) If one or more values fall outside the control limits, then either a rare event has occurred or the process is out of control. A) True B) False

88)

89) The centerline of a control chart is drawn at the level of the median of the sample. A) True B) False

89)

Answer Key Testname: CHAPTER 13 1) B 2) C 3) C 4) B 5) A 6) C 7) upper control limit: 20.56 + 1.04 = 21.60; lower control limit: 20.56 - 1.04 = 19.52 8) a. 4.389 b. 2.29 c. d2 = 2.326 and A2 = 0.577 d.

There are no patterns evident in the control chart to indicate that the process is out of control. Total defectives in all samples 178 = = .0297 9) p = Total units sampled 6000

Upper Control Limit = p +3

p(1 - p) = .0297 + 3 n

.0297(.9703) = .0591 300

Lower Control Limit = p - 3

p(1 - p) = .0297 - 3 n

.0297(.9703) = .0003 300

10) in control 11) 24.28 R1 + R2 + ... + R20 22.47 = = 1.1235 12) R = k 20

Answer Key Testname: CHAPTER 13 13) The R-chart is:

The process appears to be in control. 14) 24.28 ± 0.3460 x1 + x2 + ... + x20 485.6 15) x = = = 24.28 k 20 R=

R1 + R2 + ... + R20 k

22.47 = 1.1235 20

Upper Control Limit = x + A2 R = 24.28 + 0.308(1.1235) = 24.63 Lower Control Limit = x - A2 R = 24.28 - 0.3081(1.1235) = 23.93

The process is out of control. 16) out of control; four out of five points in a row in Zone B or beyond

17) Upper Control Limit = RD4 = 1.1235(1.777) = 1.996 Lower Control Limit = RD3 = 1.1235(0.223) = 0.251

18) x = 20.56 19) in control 20) in control 21) in control R1 + R2 + ... + R10 10.53 = = 1.053 22) R = k 10

Answer Key Testname: CHAPTER 13 23) a. b.

D3 = 0 and D4 = 2.114

c. There are no patterns evident in the control chart to indicate that the process variation is out of control. 24) out of control; fourteen points in a row alternating up and down 25) out of control; six points in a row steadily increasing 26) out of control; nine points in a row in Zone C or beyond 27)

The process is out of control as a decreasing run of seven in a row is detected.

Answer Key Testname: CHAPTER 13 28) a.

b. 5% c.

5.7 - 2.1 Cp = 6(.939363)

.64; Since this value is less than 1, the process is not capable.

29) Upper Control Limit = RD4 = 1.053(2.114) = 2.226 Lower Control Limit = RD3 = 1.053(0) = 0

30) B 31) A 32) B 33) A 34) C 35) B 36) B 37) A 38) B 39) B 40) D 41) B 42) B 43) B 44) B 45) B 46) D 47) A 48) A 49) A 50) C 51) A 52) B 53) D

Answer Key Testname: CHAPTER 13 54) B 55) A 56) D 57) B 58) C 59) A 60) A 61) B 62) B 63) A 64) A 65) B 66) D 67) B 68) A 69) A 70) D 71) B 72) A 73) D 74) B 75) C 76) C 77) D 78) A 79) A 80) A 81) A 82) A 83) B 84) A 85) B 86) A 87) B 88) A 89) B

Chapter 14 Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Which of the following statements about the Durbin-Watson d-statistic is true? A) It can assume any value between 0 and 2. B) It can assume any value between -4 and 4. C) It can assume any value between -4 and 0. D) It can assume any value between 0 and 4.

2) Consider the actual values Y and forecast values F given in the table below. Time Period 1 2 3

Y 19.5 21.5 22.6

F 19.3 20.9 22.5

Calculate the root mean squared error (RMSE) of the forecasts. A) 1.42 B) 0.90 C) 0.30

D) 0.37

(Situation O) Using data from the post-Korean war period, an economist modeled annual consumption, yt, as a function of total labor income, x 1t, and total property income, x 2t, with the following results. Assume data for n = 40 years were used in the analysis. ^

yt = 7.81 + 0.91x 1t + 0.57x 2t

s = 1.29

Durbin-Watson d = 2.09

3) For the situation above, give the rejection region for the Durbin-Watson test for autocorrelation of residuals. Use = 0.10. A) d > 1.60 or 4 - d > 1.60 B) 1.39 < d < 1.60 C) d < 1.39 D) d < 1.39 or 4 - d < 1.39

4) For the situation above, set up the null and alternative hypotheses for testing for the presence of autocorrelation of residuals. A) H0 : No first-order autocorrelation

Ha : Positive or Negative first-order autocorrelation

B) H0 : No first-order autocorrelation

Ha : Positive first-order autocorrelation

C) H0 : 1 = 2 = 0 Ha : At least one

D) H0 : No first-order autocorrelation

Ha : Negative first-order autocorrelation

Solve the problem. 5) To test for first-order autocorrelation, we use the _______ test. A) Wilcoxon B) Laspeyres C) Durbin-Watson D) Paasche 1

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. Year 1980 1981 1982 1983 1984 1985

Sales 1.740 1.444 0.896 1.289 1.455 4.882

6) Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using w = 0.4 and v = 0.5. A) 8.72 million cars B) 6.068 million cars C) 6.39 million cars D) 8.952 million cars Solve the problem. 7) Consider the actual values Y and forecast values F given in the table below. Time Period 1 2 3

Y 19.5 21.5 22.6

F 19.3 20.9 22.5

Calculate the mean absolute deviation (MAD) of the forecasts. A) 1.42 B) 0.90 C) 0.37

D) 0.30

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below. Year 1986 1987 1988 1989 1990 1991 1992

G.M. 8993 7101 6762 6244 7769 8256 9305

Ford 5810 4328 4313 4255 4934 5585 5551

Chrysler 1796 1225 1283 1182 1494 2034 2157

8) Using 1986 as the base year, and only using the Chrysler sales data, find the simple index for 1992. A) 102.49 B) 120.10 C) 83.26 D) 97.57 Solve the problem. 9) The _______ of a time series can account for fluctuations that recur during specific time periods. A) cyclical fluctuation B) residual effect C) secular trend D) seasonal effect 10) The _______ is what remains of a time series value after the secular, cyclical, and seasonal components have been removed. A) error effect B) exponential effect C) additive effect D) residual effect 2

10)

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. Year 1980 1981 1982 1983 1984 1985

Sales 1.740 1.444 0.896 1.289 1.455 4.882

11) Use the Holt forecasting model with trend to forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1990 using w = 0.6 and v = 0.5. A) 8.72 million cars B) 6.39 million cars C) 6.068 million cars D) 8.952 million cars Solve the problem. 12) A(n) _______ is a number that measures the change in a variable over time relative to the value of the variable during a base period. A) exponential smoothing constant B) index number C) residual value D) time series

11)

12)

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. Year 1980 1981 1982 1983 1984 1985

Sales 1.740 1.444 0.896 1.289 1.455 4.882

13) Using a smoothing constant of w = 0.80, calculate the value of the exponentially smoothed series in 1983. A) 1.228 B) 1.235 C) 1.289 D) 1.340

13)

14) Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars to be sold to U.S. and Canadian dealers in 1986 using w = 0.7. A) 3.427 million cars B) 4.882 million cars C) 3.834 million cars D) 2.448 million cars

14)

Solve the problem. 15) Consider the actual values Y and forecast values F given in the table below. Time Period 1 2 3

Y 19.5 21.5 22.6

15)

F 19.3 20.9 22.5

Calculate the mean absolute percentage error (MAPE) of the forecasts. A) 1.42 B) 0.37 C) 0.90

D) 0.30

(Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below. Year 1986 1987 1988 1989 1990 1991 1992

G.M. 8993 7101 6762 6244 7769 8256 9305

Ford 5810 4328 4313 4255 4934 5585 5551

Chrysler 1796 1225 1283 1182 1494 2034 2157

16) Using 1986 as the base year, find the simple composite index for 1990. A) 116.92 B) 151.95 C) 85.53

D) 65.81

16)

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. Year 1980 1981 1982 1983 1984 1985

Sales 1.740 1.444 0.896 1.289 1.455 4.882

17) Using a smoothing constant of w = 0.30, calculate the value of the exponentially smoothed series in 1982. A) 1.150 B) 1.289 C) 1.301 D) 1.425

17)

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.

Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996

Commercial Manufacturing Service Construction and Mining 1637 2262 1645 1331 1770 1360 1041 1463 1122 773 1204 1013 930 1378 1165 1594 2355 1599 2366 3614 2223 3840 4872 3683 8627 5247 4433 12,787 6936 5759 16,647 7004 5662 20,911 7035 5641

Retail Trade 4799 4139 3406 2889 3183 4910 6882 9730 11,429 13,787 13,501 13,509

Wholesale Trade 1089 1028 887 740 908 1284 1709 2783 3598 4882 4835 4808

18) Using just the wholesale trade failures and a smoothing constant w = 0.7, calculate the exponentially smoothed value for 1988. A) 845 B) 932.9 C) 1015.6 D) 798.4 Solve the problem. 19) The tendency of a series of values to increase or decrease over a long period of time is known as the _______ of a time series. A) cyclical fluctuation B) residual effect C) seasonal variation D) secular trend

18)

19)

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.

Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996

Retail Trade 4799 4139 3406 2889 3183 4910 6882 9730 11,429 13,787 13,501 13,509

Wholesale Trade 1089 1028 887 740 908 1284 1709 2783 3598 4882 4835 4808

20) Using 1985 as the base period and using just construction failures, calculate the simple index for 1992. A) 487.20 B) 46.43 C) 215.38 D) 217.88

20)

(Situation N) An economist wishes to study the monthly trend in the Dow Jones Industrial Average (DJIA). Data collected over the past 40 months were used to fit the model E(Yt) = 0 + 1 t, where y = monthly close of the DJIA and t = month (1, 2, 3, . . . , 40). The regression results appear below: ^ y = 88 + 0.25t R2 = 0.37

MSE = 144

F = 4.25

Durbin-Watson d = 0.96

21) What is the value of the test statistic for testing whether autocorrelation exists in the data? A) 0.25 B) 4.25 C) 0.37 D) 0.96

21)

(Situation L) A farmer's marketing cooperative recorded the volume of wheat harvested by its members from 1991-2004. The cooperative is interested in detecting the long-term trend of the amount of wheat harvested. The data collected is shown in the table.

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Wheat Harvested by Coop. Member (y, in thousands of bushels) 75 78 82 82 84 85 87 91 92 92 93 96 101 102 ^

22) Suppose the least squares regression equation is yt = 75 + 2t. Use the regression model to forecast the harvest in 2005. A) 110,000 bushels

B) 105,000 bushels

C) 103,000 bushels

22)

D) 102,000 bushels

MSE = 144

F = 4.25

Durbin-Watson d = 0.96

23) Since the data are recorded over time (months), there is a strong possibility that the residuals are positively correlated. How could you check for residual correlation using a graphical technique? A) Plot the residuals against t and look for long runs of positive and negative residuals. ^

B) Plot the residuals against y and look for a funnel shape. ^

C) Plot the residuals against y and look for a linear trend. ^

D) Plot the residuals against y and look for outliers.

23)

(Situation M) Fast food chains are closely watching what proposed legislation will do to consumption of "huge-sized meals" in the United States. Researchers have accumulated statistics on the annual consumption of "huge-sized meals" for the past 25 years. The goal of the analysis is to use the past data to predict future consumption and then compare the predicted consumption to the actual consumption in those years.

24) Propose a straight-line model for the long-term trend of the time series. Do not include a seasonal component. Let t = the year in which the data was collected (t = 1, 2, . . . , 25). A) E(Yt) = 0 + 1 t + 2Q1 + 3 Q2 + 4 Q3

B) E(Yt) = 0 + 1 t

C) E(Yt) = 1t

D) E(Yt) = 0 + 1 Q1 + 3 Q2 + 4 Q3

24)

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. Year 1980 1981 1982 1983 1984 1985

Sales 1.740 1.444 0.896 1.289 1.455 4.882

25) Using the exponential smoothing technique to the data from 1980 to 1985, forecast the number of Chevrolet passenger cars sold to U.S. and Canadian dealers in 1986 using w = 0.3. A) 3.834 million cars B) 2.448 million cars C) 4.882 million cars D) 3.427 million cars Solve the problem. 26) The _______ generally describes fluctuations of the time series that are attributable to business and economic conditions. A) seasonal variation B) residual effect C) cyclical effect D) secular trend

25)

26)

(Situation I) The average monthly retail prices (in cents per pound) of cotton and wool were recorded for each month of one year. This monthly time series appears in the table.

Month January February March April May June

Retail Prices Cotton Wool 60.2 294 61.0 287 63.9 287 65.8 284 66.2 275 64.0 257

Month July August September October November December

27) Using just the wool prices and a smoothing constant w = 0.8, find the exponentially smoothed value for May. A) 276.9 B) 275.0 C) 272.0 D) 287.0

27)

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Wheat Harvested by Coop. Member (y, in thousands of bushels) 75 78 82 82 84 85 87 91 92 92 93 96 101 102

28) Find the least squares prediction equation for the model yt = 0 + 1 t + . ^

B) yt = -74.2 - 1.9165t

D) yt = 1.9165 - 74.2t

28)

A) yt = 74.2 + 1.9165t

C) yt = 74.2 - 1.9165t

yt = 7.81 + 0.91x 1t + 0.57x 2t

s = 1.29

Durbin-Watson d = 2.09

29) Is there evidence of positive autocorrelation of residuals in the consumption model presented above? Test using = 0.10. A) Yes, since the standard deviation s = 1.29 is small. B) Yes, since the Durbin-Watson statistic d = 2.09 falls in the rejection region. C) No, since the Durbin-Watson statistic d = 2.09 falls in the nonrejection region. D) No, since the standard deviation s = 1.29 is small.

29)

30) Propose a straight-line model that includes both a long-term trend and a seasonal component for the time series. Let t = the year in which the data was collected (t = 1, 2, . . . , 25) and let Q1 , Q2 , and Q3 be dummy variables used to model a seasonal effect.

A) E(Yt) = 0 + 1 Q1 + 3 Q2 + 4 Q3

B) E(Yt) = 1t

C) E(Yt) = 0 + 1 t

D) E(Yt) = 0 + 1 t + 2Q1 + 3 Q2 + 4 Q3

30)

(Situation G) The number of industrial and construction failures in the United States by the type of firm for the years 1985-1996 is given in the table.

Year 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996

Retail Trade 4799 4139 3406 2889 3183 4910 6882 9730 11,429 13,787 13,501 13,509

Wholesale Trade 1089 1028 887 740 908 1284 1709 2783 3598 4882 4835 4808

31) Using 1985 as the base year and using all five types of firms, calculate the simple composite index for 1995. A) 416.80 B) 217.88 C) 476.49 D) 23.99

31)

Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Wheat Harvested by Coop. Member (y, in thousands of bushels) 75 78 82 82 84 85 87 91 92 92 93 96 101 102 ^

32) Suppose the least squares regression equation is yt = 75 + 2t. Interpret the estimate of 1 in terms of the problem. A) We expect the volume of wheat harvested to increase 2000 bushels for each additional corporate member. B) We expect the volume of wheat harvested to be 2000 bushels in any given year. C) We expect to harvest 2000 bushels of wheat in 2005. D) We expect the mean volume of wheat harvested to increase 2000 bushels from one year to the next.

32)

33) A forecast was obtained for the year 2005 and the corresponding 95% prediction interval was found to be (103, 107). Interpret this interval. A) We are 95% confident that the volume of wheat harvested in 2005 will be between 103,000 and 107,000 bushels. B) We are 95% confident that the 2005 harvest will be between 103,000 and 107,000 bushels larger than the harvest in 2004. C) We expect the volume of wheat harvested to increase between 103,000 and 107,000 bushels from one year to the next. D) We are 95% confident that the mean volume of wheat harvested in all years will be between 103,000 and 107,000 bushels.

33)

(Situation H) The prices of coffee, gasoline, and sugar for each month of 1983 are shown below in the table.

Month January February March April May June July August September October November December

Price of Coffee (per pound) $1.47 $1.47 $1.47 $1.39 $1.36 $1.36 $1.36 $1.36 $1.36 $1.36 $1.36 $1.36

Price of Gasoline (per gallon) $1.15 $1.10 $1.06 $1.13 $1.18 $1.20 $1.21 $1.20 $1.19 $1.17 $1.16 $1.15

Price of Sugar (per pound) $0.32 $0.33 $0.32 $0.33 $0.33 $0.34 $0.34 $0.34 $0.34 $0.34 $0.33 $0.33

34) Using just the price of gasoline and a smoothing constant of w = 0.4, calculate the exponentially smoothed value for March. A) $1.076 B) $1.12 C) $1.102 D) $1.084

34)

MSE = 144

F = 4.25

Durbin-Watson d = 0.96

35) Use the value of the Durbin-Watson test statistic to make a statement about autocorrelation of residuals in the regression model above. A) Since the value lies in the inconclusive region (using = 0.05), we need more information before a definite conclusion can be drawn. B) There is insufficient evidence (using = 0.05) to indicate that positive autocorrelation exists. C) Approximately 98.5% of the residuals lie within 2 standard deviations of their mean 0. D) There is sufficient evidence (using = 0.05) to indicate that positive autocorrelation exists.

35)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 36) Consider the monthly time series shown in the table. Month January February March April May June July August September October November December a. b.

t 1 2 3 4 5 6 7 8 9 10 11 12

36)

Y 185 192 189 201 195 199 206 203 208 209 218 216

Calculate the values in the exponentially smoothed series using w = 0.6. Graph the time series and the exponentially smoothed series on the same graph.

(Situation J) The table lists the number (in millions) of Chevrolet passenger cars sold to dealers in the U.S. and Canada from 1980 to 1985. Year 1980 1981 1982 1983 1984 1985

Sales 1.740 1.444 0.896 1.289 1.455 4.882

37) Using a smoothing constant of w = 0.70, calculate the value of the exponentially smoothed series in 1985. Solve the problem. 38) Consider the table below which displays the price of a commodity for six consecutive years. Year 1 2 3 4 5 6

37)

38)

Price (dollars) 250 255 253 255 259 261

a. Use the Holt model to forecast values for Years 7-10 using w = 0.6 and v = 0.5. b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 263, 267, 269, 268 respectively. c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.

39) The printout below shows a regression analysis for a time series that included 20 observations.39) Regression Analysis: C2 versus C1 The regression equation is C2 = 1.20 + 0.0362 C1

Predictor Constant C1 S = 0.182361

Coef 1.19947 0.036241

SE Coef 0.08471 0.007072

R-Sq = 59.3%

T 14.16 5.12

P 0.000 0.000

R-Sq(adj) = 57.1%

Analysis of Variance Source Regression Residual Error Total

DF 1 18 19

SS 0.87340 0.59860 1.47200

MS 0.87340 0.03326

F 26.26

P 0.000

Durbin-Watson statistic = 1.87807 Locate the Durbin-Watson d-statistic and test the null hypothesis that there is no autocorrelation of residuals. Use = 0.10.

40) Consider the monthly time series shown in the table. Month January February March April May June July August September October November December

t 1 2 3 4 5 6 7 8 9 10 11 12

Y 185 192 189 201 195 199 206 203 208 209 218 216

Use the method of least squares to fit the model E(Yt) = 0 + 1 t to the data. Write the prediction equation. b. Use the prediction equation to obtain forecasts for the next two months. c. Find 95% forecast intervals for the next two months.

40)

41) The table below shows the price of a commodity for each of ten consecutive years.

41)

Year

Price

$1.19

$1.22

$1.23

$1.45

$1.39

$1.42

$1.47

$1.55

$1.62

$1.65

Use exponential smoothing with w = 0.6 to forecast the price of the commodity in years 11 and 12. (Situation F) The sales (in thousands of dollars) of automobiles by the three largest American automakers from 1986 through 1992 are shown in the table below. Year 1986 1987 1988 1989 1990 1991 1992

G.M. 8993 7101 6762 6244 7769 8256 9305

Ford 5810 4328 4313 4255 4934 5585 5551

Chrysler 1796 1225 1283 1182 1494 2034 2157

42) Using 1986 as the base year, find the simple composite index for 1992.

42)

(Situation K) Foreign Exchange rates, the values of foreign currency in U.S. dollars, are important to investors and international travelers. The table lists the monthly foreign exchange rates of the British pound (in U.S. dollars per pound) for a certain year. Month January February March April May June July August September October November December

Exchange Rate 1.13 1.10 1.13 1.23 1.25 1.28 1.38 1.39 1.36 1.42 1.44 1.44

43) Calculate the value of the exponentially smoothed series for April using a smoothing constant of w = 0.7.

43)

Solve the problem. 44) Consider the monthly time series shown in the table. Month January February March April May June July August September October November December

t 1 2 3 4 5 6 7 8 9 10 11 12

44)

Y 185 192 189 201 195 199 206 203 208 209 218 216

a. Use the method of least squares to fit the model E(Yt) = 0 + 1 t to the data. Write the prediction equation. b. Construct a residual plot for the model. c. Is there evidence of a cyclical component? Explain.

45) Consider the table below which displays the price of a commodity for six consecutive years. Year 1 2 3 4 5 6

Price (dollars) 250 255 253 255 259 261

a. Use the method of least squares to fit the model E(Yt) = 0 + 1 t to the data. Write the prediction equation. b. Use the prediction equation to obtain forecasts of the prices in years 7 and 8. c. Find 95% prediction intervals for years 7 and 8.

45)

46) Retail sales for a home improvement store in quarters 1-4 over a five-year period are shown 46) (in millions of dollars) in the table below. Quarter Year 1 2 3 4 5

1 1.2 1.3 1.4 1.4 1.6

2 1.4 1.6 1.8 1.7 2.0

3 1.5 1.5 1.8 1.9 2.1

4 1.1 1.2 1.6 1.6 1.9

a. Write a regression model that contains trend and seasonal components to describe the sales data. b. Use least squares regression to fit the model. c. Use the regression model to forecast the quarterly sales during Year 6. Give 95% prediction intervals for the forecasts.

47) The table below shows the prices and quantities of three commodities for six consecutive years.47)

Year 1 2 3 4 5 6

Commodity A Price Quantity 250 1200 255 1500 253 2700 255 1800 259 2100 261 2000

Commodity B Price Quantity 121 3200 115 3500 128 2400 126 2800 129 2700 135 2500

Commodity C Price Quantity 675 1800 700 1900 714 2100 721 2500 725 3100 734 3900

a. Compute the Laspeyres price index for the six-year period, using Year 1 as the base period. b. Compute the Paasche price index for the six-year period, using Year 1 as the base period. c. Plot the Laspeyres and Paasche indexes on the same graph. Comment on the differences.

48) Consider the table below which displays the price of a commodity for six consecutive years. Year Price 1 250 2 255 3 253 4 255 5 259 6 261 a. Use the method of least squares to fit the model E(Yt) = 0 + 1 t to the data. Write the prediction equation. b. Calculate the residuals and construct a residual plot. c. Calculate the Durbin Watson d statistic.

48)

49) Consider the monthly time series shown in the table. Month January February March April May June July August September October November December

t 1 2 3 4 5 6 7 8 9 10 11 12

49)

Y 185 192 189 201 195 199 206 203 208 209 218 216

a. Use the values of Y in the table to forecast the values of Y for the next two months, using simple exponential smoothing with w = 0.7. b. Use the Holt model with w = 0.7 and v = 0.7 to forecast the values of Y for the next two months.

50) Consider the table below which displays the price of a commodity for six consecutive years. Year 1 2 3 4 5 6

50)

Price (dollars) 250 255 253 255 259 261

a. Calculate the values in the exponentially smoothed series using w = 0.6. b. Calculate the forecast errors for Years 7-10 if the actual values in those years are 262, 264, 263, 266 respectively. c. Calculate MAD, MAPE, and RMSE, using the forecast errors for Years 7-10.

51) The table below shows the price of a commodity for each of ten consecutive years.

51)

Year

Price

$1.19

$1.22

$1.23

$1.45

$1.39

$1.42

$1.47

$1.55

$1.62

$1.65

a. Using Year 1 as the base period, calculate the simple index for the price of the commodity for each year. b. Plot the simple indexes for years 1-10. c. Use the simple index to interpret the trend in the price of the commodity.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the statement is true or false. 52) We plot time series residuals against observed values of Y to determine whether a cyclical component is apparent. A) True B) False

52)

53) A major advantage of forecasting with smoothing techniques is that the standard deviation of the forecast errors is known prior to observing the future values. A) True B) False

53)

54) The Laspeyres index is a weighted index while the Paasche index is not weighted. A) True B) False

54)

55) The least squares model is an excellent choice for forecasting time series since it works particularly well outside the region of known observations. A) True B) False

55)

56) The value of the Durbin-Watson d-statistic always falls in the interval from 0 to 1. A) True B) False

56)

57) Since the theoretical distributional properties of the forecast errors with smoothing methods are unknown, many analysts regard smoothing methods as descriptive procedures rather than inferential procedures. A) True B) False

57)

58) The Laspeyres index uses the purchase quantities of the period as weights. A) True B) False

58)

59) It is common to use dummy variables to describe seasonal differences in a time series. A) True B) False

59)

60) Fourth-order autocorrelation in a quarterly time series may indicate seasonality. A) True B) False

60)

61) The straight-line regression model accounts for both the secular trend and the cyclical effect in a time series. A) True B) False

61)

62) The choice of exponential smoothing constant w has little or no effect on forecast values found using exponential smoothing. A) True B) False

62)

63) Smaller choices of the exponential smoothing constant w assign more weight to the current value of the series and yield a smoother series. A) True B) False

63)

64) Smaller values of the trend smoothing constant v assign more weight to the most recent trend of the series and less to past trends. A) True B) False

64)

65) The d-test requires that the residuals be normally distributed. A) True B) False

65)

66) A composite index number represents combinations of the prices or quantities of several commodities. A) True B) False

66)

67) One of the major weaknesses of exponential smoothing is that it is not easily adapted to forecasting. A) True B) False

67)

68) The Holt forecasting model consists of both an exponentially smoothed component and a trend component. A) True B) False

68)

69) Smoothing techniques are used to remove rapid fluctuations in a time series so the general trend can be seen. A) True B) False

69)

70) The exponential smoothing constant can be any number between 0 and 100. A) True B) False

70)

71) Price indexes measure changes in the price of a commodity or group of commodities over time. A) True B) False

71)

72) With N time periods in your data, a good rule of thumb is to forecast ahead no more than 2N time periods. A) True B) False

72)

73) The exponentially smoothed forecast takes into account both changes in trend and seasonality.

73)

A) True

B) False

Answer Key Testname: CHAPTER 14 1) D 2) D 3) D 4) A 5) C 6) B 7) D 8) B 9) D 10) D 11) D 12) B 13) B 14) C 15) A 16) C 17) D 18) D 19) D 20) C 21) D 22) B 23) A 24) B 25) B 26) C 27) A 28) A 29) C 30) D 31) A 32) D 33) A 34) C 35) D

Answer Key Testname: CHAPTER 14 36) a. Month January February March April May June July August September October November December

t 1 2 3 4 5 6 7 8 9 10 11 12

E 185 189.2 189.1 196.2 195.5 197.6 202.6 202.9 205.9 207.8 213.9 215.2

37) The first value of the exponentially smoothed series is equal to the first value of the time series. The rest of the values are found using the following: Et = wYt + (1 - w)Et-1 , where w is the smoothing constant For w = 0.70

1980:

1981: 1982: 1983: 1984: 1985:

E1 = Y1 = 1.740

E2 = wY2 + (1 - w)E1 = 0.7(1.444) + (1 - 0.7)(1.740) = 1.533 E3 = wY3 + (1 - w)E2 = 0.7(0.896) + (1 - 0.7)(1.533) = 1.087 E4 = wY4 + (1 - w)E3 = 0.7(1.289) + (1 - 0.7)(1.087) = 1.228 E5 = wY5 + (1 - w)E4 = 0.7(1.455) + (1 - 0.7)(1.228) = 1.387 E6 = wY6 + (1 - w)E5 = 0.7(4.882) + (1 - 0.7)(1.387) = 3.834

Answer Key Testname: CHAPTER 14 38) a. Year 7 8 9 10

Forecast 262.98 265.07 267.17 269.26

Year

Forecast Error 0.019 1.925 1.832 -1.262

7 8 9 10

c. MAD: 1.26; MAPE: 0.47; RMSE: 1.47 39) d = 1.87807; The rejection region is d < 1.2 or 4 - d < 1.2. Since d does not fall within the rejection region, we cannot reject the null hypothesis. There is insufficient evidence of autocorrelation. ^

40) a. y = 184 + 2.73t b. 219.50; 222.23 c. (211.07, 227.93); (213.50, 230.96) 41) Year 11: $1.62; Year 12: $1.62 42) The simple composite index for 1992 is: 9305 + 5551 + 2157 × 100 I1992 = 8993 + 5810 + 1796 = 102.49 43) The first value of the exponentially smoothed series is equal to the first value in the time series. The rest of the values are found using the following: Et = wYt + (1 - w)Et-1 , where w is the smoothing constant For w = .70:

Jan: Feb: March: April:

E1 = Y1 = 1.13

E2 = wY2 + (1 - w)E1 = 0.7(1.10) + (1 - 0.7)(1.13) = 1.109

E3 = wY3 + (1 - w)E2 = 0.7(1.13) + (1 - 0.7)(1.109) = 1.124 E4 = wY4 + (1 - w)E3 = 0.7(1.23) + (1 - 0.7)(1.124) = 1.198

Answer Key Testname: CHAPTER 14

44) a. b.

y = 184 + 2.73t

No, there are no long runs of either positive or negative residuals. ^

45) a. b. c.

y = 248.6 + 1.97x Year 7: 262.40; Year 8: 264.37 Year 7: (255.97, 268.83); Year 8: (257.20, 271.54)

46) a.

E(Yt) = 0 + 1 t + 2Q1 + 3 Q2 + 4 Q3

1 if Quarter 1 1 if Quarter 2 1 if Quarter 3 Q1 = , Q2 = , Q3 = 0 if not 0 if not 0 if not b. c.

y = 1.045 + 0.0362t + 0.0088Q1 + 0.293Q2 + 0.316Q3

Quarter 1: 1.815, (1.5631, 2.0669); Quarter 2: 2.135, (1.8831, 2.3869); Quarter 3: 2.195, (1.9431, 2.4469); Quarter 4: 1.915, (1.6631, 2.1669)

Answer Key Testname: CHAPTER 14 47) a. Year 1 2 3 4 5 6

Laspeyres Index 100 101.6717 105.0573 105.5094 106.6449 108.6321

Year

Paasche Index 100 101.6338 104.4819 105.5728 106.6402 108.3581

1 2 3 4 5 6 c.

Though the values of the Laspeyres index tend to be slightly higher, there is actually very little difference between the values of the indexes.

Answer Key Testname: CHAPTER 14

48) a. b.

y = 248.6 + 1.971t Year 1 2 3 4 5 6

c. 49) a. b. 50) a.

Predicted Y Residuals 250.57 -0.57 252.54 2.46 254.51 -1.51 -1.49 256.49 258.46 0.54 260.43 0.57

d = 2.53 215.72; 215.72 220.31; 223.03 Year 1 2 3 4 5 6

Smoothed Value 250 253 253 254.2 257.08 259.432

b. Year 7 8 9 10

Forecast Error 2.568 4.568 3.568 6.568

c. MAD: 4.32; MAPE: 1.63; RMSE: 4.56

Answer Key Testname: CHAPTER 14 51) a. Year 1 Index 100

2 3 4 102.52 103.36 121.85

5 6 7 116.81 119.33 123.53

8 130.25

9 10 136.13 138.66

c. In general, the price of the commodity is increasing with a sharp increase in Year 4 followed by a slight decrease in Year 5. During the ten-year period, the price of the commodity increased 38.66%. 52) B 53) B 54) B 55) B 56) B 57) A 58) B 59) A 60) A 61) B 62) B 63) B 64) B 65) A 66) A 67) B 68) A 69) A 70) B 71) A 72) B 73) B

Chapter 15 Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) Independent random samples from two populations are shown in the table.

11 3 16

Sample 1 14 9 12 5

19 22 14

Sample 2 17 10 18 8 25

Use the Wilcoxon rank sum test to determine whether the data provide sufficient evidence to indicate a shift in the locations of the probability distributions of the sampled populations. Use = 0.05.

2) Suppose you have used a randomized block design to compare the effects of four different energy drinks on running speeds. Eight athletes were randomly selected. Each Monday each athlete was assigned an energy drink and their time to run four miles was recorded. The results (in seconds) are shown below. Is there evidence of a difference in the probability distributions of the running times among the four different drinks? Apply the Friedman Fr-test to the data. Be sure to specify the null and alternative hypotheses. Use = 0.025.

Runner 1 2 3 4 5 6 7 8

1 1214 1118 1218 1245 1148 1307 1209 1232

Drink 2 3 1215 1262 1024 1140 1346 1280 1206 1261 1110 1204 1284 1337 1250 1246 1123 1260

4 1233 1148 1292 1256 1209 1333 1232 1237

3) A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten subjects are randomly selected and their cholesterol levels are recorded. The results are listed below. The subjects were placed on the drug for a period of 6 months, after which their cholesterol levels were tested again. The results are listed below. (All units are milligrams per deciliter.) Use the Wilcoxon signed rank test to test the company's claim that the drug lowers cholesterol levels. Use = 0.05. Subject Before After

1 2 3 4 5 6 7 8 9 10 240 242 231 195 194 237 248 239 218 186 225 237 239 185 189 237 218 221 216 171

4) Nine students took the SAT test. Later, they took a test preparation course and retook the SAT. Their original scores and new scores are shown below. Use the Wilcoxon signed rank test to test the claim that the test preparation had an effect on their scores. Use = 0.05.

Student 1 2 3 4 5 6 7 8 9 Before Score 1050 1110 1170 1080 970 920 900 1040 810 After Score 1070 1110 1160 1120 1000 930 890 1080 830

5) What is referred to as nonparametrics?

6) A government agency claims that the median hourly wages for workers at fast food restaurants in the western U.S. is $6.90. In a random sample of 100 workers, 68 were paid less than $6.90, 10 were paid $6.90, and the rest more than $6.90. Test the government's claim. Use = 0.05.

7) What are rank statistics (or rank tests)?

8) The drama department at a college asked professors and students in the drama department to rank 8 actors according to their performance. The data are listed below. A 10 is the highest ranking and a 1 the lowest ranking. Calculate Spearman's rank correlation coefficient. Test the claim of no correlation between the rankings. Be sure to specify the null and alternative hypotheses. Use = 0.05.

Actor Professors Students

1 2 4

2 3 3

3 6 1

4 10 4

5 8 5

6 1 7

7 5 9

8 4 6

9) The temperatures on randomly chosen days during a summer class and the number of absences from class on those days are listed below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between the temperature and the number absent? Use = 0.01.

Temp 62 75 81 80 78 88 65 90 70 Absences 11 15 18 18 16 23 12 23 13

10) Four different types of fertilizers are used on raspberry plants. The number of raspberries on each of 24 randomly selected plants is given below. Use the Kruskal-Wallis H-test to test whether the distributions of the numbers of raspberries differ among the four groups. Be sure to specify the null and alternative hypotheses. Use = 0.05. Fertilizer 1 Fertilizer 2 Fertilizer 3 Fertilizer 4 22 21 22 19 21 24 19 21 22 21 20 19 23 21 19 20 23 21 18 21 22 22 19 20

10)

11) The final exam scores of 10 randomly selected statistics students and the number of hours they studied for the exam are given below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between the scores on the test and the times spent studying? Use = 0.01. Hours 6 Scores 68

8 83

5 63

11 5 91 69

7 81

7 88

8 93

11)

9 6 93 74

12) A researcher wishes to determine whether there is a difference in the average age of elementary school, high school, and community college teachers. Teachers are randomly selected. Their ages are recorded below. Use the Kruskal-Wallis H-test to test whether the distributions of the ages of teachers differ among the three types of school. Be sure to specify the null and alternative hypotheses. Use = 0.05.

12)

Elementary School High School Community College Teachers Teachers Teachers 29 40 43 32 45 49 31 42 40 56 51 65 41 46 49 29 35 39

13) What are distribution-free tests?

13)

14) Suppose you have used a randomized block design to compare the efficacy of three different doses of an experimental drug. You used seven patients in your study. The data are listed below. Do the data indicate that a particular dosage is more effective than other dosages? Apply the Friedman Fr-test to the data. Be sure to specify the null and alternative hypotheses. Use = 0.10.

14)

Patient 1 2 3 4 5 6 7

A -40 -36 -38 -39 -40 -35 -39

Dosage B -38 -36 -37 -34 -41 -37 -37

C -31 -36 -37 -33 -39 -33 -34

15) Calculate or use a table to find the binomial probability P(x 20) when n = 25 and p = 0.5. Also use the normal approximation to calculate the probability.

15)

six 16) The median household income of a community is reported to be $62,000. A random sample of 16) households in the community yielded the following incomes. $42,000

$57,000

$65,000

$69,000

$75,000

$150,000

Does the sample provide sufficient evidence to refute the reported median household income? Perform a sign test using = 0.10.

17) The number of absences and the final grades of 9 randomly selected students from a statistics class are given below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between the final grade and the number of absences? Use = 0.01. Number of Absences 0 3 6 4 9 2 Final Grade 98 86 80 82 71 92

15 55

17)

8 5 76 82

18) A physician claims that a person's diastolic blood pressure can be lowered, if, instead of taking a drug, the person listens to a relaxation tape each evening. Ten subjects are randomly selected. Their blood pressures, measured in millimeters of mercury, are listed below. The 10 patients are given the tapes and told to listen to them each evening for one month. At the end of the month, their blood pressures are taken again. The data are listed below. Use the Wilcoxon signed rank test to test the physician's claim. Use = 0.05.

18)

Patient 1 2 3 4 5 6 7 8 9 10 Before 99 89 91 82 93 87 83 84 86 96 After 96 83 91 74 87 76 86 74 82 80

19) The grade point averages of students participating in different sports at a college are to be compared. The GPAs of students randomly selected from three different groups are listed below. Use the Kruskal-Wallis H-test to test whether the distributions of GPAs differ among the three groups. Be sure to specify the null and alternative hypotheses. Use = 0.05. Tennis 3.0 2.4 2.3 3.3 2.9 1.9

Golf 1.6 1.9 3.1 1.7 2.1 1.8

19)

Swimming 2.5 2.8 2.6 2.3 2.3 2.2

20) Calculate or use a table to find the binomial probability P(x 6) when n = 9 and p = 0.5.

20)

21) Test the hypothesis that the median age of statistics teachers is 45 years. A random sample of 60 statistics teachers found 25 above 45 years and 35 below 45 years. Use = 0.01.

21)

22) Suppose you want to compare two treatments, A and B. In particular, you wish to determine 22) whether the distribution for population B is shifted to the right of the distribution for population A. You plan to use the Wilcoxon rank sum test. a.

Specify the null and alternative hypotheses you would test.

b. Suppose you obtained the following independent random samples of observations on experimental units subjected to the two treatments. Conduct the test of hypotheses described above, using = 0.05. Sample A: 1.2, 1.5, 2.3, 3.2, 3.7, 4.1 Sample B: 2.5, 2.8, 3.6, 4.2, 4.5

23) A researcher wishes to determine whether physical exercise is effective in helping people to lose weight. 20 people were randomly selected to participate in an exercise program for 30 days. Use the Wilcoxon signed rank test to test the claim that exercise has an effect on weight. Use = 0.02.

23)

Weight Before Program (in Pounds) 178 210 156 188 193 225 190 165 168 200 Weight After Program (in Pounds) 182 205 156 190 183 220 195 155 165 200 Weight Before Program (in Pounds) 186 172 166 184 225 145 208 214 148 174 Weight After Program (in Pounds) 180 173 165 186 240 138 203 203 142 170

24) Specify the rejection region for the Wilcoxon rank sum test in the following situation. n 1 = 7, n2 = 5,

24)

= 0.05

H0 : Two probability distributions, 1 and 2, are identical

Ha : Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

25) Specify the rejection region for the Wilcoxon rank sum test in the following situation. n 1 = 6, n2 = 8,

= 0.10

H0 : Two probability distributions, 1 and 2, are identical

Ha : Probability distribution of population 1 is shifted to the right or left of the probability distribution for population 2

25)

26) A technician is interested in comparing the time it takes to assemble a certain computer component using three different machines. Workers are randomly selected and randomly assigned to one of the machines. The assembly times (in minutes) are shown in the table. Use the Kruskal-Wallis H-test to test whether the distributions of assembly times differ for the three different machines. Be sure to specify the null and alternative hypotheses. Use = 0.05.

26)

Machine 1 Machine 2 Machine 3 28 36 24 27 25 21 28 34 25 26 29 27 29 31 26 27 28 23 28 32 33

27) A local school district is concerned about the number of school days missed by its teachers due to illness. A random sample of 10 teachers is selected. An incentive program is offered in an attempt to reduce absences. The number of days of absence in the year before the incentive program and in the year after the incentive program are shown below for each teacher. Use the Wilcoxon signed rank test to test the claim that the incentive program is effective in reducing absences. Use = 0.05. Teacher 1 Days Absent Before Incentive 5 Days Absent After Incentive 3

28) Independent random samples from two populations are shown below.

27)

28)

Sample A: 35, 38, 42, 43, 45, 47, 49, 58 Sample B: 36, 41, 44, 53, 57 Calculate the rank sum for each sample. Which would be used as the test statistic in a Wilcoxon rank sum test?

29) Fading of wood is a problem with wooden decks on boats. Three varnishes used to retard this aging process were tested to see whether there were any differences among them. Samples of 10 different types of wood were treated with each of the three varnishes and the amount of fading was measured after three months of exposure to the sun. The data are listed below. Is there evidence of a difference in the probability distributions of the amounts of fading for the three different types of varnish? Apply the Friedman Fr-test to the data. Be sure to specify the null and alternative hypotheses. Use = 0.05.

Sample 1 2 3 4 5 6 7 8 9 10

1 4.2 6.8 3.0 5.8 8.0 5.2 3.3 5.7 4.5 4.8

29)

Varnish 2 3 3.8 4.5 6.0 6.6 3.1 3.0 4.9 5.5 6.4 7.4 4.4 5.3 3.3 3.7 5.8 6.0 3.7 3.8 4.6 4.7

30) Specify the rejection region for the Wilcoxon signed rank test in the following situation.

30)

n = 25, = 0.05 H0 : Two probability distributions, 1 and 2, are identical

Ha : Probability distribution of population 1 is shifted to the right of the probability distribution for population 2

31) A weight-lifting coach claims that a weight-lifter can increase strength by taking vitamin E. To test the theory, the coach randomly selects 9 athletes and gives them a strength test using a bench press. Thirty days later, after regular training supplemented by vitamin E, they are given the same test again. The weights pressed (in pounds) before and after the vitamin E regimen are shown below. Use the Wilcoxon signed rank test to test the claim that the vitamin E supplement is effective in increasing the athletes' strength. Use = 0.05.

31)

Athlete 1 2 3 4 5 6 7 8 9 Before 258 188 275 248 220 208 254 218 181 After 268 193 275 246 227 223 259 213 186

32) Verbal SAT scores for students randomly selected from two different schools are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the scores from the two schools. Use = 0.05. School 1 530 500 750 460 730 510 560 760 590 570 710 730

School 2 470 420 660 410 690 570 670 530 510 610 620 520

32)

33) A consumer protection organization claims that a new car model gets less than 27 miles per gallon of gas. Ten cars are tested. The results are given below. Test the organization's claim. Use = 0.05. 21.8

19.6

25.8

20.9

26.2

29.3

23.9 18.7 25

34) A realtor wishes to compare the square footage of houses of similar prices in 4 different cities. The data are listed below. Use the Kruskal-Wallis H-test to test whether the square-footage distributions differ for the four different cities. Be sure to specify the null and alternative hypotheses. Use = 0.05. City 1 2720 2550 2570 2780 2470 2620

City 2 2350 2110 2260 2220 2270

City 3 2100 2240 2150 2170 2070 2320 2220

34)

City 4 2970 2920 3170 2720 2570 2770 2920 2820

35) A convenience store owner believes that the median number of lottery tickets sold per day is 54. The lottery company believes the median number is smaller. A random sample of 20 days yields the following data. Test the lottery company's claim. Use = 0.05. 47 63 74 62 69 69

33)

35)

79 46 70 85 42 48 53 59 59 64 64 74 69 53

36) The table below lists the verbal and math SAT scores of 10 students selected at random. Calculate Spearman's rank correlation coefficient. Test the hypothesis of no correlation between verbal and math SAT scores. Be sure to specify the null and alternative hypotheses. Use = 0.05.

36)

Verbal 510 595 600 505 585 Math 595 665 690 625 675 Verbal 615 515 565 635 525 Math 640 725 645 515 525

37) Independent random samples from two populations are shown below.

37)

Sample A: 11, 15, 18, 21 Sample B: 9, 12, 15, 17, 20, 23 Calculate the rank sum for each sample. Which would be used as the test statistic in a Wilcoxon rank sum test?

38) Six patients were each given four different pain killers and asked to rate each pain killer's effectiveness in reducing pain on a scale of 1 to 10. A Friedman Fr-test was performed on

38)

the results. A printout is shown below.

Friedman Test: Response versus Treatment blocked by Patient S = 18.00 DF = 3 P = 0.000

Treatment 1 2 3 4

N 6 6 6 6

Est Median 7.500 4.500 6.000 2.500

Sum of Ranks 24.0 12.0 18.0 6.0

Grand median = 5.125 Is there evidence that at least two of the treatment probability distributions differ in location? Explain.

39) The ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults are given below. Calculate Spearman's rank correlation coefficient. Can you conclude that there is a correlation between age and blood pressure? Use = 0.05.

39)

Age 38 41 45 48 51 53 57 61 65 Pressure 116 120 123 131 142 145 148 150 152

40) A medical researcher wishes to try three different techniques to lower blood pressure of patients with high blood pressure. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is assigned an exercise program and Group 3 is assigned a dietary regimen. At the end of six weeks, the reduction in each subject's blood pressure is recorded. The results (in mmHg) are shown in the table. Use the Kruskal-Wallis H-test to test whether the distributions of the blood pressure reductions differ among the three groups. Be sure to specify the null and alternative hypotheses. Use = 0.05.

40)

Group 1 Group 2 Group 3 10 7 5 11 4 11 8 1 3 14 2 7 12 3 8 -1 7 3

41) Specify the rejection region for the Wilcoxon signed rank test in the following situation. n = 35, = 0.10 H0 : Two probability distributions, 1 and 2, are identical

Ha : Probability distribution of population 1 is shifted to the right or left of the probability distribution for population 2

41)

42) A real estate agent believes that the median rent for a one-bedroom apartment in a beach community in southern California is greater than $1800 per month. The rents for a random sample of 15 one-bedroom apartments are listed below. Test the agent's claim. Use = 0.01. $2100 $2550 $1795

$2050 $1975 $1800

$1500 $1470 $1875

$1675 $2190 $1800

42)

$1535 $2800 $1580

43) A researcher wants to know if the time spent in prison for a particular type of crime is the same for men and women. A random sample of men and women were each asked to give the length of sentence received. The data, in months, are listed below. Use the Wilcoxon rank sum procedure to test the claim that there is no difference in the sentences received by men and the sentences received by women. Use = 0.05.

43)

Men 10 22 16 18 19 26 Women 9 12 9 14 26 12 Men 14 22 12 19 23 24 Women 34 8 10 13 17 27

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question True or False. 44) The Wilcoxon signed rank test for large samples can be used when n 10. A) True B) False

44)

45) The sign test provides inferences about the population median rather than the population mean. A) True B) False

45)

46) When applying the Wilcoxon signed rank test, the number of ties should be small relative to the number of observations to ensure the validity of the test. A) True B) False

46)

47) For a sign test to be valid, a large sample must be selected from a population which is approximately normally distributed. A) True B) False

47)

48) The Wilcoxon rank sum test is used to test the hypothesis that the probability distributions associated with two populations are equivalent. A) True B) False

48)

49) The Wilcoxon rank sum test is recommended for comparing discrete distributions. A) True B) False

49)

Answer Key Testname: CHAPTER 15 1) H0 : Two probability distributions, 1 and 2, are identical

Ha : Probability distribution of population 1 is shifted to the right or left of the probability distribution for population 2

Since the smaller sample is sample 1, the test statistic is T = 38.5, the rank sum for sample 1. The rejection region is T 39 or T 73. Since the test statistic falls in the rejection region, we reject the null hypothesis. There is sufficient evidence to indicate that the distribution for population 1 is shifted to the right or left of the distribution for population 2. 2) H0 : The probability distributions of running times are identical for the four different types of drink

Ha : At least two of the four drinks have running-time distributions that differ in location Critical value 9.348; Fr = 11.1; reject H0

There is enough evidence to conclude that at least two of the four drinks have running-time distributions that differ in location 3) H0 : Probability distribution of cholesterol levels before drug is identical to probability distribution of cholesterol levels after

drug, Ha : Probability distribution of cholesterol levels after drug is shifted to the left of probability distribution of cholesterol levels before drug; critical value 8; test statistic T = 4; reject H0 ; There is sufficient evidence to support the claim that the drug is effective in

lowering cholesterol levels. 4) H0 : Probability distribution of before scores identical to probability distribution of after scores,

Ha : Probability distribution of scores after test prep is shifted to the right or left of probability distribution of scores before test prep; critical value 4; test statistic T = 4; reject H0 ; There is sufficient evidence to conclude that the test preparation had an effect

on scores. 5) The branch of inferential statistics devoted to distribution-free tests is called nonparametrics. 6.90; critical value 1.96; test statistic z = 3.5; reject H0 ; There is sufficient evidence to reject 6) H0 : = 6.90 versus Ha:

the claim. 7) nonparametric statistics (or tests) based on the ranks of measurements 0 8) H0 : = 0; Ha : Critical values ±0.738; test statistic rs -0.216; fail to reject H0 .

There is not enough evidence to conclude that there is a significant correlation between the rankings. 9) critical values ±0.833; test statistic rs 0.992; reject H0 ; There is enough evidence to conclude that there is a significant

correlation between the temperature and the number of absences. 10) H0 : The distributions of the numbers of raspberries are the same for the three groups Ha : At least two of the four groups have distributions that differ in location Critical value 7.815; test statistic H 12.833; reject H0

There is enough evidence to conclude that at least two of the four groups have distributions that differ in location 11) critical values ±0.794; test statistic rs 0.889; reject H0 ; There is enough evidence to conclude that there is a significant

correlation between the scores and the time spent studying. 12) H0 : The distributions of the ages of teachers are the same for the three types of school

Ha : At least two of the three types of school have age distributions that differ in location Critical value 5.991; test statistic H 4.056; fail to reject H0

There is not enough evidence to conclude that there is a difference in the distributions of teachers' ages at the three different types of school 13) Distribution-free tests are statistical tests that do not rely on any underlying assumptions about the probability distribution of the sampled population.

Answer Key Testname: CHAPTER 15 14) H0 : The probability distributions are identical for the three different dosages Ha : At least two of the three probability distributions differ in location Critical value 4.605; Fr 6.93; reject H0 .

There is enough evidence to conclude that at least two of the three probability distributions differ in location 15) P(x 20) = 0.0020; P(z 2.8) = 0.0026 62,000; p-value = 2 · P(x 4) = 0.688; Since the p-value is not less than 0.10, we do not 16) H0 : = 62,000 versus Ha :

reject the null hypothesis. There is insufficient evidence to refute the reported median income of $62,000. 17) critical values ±0.833; test statistic rs -0.996; reject H0 ; There is enough evidence to conclude that there is a significant

correlation between the final grade and the number of absences. 18) H0 : Probability distribution of blood pressures before relaxation tapes is identical to probability distribution of blood pressures after relaxation tapes, Ha : Probability distribution of blood pressures after relaxation tapes is shifted to the left of probability distribution of blood pressures before relaxation tapes; critical value 8; test statistic T = 1.5; reject H0 ; There is sufficient evidence to support the claim that the relaxation tapes

are effective in lowering blood pressure. 19) H0 : The distributions of GPAs are the same for the three groups

Ha : At least two of the three groups have GPA distributions that differ in location Critical value 5.991; test statistic H 5.108; fail to reject H0 .

There is not enough evidence to conclude that there is a difference in the GPA distributions of the three different groups 20) P(x 6) = 0.254 45; critical value 2.575; test statistic z 1.162; fail to reject H0 ; There is not sufficient evidence 21) H0 : = 45 versus Ha :

to reject the hypothesis that the median age of statistics teachers is 45 years. 22) a. H0 : Two probability distributions, A and B, are identical

Ha : Probability distribution of population A is shifted to the left of the probability distribution for population B

b. Since the smaller sample is sample B, the test statistic is T = 37, the rank sum for sample B. The rejection region is T 40. Since the test statistic does not fall in the rejection region, we do not reject the null hypothesis. There is insufficient evidence to conclude that the distribution for population B is shifted to the right of the distribution for population A. 23) H0 : Probability distribution of weights before exercise program is identical to probability distribution of weights after exercise

program, Ha : Probability distribution of weights after exercise program is shifted to the right or to the left of probability distribution of weights before exercise program; critical value 33; test statistic T = 42.5; fail to reject H0 ; There is not sufficient evidence to support the claim that exercise

has an effect on weight. 24) The test statistic is T = T2 since population 2 has the smaller sample. The rejection region is T2 22.

25) The test statistic is T = T1 since population 1 has the smaller sample. The rejection region is T 32 or T 58. 26) H0 : The distributions of assembly times are the same for the three different machines

Ha : At least two of the three machines have assembly time distributions that differ in location Critical value 5.991; test statistic H 7.482; reject H0

There is enough evidence to conclude that at least two of the three machines have assembly time distributions that differ in location.

Answer Key Testname: CHAPTER 15 27) H0 : Probability distribution of absences before incentive program is identical to probability distribution of absences after

incentive program, Ha : Probability distribution of absences after incentive program is shifted to the left of probability distribution of absences before incentive program; critical value 6; test statistic T = 2; reject H0 ; There is sufficient evidence to support the claim that the incentive program is

effective in reducing absences. 28) Population A: T1 = 1 + 3 + 5 + 6 + 8 + 9 + 10 + 13 = 55 Population B: T2 = 2 + 4 + 7 + 11 + 12 = 36

The test statistic is T2 = 36 since it corresponds to the smaller sample.

29) H0 : The probability distributions of amounts of fading are identical for the three different types of varnish Ha : At least two of the three types of varnish have distributions of fading amounts that differ in location Critical value 5.991; Fr = 6.35; reject H0

There is enough evidence to conclude that at least two of the three types of varnish have distributions of fading amounts that differ in location 30) T 101 31) H0 : Probability distribution of before weights identical to probability distribution of after weights, Ha : Probability distribution of weights after vitamin E is shifted to the right of probability distribution of weights before vitamin E; critical value 6; test statistic T = 4.5; reject H0 ; There is sufficient evidence to support the claim that the vitamin E

supplement is effective in increasing the athletes' strength. 32) H0 : the two schools have identical probability distributions,

Ha : the distribution of school 1 is shifted to the right or to the left of the distribution for school 2; critical values ±1.96;

T1 test statistic z =

n 1 (n 1 +n 2+1) 2

n 1 n 2 (n 1 +n 2 +1) 12

171.5 =

12(25) 2

12*12(25) 12

1.241;

fail to reject H0 ; There is not sufficient evidence to reject the claim.

33) H0 :

= 27 versus Ha : < 27; p-value = P(x 9) = 0.011; Since the p-value is less than 0.05, we reject the null hypothesis. There is sufficient evidence to support the organization's claim. 34) H0 : The square-footage distributions are the same for the four different cities Ha : At least two of the four cities have square-footage distributions that differ in location Critical value 7.815; test statistic H 20.657; reject H0

There is enough evidence to conclude that at least two of the four cities have square-footage distributions that differ in location 35) H0 : = 64 versus Ha : < 64; p-value = P(x 10) = 0.588; Since the p-value is not less than 0.05, we do not reject the

null hypothesis. There is insufficient evidence to reject the lottery company's claim. 0 36) H0 : = 0; Ha : Critical values ±0.648; test statistic rs -0.006; fail to reject H0

There is not enough evidence to conclude that there is a significant correlation between verbal and math SAT scores. 37) Population A: T1 = 2 + 4.5 + 7 + 9 = 22.5 Population B: T2 = 1 + 3 + 4.5 + 6 + 8 + 10 = 32.5

The test statistic is T1 = 22.5 since it corresponds to the smaller sample.

38) Yes, there is evidence that at least two of the treatment probability distributions differ in location. The p-value is 0.000. 13

Answer Key Testname: CHAPTER 15 39) critical values ±0.683; test statistic rs = 1.000; reject H0 ; There is enough evidence to conclude that there is a significant correlation between age and blood pressure. 40) H0 : The distributions of the blood pressure reductions are the same for the three groups

Ha : At least two of the three groups have blood pressure reduction distributions that differ in location Critical value 5.991; test statistic H 10.187; reject H0

The data provide ample evidence that at least two of the three groups have distributions that differ in location 41) T 214 42) H0 : = 1800 versus Ha : > 1800; p-value = P(x 7) = 0.696; Since the p-value is greater than 0.05, we do not reject the

null hypothesis. There is insufficient evidence to support the real estate agent's claim. 43) H0 : men and women have identical probability distributions,

Ha : the distribution for men is shifted to the right or to the left of the distribution for women; critical values ±1.96;

T1 test statistic z =

n 1 (n 1 +n 2 +1) 2

n 1 n 2 (n 1 +n 2 +1) 12

174.5 =

12(25) 2

12*12(25) 12

1.415;

fail to reject H0 ; There is not sufficient evidence to reject the claim.

44) B 45) A 46) A 47) B 48) A 49) B