Ch. 1 Data Collection 1.1 Introduction to the Practice of Statistics 1 Define statistics and statistical thinking. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) What is statistics? Answer: Statistics is the science of collecting, summarizing, organizing, and analyzing information in order to answer questions or draw conclusions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) Which of the following is not true of statistics? A) Statistics is used to answer questions with 100% certainty. B) Statistics involves collecting and summarizing data. C) Statistics can be used to organize and analyze information. D) Statistics is about providing a measure of confidence in any conclusions Answer: A 2 Explain the process of statistics. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the underlined value is a parameter or a statistic. 1) In a survey conducted in the town of Atherton, 23% of adult respondents reported that they had been involved in at least one car accident in the past ten years. A) statistic B) parameter Answer: A 2) 23.2% of the mayors of cities in an entire certain state are from minority groups. A) parameter B) statistic Answer: A 3) A study of 2,700 college students in the city of Pemblington found that 9% had been victims of violent crimes. A) statistic B) parameter Answer: A 4) 51.6% of all the residents of Idlington Garden City are female. A) parameter B) statistic Answer: A 5) Telephone interviews of 316 employees of a large electronics company found that 45% were dissatisfied with their working conditions. A) statistic B) parameter Answer: A 6) The average age of the 65 students in Ms. Hope's political science class is 21 years 8 months. A) parameter B) statistic Answer: A
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7) Mark retired from competitive athletics last year. In his whole career as a sprinter he had competed in the 100-meters event a total of 328 times. His average time for these 328 races was 10.25 seconds. A) parameter B) statistic Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 8) A survey of 1,805 American households found that 71% of the households own a DVD recorder. Identify the population, the sample, and the individuals in the study. Answer: population: collection of all American households; sample: collection of 1,805 American households surveyed; individuals: each household 9) A survey of 1,242 American households found that 32% of the households own at least two bicycles. Identify the population, the sample, and the individuals in the study. Answer: population: collection of all American households; sample: collection of 1,242 American households surveyed; individuals: each household MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 10) 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 190 students and carefully recorded their parking times. Identify the population of interest to the university administration. A) the parking times of the entire set of students that park at the university B) the parking times of the 190 students from whom the data were collected C) the entire set of faculty, staff, and students that park at the university D) the students that park at the university between 9 and 10 AM on Wednesdays Answer: A 11) A manufacturer of cellular phones has decided that an assembly line is operating satisfactorily if less than 0.0 1% of the phones produced per day are defective. To check the quality of a day's production, the company decides to randomly sample 60 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 60 phones sampled and tested C) the 60 responses: defective or not defective D) the 0.01% of the phones that are defective Answer: A 12) A recent 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-one hundred Florida residents were surveyed. Which of the following is the population used in the study? A) all Florida residents B) the 2,100 Florida residents surveyed C) the Florida residents who were willing to spend more tax dollars on protecting the beaches from environmental disasters D) all Florida residents who lived along the beaches Answer: A
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13) 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 130 students and carefully recorded their parking times. Identify the sample of interest to the university administration. A) parking times of the 130 students B) parking time of a student C) location of the parking spot D) type of car (import or domestic) Answer: A 14) The legal profession conducted a study to determine the percentage of cardiologists who had been sued for malpractice in the last five years. The sample was randomly chosen from a national directory of doctors. Identify the individuals in the study. A) each cardiologist selected from the directory B) the responses: have been sued/have not been sued for malpractice in the last five years C) the doctor's area of expertise (i.e., cardiology, pediatrics, etc.) D) all cardiologists in the directory Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 15) Administrators at a large university want to know the average debt incurred by their graduates. Surveys were mailed to 260 graduating seniors asking them to report their total student loan debt. Identify the population, sample, and individuals in the study. Answer: The population of interest is the student loan debt incurred by all graduates of the university. The sample is student loan debt of the 260 graduating seniors that were collected by the university administrators. The individuals are each graduating senior whose student loan debt was recorded. 16) A study was conducted to determine if listening to heavy metal music affects critical thinking. To test the claim, 120 subjects were randomly assigned to two groups. Both groups were administered a basic math skills exam. The first group took the exam while heavy metal music was piped into the exam room, while the second group took the exam in a silent room. The mean exam score for the first group was 82, and the mean exam score for the second group was 90. The researchers concluded that heavy metal music negatively affects critical thinking. Identify (a) the research objective, (b) the sample, (c) the descriptive statistics, and (d) the conclusions made in the study. Answer: (a) if listening to heavy metal music affects critical thinking (b) the 120 subjects (c) the mean exam score for the first group = 82, and the mean exam score for the second group was 90 (d) that heavy metal music negatively affects critical thinking 17) A telephone poll asked 1,122 registered voters "Would you vote for the current vice president if he ran for president?" Of these 1,122 respondents, 37% would vote for the current vice president if he ran for president. The administrators of the study concluded that 37% of all registered voters would vote for the current vice president if he ran for president. Identify (a) the research objective, (b) the sample, (c) the descriptive statistics, and (d) the conclusions made in the study. Answer: (a) to determine the percentage of registered voters who would vote for the current vice president if he ran for president (b) the 1,122 registered voters surveyed (c) 37% of the respondents supported reelection (d) that 37% of all registered voters would vote for the current vice president if he ran for president
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) Which branch of statistics deals with the organization and summarization of collected information? A) Descriptive statistics B) Inferential statistics C) Survey design D) Computational statistics Answer: A 3 Distinguish between qualitative and quantitative variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Classify the variable as qualitative or quantitative. 1) the colors of book covers on a bookshelf A) qualitative
B) quantitative
Answer: A 2) the number of calls received at a company's help desk A) quantitative
B) qualitative
Answer: A 3) the number of seats in a school auditorium A) quantitative
B) qualitative
Answer: A 4) the numbers on the shirts of a football team A) qualitative
B) quantitative
Answer: A 5) the bank account numbers of the students in a class A) qualitative
B) quantitative
Answer: A 6) the weights of cases loaded onto an airport conveyor belt A) quantitative
B) qualitative
Answer: A 7) the temperatures of cups of coffee served at a restaurant A) quantitative
B) qualitative
Answer: A 8) the native languages of students in an English class A) qualitative Answer: A
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B) quantitative
Solve the problem. 9) A bicycle manufacturer produces four different bicycle models. Information is summarized in the table below: Model Series Number Weight Style Ascension A120 32 Mountain Road Runner B640 22 Road All Terrain C300 27 Hybrid Class Above D90 17 Racing Identify the variables and determine whether each variable is quantitative or qualitative. A) series number: qualitative; weight: quantitative; style: qualitative B) series number: quantitative; weight: quantitative; style: qualitative C) series number: quantitative; weight: qualitative; style: qualitative D) series number: qualitative; weight: qualitative; style: qualitative Answer: A 10) An international relations professor is supervising four master's students. Information about the students is summarized in the table. Student Name Student Number Area of Interest Anna 914589205 Africa Pierre 981672635 Middle East Juan 906539012 Latin America Yoko 977530271 Asia
GPA 3.23 3.50 3.80 3.71
Identify the variables and determine whether each variable is quantitative or qualitative. A) student number: qualitative; area of interest: qualitative; GPA: quantitative B) student number: quantitative; area of interest: qualitative; GPA: quantitative C) student number: quantitative; area of interest: qualitative; GPA: qualitative D) student number: qualitative; area of interest: qualitative; GPA: qualitative Answer: A Provide an appropriate response. 11) Quantitative variables classify individuals in a sample according to A) numerical measure. B) physical attribute. C) personality characteristic. D) exhibited trait. Answer: A 4 Distinguish between discrete and continuous variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the quantitative variable is discrete or continuous. 1) the number of bottles of juice sold in a cafeteria during lunch A) discrete B) continuous Answer: A 2) the weight of a player on the wrestling team A) continuous
B) discrete
Answer: A 3) the cholesterol levels of a group of adults the day after Thanksgiving A) continuous B) discrete Answer: A Page 5
4) the low temperature in degrees Fahrenheit on January 1st in Cheyenne, Wyoming A) continuous B) discrete Answer: A 5) the number of goals scored in a hockey game A) discrete
B) continuous
Answer: A 6) the speed of a car on a Boston tollway during rush hour traffic A) continuous B) discrete Answer: A 7) the number of phone calls to the police department on any given day A) discrete B) continuous Answer: A 8) the age of the oldest employee in the data processing department A) continuous B) discrete Answer: A 9) the number of pills in an aspirin bottle A) discrete
B) continuous
Answer: A Provide an appropriate response. 10) The peak shopping time at a pet store is between 8-11:00 am on Saturday mornings. Management at the pet store randomly selected 100 customers last Saturday morning and decided to observe their shopping habits. They recorded the number of items that a sample of the customers purchased as well as the total time the customers spent in the store. Identify the types of variables recorded by the pet store. A) number of items - discrete; total time - continuous B) number of items - continuous; total time - continuous C) number of items - continuous; total time - discrete D) number of items - discrete; total time - discrete Answer: A 11) The number of violent crimes committed in a city on a given day in a random sample of 50 days is a __________ random variable. A) discrete B) continuous Answer: A 12) Classify the following random variable: telephone area codes A) qualitative data B) experimental data C) quantitative continuous data D) quantitative discrete data Answer: A 13) A student is asked to rate a guest speaker's ability to communicate on a scale of poor-average-good-excellent. The student is to fill in a corresponding circle on a bubble form. This is an example of collecting what type of data? A) qualitative B) continuous C) discrete D) insightful Answer: A
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5 Determine the level of measurement of a variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the level of measurement of the variable. 1) the musical instrument played by a music student A) nominal B) ratio
C) ordinal
D) interval
2) the medal received (gold, silver, bronze) by an Olympic gymnast A) ordinal B) ratio C) nominal
D) interval
Answer: A
Answer: A 3) height of a tree A) ratio
B) interval
C) nominal
D) ordinal
C) ordinal
D) interval
B) ratio
C) nominal
D) ordinal
6) an officer's rank in the military A) ordinal B) ratio
C) nominal
D) interval
C) nominal
D) ordinal
C) ordinal
D) interval
Answer: A 4) the native language of a tourist A) nominal B) ratio Answer: A 5) the day of the month A) interval Answer: A
Answer: A 7) weight of rice bought by a customer A) ratio B) interval Answer: A 8) a student's favorite sport A) nominal
B) ratio
Answer: A 9) ranking (first place, second place, etc.) of contestants in a singing competition A) ordinal B) ratio C) nominal
D) interval
Answer: A 10) weight capacity of a backpack A) ratio B) interval
C) nominal
D) ordinal
Answer: A 11) an evaluation received by a physics student (excellent, good, satisfactory, or poor). A) ordinal B) ratio C) nominal
D) interval
Answer: A 12) the year of manufacture of a car A) interval B) ratio Answer: A
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C) nominal
D) ordinal
13) time spent playing basketball A) ratio
B) interval
C) nominal
D) ordinal
C) nominal
D) interval
Answer: A 14) category of storm (gale, hurricane, etc.) A) ordinal B) ratio Answer: A
1.2 Observational Studies versus Designed Experiments 1 Distinguish between an observational study and an experiment. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the study depicts an observational study or an experiment. 1) A medical researcher obtains a sample of adults suffering from diabetes. She randomly assigns 42 people to a treatment group and 42 to a placebo group. The treatment group receives a medication over a period of three months and the placebo group receives a placebo over the same time frame. At the end of three months the patients' symptoms are evaluated. A) experiment B) observational study Answer: A 2) A poll is conducted in which professional musicians are asked their ages. A) observational study B) experiment Answer: A 3) A pollster obtains a sample of students and asks them how they will vote on an upcoming referendum. A) observational study B) experiment Answer: A 4) The personnel director at a large company would like to determine whether the company cafeteria is widely used by employees. She calls each employee and asks them whether they usually bring their own lunch, eat at the company cafeteria, or go out for lunch. A) observational study B) experiment Answer: A 5) A scientist was studying the effects of a new fertilizer on crop yield. She randomly assigned half of the plots on a farm to group one and the remaining plots to group two. On the plots in group one, the new fertilizer was used for a year. On the plots in group two, the old fertilizer was used. At the end of the year the average crop yield for the plots in group one was compared with the average crop yield for the plots in group two. A) experiment B) observational study Answer: A 6) A researcher obtained a random sample of 100 smokers and a random sample of 100 nonsmokers. After interviewing all 200 participants in the study, the researcher compared the rate of depression among the smokers with the rate of depression among nonsmokers. A) observational study B) experiment Answer: A Provide an appropriate response. 7) True or False: Observational studies allow the researcher to claim causation, not just association. A) False B) True Answer: A
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8) True or False: Experiments intentionally manipulate the value of an explanatory variable. A) True B) False Answer: A 2 Explain the various types of observational studies. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine what type of observational study is described. Explain. 1) Researchers wanted to determine whether there was an association between high blood pressure and the suppression of emotions. The researchers looked at 1800 adults enrolled in a Health Initiative Observational Study. Each person was interviewed and asked about their response to emotions. In particular they were asked whether their tendency was to express or to hold in anger and other emotions. The degree of suppression of emotions was rated on a scale of 1 to 10. Each person's blood pressure was also measured. The researchers analyzed the results to determine whether there was an association between high blood pressure and the suppression of emotions. A) cross-sectional; Information is collected at a specific point in time. B) cohort; Individuals are observed over a long period of time. C) case-control; Individuals are asked to look back in time. Answer: A 2) Researchers wanted to determine whether there was an association between city driving and stomach ulcers. They selected a sample of 900 young adults and followed them for a twenty-year period. At the start of the study none of the participants was suffering from a stomach ulcer. Each person kept track of the number of hours per week they spent driving in city traffic. At the end of the study each participant underwent tests to determine whether they were suffering from a stomach ulcer. The researchers analyzed the results to determine whether there was an association between city driving and stomach ulcers. A) cohort; Individuals are observed over a long period of time. B) cross-sectional; Information is collected at a specific point in time. C) case-control; Individuals are asked to look back in time. Answer: A 3) A researcher wanted to determine whether women with children are more likely to develop anxiety disorders than women without children. She selected a sample of 900 twenty-year old women and followed them for a twenty-year period. At the start of the study, none of the women had children. By the end of the study 53% of the women had at least one child. The level of anxiety of each participant was evaluated at the beginning and at the end of the study and the increase (or decrease) in anxiety was recorded. The researchers analyzed the results to determine whether there was an association between anxiety and having children. A) cohort; Individuals are observed over a long period of time. B) cross-sectional; Information is collected at a specific point in time. C) case-control; Individuals are asked to look back in time. Answer: A 4) Vitamin D is important for the metabolism of calcium and exposure to sunshine is an important source of vitamin D. A researcher wanted to determine whether osteoperosis was associated with a lack of exposure to sunshine. He selected a sample of 250 women with osteoperosis and an equal number of women without osteoperosis. The two groups were matched - in other words they were similar in terms of age, diet, occupation, and exercise levels. Histories on exposure to sunshine over the previous twenty years were obtained for all women. The total number of hours that each woman had been exposed to sunshine in the previous twenty years was estimated. The amount of exposure to sunshine was compared for the two groups. A) case-control; Individuals are asked to look back in time B) cross-sectional; Information is collected at a specific point in time. C) cohort; Individuals are observed over a long period of time. Answer: A Page 9
5) Can money buy happiness? A researcher wanted to determine whether there was any association between economic status and happiness. She selected a sample of 1000 adults and interviewed them. Each person was asked about their financial situation and their level of happiness was evaluated. The researcher analyzed the results to determine whether there was an association between economic status and happiness. A) cross-sectional; Information is collected at a specific point in time. B) cohort; Individuals are observed over a long period of time. C) case-control; Individuals are asked to look back in time. Answer: A 6) A researcher wanted to determine whether colon cancer was associated with eating meat. He selected a sample of 500 men with colon cancer and an equal number of men without colon cancer. The two groups were matched - in other words they were similar in terms of age, occupation, income, and exercise levels. Histories on the amount of meat consumed over the previous twenty years were obtained for all men. The total amount of meat that each man eaten in the previous twenty years was estimated. The meat consumption was compared for the two groups. A) case-control; Individuals are asked to look back in time B) cross-sectional; Information is collected at a specific point in time. C) cohort; Individuals are observed over a long period of time. Answer: A
1.3 Simple Random Sampling 1 Obtain a simple random sample. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The government of a town needs to determine if the city's residents will support the construction of a new town hall. The government decides to conduct a survey of a sample of the city's residents. Which one of the following procedures would be least appropriate for obtaining a sample of the town's residents? A) Survey the first 300 people listed in the town's telephone directory. B) Survey a random sample of persons within each geographic region of the city. C) Survey a random sample of employees at the old city hall. D) Survey every 8th person who walks into city hall on a given day. Answer: A 2) The city council of a small town needs to determine if the town's residents will support the building of a new library. The council decides to conduct a survey of a sample of the town's residents. Which one of the following procedures would be least appropriate for obtaining a sample of the town's residents? A) Survey a random sample of librarians who live in the town. B) Survey a random sample of persons within each neighborhood of the town. C) Survey 300 individuals who are randomly selected from a list of all people living in the state in which the town is located. D) Survey every 15th person who enters the old library on a given day. Answer: A
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3) The policy committee at State University has 6 members: Jose, John, Prof. Rise, Dr. Hernandez, LaToyna, and Ming. A subcommittee of two members must be formed to investigate the visitation policy in the dormitories. List all possible simple random samples of size 2. A) Jose and John, Jose and Prof. Rise, Jose and Dr. Hernandez, Jose and LaToyna, Jose and Ming, John and Prof. Rise, John and Dr. Hernandez, John and LaToyna, John and Ming, Prof. Rise and Dr. Hernandez, Prof. Rise and LaToyna, Prof. Rise and Ming, Dr. Hernandez and LaToyna, Dr. Hernandez and Ming, LaToyna and Ming B) Jose and John, Prof. Rise and Dr. Hernandez, LaToyna and Ming C) Jose and John, John and Prof. Rise, Prof. Rise and Dr. Hernandez, Dr. Hernandez and LaToyna, LaToyna and Ming D) Jose and John, Jose and Prof. Rise, Jose and Dr. Hernandez, Jose and LaToyna, Jose and Ming Answer: A 4) Select a random sample of five state capitals from the list below using the two digit list of random numbers provided. Begin with the uppermost left random number and proceed down each column. When a column is complete, use the numbers at the top of the next right column and proceed down that column. State Capitals
A) Springfield, IL; Atlanta,GA; Providence, RI; Santa Fe, NM; Columbus OH. B) Springfield, IL; Des Moines, IA; Boston, MA; Santa Fe, NM; Columbus OH. C) Carson City NV; Boise ID; Atlanta, GA; Cheyenne, WY; Boston, MA. D) Boston, MA; Concord, NH; Dover DE; Santa Fe, NM; Richmond, VA. Answer: A
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5) The top 38 cities in Wisconsin as determined by population are given below. Select a random sample of four cities from the list below using the two digit list of random numbers provided. Begin with the uppermost left random number and proceed down each column. When a column is complete, use the numbers at the top of the next right column and proceed down that column. Information was obtained from the web site http://www.citypopulation.de/USA-Wisconsin.html. Wisconsin Cities by Population
A) Manitowoc, La Crosse, Franklin, Oshkosh. B) Manitowoc, Appleton, Greenfield, Fond du Lac. C) Milwaukee, Madison, Green Bay, Kenosha. D) Milwaukee, Eau Claire, New Berlin, West Bend. Answer: A
1.4 Other Effective Sampling Methods 1 Determine the appropriate sampling type MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the type of sampling used. 1) Thirty-five math majors, 34 music majors and 65 history majors are randomly selected from 244 math majors, 453 music majors and 550 history majors at the state university. What sampling technique is used? A) stratified B) simple random C) cluster D) convenience E) systematic Answer: A 2) Every fifth adult entering an airport is checked for extra security screening. What sampling technique is used? A) systematic B) simple random C) cluster D) convenience E) stratified Answer: A
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3) At a local technical school, five auto repair classes are randomly selected and all of the students from each class are interviewed. What sampling technique is used? A) cluster B) simple random C) convenience D) systematic E) stratified Answer: A 4) A writer for an art magazine randomly selects and interviews fifty male and fifty female artists. What sampling technique is used? A) stratified B) simple random C) cluster D) convenience E) systematic Answer: A 5) A travel industry researcher interviews all of the passengers on five randomly selected cruises. What sampling technique is used? A) cluster B) simple random C) convenience D) systematic E) stratified Answer: A 6) A statisticsstudent interviews everyone in his apartment building to determine what percent of people own a cell phone. What sampling technique is used? A) convenience B) simple random C) cluster D) systematic E) stratified Answer: A 7) A lobbyist for the oil industry assigns a number to each senator and then uses a computer to randomly generate ten numbers. The lobbyist contacts the senators corresponding to these numbers. What sampling technique was used? A) simple random B) convenience C) cluster D) stratified E) systematic Answer: A 8) Based on 9,000 responses from 44,500 questionnaires sent to all its members, a major medical association estimated that the annual salary of its members was $122,500 per year. What sampling technique was used? A) simple random B) stratified C) cluster D) convenience E) systematic Answer: A Page 13
9) In a recent Twitter survey, participants were asked to answer "yes" or "no" to the question "Are you in favor of stricter gun control?" 6571 responded "yes" while 5237 responded "no". What sampling technique was used? A) convenience B) simple random C) cluster D) stratified E) systematic Answer: A 10) A sample consists of every 35th worker from a group of 5000 workers. What sampling technique was used? A) systematic B) simple random C) cluster D) stratified E) convenience Answer: A 11) A market researcher randomly selects 400 homeowners under 60 years of age and 200 homeowners over 60 years of age. What sampling technique was used? A) stratified B) simple random C) cluster D) convenience E) systematic Answer: A 12) To avoid working late, the plant foreman inspects the first 30 microwaves produced that day. What sampling technique was used? A) convenience B) simple random C) cluster D) stratified E) systematic Answer: A 13) The names of 80 employees are written on 80 cards. The cards are placed in a bag, and three names are picked from the bag. What sampling technique was used? A) simple random B) stratified C) cluster D) convenience E) systematic Answer: A 14) An education researcher randomly selects 90 of the nation's junior colleges and interviews all of the professors at each school. What sampling technique was used? A) cluster B) simple random C) stratified D) convenience E) systematic Answer: A
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Provide an appropriate response. 15) The United States can be divided into four geographical regions: Northeast, South, Midwest, and West. The Northeast region consists of 9 states; the South region consists of 16 states; the Midwest consists of 12 states; and the West consists of 13 states. If a survey is to be administered to the governors of 12 of the states and we want equal representation for the states in each of the four regions, how many states from the South should be selected? Round to the nearest whole state. A) 3 B) 4 C) 2 D) 5 Answer: A 2 Design a sampling method. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) For a poll of voters regarding a referendum calling for renewing the residential renewable energy tax credit, design a sampling method to obtain the individuals in the sample. Answer: Answers will vary. One option would be stratified sampling. Since this is a national issue, different geographical locations are likely to have similar views. 2) A pharmaceutical company wants to conduct a survey of 50 individuals who have type 1 diabetes. The company has obtained a list from doctors throughout the country of 7400 individuals who are known to have type 1 diabetes. Design a sampling method to obtain the individuals in the sample. Answer: Answers will vary. Simple random sampling will work fine here, especially because a list of 7400 individuals who meet the needs of our study already exists.
1.5 Bias in Sampling 1 Explain the sources of bias in sampling. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) An online newspaper conducted a survey by asking, "Do you support the lowering of air quality standards if it could cause the death of millions of innocent people from pollution related diseases?" Determine the type of bias. Answer: Response bias; poorly worded question 2) A local hardware store wants to know if its customers are satisfied with the customer service they receive. The store posts an interviewer at the front of the store to ask the first 140 shoppers who leave the store, "How satisfied, on a scale of 1 to 10, were you with this store's customer service?" Determine the type of bias. Answer: Sampling bias; the customers are not chosen through a random sample. 3) Before opening a new dealership, an auto manufacturer wants to gather information about car ownership and driving habits of the local residents. The marketing manager of the company randomly selects 1000 households from all households in the area and mails a questionnaire to them. Of the 1000 surveys mailed, she receives 145 back. Determine the type of bias. Answer: Nonresponse bias MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) Which type of bias occurs because the individuals tend to favor one part of the population over another? A) sampling bias B) response bias C) nonresponse bias D) no bias Answer: A
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5) A researcher wants to study the effects of advertising of accessible online college upon high school boys and sampled small Midwestern towns. The results from this study were used to project a national campaign on accessible forms of online college. What type or error may have occurred? A) Sampling error B) Data-entry error C) Question error D) Nonsampling error Answer: A
1.6 The Design of Experiments 1 Describe the characteristics of an experiment. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) What is a designed experiment? Answer: A designed experiment is a controlled study in which treatments are applied to experimental units, and the effect of varying these treatments on a response variable is observed. 2) What is a factor? Answer: A factor is the variable whose effect on the response variable is to be assessed by the experimenter. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3) Which of the following is not true about factors? A) Factors whose effect on the response variable is not of interest can be set after the experiment. B) Factors whose effect on the response variable interests us should be set at predetermined levels. C) One way to control factors is to fix their level at one predetermined value throughout the experiment. D) Any combination of the values of the factors is called a treatment. Answer: A 4) The variable measured in the experiment is called ____________ . A) the response variable B) a sampling unit C) the treatment D) the predictor variable Answer: A 5) The object upon which the treatment is applied is called ________ . A) an experimental unit B) the factor C) the predictor variable D) a treatment Answer: A 6) __________ is a combination of the values of factors in an experiment. A) A treatment B) The sampling design C) The factor level D) The design Answer: A 7) An experiment in which the experimental unit (or subject) does not know which treatment he or she is receiving is called a ________________ . A) single-blind experiment B) double-blind experiment C) randomized block design D) matched-pairs design Answer: A
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8) An experiment in which neither the experimental unit nor the researcher in contact with the experimental unit knows which treatment the experimental unit is receiving is called a ________________ . A) double-blind experiment B) single-blind experiment C) randomized block design D) matched-pairs design Answer: A 9) A salesman boasts to a farmer that his new fertilizer will increase the yield of the farmer's crops by 15%. The farmer wishes to test the effects of the new fertilizer on her corn yield. She has four equal sized plots of land one with sandy soil, one with rocky soil, one with clay-rich soil, and one with average soil. She divides each of the four plots into three equal sized portions and randomly labels them A, B and C. The four A portions are treated with her old fertilizer. The four B portions are treated with the new fertilizer. The four C portions receive no fertilizer. At harvest time, the corn yield is recorded for each section of land. What is the claim she is testing? A) The new fertilizer yielded at least a 15% improvement. B) The total yield increased at least 15%. C) The A sections had at least a 15% increase in yield. D) The average soil field had at least a 15% increase in yield. Answer: A 10) A drug company wanted to test a new indigestion medication. The researchers found 700 adults aged 25-35 and randomly assigned them to two groups. The first group received the new drug, while the second received a placebo. After one month of treatment, the percentage of each group whose indigestion symptoms decreased was recorded and compared. What is the response variable in this experiment? A) The percentage who had decreased indigestion symptoms. B) The type of drug (medication or placebo). C) The 700 adults aged 25-35. D) The one month treatment time. Answer: A 11) A drug company wanted to test a new indigestion medication. The researchers found 300 adults aged 25-35 and randomly assigned them to two groups. The first group received the new drug, while the second received a placebo. After one month of treatment, the percentage of each group whose indigestion symptoms decreased was recorded and compared. What is the treatment in this experiment? A) The drug. B) The percentage who had decreased indigestion symptoms. C) The 300 adults aged 25-35. D) The one month treatment time. Answer: A 12) A drug company wanted to test a new depression medication. The researchers found 200 adults aged 25-35 and randomly assigned them to two groups. The first group received the new drug, while the second received a placebo. After one month of treatment, the percentage of each group whose depression symptoms decreased was recorded and compared. How many levels does the treatment in this experiment have? A) 2 (medication or placebo) B) 200 (number of respondents) C) 1 (months of treatment) D) 10 (age span of respondents) Answer: A
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13) A drug company wanted to test a new depression medication. The researchers found 500 adults aged 25-35 and randomly assigned them to two groups. The first group received the new drug, while the second received a placebo. After one month of treatment, the percentage of each group whose depression symptoms decreased was recorded and compared. Identify the experimental units. A) The 500 adults aged 25-35. B) The percentage who had decreased depression symptoms. C) The drug (medication or placebo). D) The one month treatment time Answer: A 2 Explain the steps in designing an experiment, completely randomized design, matched -pairs design, or randomized block design. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A drug company wanted to test a new depression medication. The researchers found 700 adults aged 25-35 and randomly assigned them to two groups. The first group received the new drug, while the second received a placebo. After one month of treatment, the percentage of each group whose depression symptoms decreased was recorded and compared. What type of experimental design is this? A) completely randomized design B) randomized block design C) matched-pairs design D) single-blind design Answer: A 2) A medical journal published the results of an experiment on anxiety. The experiment investigated the effects of a controversial new therapy for anxiety. Researchers measured the anxiety levels of 96 adult women who suffer moderate conditions of the disorder. After the therapy, the researchers again measured the women's anxiety levels. The differences between the the pre- and post-therapy anxiety levels were reported. What is the response variable in this experiment? A) The differences between the the pre- and post-therapy anxiety levels B) The 96 adult women who suffer from anxiety. C) The disorder (anxiety or no anxiety). D) The therapy. Answer: A 3) A medical journal published the results of an experiment on anorexia. The experiment investigated the effects of a controversial new therapy for anorexia. Researchers measured the anorexia levels of 39 adult women who suffer moderate conditions of the disorder. After the therapy, the researchers again measured the women's anorexia levels. The differences between the the pre- and post-therapy anorexia levels were reported. What is the treatment in this experiment? A) the therapy B) the 39 adult women who suffer from anorexia C) the disorder (anorexia or no anorexia) D) the differences between the the pre- and post-therapy anorexia levels Answer: A
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4) A medical journal published the results of an experiment on depression. The experiment investigated the effects of a controversial new therapy for depression. Researchers measured the depression levels of 83 adult women who suffer moderate conditions of the disorder. After the therapy, the researchers again measured the women's depression levels. The differences between the the pre- and post-therapy depression levels were reported. How many levels does the treatment have in this experiment? A) 2 (pre- and post-therapy) B) 1 (therapy) C) 83 (the adult women who suffer from depression) D) 166 (the adult women who suffer from depression measured pre- and post-therapy) Answer: A 5) A medical journal published the results of an experiment on depression. The experiment investigated the effects of a controversial new therapy for depression. Researchers measured the depression levels of 72 adult women who suffer moderate conditions of the disorder. After the therapy, the researchers again measured the women's depression levels. The differences between the the pre- and post-therapy depression levels were reported. What type of experimental design is this? A) matched-pairs design B) completely randomized design C) randomized block design D) single-blind design Answer: A 6) A medical journal published the results of an experiment on anxiety. The experiment investigated the effects of a controversial new therapy for anxiety. Researchers measured the anxiety levels of 79 adult women who suffer moderate conditions of the disorder. After the therapy, the researchers again measured the women's anxiety levels. The differences between the the pre- and post-therapy anxiety levels were reported. Identify the experimental units. A) the 79 adult women who suffer from anxiety B) the differences between the pre- and post-therapy anxiety levels C) the disorder (anxiety or no anxiety) D) the therapy time period (pre or post) Answer: A 7) A farmer wishes to test the effects of a new fertilizer on her potato yield. She has four equal-sized plots of land-- one with sandy soil, one with rocky soil, one with clay-rich soil, and one with average soil. She divides each of the four plots into three equal-sized portions and randomly labels them A, B, and C. The four A portions of land are treated with her old fertilizer. The four B portions are treated with the new fertilizer, and the four C's are treated with no fertilizer. At harvest time, the potato yield is recorded for each section of land. What is the response variable in this experiment? A) the potato yield recorded for each section of land B) the type of fertilizer (old, new, or none) C) the section of land (A, B, or C) D) the four types of soil Answer: A 8) A farmer wishes to test the effects of a new fertilizer on her tomato yield. She has four equal-sized plots of land-- one with sandy soil, one with rocky soil, one with clay-rich soil, and one with average soil. She divides each of the four plots into three equal-sized portions and randomly labels them A, B, and C. The four A portions of land are treated with her old fertilizer. The four B portions are treated with the new fertilizer, and the four C's are treated with no fertilizer. At harvest time, the tomato yield is recorded for each section of land. What is the treatment in this experiment? A) the fertilizers B) the tomato yield recorded for each section of land C) the section of land (A, B, or C) D) the four types of soil Answer: A Page 19
9) A farmer wishes to test the effects of a new fertilizer on her tomato yield. She has four equal-sized plots of land-- one with sandy soil, one with rocky soil, one with clay-rich soil, and one with average soil. She divides each of the four plots into three equal-sized portions and randomly labels them A, B, and C. The four A portions of land are treated with her old fertilizer. The four B portions are treated with the new fertilizer, and the four C's are treated with no fertilizer. At harvest time, the tomato yield is recorded for each section of land. How many levels does the treatment have in this experiment? A) 3 (old, new, or no fertilizer) B) 4 (rocky, sandy, clay, or average soil) C) 12 (sections of land) D) 1 (tomato yield) Answer: A 10) A farmer wishes to test the effects of a new fertilizer on her corn yield. She has four equal-sized plots of land-one with sandy soil, one with rocky soil, one with clay-rich soil, and one with average soil. She divides each of the four plots into three equal-sized portions and randomly labels them A, B, and C. The four A portions of land are treated with her old fertilizer. The four B portions are treated with the new fertilizer, and the four C's are treated with no fertilizer. At harvest time, the corn yield is recorded for each section of land. What type of experimental design is this? A) randomized block design B) completely randomized design C) matched-pairs design D) double-blind design Answer: A 11) A farmer wishes to test the effects of a new fertilizer on her soybean yield. She has four equal-sized plots of land-- one with sandy soil, one with rocky soil, one with clay-rich soil, and one with average soil. She divides each of the four plots into three equal-sized portions and randomly labels them A, B, and C. The four A portions of land are treated with her old fertilizer. The four B portions are treated with the new fertilizer, and the four C's are treated with no fertilizer. At harvest time, the soybean yield is recorded for each section of land. Identify the experimental units. A) the soybean plants on the various plots of land B) the soybean yield at harvest time C) the three types of fertilizer D) the four types of soil Answer: A 12) When the effects of the explanatory variable upon the response variable cannot be determined, then A) confounding has occurred. B) a lurking variable is present. C) there is sampling error. D) the claim is invalid. Answer: A
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Ch. 1 Data Collection Answer Key 1.1 Introduction to the Practice of Statistics 1 Define statistics and statistical thinking. 1) Statistics is the science of collecting, summarizing, organizing, and analyzing information in order to answer questions or draw conclusions. 2) A 2 Explain the process of statistics. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) population: collection of all American households; sample: collection of 1,805 American households surveyed; individuals: each household 9) population: collection of all American households; sample: collection of 1,242 American households surveyed; individuals: each household 10) A 11) A 12) A 13) A 14) A 15) The population of interest is the student loan debt incurred by all graduates of the university. The sample is student loan debt of the 260 graduating seniors that were collected by the university administrators. The individuals are each graduating senior whose student loan debt was recorded. 16) (a) if listening to heavy metal music affects critical thinking (b) the 120 subjects (c) the mean exam score for the first group = 82, and the mean exam score for the second group was 90 (d) that heavy metal music negatively affects critical thinking 17) (a) to determine the percentage of registered voters who would vote for the current vice president if he ran for president (b) the 1,122 registered voters surveyed (c) 37% of the respondents supported reelection (d) that 37% of all registered voters would vote for the current vice president if he ran for president 18) A 3 Distinguish between qualitative and quantitative variables. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 4 Distinguish between discrete and continuous variables. 1) A 2) A 3) A 4) A Page 21
5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 5 Determine the level of measurement of a variable. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A
1.2 Observational Studies versus Designed Experiments 1 Distinguish between an observational study and an experiment. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 2 Explain the various types of observational studies. 1) A 2) A 3) A 4) A 5) A 6) A
1.3 Simple Random Sampling 1 Obtain a simple random sample. 1) A 2) A 3) A 4) A 5) A
1.4 Other Effective Sampling Methods 1 Determine the appropriate sampling type 1) A 2) A 3) A 4) A Page 22
5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 2 Design a sampling method. 1) Answers will vary. One option would be stratified sampling. Since this is a national issue, different geographical locations are likely to have similar views. 2) Answers will vary. Simple random sampling will work fine here, especially because a list of 7400 individuals who meet the needs of our study already exists.
1.5 Bias in Sampling 1 Explain the sources of bias in sampling. 1) Response bias; poorly worded question 2) Sampling bias; the customers are not chosen through a random sample. 3) Nonresponse bias 4) A 5) A
1.6 The Design of Experiments 1 Describe the characteristics of an experiment. 1) A designed experiment is a controlled study in which treatments are applied to experimental units, and the effect of varying these treatments on a response variable is observed. 2) A factor is the variable whose effect on the response variable is to be assessed by the experimenter. 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 2 Explain the steps in designing an experiment, completely randomized design, matched -pairs design, or randomized block design. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A Page 23
Ch. 2 Organizing and Summarizing Data 2.1 Organizing Qualitative Data 1 Organize qualitative data in tables. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. Round relative frequencies to thousandths. 1) Scott Tarnowski owns a pet grooming shop. His prices for grooming dogs are based on the size of the dog. His records from last year are summarized below. Construct a frequency distribution and a relative frequency distribution. Show the percentage represented by each relative frequency. Class Large Medium Small
Frequency 345 830 645
Answer: Class Large Medium Small Total
Frequency 345 830 645 1820
Relative Frequency 0.190 0.456 0.354 1.000
Percentage 19.0 45.6 35.4 100.0
2) The results of a survey about a recent judicial appointment are given in the table below. Construct a relative frequency distribution. Response Frequency Strongly Favor 23 Favor 14 Neutral 29 Oppose 5 Strongly Oppose 129 Answer: Response Frequency Relative Frequency Strongly Favor 23 0.115 Favor 14 0.07 Neutral 29 0.145 Oppose 5 0.025 Strongly Oppose 129 0.645
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3) The preschool children at Elmwood Elementary School were asked to name their favorite color. The results are listed below. Construct a frequency distribution and a relative frequency distribution. purple red red green
purple red green red
green purple blue blue
blue red blue red
red green blue yellow
Answer: Color Frequency Relative Frequency purple 3 0.15 green 4 0.20 blue 5 0.25 red 7 0.35 yellow 1 0.05
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) True or False: In fraction form, the sum of all the relative frequencies of a distribution will always add up to 1. A) True B) False Answer: A 5) True or False: Relative frequency is the proportion (or percent) of observations within a category and is found sum of all frequencies using the formula: relative frequency = . frequency A) False Answer: A
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B) True
2 Construct bar graphs. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The bar graph shows the number of tickets sold each week by the garden club for their annual flower show.
1) During which week was the most number of tickets sold? A) week 4 B) week 5
C) week 2
D) week 1
2) During which week was the fewest number of tickets sold? A) week 2 B) week 4 C) week 6
D) week 5
Answer: A
Answer: A 3) Approximately how many tickets were sold during week 3? A) 30 tickets B) 59 tickets C) 40 tickets Answer: A
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D) 19 tickets
Provide an appropriate response. 4) The payroll amounts for 26 major-league baseball teams are shown below. Approximate the percentage of payrolls that were in the $30-$40 million range. Round to the nearest whole percent.
A) 31%
B) 8%
C) 42%
D) 19%
Answer: A 5) Retailers are always interested in determining why a customer selected their store to make a purchase. A sporting goods retailer conducted a customer survey to determine why its customers shopped at the store. The results are shown below. Find the percentage of the customers responded that the merchandise was the reason they shopped at the store. Round to the nearest whole percent.
A) 43% Answer: A
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B) 30%
C) 50%
D) 29%
6) The bar graph below shows the political party affiliation of 1000 registered U.S. voters. Identify the percentage of the 1000 registered U.S. voters that belong to one of the traditional two parties (Democrat and Republican)?
A) 75%
B) 40%
C) 35%
D) 25%
Answer: A 7) The Excel horizontal frequency bar graph below describes the employment status of a random sample of U.S. adults. Find the percentage of those reporting having “no job”.
A) 15% Answer: A
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B) 20%
C) 40%
D) cannot determine
The following double-bar graph illustrates the revenue for a company for the four quarters of the year for two different years. Use the graph to answer the question.
8) In what quarter was the revenue the greatest for Year 1? A) fourth quarter B) first quarter
C) second quarter
D) third quarter
C) fourth quarter
D) third quarter
C) $25 million
D) $5 million
Answer: A 9) In what quarter was the revenue the least for Year 1? A) second quarter B) first quarter Answer: A 10) What was the revenue for the second quarter of Year 2? A) $20 million B) $4 million Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 11) The student tally of 8 evening classes are listed below. Construct a frequency bar graph and a relative frequency bar graph. Grade Point Average Frequency Math099 15 English101 21 Chem104 17 Bio100 19 Bio100L 15 Astr101 10 Ag104 12 Mktg103 9 Answer:
12) The local police recorded traffic offenses in a construction area. The results are listed below. Construct a frequency bar graph and a relative frequency bar graph. Offense Recorded Failure to slow down Failure to stop Failure to yield Disobey flagger
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Frequency 13 9 6 2
Answer:
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13) Listed below are the ACT scores of 40 randomly selected students at a major university. 18 16 26 19
22 25 26 19
13 14 25 14
15 19 25 24
24 21 19 20
24 23 17 21
20 25 18 23
19 18 15 22
19 18 13 19
12 13 21 17
a) Construct a relative frequency bar graph of the data, using eight classes. b) If the university wants to accept the top 90% of the applicants, what should the minimum score be? c) If the university sets the minimum score at 17, what percent of the applicants will be accepted? Answer: a) See graph below b) The minimum score = 14 c) The university will accept 76.57% of the applicants.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 14) Given the bar graph shown below, the Pareto chart that would best represent the data should have the bars in the following order.
A) D A E C F B Answer: A
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B) B F C E A D
C) C A D E F B
D) B F E D A C
3 Construct pie charts. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The pie chart shows the percentage of votes received by each candidate in the student council presidential election. Use the pie chart to answer the question. 1) Student Council President
Ming 20% Jim 35%
Ann 15% Ted 30%
300 total votes Who received the most votes? A) Jim B) Ted
C) Ming
D) Ann
C) Ann
D) Ming
Answer: A 2) Student Council President
Ann 24% Ming 30%
Ted 18% Jim 28%
500 total votes Who received the fewest votes? A) Ted B) Jim Answer: A
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3) Student Council President
Ben 20% Lili 35%
Ming 15% Ann 30%
400 total votes What percent of the votes did Ming and Ben receive together? A) 35% B) 65% C) 15%
D) 20%
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a pie chart for the data. Label each category with its percentage. 4) A study was conducted to determine how people get jobs. Four hundred subjects were randomly selected and the results are listed below. Round any percent to whole numbers. Job Sources of Survey Respondents Frequency Newspaper want ads 72 Online services 124 Executive search firms 69 Mailings 32 Networking 103 Answer:
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5) Scott Tarnowski owns a pet grooming shop. His prices for grooming dogs are based on the size of the dog. His records from last year are summarized below. Round any percent to whole numbers. Class Large Medium Small
Frequency 345 830 645
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 6) A two-pound bag of assorted candy contained 100 caramels, 83 mint patties, 93 chocolate squares, 80 nut clusters, and 79 peanut butter taffy pieces. To create a pie chart of this data, the angle for the slice representing each candy type must be computed. Calculate the degree measure of the slice representing the mint patties rounded to the nearest degree. A) 69° B) 19° C) 5° D) 52° Answer: A
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2.2 Organizing Quantitative Data: The Popular Displays 1 Organize discrete data in tables. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a frequency distribution for the data. 1) A random sample of 30 high school students is selected. Each student is asked how much time he or she spent on the Internet during the previous week. The following times (in hours) are obtained: 5 13 7 10 7 5 7 6 4 10 8 6 6 5 8 7 4 4 9 6 4 6 13 8 5 9 5 8 7 6 Construct a frequency distribution for the data. Answer: Hours Number of On Net HS Students 4 4 5 5 6 6 7 5 8 4 9 2 10 2 13 2 2) A sample of 25 service project scores is taken and is recorded below. Construct a frequency distribution for the data. 97 96 96 95 96 99 97 97 100 99 95 98 95 96 100 95 98 96 96 100 95 97 99 97 98 Answer:
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Measure 95 96 97 98 99 100
Frequency 5 6 5 3 3 3
2 Construct histograms of discrete data. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct the specified histogram. 1) A random sample of 30 high school students is selected. Each student is asked how much time he or she spent on the Internet during the previous week. The following times (in hours) are recorded: 6 14 8 11 8 6 8 7 5 11 9 7 7 6 9 8 5 5 10 7 5 7 14 9 6 10 6 9 8 7 Construct a frequency histogram for this data. Answer:
2) A sample of 25 community service projects is obtained and the scores are recorded. The results are shown below. Construct a frequency histogram for this data. 97 96 96 95 96 99 97 97 100 99 95 98 95 96 100 95 98 96 96 100 95 97 99 97 98 Answer:
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3 Organize continuous data in tables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The class width is the difference between A) Two consecutive lower class limits B) The high and the low data values C) The upper class limit and the lower class limit of a class D) The largest frequency and the smallest frequency Answer: A 2) Determine the number of classes in the frequency table below. Class Frequency 23-24 7 25-26 2 27-28 6 29-30 4 31-32 1
A) 5
B) 6
C) 20
D) 2
C) 1.5
D) 2.5
Answer: A 3) Find the class width for the frequency table below. Class Frequency 42-43 3 44-45 1 46-47 3 48-49 6 50-51 2 A) 2
B) 1
Answer: A 4) Use the following frequency distribution to determine the class limits of the third class. Class Frequency 2-10 4 11-19 8 20-28 5 29-37 2 38-46 6 47-55 3 A) lower limit: 20; upper limit: 28 C) lower limit: 20; upper limit: 29 Answer: A
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B) lower limit: 19.5; upper limit: 28.5 D) lower limit: 19; upper limit: 29
5) A researcher records the number of employees of each of the IT companies in the town of Westmoore. The results are summarized in the table. Number of Employees Number of IT Companies 0 - 249 37 250 - 499 21 500 - 749 7 750 - 999 5 1,000 - 1,249 7 Find the class width. A) 250
B) 1,249
C) 249.5
D) 5
Answer: A 6) A researcher records the number of employees of each of the IT companies in the town of Westmoore. The results are summarized in the table. Number of Employees Number of IT Companies 0 - 749 36 750 - 1,499 20 1,500 - 2,249 8 2,250 - 2,999 6 3,000 - 3,749 6 Find the class limits of the third class. A) lower limit: 1,500; upper limit: 2,249 C) lower limit: 1,499.5; upper limit: 2,249.5
B) lower limit: 1,500; upper limit: 2,250 D) lower limit: 1,499; upper limit: 2,250
Answer: A 7) The weights (in pounds) of babies born at St Mary's hospital last month are summarized in the table. Weight (lb) Number of Babies 5.0 - 5.8 6 5.9 - 6.7 19 6.8 - 7.6 18 7.7 - 8.5 9 8.6 - 9.4 5 Find the class width. A) 0.9 lb B) 0.8 lb C) 0.95 lb D) 0.85 lb Answer: A 8) The weights (in pounds) of babies born at St. Mary's hospital last month are summarized in the table. Weight (lb) Number of Babies 5.0 - 6 7 6.1 - 7.1 20 7.2 - 8.2 18 8.3 - 9.3 8 9.4 - 10.4 4 Find the class limits for the second class. A) lower limit: 6.1; upper limit: 7.1 C) lower limit: 6.05; upper limit:7.15 Answer: A
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B) lower limit: 6.1; upper limit: 7.2 D) lower limit: 6; upper limit: 7.2
9) The table below summarizes the weights of the almonds (in grams) in a one-pound bag. What is the class width? Weight (g) 0.7585-0.8184 0.8185-0.8784 0.8785-0.9384 0.9385-0.9984 0.9985-1.0584 1.0585-1.1184 1.1185-1.1784
Frequency 1 1 1 3 157 171 8
A) 0.06
B) 0.059
C) 0.408
D) 0.4
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct the requested frequency distribution. 10) The June precipitation amounts (in inches) for 40 cites are listed below. Construct a frequency distribution and a relative frequency distribution using eight classes. 2.0 3.1 2.2 3.0
3.2 2.4 2.2 4.0
1.8 2.4 1.7 4.0
2.9 2.3 0.5 2.1
0.9 1.6 3.6 1.9
4.0 1.6 3.4 1.1
3.3 4.0 1.9 0.5
2.9 3.1 2.0 3.2
3.6 3.2 3.0 3.0
0.8 1.8 1.1 2.2
Answer: Precip. Frequency 0.5-0.9 4 1.0-1.4 2 1.5-1.9 7 2.0-2.4 9 2.5-2.9 2 3.0-3.4 10 3.5-3.9 2 4.0-4.4 4
Relative Frequency 0.10 0.05 0.175 0.225 0.05 0.25 0.05 0.10
11) The commute times (in minutes) of 30 executives are listed below. Construct a frequency distribution and a relative frequency distribution using five classes. Round relative frequency values to three decimal places. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68 Answer: Commute Time (in min) 67.0-68.4 68.5-69.9 70.0-71.4 71.5-72.9 73.0-74.4
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Frequency 6 5 13 2 4
Relative Frequency 0.200 0.167 0.433 0.067 0.133
12) The March utility bills (in dollars) of 30 homeowners are listed below. Construct a frequency distribution and a relative frequency distribution using six classes. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer: Util. Bill (dollars) 33-35 36-38 39-41 42-44 45-47 48-50
Frequency 3 6 6 6 3 6
Relative Frequency 0.10 0.20 0.20 0.20 0.10 0.20
Provide an appropriate response. 13) A sample of 15 Boy Scouts was selected and their weights (in pounds) were recorded as follows: 97 120 137 124 117 108 134 126 123 106 130 110 100 120 140 a. Using a class width of 10, give the upper and lower limits for five classes, starting with a lower limit of 95 for the first class. b. Construct a frequency distribution for the data Answer: a.
95-104, 105-114, 115-124, 125-134, 135-144
b. Weight (lb) Tally Frequency 95-104 ll 2 105-114 lll 3 115-124 lllll 5 125-134 lll 3 135-144 ll 2
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4 Construct histograms of continuous data. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct the specified histogram. 1) For the data below, construct a frequency histogram and a relative frequency histogram. Height (in inches) Frequency 50 - 52 5 53 - 55 8 56 - 58 12 59 - 61 13 62 - 64 11 Answer:
2) For the data below, construct a frequency histogram and a relative frequency histogram. Weight (in pounds) Frequency 135 - 139 6 140 - 144 4 145 - 149 11 150 - 154 15 155 - 159 8 Answer: Frequency Histogram:
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Relative Frequency Histogram:
3) The 30 students in Mrs. Harrison's literature class were asked how many cousins they had. The results are shown below. Create a frequency histogram for the data using a class width of 2. 10 1 3 5 4 7 5 1 0 9 11 1 5 4 1 7 7 11 0 6 6 1 5 7 10 1 1 5 6 0 Answer:
Number of Cousins 4) The 30 students in Mrs. Harrison's literature class were asked how many cousins they had. The results are shown below. Construct a relative-frequency histogram using a class width of 2. 10 1 3 5 4 7 5 1 0 9 11 1 5 4 1 7 7 11 0 6 6 1 5 7 10 1 1 5 6 0 Answer:
Number of Cousins
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5) A sample of 15 Girl Scouts was selected and the number of boxes of cookies they sold were recorded. The results are listed below. Construct a frequency histogram for the data using a class width of 10 and using 95 as the lower limit of the first class. 97 120 137 124 117 108 134 126 123 106 130 110 100 120 140 Answer:
Weight (pounds) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 6) What is the difference between a bar graph and a histogram? A) The bars on a bar graph do not touch while the bars of a histogram do touch. B) The bars in a bar graph may be of various widths while the bars of a histogram are all the same width. C) The bars in a bar graph are all the same width while the bars of a histogram may be of various widths. D) There is no difference between these two graphical displays. Answer: A
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5 Draw dot plots. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a dot plot for the data. 1) The local police, using radar, checked the speeds (in mph) of 30 motorists at a busy intersection. The results are listed below. Construct a dot plot for the data. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer:
2) The heights (in inches) of 30 mechanics are listed below. Construct a dot plot for the data. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68 Answer:
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6 Identify the shape of a distribution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Construct a frequency distribution for the data using five classes. Describe the shape of the distribution. 1) The data set: Pick Three Lottery Outcomes for 10 Consecutive Weeks 3 6 7 6 0 6 1 7 8 4 1 5 7 5 9 1 5 3 9 9 2 2 3 0 8 8 4 0 2 4 A) uniform B) bell shaped C) skewed to the left D) skewed to the right Answer: A 2) The data set: ages of dishwashers (in years) in 20 randomly selected households 12 6 4 9 11 1 7 8 9 8 9 13 5 15 7 6 8 8 2 1 A) bell shaped B) uniform C) skewed to the left D) skewed to the right Answer: A 3) The data set: weekly grocery bills (in dollars) for 20 randomly selected households 135 120 115 132 136 124 119 145 98 110 125 120 115 130 140 105 116 121 125 108 A) bell shaped B) uniform C) skewed to the left D) skewed to the right Answer: A Describe the shape of the distribution. 4)
A) skewed to the right C) uniform
B) skewed to the left D) bell shaped
Answer: A 5)
A) skewed to the left C) uniform Answer: A
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B) skewed to the right D) bell shaped
Use the histograms shown to answer the question. 6)
Is either histogram symmetric? A) Neither is symmetric. B) The first is symmetric, but the second is not symmetric. C) The second is symmetric, but the first is not symmetric. D) Both are symmetric. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Describe the shape of the distribution. 7) A sample of 15 Little League players was selected and their number of at-batsweights (in pounds) over three seasons were recorded as follows: 97 120 137 124 117 108 134 126 123 106 130 110 100 120 140 Answer: symmetric
2.3 Additional Displays of Quantitative Data 1 Draw stem-and-leaf plots. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) For the stem-and-leaf plot below, what are the maximum and minimum entries? 1 1 2 2 3 3 4
89 666789 0112344566 77788999 011234455 66678899 12 A) max: 42; min: 18
Answer: A
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B) max: 38; min: 7
C) max: 47; min: 19
D) max: 41; min: 18
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Determine the original set of data. 2) Stem Leaves 7 4 8 1 9 0 8 10 4 11 3 4 12 6 9 13 6 7 9 14 2 3 8 9 15 6 9 Legend: 5 4 represents 54 Answer: 74, 81, 90, 98, 104, 113, 114, 126, 129, 136, 137, 139, 142, 143, 148, 149, 156, 159 3) Stem Leaves 5 5 6 7 7 0 3 8 8 9 5 8 10 6 9 11 6 7 9 12 2 3 8 9 13 0 9 Legend: 5 5 represents 5.5 Answer: 5.5, 6.7, 7.0, 7.3, 8.8, 9.5, 9.8, 10.6, 10.9, 11.6, 11.7, 11.9, 12.2, 12.3, 12.8, 12.9, 13.0, 13.9 Construct a stem-and-leaf plot for the data. 4) The number of home runs that Mark McGwire hit in the first 13 years of his major league baseball career are listed below. (Source: Major League Handbook) Construct a stem-and-leaf plot for this data. 3 49 32 33 39 22 42 9 9 39 52 58 70 Answer: 0 1 2 3 4 5 6 7
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399 2 2399 29 28 0
5) The numbers of runs batted in by Mark McLemore in the first 13 years of his major league baseball career are listed below. (Source: Major League Handbook) Construct a stem-and-leaf plot for this data. 0 102 56 25 9 9 56 165 88 122 150 91 114 Answer: 0 099 1 2 5 3 4 5 66 7 8 8 9 1 10 2 11 4 12 2 13 14 15 0 16 5 6) The heights (in inches) of 30 mechanics are listed below. Construct a stem-and-leaf plot for the data. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68 Answer: 6 77888899999 7 0000011111111223344 7) The March utility bills (in dollars) of 30 homeowners are listed below. Construct a stem-and-leaf plot for the data. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer: 3 3556667889 4 00111223334557889 5 000
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8) The scores for an economics test are listed below. Create a stem-and-leaf plot for the data. 87 76 92 77 98 94 88 85 66 89 79 90 50 96 83 88 82 58 15 69 Answer: The stem will consist of the tens digit and range from 1 to 9. The leaves will be drawn in the appropriate stems based on the data values. Stem Leaves 15 2 3 4 50 8 66 9 76 7 9 87 8 5 9 3 8 2 92 8 4 0 6 2 Construct frequency polygons. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a frequency polygon for the data. 1) Height (in inches) Frequency 50 - 52 5 53 - 55 8 56 - 58 12 59 - 61 13 62 - 64 11 Answer:
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2) Weight (in pounds) Frequency 135 - 139 6 140 - 144 4 145 - 149 11 150 - 154 15 155 - 159 8 Answer:
3) The grade point averages for 40 evening students are listed below. Construct a frequency polygon using eight classes. 2.0 3.1 2.2 3.0
3.2 2.4 2.2 4.0
Answer:
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1.8 2.4 1.7 4.0
2.9 2.3 0.5 2.1
0.9 1.6 3.6 1.9
4.0 1.6 3.4 1.1
3.3 4.0 1.9 0.5
2.9 3.1 2.0 3.2
3.6 3.2 3.0 3.0
0.8 1.8 1.1 2.2
4) The local police, using radar, checked the speeds (in mph) of 30 motorists in a construction area. The results are listed below. Construct a frequency polygon using six classes and a class width of 3. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the statement is true or false. 5) A frequency polygon always begins and ends with a frequency of zero. A) True B) False Answer: A 6) The class midpoint can be determined by adding to the lower class limit one-half of the class width. A) True B) False Answer: A 3 Create cumulative frequency and relative frequency tables. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct the requested frequency distribution. 1) The April precipitation amounts (in inches) for 40 cities are listed below. Construct a frequency distribution, a relative frequency distribution, a cumulative frequency distribution, and a relative cumulative frequency distribution using eight classes. 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 3.1 2.4 2.4 2.3 1.6 1.6 4.0 3.1 3.2 1.8 2.2 2.2 1.7 0.5 3.6 3.4 1.9 2.0 3.0 1.1 3.0 4.0 4.0 2.1 1.9 1.1 0.5 3.2 3.0 2.2 Answer: Precip (in.) Frequency 0.5-0.9 4 1.0-1.4 2 1.5-1.9 7 2.0-2.4 9 2.5-2.9 2 3.0-3.4 10 3.5-3.9 2 4.0-4.4 4
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Relative Cumulative Cumulative Frequency Frequency Relative Frequency 0.10 4 0.10 0.05 6 0.15 0.175 13 0.325 0.225 22 0.55 0.05 24 0.60 0.25 34 0.85 0.05 36 0.90 0.10 40 1
2) The commute time (in minutes) of 30 executives are listed below. Construct a frequency distribution, a relative frequency distribution, a cumulative frequency distribution, and a relative cumulative frequency distribution using five classes. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68 Answer: Commute Time (in min) Frequency 67.0-68.4 6 68.5-69.9 5 70.0-71.4 13 71.5-72.9 2 73.0-74.4 4
Relative Cumulative Cumulative Frequency Frequency Relative Frequency 0.20 6 0.20 0.167 11 0.367 0.433 24 0.80 0.067 26 0.867 0.133 30 1
3) The local police, using radar, checked the speeds (in mph) of 30 motorists in a construction area. The results are listed below. Construct a frequency distribution, a relative frequency distribution, a cumulative frequency distribution, and a relative cumulative frequency distribution using six classes. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer: Speed (in mph) 33-35 36-38 39-41 42-44 45-47 48-50
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Frequency 3 6 6 6 3 6
Relative Cumulative Cumulative Frequency Frequency Relative Frequency 0.10 3 0.10 0.20 9 0.30 0.20 15 0.50 0.20 21 0.70 0.10 24 0.80 0.20 30 1
4 Construct frequency and relative frequency ogives. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct the requested ogive. 1) The grade point averages for 40 evening students are listed below. Construct a frequency ogive using eight classes. 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 3.1 2.4 2.4 2.3 1.6 1.6 4.0 3.1 3.2 1.8 2.2 2.2 1.7 0.5 3.6 3.4 1.9 2.0 3.0 1.1 3.0 4.0 4.0 2.1 1.9 1.1 0.5 3.2 3.0 2.2 Answer:
2) The heights (in inches) of 30 lawyers are listed below. Construct a frequency ogive using five classes. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68 Answer:
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3) The local police, using radar, checked the speeds (in mph) of 30 motorists on a rural road. The results are listed below. Construct a frequency ogive using six classes. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer:
4) The grade point averages for 40 evening students are listed below. Construct a relative frequency ogive using eight classes. 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 3.1 2.4 2.4 2.3 1.6 1.6 4.0 3.1 3.2 1.8 2.2 2.2 1.7 0.5 3.6 3.4 1.9 2.0 3.0 1.1 3.0 4.0 4.0 2.1 1.9 1.1 0.5 3.2 3.0 2.2 Answer:
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5) The heights (in inches) of 30 lawyers are listed below. Construct a relative frequency ogive using five classes. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 69 71 68 67 73 74 70 71 69 68 Answer:
6) The local police, using radar, checked the speeds (in mph) of 30 motorists on a rural road. The results are listed below. Construct a relative frequency ogive using six classes. 44 38 41 50 36 36 43 42 49 48 35 40 37 41 43 50 45 45 39 38 50 41 47 36 35 40 42 43 48 33 Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 7) An ogive is a graph that represents cumulative frequencies or cumulative relative frequencies. The points labeled on the horizontal axis are the A) Upper class limits B) Lower class limits C) Midpoints D) Frequencies Answer: A
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5 Draw time-series graphs. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use a time series plot to display the data. Comment on the trend, 1) The data below represent the consumption of high-energy drinks (in gallons) by adult Americans over a nine-year period. Year 1 2 3 4 5 6 7 8 9 Consumption (gal) 10 11 11 12 13 14 15 15 13 Answer: In general, there is an increasing trend in high-energy drinks consumption of adult Americans. However, beginning in Year 9, there is sign of a decreasing trend.
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2) A transportation engineer wishes to use the following data to illustrate the number of deaths from the collision of passenger cars with motorcycles on a particular highway. Year Number of Deaths 1 12 2 17 3 22 4 21 5 16 6 13 7 11 Answer:
From Year 1 to Year 3, there was an increasing trend in the number of collision deaths. Subsequently, there was a decreasing trend. 3) Women were allowed to enter the Boston Marathon for the first time in 1972. Listed below are the winning women's times (in minutes) for the first 10 years. Year 1 2 3 4 5 6 7 8 9 Time 190 186 167 162 167 168 165 155 154
10 147
Answer:
In general, there was a decreasing trend in women's Boston marathon times.
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2.4 Graphical Misrepresentations of Data 1 Describe what can make a graph misleading or deceptive. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Explain what is misleading about the graphic. 1)
"The volume of our sales has doubled!" A) The length of a side has doubled, but the area has been multiplied by 4. B) The length of a side has doubled, but the area has been multiplied by 8. C) The length of a side has doubled, but the area has been unchanged. D) The graphic is not misleading. Answer: A 2)
2004 2006 2008 2010 2012 A) The vertical scale does not begin at zero. C) The trend is depicted in the wrong direction.
B) The horizontal label is incomplete. D) The graphic is not misleading.
Answer: A 3)
2012 DUI Figures for State County
A) The graphic may give the impression that drivers over age 65 had no DUI's in 2012. B) The graphic only includes information for one year. C) The horizontal scale does not begin at zero. D) The graphic is not misleading. Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 4) The following graph shows the number of car accidents occurring in one city in each of the years 2014 through 2019 (Year 1 = 2014, Year 2 = 2015 etc). The number of accidents dropped in 2016 after a new speed limit was imposed. How is the bar graph misleading? How would you redesign the graph to be less misleading?
Answer: The bar graph is misleading because the vertical axis starts at 60 instead of 0. This tends to indicate that the number of accidents decreased at a faster rate than they actually did. The graph would be less misleading if the vertical scale began at 0 or if a symbol were used to clearly indicate that the vertical scale is truncated and has a gap. 5) A parcel delivery store finds that their delivery rates increased over the past year. Last year it delivered 3402 parcels. This year it delivered 8942 parcels.
Discuss why the graphics may be misleading. Answer: roughly 3 times larger
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Ch. 2 Organizing and Summarizing Data Answer Key 2.1 Organizing Qualitative Data 1 Organize qualitative data in tables. 1) Class Frequency Relative Frequency Large 345 0.190 Medium 830 0.456 Small 645 0.354 Total 1820 1.000 2) Response Frequency Relative Frequency Strongly Favor 23 0.115 Favor 14 0.07 Neutral 29 0.145 Oppose 5 0.025 Strongly Oppose 129 0.645 3) Color Frequency Relative Frequency purple 3 0.15 green 4 0.20 blue 5 0.25 red 7 0.35 yellow 1 0.05 4) A 5) A 2 Construct bar graphs. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11)
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Percentage 19.0 45.6 35.4 100.0
12) 13) a) See graph below b) The minimum score = 14 c) The university will accept 76.57% of the applicants.
14) A 3 Construct pie charts. 1) A 2) A 3) A
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4)
5)
6) A
2.2 Organizing Quantitative Data: The Popular Displays 1 Organize discrete data in tables. 1) Hours Number of On Net HS Students 4 4 5 5 6 6 7 5 8 4 9 2 10 2 13 2 2) Measure Frequency 95 5 96 6 97 5 98 3 99 3 100 3
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2 Construct histograms of discrete data. 1)
2)
3 Organize continuous data in tables. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) Precip. Frequency Relative Frequency 0.5-0.9 4 0.10 1.0-1.4 2 0.05 1.5-1.9 7 0.175 2.0-2.4 9 0.225 2.5-2.9 2 0.05 3.0-3.4 10 0.25 3.5-3.9 2 0.05 4.0-4.4 4 0.10
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11) Commute Time (in min) 67.0-68.4 68.5-69.9 70.0-71.4 71.5-72.9 73.0-74.4
Frequency 6 5 13 2 4
Relative Frequency 0.200 0.167 0.433 0.067 0.133
12) Util. Bill (dollars) Frequency Relative Frequency 33-35 3 0.10 36-38 6 0.20 39-41 6 0.20 42-44 6 0.20 45-47 3 0.10 48-50 6 0.20 13) a. 95-104, 105-114, 115-124, 125-134, 135-144 b. Weight (lb) Tally Frequency 95-104 ll 2 105-114 lll 3 115-124 lllll 5 125-134 lll 3 135-144 ll 2 4 Construct histograms of continuous data. 1)
2) Frequency Histogram:
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Relative Frequency Histogram:
3)
Number of Cousins 4)
Number of Cousins 5)
Weight (pounds) 6) A
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5 Draw dot plots. 1)
2)
6 Identify the shape of a distribution. 1) A 2) A 3) A 4) A 5) A 6) A 7) symmetric
2.3 Additional Displays of Quantitative Data 1 Draw stem-and-leaf plots. 1) A 2) 74, 81, 90, 98, 104, 113, 114, 126, 129, 136, 137, 139, 142, 143, 148, 149, 156, 159 3) 5.5, 6.7, 7.0, 7.3, 8.8, 9.5, 9.8, 10.6, 10.9, 11.6, 11.7, 11.9, 12.2, 12.3, 12.8, 12.9, 13.0, 13.9
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4) 0 1 2 3 4 5 6 7
399 2 2399 29 28 0
5) 0 099 1 2 5 3 4 5 66 7 8 8 9 1 10 2 11 4 12 2 13 14 15 0 16 5 6) 6 77888899999 7 0000011111111223344 7) 3 3556667889 4 00111223334557889 5 000 8) The stem will consist of the tens digit and range from 1 to 9. The leaves will be drawn in the appropriate stems based on the data values. Stem Leaves 15 2 3 4 50 8 66 9 76 7 9 87 8 5 9 3 8 2 92 8 4 0 6
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2 Construct frequency polygons. 1)
2)
3)
4)
5) A 6) A
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3 Create cumulative frequency and relative frequency tables. 1) Relative Cumulative Cumulative Precip (in.) Frequency Frequency Frequency Relative Frequency 0.5-0.9 4 0.10 4 0.10 1.0-1.4 2 0.05 6 0.15 1.5-1.9 7 0.175 13 0.325 2.0-2.4 9 0.225 22 0.55 2.5-2.9 2 0.05 24 0.60 3.0-3.4 10 0.25 34 0.85 3.5-3.9 2 0.05 36 0.90 4.0-4.4 4 0.10 40 1 2) Relative Cumulative Cumulative Commute Time (in min) Frequency Frequency Frequency Relative Frequency 67.0-68.4 6 0.20 6 0.20 68.5-69.9 5 0.167 11 0.367 70.0-71.4 13 0.433 24 0.80 71.5-72.9 2 0.067 26 0.867 73.0-74.4 4 0.133 30 1 3) Relative Cumulative Cumulative Speed (in mph) Frequency Frequency Frequency Relative Frequency 33-35 3 0.10 3 0.10 36-38 6 0.20 9 0.30 39-41 6 0.20 15 0.50 42-44 6 0.20 21 0.70 45-47 3 0.10 24 0.80 48-50 6 0.20 30 1 4 Construct frequency and relative frequency ogives. 1)
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2)
3)
4)
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5)
6)
7) A
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5 Draw time-series graphs. 1) In general, there is an increasing trend in high-energy drinks consumption of adult Americans. However, beginning in Year 9, there is sign of a decreasing trend.
2)
From Year 1 to Year 3, there was an increasing trend in the number of collision deaths. Subsequently, there was a decreasing trend.
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3)
In general, there was a decreasing trend in women's Boston marathon times.
2.4 Graphical Misrepresentations of Data 1 Describe what can make a graph misleading or deceptive. 1) A 2) A 3) A 4) The bar graph is misleading because the vertical axis starts at 60 instead of 0. This tends to indicate that the number of accidents decreased at a faster rate than they actually did. The graph would be less misleading if the vertical scale began at 0 or if a symbol were used to clearly indicate that the vertical scale is truncated and has a gap. 5) roughly 3 times larger
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Ch. 3 Numerically Summarizing Data 3.1 Measures of Central Tendency 1 Determine the arithmetic mean of a variable from raw data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the population mean or sample mean as indicated. 1) Sample: 7, 9, 15, 17, 22 A) x = 14
B) x = 15
C) μ = 13
D) μ = 14
B) x = 10
C) x = 9
D) μ = 11
Answer: A 2) Population: 4, 7, 3, 11, 13, 10 A) μ = 8 Answer: A Provide an appropriate response. 3) If X 1 , X 2 , X 3 , ..., X N are the N observations of a variable from a population, then the population mean is symbolized by A) μ
B) X
C) Σ
~
D) X
Answer: A 4) A numerical summary of a population is a A) Parameter C) Variable
B) Statistic D) Qualitative response
Answer: A 5) The ________ of a variable is computed by determining the sum of all the values of the variable in the data set and dividing this sum by the number of observations in the data set. A) Arithmetic mean B) Median C) Mode D) Geometric mean Answer: A 6) The heights of ten female students (in inches) in a college math class are listed below. Find the mean. 65 66 67 66 67 70 67 70 71 68 A) 67.7 inches B) 65.5 inches C) 71.1 inches D) 70.0 inches Answer: A 7) The heights of ten male students (in inches) in a college biology class are listed below. Find the mean. 71 67 67 72 76 72 73 68 72 72 A) 71 inches B) 67 inches C) 68 inches D) 72 inches Answer: A 8) The repair costs for five cars which were crashed by a safety testing organization were as follows: $100, $150, $200, $250, and $150. Find the mean cost of repair. A) $170 B) $160 C) $180 D) $140 Answer: A
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9) Each year advertisers spend billions of dollars purchasing commercial time on network sports television. In the first 6 months of 1988, advertisers spent $1.1 billion. A recent article listed the top 10 leading spenders (in millions of dollars): Company A $73.2 Company B 62.6 Company C 57.1 Company D 57 Company E 28.7
Company F $27.9 Company G 25.5 Company H 23.6 Company I 21.4 Company J 19
Calculate the mean amount spent. A) 39.60 million dollars C) 414.79 million dollars
B) 20.37 million dollars D) 54.20 million dollars
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) Find the mean of the following Statistic students' test scores: 91 94 87 91 84 92 91 85 86 89 Answer: mean 89 11) Calculate the mean for the following sample: 5, 10, 6, 11, 4, 4, 10, 4. Answer: mean = 6.75 12) Calculate the mean for the following sample: 5, 10, 6, 100, 0, 0, 10, 0. Answer: mean = 6.375 13) The high temperatures (in degrees Celsius) each day over a three week period were as follows: 17, 18, 20, 22, 21, 19, 16, 15, 18, 20, 21, 21, 22, 21, 19, 20, 19, 17, 16, 16, 17. Compute the mean. Answer: mean = 18.81°C 14) In a sample of 18 students at East High School the following number of days of absences were recorded for the previous semester: 4, 3, 1, 0, 4, 2, 3, 0, 1, 2, 3, 0, 4, 1, 1, 5, 1, 1. Compute the mean. Answer: mean = 2 days 15) The number of goals scored by a random sample of 16 hockey players for a given season are 5, 3, 21, 10, 7, 2, 0, 30, 19, 6, 4, 7, 10, 5, 7, and 24. Compute the mean. Answer: mean = 10 goals 2 Determine the median of a variable from raw data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Which measure of central tendency may have more than one value in a numeric data set? A) Mode B) Median C) Mean D) Midrange Answer: A 2) Which measure of central tendency is not resistant to extreme values in a numeric data set? A) Mean B) Mode C) Median D) Parameters Answer: A
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3) The number of students enrolled in a physics class for the last ten semesters are listed below. Find the median number of students. 65 66 67 66 67 70 67 70 71 68 A) 67 students B) 66 students C) 68 students D) 70 students Answer: A 4) The commuting times (in minutes) of an employee for ten consecutive days are listed below. Find the median commute. 71 67 67 72 76 72 73 68 72 72 A) 72 minutes B) 67 minutes C) 71 minutes D) 73 minutes Answer: A 5) The following data represents a random sample of 15 complaints registered with the customer service department of a store. Determine the median complaint. Other defective productexcessive waiting time Messy store other other Messy store other messy store Other messy store messy store Defective product other messy store A) No median B) Messy store C) Defective product D) Excessive waiting time Answer: A 6) Each year advertisers spend billions of dollars purchasing commercial time on network sports television. In the first 6 months of the year, advertisers spent $1.1 billion. A recent article listed the top 10 leading spenders (in millions of dollars): Company A Company B Company C Company D Company E
$70 63.7 54.3 54 28.4
Company F $27.4 Company G 26.7 Company H 21.2 Company I 22.5 Company J 20.3
Calculate the median. A) 27.90 million dollars C) 38.85 million dollars
B) 5.28 million dollars D) 49.70 million dollars
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) The number of yards that a football player rushed in the first 13 games of his NFL career are listed below. Find the mean and median number of yards rushed. Round the mean to the nearest whole number. Which measure of central tendency-the mean or the median-better represents the data? Explain your reasoning. 3 49 32 33 39 22 42 9 9 39 52 58 70 Answer: mean: 35 yd; median 39 yd; The median better represents the data since the data is not symmetric. 8) The number of homework points earned by twelve students in a history class over one term are listed below. Find the mean and median number of points. Round the mean to the nearest whole number. Which measure of central tendency- the mean or the median- best represents the data? Explain your reasoning. 102 56 25 9 9 56 165 88 122 150 91 114 Answer: mean: 82; median: 89.5; the median
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9) The annual profits of ten internet businesses are listed below. Find the mean and median profits. Round the median to the nearest dollar. Which measure- the mean or the median- best represents the data? Explain your reasoning. $1,172,246 $163,659 $440,584 $350,634 $290,596 $186,731 $145,809 $143,209 $139,096 $125,106 Answer: mean: $315,767; median: $175,195; the median 3 Explain what it means for a statistic to be resistant. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Describe the shape of the histogram. The data set: Pick-Three lottery results for 10 consecutive weeks 3 6 7 6 0 6 1 7 8 4 1 5 7 5 9 1 5 3 9 9 2 2 3 0 8 8 4 0 2 4 A) uniform B) symmetric C) skewed to the left D) skewed to the right Answer: A 2) Describe the shape of the histogram. The data set: age of 20 household stereo systems randomly selected from a neighborhood 12 6 4 9 11 1 7 8 9 8 9 13 5 15 7 6 8 8 2 1 A) symmetric B) uniform C) skewed to the left D) skewed to the right Answer: A 3) Describe the shape of the histogram. The data set: round-trip commuting times (in minutes) of 20 randomly selected employees 135 120 115 132 136 124 119 145 98 110 125 120 115 130 140 105 116 121 125 108 A) skewed to the right B) uniform C) skewed to the left D) symmetric Answer: A 4) 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) median B) mode C) mean D) frequency Answer: A
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5)
For the distribution drawn here, identify the mean, median, and mode. A) A = mode, B = median, C = mean B) A = median, B = mode, C = mean C) A = mode, B = mean, C = median D) A = mean, B = mode, C = median Answer: A 6) In distributions that are skewed to the right, what is the relationship of the mean, median, and mode? A) mean > median > mode B) median > mean > mode C) mode > median > mean D) mode > mean > median Answer: A 7) In distributions that are skewed to the left, what is the relationship of the mean, median, and mode? A) mode > median > mean B) mean > median > mode C) mode < mean < median D) mode > mean > median Answer: A 8) Many firms use on-the-job training to teach their employees new software. Suppose you work in the personnel department of a firm that just finished training a group of its employees in new software, 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 74. Additional information indicated that the median of the test scores was 81. What type of distribution most likely describes the shape of the test scores? A) skewed to the left B) symmetric C) skewed to the right D) unable to determine with the information given Answer: A 9) A severe drought affected several western states for 3 years. A Christmas tree farmer is worried about the drought's effect on the size of his trees. To decide whether the growth of the trees has been retarded, the farmer decides to take a sample of the heights of 25 trees and obtains the following results (recorded in inches): 60 57 62 69 46 54 64 60 59 58 75 51 49 67 65 44 58 55 48 62 63 73 52 55 50 Which measure of central tendency would be considered the best measure to use in this problem? A) mean B) median C) mode D) range Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) 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 150 students and carefully recorded their parking times. The parking times recorded followed a distribution that was skewed to the right. Based on this information, discuss the relationship between the mean and the median for the 150 student parking times collected. Answer: Since the distribution of parking times is skewed to the right, we know that the mean parking time will exceed the median parking time. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 11) If the graph of a distribution of data shows that the graph is skewed to the right then the A) Mean > Median B) Mean ≈ Median C) Mean < Median D) No conclusion about the relative position of the mean and the median can be made Answer: A 12) If the graph of a distribution of data shows that the graph is skewed to the left then the A) Median > Mean B) Mean ≈ Median C) Mean > Median D) No conclusion about the relative position of the mean and the median can be made Answer: A 13) If the graph of a distribution of data shows that the graph is symmetric then the A) Mean is a better measure of central tendency B) Median is a better measure of central tendency C) Mode is a better measure of central tendency D) Midrange is a better measure of central tendency Answer: A 14) Which measure of central tendency is more representative of the typical observation if the graph of the data is skewed to the right? A) Median B) Mean C) Mode D) Midrange Answer: A 15) Which measure of central tendency is more representative of the typical observation if the graph of the data is skewed to the left? A) Median B) Mean C) Mode D) Midrange Answer: A 4 Determine the mode of a variable from raw data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The commuting times of ten employees (in minutes) are listed below. Find the mode score. 65 66 67 66 67 70 67 70 71 68 A) 67 minutes B) 65 minutes C) 66 minutes D) 68 minutes Answer: A
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2) The February utility bills (in dollars) for ten residents of a large city are listed below. Find the mode amount. 71 67 67 72 76 72 73 68 72 72 A) $72 B) $67 C) $76 D) $73 Answer: A 3) The annual profits of five large corporations in a certain area are given below. Which measure of central tendency should be used? $192,000 $200,000 $220,000 $190,000 $1,270,000 A) median B) mean C) mode D) midrange Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 4) The accompanying data set contains quiz scores for 12 students in a chemistry class: 18, 15, 5, 8, 15, 20, 2, 16, 10, 12, 20, 15. a. Find a measure of central tendency that separates the data into two groups such that each group consists of 50% of the scores above and 50% of the scores below that measure. b. Find a measure of central tendency that represents the quiz score that occurs most often. c. Find a measure of central tendency that represents the average of the 12 quiz scores. Answer: a. median = 15 b. mode = 15 c. mean = 13 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) The following data represent the bachelor degrees of CEO's at area small businesses. Determine the mode degree. Degree Number Accounting 29 Business 41 Liberal Arts 13 Marketing 30 Other 5 A) business B) accounting C) marketing D) no mode Answer: A 6) The following data represent a random sample of 15 complaints registered with the customer service department of a store. Determine the mode complaint. rude personnel excessive waiting time other defective product rude personnel rude personnel defective product messy store defective product rude personnel defective product defective product excessive waiting time rude personnel defective product A) defective product B) rude personnel C) other D) no mode Answer: A 7) Which measure of central tendency may not exist for all numeric data sets? A) Mode B) Median C) Mean Answer: A
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D) Midrange
3.2 Measures of Dispersion 1 Determine the range of a variable from raw data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute the range for the set of data. 1) 7, 8, 9, 10, 11 A) 4
B) 11
C) 0.8
D) 7
B) 17.9
C) 20
D) 13.1
B) 17
C) 12.5
D) 18
B) 16
C) 2
D) 3
B) 56
C) 13
D) 36.2
B) 13
C) 6.4
D) 14
B) 162
C) 33
D) 96.6
B) 597
C) 119
D) 356.3
B) 6.8
C) 1.4
D) 4.0
C) 0.117
D) 0.337
Answer: A 2) 14, 14, 14, 20, 21, 21, 21 A) 7 Answer: A 3) 8, 17, 8, 17, 8, 17, 8, 17 A) 9 Answer: A 4) 6, 16, 2, 14, 9 A) 14 Answer: A 5) 25, 40, 13, 47, 56 A) 43 Answer: A 6) 9, 4, 7, 1, 4, 13, 7, 7, 6 A) 12 Answer: A 7) 56, 134, 33, 98, 162 A) 129 Answer: A 8) 119, 543, 215, 597, 411, 253 A) 478 Answer: A 9) 2.6, 5.1, 1.4, 4.5, 6.8, 3.5 A) 5.4 Answer: A 10) 0.133, 0.117, 0.54, 0.383, 0.573, 0.276 A) 0.456 B) 0.573 Answer: A Provide an appropriate response. 11) The April precipitation amounts (in inches) for 10 cities are listed below. Find the range of the data. 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 A) 3.2 inches B) 2.45 inches C) 1.4 inches D) 2.8 inches Answer: A
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12) The January utility bills (in dollars) for 20 residents of a large city are listed below. Find the range of the data. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 A) $7 B) $2.98 C) $2 D) $1.73 Answer: A 2 Determine the standard deviation of a variable from raw data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the sample standard deviation. 1) 9, 10, 11, 12, 13 A) 1.6
B) 2.5
C) 1.3
D) 1.5
B) 2.8
C) 7.7
D) 9.0
B) 3.9
C) 16.9
D) 15.0
B) 22.3
C) 550.1
D) 495.1
C) 5,217.4
D) 4,637.7
B) 5.1
C) 1.3
D) 5.0
B) 5.4
C) 32.4
D) 28.8
Answer: A 2) 15, 15, 15, 18, 21, 21, 21 A) 3 Answer: A 3) 16, 16, 10, 13, 6, 13, 8, 18, 9 A) 4.1 Answer: A 4) 24, 88, 65, 54, 50, 89, 78, 76, 24 A) 24.9 Answer: A 5) 128, 185, 281, 280, 103, 109, 245, 256, 223 A) 72.2 B) 68.1 Answer: A 6) 18, 9, 11, 8, 20, 9, 6, 18, 5, 16 A) 5.5 Answer: A 7) 3, 4, 18, 14, 18, 9, 12, 5, 9 A) 5.7 Answer: A Provide an appropriate response. 8) The costs (in dollars) of 10 college math textbooks are listed below. Find the sample standard deviation. 70 72 71 70 69 73 69 68 70 71 A) $1.49 B) $70.30 C) $5.00 D) $2.23 Answer: A 9) The top speeds (in mph) for a sample of five new automobile brands are listed below. Calculate the standard deviation of the speeds. 165, 130, 200, 180, 100 A) 40.0 mph B) 31,000.00 mph C) 155.00 mph D) 100 mph Answer: A
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10) Over the last 10 years four mutual funds all had the same mean rate of return, 12%. These mutual funds had different standard deviations as follows: Mutual Money 8%, Co-joined Investments 6%, Together Fund 4%, All for One Fund 9%. Which mutual fund investment is the most consistent in rate of return? A) Together Fund B) Mutual Money C) Co-joined Investments D) All for One Fund Answer: A 3 Determine the variance of a variable from raw data. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The ages of five randomly selected students in the biology department at a private university are 24, 31, 35, 19, and 25. Calculate the sample variance of these ages. Answer: (x - x)2 s2 = ∑ n -1 x=
∑x = 24 + 31 + 35 + 19 + 25 = 26.8 n
5
(24 - 26.8)2 + (31 - 26.8)2 + (35 - 26.8)2 + (19 - 26.8)2 + (25 - 26.8)2 s2 = 5-1 = 39.20 2) The costs (in dollars) of 10 college math textbooks are listed below. Find the population standard deviation and the population variance. 70 72 71 70 69 73 69 68 70 71 Answer: σ = $1.42, σ2 = 2.01 3) In a random sample, 10 employees at a local plant were asked to compute the distance they travel to work to the nearest tenth of a mile. The data is listed below. Compute the range, sample standard deviation and sample variance of the data. 1.1 5.2 3.6 5.0 4.8 1.8 2.2 5.2 1.5 0.8 Answer: range = 4.4 miles, s = 1.8 miles, s2 = 3.324 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) Each year advertisers spend billions of dollars purchasing commercial time on network sports television. In the first 6 months of 1988, advertisers spent $1.1 billion. Who were the largest spenders? In a recent article, listed the top 10 leading spenders (in million of dollars): Company A $70.2 Company B 61 Company C 55 Company D 54.9 Company E 32
Company F $27.7 Company G 27.1 Company H 21.1 Company I 21.9 Company J 20.1
Calculate the sample variance. A) 361.61 B) 19.02 Answer: A
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C) 39.10
D) 50.10
5) Last year batting averages in professional baseball averaged 0.254 with a high of 0.332 and a low of 0.228 (minimum 250 at-bats). Based on this information, which measure of variation could be calculated? A) range B) variance C) standard deviation D) percentile Answer: A 6) Which is not a measure of dispersion? A) Mean C) Variance
B) Standard deviation D) Range
Answer: A 7) True or False: The variance of a population is the arithmetic average of the squared deviations about the population mean. A) True B) False Answer: A 8) True or False: Variance is the square root of standard deviation. A) False B) True Answer: A 9) The book cost (in dollars) for one semester's books are given below for a sample of five college students. Calculate the sample variance of the book costs. 265, 135, 375, 485, 310 A) 16,855.00 B) 129.83 C) 350.00 D) 314.00 Answer: A 4 Use the Empirical Rule to describe data that are bell shaped. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) At a tennis tournament a statistician keeps track of every serve. 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 11 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bell shaped. What proportion of the player's serves are expected to be between 111 mph and 122 mph? A) 0.135,0 B) 0.270 C) 0.95 D) 0.68 Answer: A 2) 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 watched television. The mean and the standard deviation for their responses were 15 and 2, respectively. PAWT constructed a stem-and-leaf display for the data that showed that the distribution of times was a bell-shaped distribution. Give an interval around the mean where you believe most (approximately 95%) of the television viewing times fell in the distribution. A) between 11 and 19 hours per week B) less than 13 and more than 17 hours per week C) between 9 and 21 hours per week D) between 13 and 17 hours per week Answer: A
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3) Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify for government health care but who have no private health insurance. The ages of 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 74.0 and 9.7, respectively. If we assume that the distribution of ages is bell shaped, what percentage of the respondents will be between 64.3 and 93.4 years old? A) approximately 81.5% B) approximately 68% C) approximately 95% D) approximately 83.9% Answer: A 4) A small computing center has found that the number of jobs submitted per day to its computers has a distribution that is approximately bell shaped, with a mean of 100 jobs and a standard deviation of 11. Where do we expect most (approximately 95%) of the distribution to fall? A) between 78 and 122 jobs per day B) between 89 and 111 jobs per day C) between 67 and 133 jobs per day D) between 78 and 133 jobs per day Answer: A 5) 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 20 on a standardized test. Assuming a bell-shaped distribution, what percentage of scores exceeded 260? A) approximately 97.5% B) approximately 95% C) approximately 2.5% D) approximately 84% Answer: A 6) 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 320 with a standard deviation of 30 on a standardized test. Assuming a bell-shaped distribution, where would approximately 95% of the students score? A) between 260 and 380 B) between 230 and 410 C) below 230 or above 410 D) below 260 or above 380 Answer: A 7) Solar energy is considered by many to be the energy of the future. A recent survey was taken 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 $132 and a standard deviation of $10. If the distribution can be considered bell shaped, what percentage of homes will have a monthly utility bill of more than $122? A) approximately 84% B) approximately 95% C) approximately 16% D) approximately 32% Answer: A 8) Many firms use on-the-job training to teach their employees new software. Suppose you work in the personnel department of a firm that just finished training a group of its employees in new software, 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 82 and 3, respectively, and the distribution of scores is bell shaped. What percentage of test-takers scored better than a trainee who scored 73? A) approximately 99.85% B) approximately 84% C) approximately 95% D) approximately 97.5% Answer: A
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9) A severe drought affected several western states for 3 years. A Christmas tree farmer is worried about the drought's effect on the size of his trees. To decide whether the growth of the trees has been retarded, the farmer decides to take a sample of the heights of 25 trees. Typically trees of this age have a mean height of 65 inches with a standard deviation of 9 inches. Assuming the distribution is bell shaped, where do you expect middle 95% of the tree heights to fall? A) between 47 and 83 inches tall B) between 56 and 74 inches tall C) between 38 and 92 inches tall D) over 56 inches tall Answer: A 10) The scores from a state standardized test have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. Use the Empirical Rule to find the percentage of students with scores between 70 and 130. A) 95% B) 68% C) 99.7% D) 100% Answer: A 11) The scores from a state standardized test have a mean of 80 and a standard deviation of 10. The distribution of the scores is roughly bell shaped. Use the Empirical Rule to find the percentage of scores that lie between 60 and 80. A) 47.5% B) 68% C) 34% D) 95% Answer: A 12) The average score of local students on a college entrance exam is 110, with a standard deviation of 5. The distribution is roughly bell shaped. Use the Empirical Rule to find the percentage of local students with scores above 120. A) 2.5% B) 5% C) 95% D) 97.5% Answer: A 5 Use Chebyshev's inequality to describe any set of data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 95 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, give an interval that will contain the speeds of at least eight-ninths of the player's serves. A) 50 mph to 140 mph B) 65 mph to 125 mph C) 35 mph to 155 mph D) 140 mph to 185 mph Answer: A 2) 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 250 with a standard deviation of 30 on a standardized test. Assuming no information concerning the shape of the distribution is known, what percentage of the students scored between 190 and 310? A) at least 75% B) approximately 95% C) at least 88.9% D) approximately 68% Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) Commuting times for employees of a local company have a mean of 63.6 minutes and a standard deviation of 2.5 minutes. What does Chebyshev's Theorem say about the percentage of employees with commuting times between 58.6 minutes and 68.6 minutes? Answer: At least 75% of the commuting times should fall between 58.6 minutes and 68.6 minutes.
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4) Commuting times for employees of a local company have a mean of 63.6 minutes and a standard deviation of 2.5 minutes. What does Chebyshev's Theorem say about the percentage of employees with commuting times between 56.1 minutes and 71.1 minutes? Answer: At least 89% of the commuting times should fall between 56.1 minutes and 71.1 minutes. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) Fill in the blank. ____________ gives us a method of interpreting the standard deviation that applies to any data set, regardless of the shape of the distribution. A) Chebyshev's rule B) The Empirical Rule C) Chebyshev's rule and the empirical rule D) none of these Answer: A 6) Fill in the blank. ____________ is a method of interpreting the standard deviation that applies to data that have a bell-shaped distribution. A) The Empirical Rule B) Chebyshev's rule C) Chebyshev's rule and the empirical rule D) none of these Answer: A
3.3 Measures of Central Tendency and Dispersion from Grouped Data 1 Approximate the mean of a variable from grouped data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) For the following data, approximate the mean number of unused vacation days at the end of the year. Days Frequency 1-3 15 4-6 12 7-9 27 10-12 13 13-15 7 A) 7.9
B) 6.4
C) 8.4
D) 9.1
Answer: A 2) For the following data, approximate the mean number of emails received per day. Emails (per day) Frequency 8-11 21 12-15 25 16-19 6 20-23 15 24-27 49 A) 19.6 Answer: A
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B) 18.6
C) 17.6
D) 20.6
3) For the following data, approximate the mean weekly grocery bill. Bill (in dollars) Frequency 135-139 7 140-144 6 145-149 8 150-154 13 155-159 5 A) $147.90
B) $145.40
C) $149.40
D) $146.40
Answer: A 4) On a recent day during the flu season, the Midtown Medical Clinic saw many patients. Patients had their temperatures (in °F) taken. The distribution of temperatures is given below. Determine the mean temperature of a sample of 100 patients. Temperatures 95.6 - 96.49 96.5 - 97.39 97.4 - 98.29 98.3 - 99.19 99.2 - 100.09 100.1- 100.99 101.0- 101.89
Frequency 1 3 19 28 35 12 2
A) 99.08°F
B) 99.52°F
C) 98.75°F
D) 98.63°F
Answer: A 5) A 1-pound bag of peanuts contains 430 peanuts. The distribution of weights in grams of the peanuts is given below . What is the mean weight of a peanut? Weights (grams) 0.755- 0.814 0.815- 0.874 0.875- 0.934 0.935- 0.994 0.995- 1.054 1.055- 1.114 1.115- 1.174
Frequency 3 2 2 2 168 241 12
A) 1.059 g
B) 1.088 g
C) 0.965 g
D) 1.029 g
Answer: A 2 Compute the weighted mean. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A student receives test scores of 62, 83, and 91. The student's term project score is 88 and her homework score is 76. Each test is worth 20% of the final grade, the term project is 25% of the final grade, and the homework grade is 15% of the final grade. What is the student's mean score in the class? A) 80.6 B) 76.6 C) 83.3 D) 80.0 Answer: A Page 15
2) The grades are given for a student for a particular term. Find the grade point average. The point values of grades are given below. A : 4, B : 3, C : 2, D : 1, F : 0 Grade
Credit Hours
A C D A C
2 3 1 4 2
A) 2.92 C) 2.6
B) 2.4 D) 3.99
Answer: A 3) In a health food store, Jenny and Kevin create a trail mix from dried fruit, nuts, and granola. They buy 2 pounds of dried fruit at $4.00 per pound, 4 pounds of nuts at $5.00 per pound, and 5 pounds of granola at $3.50 per pound. Determine the cost per pound of the mix. A) $4.14 B) $3.64 C) $4.34 D) $4.26 Answer: A 4) Jim buys his school supplies in bulk. On one particular shopping trip he bought 5 red pens at $1.29 each, 3 blue pens at $1.49 each, 6 green pens at $1.79 each and 9 black pens at $0.99 each. What was the average cost of a pen? A) $1.33 B) $1.29 C) $0.99 D) $1.39 Answer: A 3 Approximate the standard deviation of a variable from grouped data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) For the following data set, approximate the sample standard deviation of unused vacation days . Days Frequency 1-2 9 3-4 22 5-6 28 7-8 15 9-10 14 A) 2.4 days
B) 5.5 days
C) 3.5 days
D) 5.9 days
Answer: A 2) For the following data set, approximate the sample standard deviation of emails per day. Emails (per day) Frequency 8-11 18 12-15 23 16-19 38 20-23 47 24-27 32 A) 5.1 emails Answer: A
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B) 4.0 emails
C) 25.8 emails
D) 19.3 emails
3) For the following data set, approximate the sample standard deviation of commuting times per day. Commute (in min) Frequency 50-52 5 53-55 8 56-58 12 59-61 13 62-64 11 A) 3.9 min B) 6.6 min
C) 2.5 min
D) 55.7 min
Answer: A 4) For the following data set, approximate the sample standard deviation of unused vacation days. Days Frequency 1-2 9 3-4 12 5-6 8 7-8 5 9-10 4 A) 2.6 days
B) 2.5 days
C) 6.5 days
D) 6.6 days
Answer: A 5) For the following data set, approximate the sample standard deviation of distances from work (in miles). Distance(miles) Frequency 8-11 15 12-15 21 16-19 36 20-23 39 24-27 23 A) 4.9 miles
B) 24.5 miles
C) 5.2 miles
D) 24.3 miles
Answer: A 6) For the following data set, approximate the sample standard deviation of monthly telephone bills (in dollars). Bill (in $) Frequency 50-52 2 53-55 5 56-58 12 59-61 19 62-64 7 A) $3.11 Answer: A
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B) $58.60
C) $93.12
D) $9.65
7) On a recent day during the flu season, the Midtown Medical Clinic saw many patients. Patients had their temperatures taken (in °F). The distribution of temperatures is given below. Determine the standard deviation of the temperatures of a sample of 100 patients. Temperatures 95.6 - 96.49 96.5 - 97.39 97.4 - 98.29 98.3 - 99.19 99.2 - 100.09 100.1- 100.99 101.0- 101.89
Frequency 1 3 19 28 35 12 2
A) 1.021°F
B) 1.031°F
C) 1.015°F
D) 1.04°F
Answer: A 8) A 1-pound bag of peanuts contained 430 peanuts. The distribution of weights (in grams) of the peanuts is given below . What is the sample standard deviation of the weight of a peanut? Weights (grams) 0.755- 0.814 0.815- 0.874 0.875- 0.934 0.935- 0.994 0.995- 1.054 1.055- 1.114 1.115- 1.174
Frequency 3 2 2 2 168 241 12
A) 0.045 g
B) 0.002 g
C) 0.209 g
D) 0.000004 g
Answer: A
3.4 Measures of Position and Outliers 1 Determine and interpret z-scores. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Test scores for a statistics class had a mean of 79 with a standard deviation of 4.5. Test scores for a calculus class had a mean of 69 with a standard deviation of 3.7. Suppose a student gets a 75 on the statistics test and a 87 on the calculus test. Calculate the z-score for each test. On which test did the student perform better relative to the other students in each class? Answer: statistics z-score = -0.89; calculus z-score = 4.86; The student performed better on the calculus test.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) A student scores 68 on a geography test and 249 on a mathematics test. The geography test has a mean of 80 and a standard deviation of 10. The mathematics test has a mean of 300 and a standard deviation of 34. If the data for both tests are normally distributed, on which test did the student score better relative to the other students in each class? A) The student scored better on the geography test. B) The student scored better on the mathematics test. C) The student scored the same on both tests. Answer: A 3) Many firms use on-the-job training to teach their employees new software. Suppose you work in the personnel department of a firm that just finished training a group of its employees in new software, 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 84 and 4, respectively, and the distribution of scores is mound-shaped and symmetric. Suppose the trainee in question received a score of 71. Compute the trainee's z-score. A) z = -3.25 B) z = 3.25 C) z = -0.8 D) z = 0.80 Answer: A 4) A severe drought affected several western states for 3 years. A Christmas tree farmer is worried about the drought's effect on the size of his trees. To decide whether the growth of the trees has been retarded, the farmer decides to take a sample of the heights of 25 trees and obtains the following results (recorded in inches): 60 57 62 69 46 54 64 60 59 58 75 51 49 67 65 44 58 55 48 62 63 73 52 55 50 The tree farmer feels the normal height of a tree that was unaffected by the drought would be 65 inches. Find the z-score for a tree that is 65 inches tall. A) z = 0.84 B) z = 0.98 C) z = 0.77 D) z = 0.98 Answer: A 5) A television station claims that the amount of advertising per hour of broadcast time has an average of 13 minutes and a standard deviation equal to 2.2 minutes. You watch the station for 1 hour, at a randomly selected time, and carefully observe that the amount of advertising time is equal to 16 minutes. Calculate the z-score for this amount of advertising time. A) z = 1.36 B) z = -1.36 C) z = -1.06 D) z = 1.06 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) 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 270 and a standard deviation of 30 on a standardized test. Find and interpret the z-score of a student who scored 490 on the standardized test. Answer: The z-score is z =
x-μ . σ
For a score of 49, z =
490 - 270 = 7.33. 30
This student's score falls 7.33 standard deviations above the mean score of 270.
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7) Solar energy is considered by many to be the energy of the future. A recent survey was taken 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 $114 and a standard deviation of $14. 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 $212.00? Explain. Answer: The z-score for the value $212.00 is: z=
x - x 212 - 114 = =7 s 14
An observation that falls 7 standard deviations above the mean is very unlikely. We would not expect to see a monthly utility bill of $212.00 for this home. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 8) Find the z-score for the value 50, when the mean is 78 and the standard deviation is 9. A) z = -3.11 B) z = -3.22 C) z = -0.53
D) z = 0.53
Answer: A 9) A highly selective boarding school will only admit students who place at least 1.5 z-scores above the mean on a standardized test that has a mean of 110 and a standard deviation of 12. What is the minimum score that an applicant must make on the test to be accepted? A) 128 B) 92 C) 122 D) 98 Answer: A 10) A pharmaceutical testing company wants to test a new cholesterol drug. The average cholesterol of the target population is 200 mg and they have a standard deviation of 25 mg. The company wished to test a sample of people who fall between 1.5 and 3 z-scores above the mean. Into what range must a candidate's cholesterol level be in order for the candidate to be included in the study? A) 237.5 - 275 B) 225 - 237.5 C) 125 - 162.5 D) 162.5 - 275 Answer: A 2 Interpret percentiles. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) When results from a scholastic assessment test are sent to test-takers, the percentiles associated with their scores are also given. Suppose a test-taker scored at the 79th percentile for their verbal grade and at the 44th percentile for their quantitative grade. Interpret these results. A) This student performed better than 79% of the other test-takers in the verbal part and better than 44% in the quantitative part. B) This student performed better than 79% of the other test-takers in the verbal part and better than 56% in the quantitative part. C) This student performed better than 21% of the other test-takers in the verbal part and better than 56% in the quantitative part. D) This student performed better than 21% of the other test-takers in the verbal part and better than 44% in the quantitative part. Answer: A 2) The percentage of measurements that are above the 39th percentile is A) 61% B) 39% C) 71% Answer: A
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D) cannot determine
3) The percentage of measurements that are below the 88th percentile is A) 88% B) 12% C) 22%
D) cannot determine
Answer: A Use the ogive to solve the problem. 4) The graph below is an ogive of scores on a math test. The vertical axis in an ogive is the cumulative relative frequency and can also be interpreted as a percentile. Percentile Ranks of Math Test Scores 100 90 80
Percentile
70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 Test Score
Use the graph to approximate the percentile rank of an individual whose test score is 30. A) 4 B) 55 C) 9 D) 50 Answer: A
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5) The graph below is an ogive of scores on a math test. The vertical axis in an ogive is the cumulative relative frequency and can also be interpreted as a percentile. Percentile Ranks of Math Test Scores 100 90 80
Percentile
70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 Test Score
Use the graph to approximate the test score that corresponds to the 30th percentile? A) 56 B) 4 C) 50
D) 9
Answer: A 3 Determine and interpret quartiles. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The commute times (in minutes) of 30 employees are listed below. Find Q2 . 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) 72 min B) 83 min
C) 88 min
D) 71 min
Answer: A 2) The one way distances from work (in miles) of 30 employees are listed below. Find Q3 . 25 25 26 26.5 27 27 27.5 28 28 28.5 29 29 30 30 30.5 31 31 32 32.5 32.5 33 33 34 34.5 35 35 37 37 38 38 A) 34 mi B) 28 mi Answer: A
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C) 30.75 mi
D) 35 mi
3) Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify for government health care but who have no private health insurance. The ages of 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 Q3 of the data. A) 81.5
B) 73
C) 80.5
D) 82.5
Answer: A 4) Health care issues are receiving much attention in both academic and political arenas. A sociologist recently conducted a survey of senior citizens whose net worth is too high to qualify for government health care but who have no private health insurance. The ages of 25 uninsured senior citizens were as follows: 70 63 78 72
75 91 64 75
68 67 83 62
78 92 65 89
88 71 70 77
76 94 83 66 84
Find Q1 of the data. A) 67.5
B) 67
C) 68
D) 75.5
Answer: A 5) A group of 79 students were asked how far they commute to work from home each time they go to work from home. The results are given below. Determine the first quartile. Miles traveled 1 2 3 4 5 6 7 8 9 10 A) 4 mi Answer: A
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Frequency 1 2 12 18 7 10 10 11 5 3 B) 3 mi
C) 5 mi
D) 6 mi
4 Determine and interpret the interquartile range . SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The one way distances from work (in miles) of 30 employees are listed below. Find the interquartile range of the 30 distances listed below. 25 25 26 26.5 27 27 27.5 28 28 28.5 29 29 30 30 30.5 31 31 32 32.5 32.5 33 33 34 34.5 35 35 37 37 38 38 Answer: IQR = Q3 - Q1 = 34 - 28 = 6 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) The monthly telephone usage (in minutes) of 30 adults is listed below. Find the interquartile range for the telephone usage of the 30 adults. 154 156 165 165 170 171 172 180 184 185 189 189 190 192 195 198 198 200 200 200 205 205 211 215 220 220 225 238 255 265 A) 31 B) 30
C) 32
D) 29
Answer: A 3) The following data are the yields, in bushels, of hay from a farmer's last 10 years: 375, 210, 150, 147, 429, 189, 320, 580, 407, 180. Find the IQR. A) 227 B) 265 C) 253
D) 279
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 4) To study the physical fitness of a sample of 28 people, the data below were collected representing the number of sit-ups that a person could do in one minute. 10 20 32 47
12 22 33 48
12 25 40 48
15 25 40 50
15 26 40 52
15 29 45 53
18 30 46 56
Determine the lower and upper fences. Are there any outliers according to this criterion? Answer: lower fence = -22.25; upper fence = 87.75; outliers: none 5) The selling prices of mutual funds change daily. In order to study these changes, a sample of mutual funds was examined and the daily changes in price are listed below. 0.05, 0.00, -0.03, -0.01, 0.18, 0.00, 0.02, 0.29, 0.00, -0.07, 0.10, 0.07, 0.03 Determine the lower and upper fences. Are there any outliers according to this criterion? Answer: lower fence = -0.14; upper fence = 0.22; outliers: 0.29
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5 Check a set of data for outliers. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A group of 79 students were asked how far they commute to work from home each time they go to work from home. The results are given below. Would a drive of 15 miles be considered an outlier? Answer Yes or No. Miles traveled 1 2 3 4 5 6 7 8 9 10
Frequency 1 2 12 18 7 10 10 11 5 3
A) Yes
B) No
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) The normal monthly precipitation (in inches) for August is listed for 20 different U.S. cities. Find any outliers and provide an explanation for them. 0.4 2.2 3.5 3.9
1.0 2.4 3.6 4.1
1.5 2.7 3.6 4.2
1.6 3.4 3.7 4.2
2.0 3.4 3.7 7.0
Answer: 7.0 inches is an outlier. Perhaps a U.S. city with a tropical climate was included.
3.5 The Five-Number Summary and Boxplots 1 Compute the five-number summary. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Given the following five-number summary, find the interquartile range. 29, 37, 50, 66, 94 A) 29 B) 50 C) 65
D) 32.5
Answer: A 2) Given the following five-number summary, find Q3 . 2.9, 5.7, A) 13.2 Answer: A
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10.0,
13.2,
21.1. B) 5.7
C) 10.0
D) 21.1
3) Given the following five-number summary, find the IQR. 2.9, 5.7, 10.0, 13.2, 21.1. A) 7.5 B) 7.1
C) 11.1
D) 18.2
Answer: A 4) An Excel printout of some descriptive statistics for a set of data is shown below. What is the IQR?
A) 15.5
B) 15
C) 38
D) 5.5
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) The following is a sample of 19 November utility bills (in dollars) from a neighborhood: 52, 62, 66, 68, 72, 74, 74, 76, 76, 76, 78, 78, 82, 84, 84, 86, 88, 92, 96. Find the five-number summary. Answer: 52, 72, 76, 84, 96 6) Eleven high school teachers were asked to give the numbers of students in their classes. The sample data follows: 36, 31, 30, 31, 20, 19, 24, 34, 21, 28, 24. Find the five-number summary. Answer: 19, 21, 28, 31, 36 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) The following is a sample of 19 November utility bills (in dollars) from a neighborhood. What is the largest bill in the sample that would not be considered an outlier? 52, 62, 66, 68, 72, 74, 76, 76, 76, 78, 78, 82, 84, 84, 86, 88, 92, 96, 110 A) $96 B) $95 C) $88 D) $86 Answer: A 8) Eleven high school teachers were asked to give the number of students in their classes. The sample data follows. Would any of the class sizes be considered an outlier? Answer Yes or No. 36, 31, 30, 31, 20, 19, 24, 34, 21, 28, 24 A) No B) Yes Answer: A 9) A random sample of sale prices of homes yielded the following summary information: MIN $41,000 MAX $276,000
25%: $89,000 75%: $166,000
Median: $123,000
Comment on a home that had a sale price of $427,000. A) This value falls outside the upper fence and is considered an outlier. B) This sale price would be expected since it falls inside the lower and upper fences. C) This sale price falls between the lower and upper fences. It can be considered a potential outlier. D) This value falls outside of the third quartile, but cannot be considered an outlier. Answer: A
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2 Draw and interpret boxplots. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The boxplot shown below was constructed in Excel for the amount of soda that was poured by a filling machine into 12-ounce soda cans at a local bottling company.
Based on the information given in the boxplot below, what shape do you believe the data to have? A) skewed to the left B) approximately symmetric C) skewed to the right D) cannot be determined Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) The test scores of 30 students are listed below. Draw a boxplot that represents the data. 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 Answer:
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3) The cholesterol levels (in milligrams per deciliter) of 30 adults are listed below. Draw a boxplot that represents the data. 154 156 165 165 170 171 172 180 184 185 189 189 190 192 195 198 198 200 200 200 205 205 211 215 220 220 225 238 255 275 Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) SAS was used to compare the high school dropout rates for the 30 school districts in one city in 2010 and 2012. The box plots generated for these dropout rates are shown below. Compare the center of the distributions and the variation of the distributions for the two years.
YEAR
2010
2012
A) Dropout rates had a higher average with less variability in 2010 than in 2012. B) Dropout rates had a higher average with more variability in 2010 than in 2012. C) Dropout rates had a lower average with more variability in 2010 than in 2012. D) Dropout rates had a lower average with less variability in 2010 than in 2012. Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) A survey of 200 public universities indicated that the 25th percentile of the yearly tuition cost of the universities was $4800 and the 75th percentile was $7200. The minimum value was $2000, the median was $6000, and the maximum was $10,000. Use this information to construct a boxplot for the yearly tuition costs. Answer: To construct a boxplot, the first step is to find the interquartile range (IQR). IQR = Q3 - Q1 = 7200 - 4800 = 2400 Next, determine the fences. Lower fence = Q1 - 1.5 IQR = 4800 - 1.5(2400) = 1,200 Upper fence = Q3 + 1.5 IQR = 7200 + 1.5(2400) = 10,800 The boxplot is: 1,200
10,800
4800
7200
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6) In interpreting a boxplot of a data set we note that the median is to the left of the center of the box and the right line is longer than the left line. We can conclude that A) The data is skewed right. B) The data is skewed left. C) The data is symmetric. D) Skewness or symmetry cannot be determined by a box plot. Answer: A
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Ch. 3 Numerically Summarizing Data Answer Key 3.1 Measures of Central Tendency 1 Determine the arithmetic mean of a variable from raw data. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) mean 89 11) mean = 6.75 12) mean = 6.375 13) mean = 18.81°C 14) mean = 2 days 15) mean = 10 goals 2 Determine the median of a variable from raw data. 1) A 2) A 3) A 4) A 5) A 6) A 7) mean: 35 yd; median 39 yd; The median better represents the data since the data is not symmetric. 8) mean: 82; median: 89.5; the median 9) mean: $315,767; median: $175,195; the median 3 Explain what it means for a statistic to be resistant. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) Since the distribution of parking times is skewed to the right, we know that the mean parking time will exceed the median parking time. 11) A 12) A 13) A 14) A 15) A 4 Determine the mode of a variable from raw data. 1) A 2) A 3) A 4) a. median = 15 b. mode = 15 c. mean = 13 Page 30
5) A 6) A 7) A
3.2 Measures of Dispersion 1 Determine the range of a variable from raw data. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 2 Determine the standard deviation of a variable from raw data. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 3 Determine the variance of a variable from raw data. 1) (x - x)2 s2 = ∑ n -1 x=
∑x = 24 + 31 + 35 + 19 + 25 = 26.8 n
5
(24 - 26.8)2 + (31 - 26.8)2 + (35 - 26.8)2 + (19 - 26.8)2 + (25 - 26.8)2 s2 = 5-1 = 39.20 2) σ = $1.42, σ2 = 2.01 3) range = 4.4 miles, s = 1.8 miles, s2 = 3.324 4) A 5) A 6) A 7) A 8) A 9) A 4 Use the Empirical Rule to describe data that are bell shaped. 1) A 2) A 3) A 4) A Page 31
5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 5 Use Chebyshev's inequality to describe any set of data. 1) A 2) A 3) At least 75% of the commuting times should fall between 58.6 minutes and 68.6 minutes. 4) At least 89% of the commuting times should fall between 56.1 minutes and 71.1 minutes. 5) A 6) A
3.3 Measures of Central Tendency and Dispersion from Grouped Data 1 Approximate the mean of a variable from grouped data. 1) A 2) A 3) A 4) A 5) A 2 Compute the weighted mean. 1) A 2) A 3) A 4) A 3 Approximate the standard deviation of a variable from grouped data. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A
3.4 Measures of Position and Outliers 1 Determine and interpret z-scores. 1) statistics z-score = -0.89; calculus z-score = 4.86; The student performed better on the calculus test. 2) A 3) A 4) A 5) A x-μ 6) The z-score is z = . σ For a score of 49, z =
490 - 270 = 7.33. 30
This student's score falls 7.33 standard deviations above the mean score of 270.
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7) The z-score for the value $212.00 is: z=
x - x 212 - 114 = =7 s 14
An observation that falls 7 standard deviations above the mean is very unlikely. We would not expect to see a monthly utility bill of $212.00 for this home. 8) A 9) A 10) A 2 Interpret percentiles. 1) A 2) A 3) A 4) A 5) A 3 Determine and interpret quartiles. 1) A 2) A 3) A 4) A 5) A 4 Determine and interpret the interquartile range . 1) IQR = Q3 - Q1 = 34 - 28 = 6 2) A 3) A 4) lower fence = -22.25; upper fence = 87.75; outliers: none 5) lower fence = -0.14; upper fence = 0.22; outliers: 0.29 5 Check a set of data for outliers. 1) A 2) 7.0 inches is an outlier. Perhaps a U.S. city with a tropical climate was included.
3.5 The Five-Number Summary and Boxplots 1 Compute the five-number summary. 1) A 2) A 3) A 4) A 5) 52, 72, 76, 84, 96 6) 19, 21, 28, 31, 36 7) A 8) A 9) A 2 Draw and interpret boxplots. 1) A 2)
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3)
4) A 5) To construct a boxplot, the first step is to find the interquartile range (IQR). IQR = Q3 - Q1 = 7200 - 4800 = 2400 Next, determine the fences. Lower fence = Q1 - 1.5 IQR = 4800 - 1.5(2400) = 1,200 Upper fence = Q3 + 1.5 IQR = 7200 + 1.5(2400) = 10,800 The boxplot is: 1,200
10,800
4800 6) A
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7200
Ch. 4 Describing the Relation between Two Variables 4.1 Scatter Diagrams and Correlation 1 Draw and interpret scatter diagrams. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Construct a scatter diagram for the data. 1) The data below are the final exam scores of 10 randomly selected history students and the number of hours they studied for the exam. Hours, x 3 5 2 8 2 4 4 5 6 3 Scores, y 65 80 60 88 66 78 85 90 90 71 y
x
Answer: y 95 90 85
Exam Scores
80 75 70 65 60 1
2
3
4
5
6
7
Hours Studied
Page 1
8
x
2) The data below are the temperatures on randomly chosen days during a summer class and the number of absences on those days. Temperature, x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 y
x
Answer: y 16 14 12
Number 10 of Absences 8 6 4 2 70
75
80
85
90
Temperature
Page 2
95
100
x
3) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. Age, x 38 41 45 48 51 53 57 61 65 Pressure, y 116 120 123 131 142 145 148 150 152 y
x
Answer: y 155 150 145
Blood 140 Pressure 135 (mm of mercury) 130 125 120 115 35
40
45
50
55
Age
Page 3
60
65
x
4) The data below are the number of absences and the final grades of 9 randomly selected students from a literature class. Number of absences, x 0 3 6 4 9 2 15 8 5 Final grade, y 98 86 80 82 71 92 55 76 82 y
x
Answer: y 100 90
Final Grade
80 70 60 50 2
4
6
8
10
12
14
Number of Absences
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16 x
5) A manager wishes to determine the relationship between the number of miles (in hundreds of miles) the manager's sales representatives travel per month and the amount of sales (in thousands of dollars) per month. Miles traveled, x 2 3 10 7 8 15 3 1 11 Sales, y 31 33 78 62 65 61 48 55 120 y
x
Answer: 120
y
110 100
Sales (in thousands)
90 80 70 60 50 40 30 2
4
6
8
10
12
14
16 x
Miles traveled (in hundreds)
Page 5
6) In order for employees of a company to work in a foreign office, they must take a test in the language of the country where they plan to work. The data below show the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Number of years, x 3 4 4 5 3 6 2 7 3 Grades on test, y 61 68 75 82 73 90 58 93 72 y
x
Answer: y 100 90
Final Grade
80 70 60 50 1
2
3
4
5
6
7
Number of years studied
Page 6
x
7) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 y
x
Answer: y 90 80
Yield 70 (bushels) 60 50 40 30 6
8
10
12
14
16
18
Rainfall (inches)
Page 7
20
x
8) Five brands of cigarettes were tested for the amounts of tar and nicotine they contained. All measurements are in milligrams per cigarette. y
Cigarette Tar Nicotine Brand A 16 1.2 Brand B 13 1.1 Brand C 16 1.3 Brand D 18 1.4 Brand E 6 0.6
x
Answer: y 1.6 1.4 1.2
Nicotine 1 (mg) 0.8 0.6 0.4 0.2 6
9
12
Tar (mg)
Page 8
15
18
x
9) The scores of nine members of a local community college women's golf team in two rounds of tournament play are listed 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 y
x
Answer: y 95
90
Round 2 85
80
75 75
80
85
90
95
x
Round 1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Make a scatter diagram for the data. Use the scatter diagram to describe how, if at all, the variables are related. 2 7 5 4 8 3 10) x y 1 4 2 2 3 0 y 20 18 16 14 12 10 8 6 4 2 2 4
Page 9
6 8 10 12 14 16 18 20
x
A) The variables appear to be positively, linearly related.
B) The variables do not appear to be linearly related.
y
y
20 18
20 18
16 14 12
16 14 12
10 8 6
10 8 6
4 2
4 2 6 8 10 12 14 16 18 20 x
2 4
C) The variables appear to be negatively, linearly related.
2 4
D) The variables do not appear to be linearly related.
y
y
20 18
20 18
16 14 12
16 14 12
10 8 6
10 8 6
4 2
4 2 6 8 10 12 14 16 18 20 x
2 4
Answer: A 11) x y
7 5 0 4 3 9 2 12 14 10 13 16 12 19 28
y
24 20 16 12 8 4 -12 -8
-4
4 -4
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6 8 10 12 14 16 18 20 x
8
12
16 20
x
2 4
6 8 10 12 14 16 18 20 x
A) The variables do not appear to be linearly related.
B) The variables appear to be negatively, linearly related.
y
28
28
-12 -8
24
24
20
20
16
16
12
12
8
8
4
4
-4
4
8
12 16
20
x
-12 -8
-4
-4
4
8
12 16
D) The variables appear to be positively, linearly related.
y
28
-12 -8
24
24
20
20
16
16
12
12
8
8
4
4
-4
4
8
12 16
20
x
-4
Answer: A Subject A B C D E F G 12) x Time watching TV 13 9 7 12 12 10 11 y Time on Internet 8 6 2 11 12 3 12 20
y
18 16 14 12 10 8 6 4 2 2
4
6
20
x
20
x
-4
C) The variables do not appear to be linearly related. 28
Page 11
y
8 10 12 14 16 18 20 x
-12 -8
-4
y
4 -4
8
12 16
A) The variables appear to be positively, linearly related. 20
y
B) The variables do not appear to be linearly related. 20
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2 2
4
6
8 10 12 14 16 18 20 x
C) The variables appear to be negatively, linearly related. 20
y
2
18
18
16
16
14
14
12
12
10
10
8
8
6
6
4
4
2
2 4
6
8 10 12 14 16 18 20 x
4
6
8 10 12 14 16 18 20 x
D) The variables do not appear to be linearly related. 20
2
y
y
2
4
6
8 10 12 14 16 18 20 x
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 13) An agricultural business wants to determine if the rainfall in inches can be used to predict the yield per acre on a wheat farm. Identify the predictor variable and the response variable. Answer: predictor variable: rainfall in inches; response variable: yield per acre 14) A college counselor wants to determine if the number of hours spent studying for a test can be used to predict the grades on a test. Identify the predictor variable and the response variable. Answer: predictor variable: hours studying; response variable: grades on the test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 15) The
variable is the variable whose value can be explained by the
A) response; predictor C) lurking; response Answer: A
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B) response; lurking D) predictor Response
variable.
2 Describe the properties of the linear correlation coefficient. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the scatter diagrams shown, labeled a through f to solve the problem. 1) In which scatter diagram is r = 0.01? a b y
y
12
12
10
10
8
8
6
6
4
4
2
2 1
2
3
4
5
6
x
1
2
c
3
5
6
x
4
5
6
7
4
5
6
x
d
y
y
12
12
10
10
8
8
6
6
4
4
2
2 1
2
3
4
5
6
x
1
2
3
e
x
f
y
y
12
12
10
10
8
8
6
6
4
4
2
2 1
A) e Answer: A
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4
2
3
4
5
6
7
x
B) c
1
2
3
C) f
D) d
2) In which scatter diagram is r = 1? a
b
y
y
12
12
10
10
8
8
6
6
4
4
2
2 1
2
3
4
5
6
x
1
2
c
3
5
6
x
4
5
6
7
4
5
6
x
d
y
y
12
12
10
10
8
8
6
6
4
4
2
2 1
2
3
4
5
6
x
1
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A) b Answer: A
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3) In which scatter diagram is r = -1? a
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A) a Answer: A
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4) Which scatter diagram indicates a perfect positive correlation? a b y
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A) b Answer: A
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The scatter diagram shows the relationship between average number of years of education and births per woman of child bearing age in selected countries. Use the scatter plot to determine whether the statement is true or false. 5) There is a strong positive correlation between years of education and births per woman. 10 9 8
Births per Woman
7 6 5 4 3 2 1 2
A) False
4
6
8
10 12 14
Average number of years of education of Married Women of Child-Bearing Age B) True
Answer: A 6) There is no correlation between years of education and births per woman. 10 9 8
Births per Woman
7 6 5 4 3 2 1 2
A) False Answer: A
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Average number of years of education of Married Women of Child-Bearing Age B) True
7) There is a negative correlation between years of education and births per woman. 10 9 8
Births per Woman
7 6 5 4 3 2 1 2
A) True
4
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Average number of years of education of Married Women of Child-Bearing Age B) False
Answer: A 8) There is a causal relationship between years of education and births per woman. 10 9 8
Births per Woman
7 6 5 4 3 2 1 2
A) False Answer: A
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Average number of years of education of Married Women of Child-Bearing Age B) True
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 9) Construct a scatter diagram for the given data. Determine whether there is a positive linear correlation, negative linear correlation, or no linear correlation. x -5 -3 4 1 -1 -2 0 2 3 -4 y -10 -8 9 1 -2 -6 -1 3 6 -8 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2
1
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6 x
-4 -6 -8 -10 -12
Answer: There appears to be a positive linear correlation. y 10 8 6 4 2 -5 -4 -3 -2 -1 -2 -4 -6 -8 -10
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1
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x
10) Construct a scatter diagram for the given data. Determine whether there is a positive linear correlation, negative linear correlation, or no linear correlation. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 6 -6 -1 3 4 1 -4 -5 8 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2
1
2
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6 x
-4 -6 -8 -10 -12
Answer: There appears to be a negative linear correlation. 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2 -4 -6 -8 -10 -12
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1
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11) Construct a scatter diagram for the given data. Determine whether there is a positive linear correlation, negative linear correlation, or no linear correlation. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 -6 8 -3 -2 1 5 -5 6 7 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2
1
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-4 -6 -8 -10 -12
Answer: There appears to be no linear correlation. 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2 -4 -6 -8 -10 -12
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1
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6 x
12) The numbers of home runs that Mark McGwire hit in the first 13 years of his major league baseball career are listed below. (Source: Major League Handbook) Construct a scatter diagram for the data. Is there a relationship between the home runs and the batting averages? Home Runs 33 39 22 42 9 9 39 52 58 70 Batting Average .231 .235 .201 .268 .33 .252 .274 .312 .274 .299 y 0.35
0.3
0.25
0.2
0.15 15
30
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75
Answer: In general, there appears to be a relationship between the home runs and batting averages. As the number of home runs increased, the batting averages increased. y 0.35
0.3
Batting Average
0.25
0.2
0.15 15
30
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Home Runs
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75
x
13) The data below represent the numbers of absences and the final grades of 15 randomly selected students from an astronomy class. Construct a scatter diagram for the data. Decide if a linear trend exists. Student Number Final Grade of Absences as a Percent 100 y 1 5 79 2 6 78 90 3 2 86 4 12 56 80 5 9 75 6 5 90 70 7 8 78 8 15 48 60 9 0 92 50 10 1 78 11 9 81 40 12 3 86 x 13 10 75 5 10 15 14 3 89 15 11 65 Answer: There appears to be a trend in the data. As the number of absences increases, the final grade decreases. 100
y
90 80
Final Grade (%)
70 60 50 40 5
10
15
x
Number of Absences MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 14) A researcher determines that the linear correlation coefficient is 0.85 for a paired data set. This indicates that there is A) a strong positive linear correlation. B) a strong negative linear correlation. C) no linear correlation but that there may be some other relationship. D) insufficient evidence to make any decision about the correlation of the data. Answer: A 15) An instructor wishes to determine if there is a relationship between the number of absences from his class and a student's final grade in the course. What is the explanatory variable? A) Absences B) Final Grade C) The instructor's point scale for attendance D) Student's performance on the final examination Answer: A Page 23
16) A medical researcher wishes to determine if there is a relationship between the number of prescriptions written by pediatricians and the ages of the children for whom the prescriptions are written. She surveys all the pediatricians in a geographical region to collect her data. What is the response variable? A) Number of prescriptions written B) Pediatricians surveyed C) Age of the children for whom prescriptions were written D) Number of children for whom prescriptions were written Answer: A 17) A doctor wishes to determine the relationship between a male's age and that male's total cholesterol level. He tests 200 males and records each male's age and that male's total cholesterol level. True or False: The males cholesterol level is the explanatory variable. A) False B) True Answer: A 18) A scatter diagram locates a point in a two dimensional plane. The diagram locates the variable on the horizontal axis and the A) explanatory; response C) response; study
variable on the vertical axis. B) response; explanatory D) study; explanatory
Answer: A 19) A history instructor has given the same pretest and the same final examination each semester. He is interested in determining if there is a relationship between the scores of the two tests. He computes the linear correlation coefficient and notes that it is 1.15. What does this correlation coefficient value tell the instructor? A) The history instructor has made a computational error. B) There is a strong positive correlation between the tests. C) There is a strong negative correlation between the tests. D) The correlation is something other than linear. Answer: A 20) A traffic officer is compiling information about the relationship between the hour of the day and the speed over the limit at which the motorist is ticketed. He computes a correlation coefficient of 0.12. What does this tell the officer? A) There is little evidence of a linear correlation. B) There is a strong positive linear correlation. C) There is a strong negative linear correlation. D) There is a quadratic relationship between the variables. Answer: A 3 Compute and interpret the linear correlation coefficient. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Compute the linear correlation coefficient for the data below. x 2 4 11 8 6 5 7 9 10 3 y 12 4 1 9 -5 -7 -5 -3 2 6 A) 0.990 B) 0.881 C) 0.819 Answer: A
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D) 0.792
2) Compute the linear correlation coefficient for the data below. x 3 5 12 9 7 6 8 10 11 4 y 15 10 -2 3 7 8 5 0 -1 12 A) -0.995 B) -0.671 C) -0.778
D) -0.885
Answer: A 3) Compute the linear correlation coefficient for the data below. x 8 5 3 2 4 6 7 0 -1 1 y 12 -5 9 -2 -1 2 6 -4 7 8 A) -0.104 B) -0.132 C) -0.549
D) -0.581
Answer: A 4) The data below are the final exam scores of 10 randomly selected calculus students and the number of hours they slept the night before the exam. Compute the linear correlation coefficient. Hours, x 7 9 6 12 6 8 8 9 10 7 Scores, y 61 76 56 84 62 74 81 86 86 67 A) 0.847 B) 0.991 C) 0.761 D) 0.654 Answer: A 5) The data below are the average one-way commute times (in minutes) of selected students during a summer literature class and the number of absences for those students for the term. Compute the linear correlation coefficient. Commute time (min), x 75 88 94 93 91 101 78 103 83 Number of absences, y 2 6 9 9 7 14 3 14 4 A) 0.980 B) 0.890 C) 0.881 D) 0.819 Answer: A 6) The data below are the ages and annual pharmacy b ills (in dollars) of 9 randomly selected employees. Compute the linear correlation coefficient. Age, x 34 37 41 44 47 49 53 57 61 Pharmacy bill ($), y 112 116 119 127 138 141 144 146 148 A) 0.960 B) 0.998 C) 0.890 D) 0.908 Answer: A 7) The data below are the number of hours worked (per week) and the final grades of 9 randomly selected students from a drama class. Compute the linear correlation coefficient. Hours worked, x 5 8 11 9 14 7 20 13 10 Final Grade, y 90 78 72 74 63 84 47 68 74 A) -0.991 B) -0.888 C) -0.918 D) -0.899 Answer: A 8) A manager wishes to determine the relationship between the number of years the manager's sales representatives have been with the company and their average monthly sales (in thousands of dollars). Compute the linear correlation coefficient. Years with company, x 6 7 14 11 12 19 7 5 15 Sales, y 36 38 83 67 70 66 53 60 125 A) 0.632 B) 0.561 C) 0.717 D) 0.791 Answer: A
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9) In order for a company's employees to work in a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Compute the linear correlation coefficient. Number of years, x 4 5 5 6 4 7 3 8 4 Grades on test, y 65 72 79 86 77 94 62 97 76 A) 0.934 B) 0.911 C) 0.891 D) 0.902 Answer: A 10) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Compute the linear correlation coefficient. Rainfall (in inches), x 12.9 11.2 15.8 14.9 21.2 12.7 9.4 18 18.4 Yield (bushels per acre), y 47.5 43.2 55.8 56 79.4 46.2 28.9 73 75.8 A) 0.981 B) 0.998 C) 0.900 D) 0.899 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 11) Compute the coefficient of correlation, r, letting Row 1 represent the x-values and Row 2 represent the y-values. Then, compute the coefficient of correlation, r, letting Row 2 represent the x-values and Row 1 represent the y-values. What effect does switching the explanatory and response variables have on the linear correlation coefficient? Row 1 2 4 11 8 6 5 7 9 10 3 Row 2 -3 15 16 8 5 1 6 10 13 15 Answer: The linear correlation coefficient remains unchanged. 4 Determine whether a linear relation exists between two variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute the linear correlation coefficient between the two variables and determine whether a linear relation exists. 3 5 5 6 1) x 2 y 1.3 1.6 2.1 2.2 2.7 A) r = 0.983; linear relation exists B) r = 0.983; no linear relation exists C) r = 0.883; linear relation exists D) r = 0.883; no linear relation exists Answer: A 3 0 -2 -3 -1 2) x -6 -4 y -9 -7 10 2 0 -1 -5 A) r = 0.990; linear relation exists C) r = 0.819; linear relation exists
1 4
2 7
-5 -7 B) r = 0.881; no linear relation exists D) r = 0.792; no linear relation exists
Answer: A 3) x y
-13 -11 -4 -7 -9 -10 -8 21 16 4 9 13 14 11 A) r = -0.995; linear relation exists C) r = -0.885; no linear relation exists
-6 6
-5 5
-12 18 B) r = -0.995; no linear relation exists D) r = -0.885; linear relation exists
Answer: A 2 3 4 2 5 9 10 4) x 9 y 85 52 55 68 67 86 83 73 A) r = 0.708; linear relation exists C) r = -0.708; linear relation exists Answer: A
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B) r = 0.235; no linear relation exists D) r = 0.708; no linear relation exists
5) x 10 11 16 9 7 15 16 10 y 96 51 62 58 89 81 46 51 A) r = -0.335; no linear relation exists C) r = -0.335; linear relation exists
B) r = 0.462; linear relation exists D) r = -0.284; no linear relation exists
Answer: A 6) The table below shows the scores on an end-of-year project of 10 randomly selected architecture students and the number of days each student spent working on the project. Days, x 5 7 4 10 4 6 6 7 8 5 Score, y 61 76 56 84 62 74 81 86 86 67 A) r = 0.847; linear relation exists B) r = 0.847; no linear relation exists C) r = 0.761; linear relation exists D) r = 0.761; no linear relation exists Answer: A 7) The table below shows the ages and weights (in pounds) of 9 randomly selected tennis coaches. Age, x 41 44 48 51 54 56 60 64 68 Weight (pounds), y 111 115 118 126 137 140 143 145 147 A) r = 0.960; linear relation exists B) r = 0.960; no linear relation exists C) r = 0.908; no linear relation exists D) r = 0.908; linear relation exists Answer: A 8) The table shows the number of days off last year and the earnings for the year (in thousands of dollars) for nine randomly selected insurance salesmen. Number of days off, x 4 7 10 8 13 6 19 12 9 Earnings for the year (thousands of dollars), y 89 77 71 73 62 83 46 67 73 A) r = -0.991; linear relation exists B) r = -0.991; no linear relation exists C) r = -0.899; linear relation exists D) r = -0.899; no linear relation exists Answer: A 9) A manager wishes to determine whether there is a relationship between the number of years her sales representatives have been with the company and their average monthly sales. The table shows the years of service for each of her sales representatives and their average monthly sales (in thousands of dollars). Years with company, x 3 4 11 8 9 16 4 2 12 Sales , y 37 39 84 68 71 67 54 61 126 A) r = 0.632; no linear relation exists B) r = 0.632; linear relation exists C) r = 0.717; linear relation exists D) r = 0.717; no linear relation exists Answer: A 10) To investigate the relationship between yield of soybeans and the amount of fertilizer used, a researcher divides a field into eight plots of equal size and applies a different amount of fertilizer to each plot. The table shows the yield of soybeans and the amount of fertilizer used for each plot. Amount of fertilizer (pounds) ,x 1 1.5 2 2.5 3 3.5 4 4.5 Yield of soybeans (pounds), y 25 21 27 28 36 35 32 34 A) r = 0.819; linear relation exists B) r = 0.729; no linear relation exists C) r = 0.683; linear relation exists D) r = 0.683; no linear relation exists Answer: A
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5 Explain the difference between correlation and causation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A variable that is related to either the response variable or the predictor variable or both, but which is excluded from the analysis is a A) lurking variable. B) random variable. C) discrete variable. D) qualitative variable. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) For a random sample of 100 American cities, the linear correlation coefficient between the number of robberies last year and the number of schools in the city was found to be r = 0.725. What does this imply? Does this suggest that building more schools in a city could lead to more robberies? Why or why not? What is a likely lurking variable? Answer: A positive correlation exists between the number of schools in a city and the number of robberies but this is an example of correlation not causation. Building more schools is unlikely to lead to an increase in robberies. A likely lurking variable is the population of the city and this lurking variable accounts for the positive correlation. Larger cities tend to have both more schools and more robberies. 3) For a random sample of 30 countries, the linear correlation coefficient between the infant mortality rate and the average number of cars per capita was found to be r = -0.717. What does this imply? Does this suggest that if people buy more cars, this could lower the infant mortality rate? Why or why not? What is a likely lurking variable? Answer: A negative correlation exists between the infant mortality rate and the number of cars per capita but this is an example of correlation not causation. If people buy more cars, this is unlikely to lead to a decrease in the infant mortality rate. A likely lurking variable is wealth and this lurking variable accounts for the negative correlation. More affluent countries tend to have both more cars per capita and lower infant mortality rates. 4) A random sample of 200 men aged between 20 and 60 was selected from a certain city. The linear correlation coefficient between income and blood pressure was found to be r = 0.807. What does this imply? Does this suggest that if a man gets a salary raise his blood pressure is likely to rise? Why or why not? What is likely alurking variable? Answer: A positive correlation exists between income and blood pressure but this is an example of correlation not causation. An increase in salary is unlikely to lead to an increase in blood pressure. Age and level of job stress are possible lurking variables and these lurking variables account for the positive correlation. Older men tend to have both higher blood pressures and higher incomes. Also men in high stress jobs tend to have both higher blood pressures and higher incomes.
4.2 Least-Squares Regression 1 Find the least-squares regression line and use the line to make predictions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Find the equation of the least-squares regression line for the given data. Round values to the nearest thousandth. x -5 -3 4 1 -1 -2 0 2 3 -4 y -10 -8 9 1 -2 -6 -1 3 6 -8 ^
B) y = 0.522x - 2.097
^
D) y = -0.552x + 2.097
A) y= 2.097x - 0.552 C) y = 2.097x + 0.552 Answer: A Page 28
^ ^
2) Find the equation of the least-squares regression line for the given data. Round values to the nearest thousandth. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 6 -6 -1 3 4 1 -4 -5 8 ^
A) y= -1.885x + 0.758
^
B) y = 0.758x + 1.885
^
C) y= -0.758x - 1.885
^
D) y= 1.885x - 0.758
Answer: A 3) Find the equation of the least-squares regression line for the given data. Round values to the nearest thousandth. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 -6 8 -3 -2 1 5 -5 6 7 ^
A) y= -0.206x + 2.097
^
B) y= 2.097x - 0.206
^
C) y= 0.206x - 2.097
^
D) y= -2.097x + 0.206
Answer: A 4) The data below are the final exam scores of 10 randomly selected history students and the number of hours they slept the night before the exam. Find the least-squares regression line for the given data. What would be the predicted score for a history student who slept 7 hours the previous night? Is this a reasonable question? Round the least-squares regression line values to the nearest hundredth, and round the predicted score to the nearest whole number. Hours, x 3 5 2 8 2 4 4 5 6 3 Scores, y 65 80 60 88 66 78 85 90 90 71 ^
A) y= 5.04x + 56.11; 91; Yes, it is reasonable. ^
B) y= 5.04x + 56.11; 91; No, it is not reasonable. 7 hours is well outside the scope of the model. ^
C) y= -5.04x + 56.11; 21; No, it is not reasonable. 7 hours is well outside the scope of the model. ^
D) y= -5.04x + 56.11; 21; Yes, it is reasonable. Answer: A 5) The data below are the final exam scores of 10 randomly selected history students and the number of hours they slept the night before the exam. Find the least-squares regression line for the given data. What would be the predicted score for a history student who slept 15 hours the previous night? Is this a reasonable question? Round your predicted score to the nearest whole number. Round the least-squares regression line values to the nearest hundredth. Hours, x 3 5 2 8 2 4 4 5 6 3 Scores, y 65 80 60 88 66 78 85 90 90 71 ^
A) y= 5.04x + 56.11; 132; No, it is not reasonable. 15 hours is well outside the scope of the model. ^
B) y= 5.04x + 56.11; 132; Yes, it is reasonable. ^
C) y= -5.04x + 56.11; -20; No, it is not reasonable. ^
D) y= -5.04x + 56.11; -20; Yes, it is reasonable. Answer: A 6) The data below are the average one-way commute times (in minutes) for selected students and the number of absences for those students during the term. Find the least-squares regression line for the given data. What would be the predicted number of absences if the commute time was 95 minutes? Is this a reasonable question? Round the predicted number of absences to the nearest whole number. Round the least-squares regression line values to the nearest hundredth. Commute time (min), x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 ^
A) y = 0.45x - 30.27; 12 absences; Yes, it is reasonable. ^
B) y = 0.45x - 30.27; 12 absences; No, it is not reasonable. 95 minutes is well outside the scope of the model. ^
C) y = 0.45x + 30.27; 73 absences; Yes, it is reasonable. ^
D) y = 0.45x + 30.27; 73 absences; No, it is not reasonable. 95 minutes is well outside the scope of the model. Answer: A Page 29
7) The data below are the average one-way commute times (in minutes) for selected students and the number of absences for those students during the term. Find the e least-squares regression line for the given data. What would be the predicted number of absences if the commute time was 40 minutes? Is this a reasonable question? Round the predicted number of absences to the nearest whole number. Round the least-squares regression line values to the nearest hundredth. Commute time (min), x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 ^
A) y = 0.45x - 30.27; -12 absences; No, it is not reasonable. 40 minutes is well outside the scope of the model. ^
B) y = 0.45x - 30.27; -12 absences; Yes, it is reasonable. ^
C) y = 0.45x + 30.27; 48 absences; Yes, it is reasonable. ^
D) y = 0.45x + 30.27; 48 absences; No, it is not reasonable. 40 minutes is well outside the scope of the model. Answer: A 8) The data below are ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. Find the least-squares regression line for the given data. What would be the predicted pressure if the age was 60? Round the predicted pressure to the nearest whole number. Round the least-squares regression line values to the nearest hundredth. Age, x 38 41 45 48 51 53 57 61 65 Pressure, y 116 120 123 131 142 145 148 150 152 ^
B) y= 60.46x - 1.49; 3626 mm
^
D) y= 60.46x + 1.49; 3629 mm
A) y= 1.49x + 60.46; 150 mm C) y= 1.49x - 60.46; 29 mm
^ ^
Answer: A 9) The data below are the number of absences and the final grades of 9 randomly selected students from a literature class. Find the least-squares regression line for the given data. What would be the predicted final grade if a student was absent 14 times? Round the least-squares regression line values to the nearest hundredth. Round the predicted grade to the nearest whole number. Number of absences, x 0 3 6 4 9 2 15 8 5 Final grade, y 98 86 80 82 71 92 55 76 82 ^
B) y = 96.14x - 2.75; 1343
^
D) y= -96.14x + 2.75; 1343
A) y = -2.75x + 96.14; 58 C) y= -2.75x - 96.14; 134.64
^ ^
Answer: A 10) A manager wishes to determine the relationship between the number of miles traveled (in hundreds of miles) by her sales representatives and their amount of sales (in thousands of dollars) per month. Find the least-squares regression line for the given data. What would be the predicted sales if the sales representative traveled 0 miles? Is this reasonable? Why or why not? Round the least-squares regression line values to the nearest hundredth. Miles traveled, x 2 3 10 7 8 15 3 1 11 Sales, y 31 33 78 62 65 61 48 55 120 ^
A) y = 3.53x + 37.92; $37,920; No; it is not reasonable for a representative to travel 0 miles and have a positive amount of sales. ^
B) y = 3.53x + 37.92; $3792; No; it is not reasonable for a representative to travel 0 miles and have a positive amount of sales. ^
C) y = 3.53x + 37.92; $37,920; Yes, it is reasonable. ^
D) y= 37.92x + 3.53; $3792; Yes, it is reasonable. Answer: A
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11) A manager wishes to determine the relationship between the number of years her sales representatives have been employed by the firm and their amount of sales (in thousands of dollars) per month. Find the least-squares regression line for the given data. What would be the predicted sales if the sales representative was employed by the firm for 30 years Is this reasonable? Why or why not? Round the least-squares regression line values to the nearest hundredth. Years employed, x 2 3 10 7 8 15 3 1 11 Sales, y 31 33 78 62 65 61 48 55 120 ^
A) y = 3.53x + 37.92; $143,820; No; it is not reasonable. 30 years of employment is well outside the scope of the model. ^
B) y = 3.53x + 37.92; $143,820;; Yes, it is reasonable. ^
C) y = 3.53x - 37.92; $67,980; No; it is not reasonable. 30 years of employment is well outside the scope of the model. ^
D) y = 3.53x - 37.92; $67,980; Yes; it is reasonable. Answer: A 12) In order for a company's employees to work in a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Find the least-squares regression line for the given data. Round the least-squares regression line values to the nearest hundredth. Number of years, x 3 4 4 5 3 6 2 7 3 Grades on test, y 61 68 75 82 73 90 58 93 72 ^
A) y = 6.91x + 46.26
^
B) y= 6.91x - 46.26
^
C) y= 46.26x - 6.91
^
D) y = 46.26x + 6.91
Answer: A 13) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Find the least-squares regression line for the given data. Round the least-squares regression line values to the nearest thousandth. Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 ^
A) y = 4.379x + 4.267
^
B) y= -4.379x + 4.267
^
C) y= 4.267x + 4.379
^
D) y = 4.267x - 4.379
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 14) Find the least-squares regression line for the data by letting Row 1 represent the x-values and Row 2 represent the y-values. Then, find the least-squares regression line letting Row 2 represent the x-values and Row 1 represent the y-values. What effect does switching the explanatory and response variables have on the regression line? Row 1 -5 -3 4 1 -1 -2 0 2 3 -4 Row 2 -10 -8 9 1 -2 -6 -1 3 6 -8 Answer: The regression lines are not necessarily the same.
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15) Is the number of games won by a major league baseball team in a season related to the team's batting average? Data from 14 teams were collected and the summary statistics yield: y = 1,134, x = 3.642, y2 = 93,110, x2 = 0.948622, and xy = 295.54
∑
∑
∑
∑
∑
Find the least squares regression equation for predicting the number of games won, y, using a straight-line relationship with the team's batting average, x. x2 -
Answer: SSxx = ∑
SSxy = ∑xy -
∑x
2
n
= 0.948622 -
∑x ∑y = 295.54 - (3.642)(1,134) = 0.538 n
y=
∑y = 1,134 = 81
x=
∑x = 3.642 = 0.26014
n
(3.642)2 = 0.00118171 14
14
14
n
14
SSxy ^ 0.538 b1 = = = 455.27 SSxx 0.00118171 ^
^
b0 = y - b1 x = 81 - 455.27(0.26014) = -37.434 ^
The least squares equation is y= -37.434 + 455.27x. 16) The table shows, for the years 1997-2012, the mean hourly wage for residents of the town of Pity Me and the mean weekly rent paid by the residents. Year Mean weekly rent Mean hourly wage Year Mean weekly rent Mean hourly wage (dollars) (dollars) (dollars) (dollars) 1997 57 10.38 2005 116 28.99 1998 59 10.89 2006 113 28.63 1999 62 11.96 2007 112 36.75 2000 63 12.46 2008 86 14.55 2001 86 17.72 2009 90 17.90 2002 119 28.07 2010 90 14.67 2003 131 35.24 2011 100 17.97 2004 122 31.87 2012 115 22.23 Summary statistics yield: SSxx = 1222.2771, SSxy = 3031.7125, SSyy = 9144.9375, x = 21.2675, and y = 95.0625. Find the least squares line that uses mean hourly wage to predict mean weekly rent. Round values to the nearest ten-thousandth. ^
Answer: b1 = ^
SSxy SSxx
=
3031.7125 = 2.4804 1222.2771
^
b0 = y - b1 x = 95.0625 - 2.4804(21.2675) = 42.3106 ^
The least squares prediction equation is y = 42.3106 + 2.4804x.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 17) A residual is the difference between A) the observed value of y and the predicted value of y. B) the observed value of x and the predicted value of x. C) the observed value of y and the predicted value of x. D) the observed value of x and the predicted value of y. Answer: A 18) The least squares regression line A) minimizes the sum of the residuals squared. B) maximizes the sum of the residuals squared. C) minimizes the mean difference between the residuals squared. D) maximizes the mean difference between the residuals squared. Answer: A 19) For a given data set, the equation of the least-squares regression line will always pass through A) (x, y).
B) every point in the given data set.
C) at least two point in the given data set.
D) the y-intercept and the slope.
Answer: A 2 Interpret the slope and the y-intercept of the least-squares regression line. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 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, y = b1 x + b0 where y = appraised value of the house (in $thousands) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were obtained: ^
y= 17.80x + 74.80 sβ = 71.24, t = 1.05 (for testing b0 ) sβ = 2.63, t = 7.49 (for testing b1 ) SSE = 60,775, MSE = 841, s = 29, r2 = .44 Range of the x-values: 5 - 11 Range of the y-values: 160 - 300 Give a practical interpretation of the estimate of the slope of the least-squares regression line. A) For each additional room in the house, we estimate the appraised value to increase $17,800 B) For each additional room in the house, we estimate the appraised value to increase $74,800. C) For each additional dollar of appraised value, we estimate the number of rooms in the house to increase by 17.80 rooms. D) For a house with 0 rooms, we estimate the appraised value to be $74,800. Answer: A
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2) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses 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, y = b1 x + b0 , where y = appraised value of the house (in $thousands) and x = number of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were obtained: ^
y= 19.72x+74.80 sβ = 71.24, t = 1.05 (for testing b0 ) sβ = 2.63, t = 7.49 (for testing b1 ) SSE = 60,775, MSE = 841, s = 29, r2 = 0.44 Range of the x-values: 5 - 11 Range of the y-values: 160 - 300 Give a practical interpretation of the estimate of the y-intercept of the least-squares regression line. A) There is no practical interpretation, since a house with 0 rooms is not possible. B) For each additional room in the house, we estimate the appraised value to increase $74,800. C) For each additional room in the house, we estimate the appraised value to increase $19,720. D) We estimate the base appraised value for any house to be $74,800. Answer: A 3) Is there a relationship between the raises administrators at State 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 straight-line regression model, y= b1 x +b0 . Using the method of ^
least-squares regression, the faculty group obtained the following prediction equation, y= 2,000x+14,000. Interpret the estimated slope of the line. A) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to increase $2,000. B) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to decrease $2,000. C) For an administrator with a rating of 1.0, we estimate his/her raise to be $2,000. D) For a $1 increase in an administrator's raise, we estimate the administrator's rating to decrease 2,000 points. Answer: A 4) Is there a relationship between the raises administrators at State 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 straight-line regression model, y = b1 x+ b0 . Using the method of ^
least-squares regression, the faculty group obtained the following prediction equation, y=2,000x+ 14,000. Interpret the estimated y-intercept of the line. A) For an administrator who receives a rating of zero, we estimate his or her raise to be $14,000. B) The base administrator raise at State University is $14,000. C) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to increase $14,000. D) There is no practical interpretation, since rating of 0 is not likely and outside the range of the sample data. Answer: A
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5) 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 service charge to a company is the company's sales revenue (x), measured in millions of dollars. Data for 21 companies who use the bank's ^
services were used to fit the model, y = b1 x+ b0 . The results of the least-squares linear regression are provided below. ^
y= 20x + 2,700 , s = 65, 2-tailed p-value = 0.064 (for testing b1 ) Interpret the estimate of b0 , the y-intercept of the line. A) There is no practical interpretation since a sales revenue of $0 is not likely to be a value. B) All companies will be charged at least $2,700 by the bank. C) About 95% of the observed service charges fall within $2,700 of the least-squares regression line. D) For every $1 million increase in sales revenue, we expect a service charge to increase $2,700. Answer: A
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^
6) Civil engineers often use the least-squares regression line equation, y = b1 x+ b0 , to model the relationship between the mean shear strength of masonry joints and precompression stress, x. To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis. Triplet Test 1 2 3 4 5 6 7 Shear Strength (tons), y 1.00 2.18 2.24 2.41 2.59 2.82 3.06 Precomp. Stress (tons), x 0 0.60 1.20 1.33 1.43 1.75 1.75
Source
DF
Analysis of Variance Sum of Mean Squares Square
Model Error C Total
1 5 6
2.39555 0.25094 2.64649
2.39555 0.05019
0.22403 2.32857 9.62073
R-square Adj R-sq
Root MSE Dep Mean C.V.
F Value
Prob > F
47.732
0.0010
0.9052 0.8862
Parameter Estimates
Variable INTERCEP X
DF 1 1
Parameter Estimate Error
Standard T for HO: Parameter=0Prob > |T|
1.191930 0.987157
6.442 6.909
0.18503093 0.14288331
0.0013 0.0010
Give a practical interpretation of the estimate of the slope of the least-squares regression line. A) For every 1 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 0.987 ton. B) For a triplet test with a precompression stress of 1 ton, we estimate the shear strength of the joint to be 0.987 ton. C) For every 0.987 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 1 ton. D) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to be 1.19 tons. Answer: A ^
7) Civil engineers often use the straight-line equation,y = b1 x+ b0 , to model the relationship between the mean shear strength of masonry joints and precompression stress, x. To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis. Triplet Test 1 2 3 4 5 6 7 Shear Strength, y 1.00 2.18 2.24 2.41 2.59 2.82 3.06 (tons) Precomp. Stress, x 0 0.60 1.20 1.33 1.43 1.75 1.75 (tons) Analysis of Variance Page 36
Source
DF
Sum of Squares
Mean Square
Model Error C Total
1 5 6
2.39555 0.25094 2.64649
2.39555 0.05019
0.22403 2.32857 9.62073
R-square Adj R-sq
Root MSE Dep Mean C.V.
F Value
Prob > F
47.732
0.0010
0.9052 0.8862
Parameter Estimates
Variable INTERCEP X
DF 1 1
Parameter Estimate Error
Standard T for HO: Parameter=0Prob > |T|
1.191930 0.987157
6.442 6.909
0.18503093 0.14288331
0.0013 0.0010
Give a practical interpretation of the estimate of the y-intercept of the least-squares regression line. A) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to be 1.19 tons. B) For every 1 ton increase in precompression stress, we estimate the shear strength of the joint to increase by 0.987 ton. C) There is no practical interpretation since a triplet test with a precompression stress of 0 tons is outside the range of the sample data. D) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to increase 1.19 tons. Answer: A 8) Each year a nationally recognized publication conducts its "Survey of America's Best Graduate and Professional Schools." An 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. Total GMAT scores range from 200 to 800. A simple linear regression of SALARY versus GMAT using 25 data points shown below. ^ ^ b0 = -92040 b1 = 228 s = 3213 R2 = 0.66 r = 0.81 df = 23 t = 6.67 ^
Give a practical interpretation of b0 = -92040. A) The value has no practical interpretation since a GMAT of 0 is not likely and outside the range of the sample data. B) We expect to predict SALARY to within 2(92040) = $184,080 of its true value using GMAT in a least-square regression line model. C) We estimate SALARY to decrease $92,040 for every 1-point increase in GMAT. D) We estimate the base SALARY of graduates of a top business school to be $-92,040. Answer: A
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9) Each year a nationally recognized publication conducts its "Survey of America's Best Graduate and Professional Schools." An 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. Total GMAT scores range from 200 to 800. A least-square regression line of SALARY versus GMAT using 25 data points shown below. ^ ^ b0 = -92040 b1 = 228 s = 3213 R2 = 0.66 r = 0.81 df = 23 t = 6.67 ^
Give a practical interpretation of b1 = 228. A) We estimate SALARY to increase $228 for every 1-point increase in GMAT. B) We expect to predict SALARY to within 2(228) = $456 of its true value using GMAT in a least-square regression line model. C) We estimate GMAT to increase 228 points for every $1 increase in SALARY. D) The value has no practical interpretation since a GMAT of 0 is not likely and outside the range of the sample data. Answer: A 10) A real estate magazine reported the results of a regression analysis designed to predict the price (y), measured in dollars, of residential properties recently sold in a northern Virginia subdivision. One independent variable used to predict sale price is GLA, gross living area (x), measured in square feet. Data for 157 properties were ^
used to fit the model, y = b1 x+ b0,. The results of the simple linear regression are provided below. ^ y = 96,600 + 22.5x s = 6500 r2 = -0.77 t = 6.1 (for testing b1 )
Interpret the estimate of b0 , the y-intercept of the line. A) There is no practical interpretation, since a gross living area of 0 is not possible as a data value. B) All residential properties in Virginia will sell for at least $96,600. C) About 95% of the observed sale prices fall within $96,600 of the least squares regression line. D) For every 1-sq ft. increase in GLA, we expect a property's sale price to increase $96,600. Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 11) In a comprehensive road test on all new car models, one variable measured is the time it takes a car to accelerate from 0 to 60 miles per hour. To model acceleration time, a least-squares 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: x1 = Maximum speed attained (miles per hour) Initially, the simple linear model E(y) = b1 x + b0 was fit to the data. Computer printouts for the analysis are given below: UNWEIGHTED 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 Find and interpret the estimate b1 in the printout above. Answer: b1 = -0.08365. For every 1 mile per hour increase in the maximum attained speed of a new car, we estimate the elapsed 0 to 60 acceleration time to decrease by .08365 seconds.
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12) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the 15 minute period following the addition of food. The data showing the weekly number of grunts and the age of the warthog (in days) are listed below: Week Number of Grunts Age (days) 1 81 116 2 59 132 3 30 146 4 35 151 5 54 158 6 31 165 7 53 174 8 8 180 9 11 186 a. Write the equation of a straight-line model using two points relating number of grunts (y) to age (x). b. Give the least-squares regression equation. ^
c. Give a practical interpretation of the value of b0 if possible. ^
d. Give a practical interpretation of the value of b1 if possible. Answer: a. E(y) = b0 + b1 x ^
^
^
b. y= b0 + b1x = 168.43 - 0.8195x c. We would expect approximately 168 grunts after feeding a warthog that was just born. However, since the value 0 in 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 0.8195. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ^
13) Given the following least squares regression equation, y = -173 + 74x, we estimate y to
by
with each 1-unit increase in x. A) increase; 74
B) decrease; 74
C) decrease; 173
D) increase; 173
Answer: A ^
14) Given that the least-squares regression line is y = 3x - 2, what is the best predicted value for y given x = 9? A) 25 B) 29 C) 15 D) 10 Answer: A ^
15) Given that the least-squares regression line is y= -3.5x- 7.0, what is the best predicted value for y given x = 5.1? A) -24.85 B) -10.85 C) 10.85 D) 24.85 Answer: A 16) Use the least-squares regression equation to predict the value of y for x = 1.3. x -5 -3 4 1 -1 -2 0 2 3 -4 y -10 -8 9 1 -2 -6 -1 3 6 -8 A) 2.174 B) 3.278 C) 1.379
D) 2.815
Answer: A 17) Use the least-squares regression equation to predict the value of y for x = 2.9. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 6 -6 -1 3 4 1 -4 -5 8 A) -4.708 B) 6.225 C) 0.313 Answer: A Page 40
D) 4.083
18) In order for a company's employees to work for the foreign office, they must take a test in the language of the country where they plan to work. The data below show the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Use the least-squares regression line to predict y given x = 5? Number of years, x 3 4 4 5 3 6 2 7 3 Grades on test, y 61 68 75 82 73 90 58 93 72 A) 81 B) 79 C) 77 D) 83 Answer: A 19) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Use the least-squares regression line to predict y given x = 11.6? Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) 55.1 B) 55.4 C) 54.9 D) 55.6 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 20) A calculus instructor is interested in finding the strength of a relationship between the final exam grades of students enrolled in Calculus I and Calculus II at his college. The data (in percentages) are listed below. Calculus I 88 78 62 75 95 91 83 86 98 Calculus II 81 80 55 78 90 90 81 80 100 a) Graph a scatter diagram of the data. b) Find the least-squares regression line. c) Predict a Calculus II exam score for a student who receives an 80 in Calculus I. Answer: a) y 100 90 80
Calculus II
70 60 50 50
60
70
80
Calculus I ^
b) y= 1.044x - 5.990 ^
c) When x = 80, y = 78.
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90
100 x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 21) In an area of Russia, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). The data for a 9 year period is as follows: Rain Fall, x 13.1 11.4 16.0 15.1 21.4 12.9 9.6 18.2 18.6 Yield, y 48.5 44.2 56.8 80.4 47.2 29.9 74.0 74.0 76.8 ^
The least-square regression line is given as y = -9.12 + 4.38x. How many bushels of wheat per acre can be predicted if it is expected that there will be 30 inches of rain? A) Cannot be certain of the result because 30 inches of rain exceeds the observed data. B) 122.28 C) 140.52 D) 8.93 Answer: A 3 Compute the sum of squared residuals. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. ^
1) The least-square regression line for the given data is y = 2.097x - 0.552. Determine the residual of a data point for which x = -5 and y = -10. x -5 -3 4 1 -1 -2 0 2 3 -4 y -10 -8 9 1 -2 -6 -1 3 6 -8 A) 1.037 B) -21.037 C) -11.037 D) 16.522 Answer: A ^
2) The least-square regression line for the given data is y = -1.885x + 0.758. Determine the residual of a data point for which x = 3 and y = -5. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 6 -6 -1 3 4 1 -4 -5 8 A) -0.103 B) -9.897 C) -4.897 D) -7.183 Answer: A ^
3) The least-square regression line for the given data is y = -0.206x + 2.097. Determine the residual of a data point for which x = -4 and y = 7. x -5 -3 4 1 -1 -2 0 2 3 -4 y 11 -6 8 -3 -2 1 5 -5 6 7 A) 4.079 B) 9.921 C) 2.921 D) -4.655 Answer: A ^
4) The least-square regression line for the given data is y = 5.044x + 56.11. Determine the residual of a data point for which x = 8 and y = 88. Hours, x 3 5 2 8 2 4 4 5 6 3 Scores, y 65 80 60 88 66 78 85 90 90 71 A) -8.462 B) 184.462 C) 96.462 D) -491.982 Answer: A ^
5) The least-square regression line for the given data is y = 0.449x - 30.27. Determine the residual of a data point for which x = 100 and y = 15. Temperature, x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 A) 0.37 B) 29.63 C) 14.63 D) 123.535 Answer: A Page 42
^
6) The least-square regression line for the given data is y = 1.488x + 60.46. Determine the residual of a data point for which x = 38 and y = 116. Age, x 38 41 45 48 51 53 57 61 65 Pressure, y 116 120 123 131 142 145 148 150 152 A) -1.004 B) 233.004 C) 117.004 D) -195.068 Answer: A ^
7) The least-square regression line for the given data is y = -2.75x + 96.14. Determine the residual of a data point for which x = 5 and y = 82. Number of absences, x 0 3 6 4 9 2 15 8 5 Final grade, y 98 86 80 82 71 92 55 76 82 A) -0.39 B) 164.39 C) 82.39 D) 134.36 Answer: A ^
8) The least-square regression line for the given data is y = 6.91x + 46.26. Determine the residual of a data point for which x = 3 and y = 73. Number of years, x 3 4 4 5 3 6 2 7 3 Grades on test, y 61 68 75 82 73 90 58 93 72 A) 6.01 B) 139.99 C) 66.99 D) -547.69 Answer: A ^
9) The regression line for the given data is y = 4.379x + 4.267. Determine the residual of a data point for which x = 18.8 and y = 82.4. Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) -4.192,2 B) 168.992,2 C) 86.592,2 D) -346.296,6 Answer: A 10) Compute the sum of the squared residuals of the least-squares regression line for the given data. x -5 -3 4 1 -1 -2 0 2 3 -4 y -10 -8 9 1 -2 -6 -1 3 6 -8 A) 7.624 B) 1.036 C) 2.097 D) 0 Answer: A 11) The data below are the final exam scores of 10 randomly selected statistics students and the number of hours they slept the night before the exam. Compute the sum of the squared residuals of the least-squares regression line for the given data. Hours, x 3 5 2 8 2 4 4 5 6 3 Scores, y 65 80 60 88 66 78 85 90 90 71 A) 318.038 B) 804.062 C) 1122.1 D) 39.755 Answer: A 12) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Compute the sum of the squared residuals of the least-squares regression line for the given data. Rain fall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) 87.192 B) 2207.628 C) 4.379 D) 0 Answer: A
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13) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in a 15 minute time period following the addition of food. The data showing the weekly number of grunts and the age of the warthog (in days) are listed below. Compute the sum of the squared residuals of the least squares regression line for the given data. Week 1 2 3 4 5 6 7 8 9
Number of Grunts 90 68 39 44 63 40 62 17 20
A) 5533.53
Age (days) 125 141 155 160 167 174 183 189 195 B) 188.84
C) 74.39
D) 13.74
Answer: A 14) A calculus instructor is interested the performance of his students from Calculus I that go on to Calculus II. Their final grades in each course (in percent) are given below. Compute the sum of the squared residuals of the least squares regression line for the given data. Calculus I Calculus II
88 81
A) 130.14
78 80
62 55
75 78
95 90
B) 30.85
91 90
83 81
C) 11.41
86 80
98 100 D) 1075.9
Answer: A
4.3 Diagnostics on the Least-Squares Regression Line 1 Compute and interpret the coefficient of determination. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Choose the coefficient of determination that matches the scatterplot. Assume that the scales on the horizontal and vertical axes are the same. 1) Response
Explanatory A) R2 = 0.77 Answer: A
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B) R 2 = 0.38
C) R2 = 0.96
D) R2 = -0.51
2) Response
Explanatory A) R2 = 0.43
B) R 2 = -0.43
C) R2 = 0.92
D) R2 = 0.02
B) R 2 = -0.31
C) R2 = 0.76
D) R2 = 0.91
Answer: A 3) Response
Explanatory A) R2 = 0.097 Answer: A Use the linear correlation coefficient given to determine the coefficient of determination, R2 . 4) r = 0.62 A) R2 = 38.44% B) R 2 = 78.74% C) R2 = 7.87%
D) R2 = 3.844%
Answer: A 5) r = -0.32 A) R2 = 10.24%
B) R 2 = 56.57%
C) R2 = -10.24%
D) R2 = -56.57%
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 6) Calculate the coefficient of determination to the nearest thousandth, given that the linear correlation coefficient, r, is 0.837. What does this tell you about the explained variation and the unexplained variation of the data about the regression line? Answer: The coefficient of determination, R2 , = 0.701. That is, 70.1% of the variation is explained and 29.9% of the variation is unexplained. 7) Calculate the coefficient of determination to the nearest thousandth, given that the linear correlation coefficient, r, is -0.625. What does this tell you about the explained variation and the unexplained variation of the data about the regression line? Answer: The coefficient of determination, R2 , = 0.391. That is, 39.1% of the variation is explained and 60.9% of the variation is unexplained.
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8) Calculate the coefficient of determination, given that the linear correlation coefficient, r, is 1. What does this tell you about the explained variation and the unexplained variation of the data about the regression line? Answer: The coefficient of determination, R2 , = 1. That is, 100% of the variation is explained and there is no variation that is unexplained. 9) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the 15 minute period following the addition of food. The data showing the weekly number of grunts and the age of the warthog (in days) are listed below. Find and interpret the value of R2 . Round R2 to the nearest thousandth. Number of Grunts 86 64 35 40 59 36 58 13 16
Age (days) 121 137 151 156 163 170 179 185 191
Answer: R2 = 0.627; Approximately 62.7% of the variation in the number of grunts is explained by age. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 10) In a comprehensive road test on all new car models, one variable measured is the time it takes a 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) Initially, the simple linear model E(y) = b1x + b0 was fit to the data. Computer printouts for the analysis are given below: UNWEIGHTED 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, rounded to the nearest whole percent, of the sample variation in acceleration time can be explained by the simple linear model? A) 70% B) 0% C) -17% D) 8% Answer: A
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11) 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: y = Number of man-hours required to erect the drum PRESSURE: x= Boiler design pressure (pounds per square inch, i.e., psi) Initially, the simple linear model E(y) = b1x + b0 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 . Express R 2 to the nearest whole percent. A) About 43% of the sample variation in number of man-hours can be explained by the simple linear model. ^
B) y = 0.00321x + 1.88 will be correct 43% of the time. C) Man hours needed to erect drums will be associated with boiler design pressure 43% of the time. D) About 2.06% of the sample variation in number of man-hours can be explained by the least-square regression model. Answer: A
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12) Civil engineers often use the straight-line equation, E(y) = b1x + b0 , to model the relationship between the mean shear strength E(y) of masonry joints and precompression stress, x. To test this theory, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the regression analysis. Triplet Test 1 2 3 4 5 6 7 Shear Strength (tons), y 1.00 2.18 2.24 2.41 2.59 2.82 3.06 Precomp. Stress (tons), x 0 0.60 1.20 1.33 1.43 1.75 1.75
Source
DF
Analysis of Variance Sum of Mean Squares Square
Model Error C Total
1 5 6
2.39555 0.25094 2.64649
2.39555 0.05019
0.22403 2.32857 9.62073
R-square Adj R-sq
Root MSE Dep Mean C.V.
F Value
Prob > F
47.732
0.0010
0.9052 0.8862
Parameter Estimates
Variable INTERCEP X
DF 1 1
Parameter Estimate Error
Standard T for HO: Parameter=0Prob > |T|
1.191930 0.987157
6.442 6.909
0.18503093 0.14288331
0.0013 0.0010
Give a practical interpretation of R2 , the coefficient of determination for the least squares model. Express R2 to the nearest whole percent. A) About 91% of the total variation in the sample of y-values can be explained by (or attributed to) the linear relationship between shear strength and precompression stress. B) In repeated sampling, approximately 91% of all similarly constructed regression lines will accurately predict shear strength. C) We expect to predict the shear strength of a triplet test to within about .91 ton of its true value. D) We expect about 91% of the observed shear strength values to lie on the least squares regression line. Answer: A
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13) 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 GPA's 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) = b1x + b0 . The results of the least-squares 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 Calculate the value of R2 , the coefficient of determination. A) 0.934
B) 0.661
C) 0.872
D) 0.339
Answer: A 14) Each year a nationally recognized publication conducts its "Survey of America's Best Graduate and Professional Schools." An 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 25 data points shown below. b0 = -92040 b1 = 228 s = 3213 R2 = 0.66 r = 0.81 df = 23 t = 6.67 Give a practical interpretation of R2 = 0.66. A) 66% of the sample variation in SALARY can be explained by using GMAT in a straight-line model. B) 66% of the differences in SALARY are caused by differences in GMAT scores. C) We estimate SALARY to increase $.66 for every 1-point increase in GMAT. D) We can predict SALARY correctly 66% of the time using GMAT in a straight-line model. Answer: A 15) A real estate magazine reported the results of a regression analysis designed to predict the price (y), measured in dollars, of residential properties recently sold in a northern Virginia subdivision. One independent variable used to predict sale price is GLA, gross living area (x), measured in square feet. Data for 157 properties were used to fit the model, E(y) = b1x + b0 . The results of the simple linear regression are provided below. y = 96,600 + 22.5x s = 6500 R2 = 0.77 t = 6.1 (for testing β 1 ) Interpret the value of the coefficient of determination, R2 . A) 77% of the total variation in the sample sale prices can be attributed to the linear relationship between GLA (x) and (y). B) GLA (x) is linearly related to sale price (y) 77% of the time. C) 77% of the observed sale prices (y's) will fall within 2 standard deviations of the least squares regression line. D) There is a moderately strong positive correlation between sale price (y) and GLA (x). Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 16) A company keeps extensive records on its new salespeople on the premise that sales should increase with experience. A random sample of seven new salespeople produced the data on experience and sales shown in the table. Months on Job Monthly Sales y ($ thousands) 2 2.4 4 7.0 8 11.3 12 15.0 1 0.8 5 3.7 9 12.0 Summary statistics yield SSxx = 94.8571, SSxy = 124.7571, SSyy = 176.5171, x = 5.8571, and y = 7.4571. Find and interpret the coefficient of determination. Round R2 to the nearest hundredth of a percent. Answer: R2 = 92.96% of the variation in the sample monthly sales values about their mean can be explained by using months on the job in a linear model. 17) To investigate the relationship between yield of potatoes, y, and level of fertilizer application, x, an experimenter 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 1 1.5 2 2.5 3 3.5 4 4.5 y 25 31 27 28 36 35 32 34 Summary statistics yield SSxx = 10.5, SSyy = 112, and SSxy = 25. Calculate the coefficient of determination rounded to the nearest ten-thousandth. Answer: R2 = 0.5315 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) The correlation coefficient from a data set is r = 0.59. Calculate the coefficient of determination R2 . Round R2 to the nearest hundredth. A) 0.35 B) 0.59 C) 0.41 D) 0.65 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 19) The coefficient of determination for a straight-line model relating selling price y to manufacturing cost x for a particular item is R2 = 0.83. Interpret this value. Answer: The model explains 83% of sample variation in cost. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 20) The
measures the proportion of total variation in the response variable that is explained
by the least squares regression line. A) coefficient of determination C) sum of the residuals squared Answer: A
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B) linear correlation coefficient D) slope of the regression line
21) If the coefficient of determination is close to 1, then A) the least squares regression line equation explains most of the variation in the response variable. B) the least squares regression line equation has no explanatory value. C) the sum of the square residuals is large compared to the total variation. D) the linear correlation coefficient is close to zero. Answer: A 22) The coefficient of determination is the A) square
B) square root
of the linear correlation coefficient. C) opposite
D) reciprocal
Answer: A 2 Perform residual analysis on a regression model. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Analyze the residual lot below. Does it violate any of the conditions for an adequate linear model? 1)
A) No, the plot of residuals is random. B) Yes, there is a discernable pattern in the residuals. C) Yes, the residuals do not display constant error variance. Answer: A 2)
A) Yes, the residuals do not display constant error variance. B) Yes, there is a discernable pattern in the residuals. C) No, the plot of residuals is random. Answer: A
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3)
A) Yes, there is a discernable pattern in the residuals. B) Yes, the residuals do not display constant error variance. C) No, the plot of residuals is random. Answer: A Provide an appropriate response. 4) True or False: Residual analysis cannot be used to check for outliers. A) False B) True Answer: A 5) True or False: If a residual plot shows an almost straight line then a linear model is appropriate. A) False B) True Answer: A 6) To determine if there are outliers in a least-squares regression model's data set, we could construct a boxplot of the A) residuals. B) response variables. C) predictor variables. D) lurking variables. Answer: A
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3 Identify influential observations. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A scatter diagram is given with one of the points labeled "A." In addition, there are two least-squares regression lines drawn. The solid line excludes the point A. The dashed line includes the point A. Based on the graph, is the point A influential? 1) 10
y
9 8 7 6 5 4 3 2
A
1 1
2
3
4
5
6
7
8
9
10 x
A) yes
B) no
Answer: A 2) 10
y
9 8 7 6 5 4 3
A
2 1 1
A) no
2
3
4
5
6
7
8
9
10 x
B) yes
Answer: A Provide an appropriate response. 3) An influential observation is an observation that significantly affects the value of the A) the slope of the least squares regression line. B) the mean of the response variable. C) the median of the predictor variable. D) the median of the response variable. Answer: A
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4) What effect will an influential observation have upon the graph of the least squares regression line? A) It will pull the graph toward the observation. B) It will have no effect. C) It will push the graph away from the observation. D) It will lower the value of the correlation coefficient to make further analysis meaningless. Answer: A
4.4 Contingency Tables and Association 1 Compute the marginal distribution of a variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The following data represent the living situation of newlyweds in a large metropolitan area and their annual household income. Find the marginal frequency for newlyweds who rent their home. <$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000 Own home 31 52 202 355 524 Rent home 67 66 52 23 11 Live w/family 89 69 30 4 2 A) 219 B) 67 C) 11 D) 52 Answer: A 2) The following data represent the living situation of newlyweds in a large metropolitan area and their annual household income. Find the marginal frequency for newlyweds who make more than $75,000 per year. <$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000 Own home 31 52 202 355 524 Rent home 67 66 52 23 11 Live w/family 89 69 30 4 2 A) 537 B) 2 C) 524 D) 11 Answer: A 3) The following data represent the living situation of newlyweds in a large metropolitan area and their annual household income. What percent of people who make less than $20,000 per year own their own home? Round to the nearest tenth of a percent. <$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000 Own home 31 52 202 355 524 Rent home 67 66 52 23 11 Live w/family 89 69 30 4 2 A) 16.6% B) 35.8% C) 47.6% D) 2.7% Answer: A 4) The following data represent the living situation of newlyweds in a large metropolitan area and their annual household income. What percent of people who live with family make between $35,000 and $50,000 per year? Round to the nearest tenth of a percent. <$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000 Own home 31 52 202 355 524 Rent home 67 66 52 23 11 Live w/family 89 69 30 4 2 A) 15.5% B) 35.6% C) 2.1% D) 10.6% Answer: A
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5) Construct a frequency marginal distribution for the given contingency table. x 1 x 2 x3 y1 25 y2 80
25
35
50
80
A) x1
x2
x3
Marginal Distribution
25
25
35
85
80
50
80
210
Marginal Distribution 105
75
115
295
x1
x2
x3
Marginal Distribution
25
25
35
85
80
50
80
210
Marginal Distribution 105
75
115
590
x1
x2
x3
Marginal Distribution
25
25
35
85
80
50
80
210
Marginal Distribution 55
25
45
590
x1
x2
x3
Marginal Distribution
25
25
35
105
80
50
80
75
Marginal Distribution 85
210
115
590
y1 y2 B) y1 y2 C) y1 y2 D) y1 y2
Answer: A
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6) Construct a relative frequency marginal distribution for the given contingency table. Round values to the nearest thousandth. x1 x2 x3 y1 15 15 20 y2 40 45 35 A)
y1 y2
x1
x2
Relative Frequency x3 Marginal Distribution
15
15
20
0.294
40
45
35
0.706
Relative Frequency Marginal Distribution0.324 0.353 0.324
1
B)
y1 y2
x1
x2
15
15
Relative Frequency x3 Marginal Distribution 20 0.294
40
45
35
0.706
Relative Frequency Marginal Distribution0.147 0.176 0.088
1
C) x1
x2
Relative Frequency x3 Marginal Distribution
15
15
20
0.50
40
45
35
1.20
Relative Frequency Marginal Distribution 0.55 0.60 0.55
1
y1 y2
D) Relative Frequency Marginal Distribution x1 y1 y2 Answer: A
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x2
x3
0.088 0.088 0.118 0.235 0.265 0.206
7) Construct a conditional distribution by x for the given contingency table. Round values to the nearest thousandth. x1 x2 x3 y1 30 35 20 y2 60 55 80 A)
B) x1 y1 y2
x2
x3
x1
0.333 0.389 0.200
y1 y2
0.667 0.611 0.800
Total 1
1
1
x1
x2
x3
x2
0.333 0.389 0.200 0.308 0.282 0.410
Total 1
C)
x3
1
1
D) x1
y1 0.107 0.125 0.071 y2 0.214 0.196 0.286
x2
x3
Total
y1 0.353 0.412 0.235 1 y2 0.308 0.282 0.410 1
Answer: A 8) A contingency table relates A) two categories of data. B) the difference in the means of two random variables. C) a particular response with order in which that response should be applied. D) only continuous random variables. Answer: A 9) To eliminate the effects of either the row or the column variables in a contingency table, a distribution is created. A) marginal
B) normalized
C) χ 2
D) Student's t
Answer: A 2 Use the conditional distribution to identify association among categorical data. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The data below show the age and favorite type of music of 779 randomly selected people. What, if any, association exists between favorite music and age? Discuss the association. Age Country Rock Pop Classical 15 - 21 21 45 90 33 21 - 30 60 55 42 48 30 - 40 65 47 31 57 40 - 50 68 39 25 53 Answer: The proportion who prefers country music increases as age increases. The proportion who prefers rock is roughly constant as age increases. The proportion who prefers pop decreases as age increases. The proportion who prefers classical increases as age increases.
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2) The following data represent the living situation of newlyweds in a large metropolitan area and their annual household income. What, if any, association exists between living situation and household income? Discuss the association. < $20,000 $20-35,000 $35-50,000 $50-75,000 > $75,000 Own home 31 52 202 355 524 Rent home 67 66 52 23 11 Live w/family 89 69 30 4 2 Answer: The proportion of home owners increases as the household income increases. The proportions of renters and of those living with family decrease as household income increases. 3 Explain Simpson's Paradox. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Researchers conducted a study to determine which of two different treatments, A or B, is more effective in the treatment of atherosclerosis. The results of their experiment are given in the table. (a) Which treatment appears to be more effective? Why? Treatment A Treatment B Effective 420 435 Not effective 130 140 The data in the table do not take into account the seriousness of the case. The data shown in the next table show the effectiveness of each treatment for both mild and advanced cases of atherosclerosis. Mild Advanced atherosclerosis atherosclerosis Treatment A Treatment B Treatment A Treatment B Effective 310 95 110 340 Not effective 80 20 50 120 (b) Determine the proportion of mild cases of atherosclerosis that were effectively dealt with using treatment A. Determine the proportion of mild cases of atherosclerosis that were effectively dealt with using treatment B. (c) Repeat part (b) for advanced cases of atherosclerosis to create a conditional distribution of effectiveness by treatment for each category of the disease. (d) Write a short report detailing and explaining your findings. Answer: (a) Treatment A appears to be more effective: treatment A was effective in 76.4% of cases treatment B was effective in 75.7% of cases. (b) Mild cases: treatment A was effective in 79.5% of cases treatment B was effective in 82.6% of cases. (c) Advanced cases: treatment A was effective in 68.8% of cases treatment B was effective in 73.9% of cases. (d) Within each category of disease, B has a higher rate of effectiveness and yet overall it has a lower rate of effectiveness. The initial analysis failed to take into account the lurking variable - seriousness of the case. Treatment B is used more often in the more serious cases where success rates are lower for both methods.
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2) A company encourages applications from minority groups who they feel are under-represented in the company. The table shows the number of applications that were accepted last year from people belonging to minority groups and the number of applications that were accepted from people not belonging to minority groups. Only applications from well qualified applicants are included in the analysis. (a) Does the acceptance rate appear to be higher for those belonging to minority groups or for those not belonging to minority groups ? Why? Minority Not minority Accepted 70 79 Rejected 460 500 The data in the table do not take into account the department of the company. The data shown in the next table show the number of applications accepted from each group within each department. Department A Department B Department C Minority Not minority Minority Not minority Minority Not minority Accepted 27 10 22 34 21 35 Rejected 260 110 80 150 120 240 (b) Determine the proportion of minority applications that were accepted within department A. Determine the proportion of non-minority applications that were accepted within department A. (c) Repeat part (b) for departments B and C to create a conditional distribution of acceptance rate by group for each department of the company. (d) Write a short report detailing and explaining your findings. Answer: (a) The acceptance rate appears to be higher for those not belonging to minority groups: 13.2% of applications from people belonging to minority groups were accepted 13.6% of applications from people not belonging to minority groups were accepted (b) Department A: minority applications: 9.4% accepted non-minority applications: 8.3% accepted (c) Department B: minority applications: 21.6% accepted non-minority applications: 18.5% accepted Department C: minority applications: 14.9% accepted non-minority applications: 12.7% accepted. (d) Within each department, the rate of acceptance is higher for people belonging to minority groups and yet overall the acceptance rate is higher for people not belonging to minority groups. The initial analysis failed to take into account the lurking variable - the department of the company. In department A, there are more applications from minorities and this department has the lowest acceptance rates. Department B has the fewest applications from minorities and this department has the highest acceptance rates.
4.5 Nonlinear Regression: Transformations (online) 1 Convert between exponential and logarithmic expressions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Change the exponential expression to an equivalent expression involving a logarithm. 1) 62 = 36 A) log6 36 = 2
B) log 366 = 2
C) log2 36 = 6
D) log6 2 = 36
B) log x3 = 2
C) log2 x = 3
D) log3 2 = x
Answer: A 2) 32 = x A) log3 x = 2 Answer: A
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3) 72 = y A) log7 y = 2
B) log 2 y = 7
C) logy7 = 2
D) logy2 = 7
B) log x1,000 = 10
C) log1,00010 = x
D) log1,000x = 10
B) log 2857 = x
C) log285x = 7
D) log7 x = 285
Answer: A 4) 10x = 1,000 A) log101,000 = x Answer: A 5) 7x = 285 A) log7 285 = x Answer: A Change the logarithmic expression to an equivalent expression involving an exponent. 6) log4 64 = 3 A) 4 3 = 64
B) 3 4 = 64
C) 4 64 = 3
D) 643 = 4
B) 2 4 = x
C) 4 x = 2
D) x2 = 4
B) 3 b = 125
C) 1253 = b
D) 125b = 3
B) x2 = 4
C) 4 x = 2
D) 4 2 = x
Answer: A 7) log4 x = 2 A) 4 2 = x Answer: A 8) logb125 = 3 A) b3 = 125 Answer: A 9) log2 4 = x A) 2 x = 4 Answer: A 2 Simplify logarithmic expressions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the expression as a sum of logs. Express powers as factors. 1) log8 xy A) log8 x + log8 y
B) log 8 x - log8 y
C) log4 x + log4 y
D) log4 x - log4 y
B) 3log 8x
C) 3 log x
D) 8log3 x8
B) 2log by + 2logbz
C) 2logbyz
D) logby + logb2z
Answer: A 2) log3 x8 A) 8log3 x Answer: A 3) logbyz2 A) logby + 2log bz Answer: A
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4) logby5z 2 A) 5logby + 2logbz
B) 10logb yz
C) logb (yz)10
D) logb 10yz
Answer: A Use a calculator to evaluate the expression. Round your answer to three decimal places. 5) log 280 A) 2.447 B) 5.635 C) 2.449
D) 2.446
Answer: A 6) log 3.50 A) 0.544
B) 1.253
C) 0.556
D) 0.531
B) 100.000
C) 6.686
D) 125.893
B) 0.132
C) 0.415
D) 1.224
Answer: A 7) 101.9 A) 79.433 Answer: A 8) 100.879,4 A) 7.575 Answer: A 3 Use logarithmic transformations to linearize exponential relations. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The following data represent the bacteria population in a laboratory experiment. The researchers suspect that the population is growing exponentially. Determine the logarithm of the y-values so that Y = log y. Day, x Population, y 0 1,929 1 5,257 2 14,240 3 38,816 4 105,586 5 287,189 Answer: Day, x Population, y Log y 0 1,929 3.285,3 1 5,257 3.720,7 2 14,240 4.153,5 3 38,816 4.589 4 105,586 5.023,6 5 287,189 5.458,2
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) The following data represent the bacteria population in a laboratory experiment. The researchers suspect that the population is growing exponentially. Find the least-squares regression line of the transformed data by determining the logarithm of the y-values so that Y = log y. Day, x Population, y 0 1952 1 5319 2 14410 3 39279 4 106845 5 290613 ^
B) Y= 3.458 + 1.151x
^
D) Y = -50,222.143 + 50,650.057x
A) Y = 3.291 + 0.435x C) Y = -7.537 + 2.301x
^ ^
Answer: A 4 Use logarithmic transformations to linearize power relations. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The following data represent the periods (in seconds) of simple pendulums of various lengths (in feet). Determine the logarithm of both the x- and y-values so that X = log x and Y = log y. Length, x Period, y 1 1.1 2 1.6 3 1.9 4 2.2 5 2.5 6 2.7 Answer: Length, x Log x Period, y Log y 1 0 1.1 0.041,4 2 0.3010 1.6 0.204,1 3 0.4771 1.9 0.278,8 4 0.6021 2.2 0.342,4 5 0.6990 2.5 0.397,9 6 0.7782 2.7 0.431,4
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) The following data represent the periods (in seconds) of simple pendulums of various lengths (in feet). Find the power equation of best fit. Length, x Period, y 1 1.3 2 1.9 3 2.3 4 2.7 5 3.0 6 3.3 ^
B) y = -0.2258 + 1.9309x
^
D) y = 1.0467 + 0.3914x
A) y = 0.1171 + 0.5176x C) y = 0.0948 + 0.0768x Answer: A Page 62
^ ^
3) The following data represent the infection rate for a particular disease in a third world country x years after a vaccine became widely available. Determine the "best" model to describe the relationship between years past and infection rate. Years After Vaccine, x Infection Rate (%), y 1 32 2 11 3 5 4 2 5 0.5 ^
A) exponential model: y= 1.955 - 0.435x ^
C) linear model: y= 31.7 - 7.2x
^
B) power model: y= 1.648 - 2.402x ^
D) linear model: y= 31.7 + 7.2x
Answer: A 4) The following data represent the compound yield in grams for a chemical reaction for various temperatures of the reaction. Determine the "best" model to describe the relationship between temperature and compound yield. Temperature (° C), x Compound Yield (grams), y 50 4 60 12 70 14 80 21 90 26 ^
B) exponential model: y= -0.195 + 0.019x
^
D) power model: y= 4.392 + 2.999x
A) linear model: y = -21.7 + 0.53x C) power model: y= -4.392 + 2.999x
^
^
Answer: A 5) The following data represent the height (relative to the ground) of a projectile shot into the air after x seconds of travel. Determine the "best" model to describe the relationship between travel time and height. Travel Time (seconds), x Height (feet), y 8 890 9 869 10 798 11 708 12 571 A) power model: y = 3.930 - 1.056x
^
B) exponential model: y = 3.354 - 0.047x
^
D) exponential model: y = 3.354 + 0.047x
C) linear model: y = 1566.2 - 79.9x Answer: A
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^ ^
Ch. 4 Describing the Relation between Two Variables Answer Key 4.1 Scatter Diagrams and Correlation 1 Draw and interpret scatter diagrams. 1) y 95 90 85
Exam Scores
80 75 70 65 60 1
2
3
4
5
6
7
8
x
Hours Studied 2) y 16 14 12
Number 10 of Absences 8 6 4 2 70
75
80
85
90
95
100
x
Temperature 3) y 155 150 145
Blood 140 Pressure 135 (mm of mercury) 130 125 120 115 35
40
45
50
55
Age
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60
65
x
4) y 100 90
Final Grade
80 70 60 50 2
4
6
8
10
12
14
16 x
Number of Absences 5) 120
y
110 100 90
Sales (in thousands) 80 70 60 50 40 30 2
4
6
8
10
12
14
16 x
Miles traveled (in hundreds) 6) y 100 90
Final Grade
80 70 60 50 1
2
3
4
5
6
7
Number of years studied
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x
7) y 90 80
Yield 70 (bushels) 60 50 40 30 6
8
10
12
14
20
x
15
18
x
90
95
16
18
Rainfall (inches) 8) y 1.6 1.4 1.2
Nicotine 1 (mg) 0.8 0.6 0.4 0.2 6
9
12
Tar (mg) 9) y 95
90
Round 2 85
80
75 75
80
85
x
Round 1 10) A 11) A 12) A 13) predictor variable: rainfall in inches; response variable: yield per acre 14) predictor variable: hours studying; response variable: grades on the test 15) A 2 Describe the properties of the linear correlation coefficient. 1) A Page 66
2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) There appears to be a positive linear correlation. y 10 8 6 4 2 -5 -4 -3 -2 -1 -2
1
2
3
4
5
x
-4 -6 -8 -10
10) There appears to be a negative linear correlation. 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2
1
2
3
4
5
6 x
-4 -6 -8 -10 -12
11) There appears to be no linear correlation. 12
y
10 8 6 4 2 -6 -5 -4 -3 -2 -1 -2 -4 -6 -8 -10 -12
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1
2
3
4
5
6 x
12) In general, there appears to be a relationship between the home runs and batting averages. As the number of home runs increased, the batting averages increased. y 0.35
0.3
Batting Average
0.25
0.2
0.15 15
30
45
60
75
x
Home Runs 13) There appears to be a trend in the data. As the number of absences increases, the final grade decreases. 100
y
90 80
Final Grade (%)
70 60 50 40 5
10
15
x
Number of Absences 14) A 15) A 16) A 17) A 18) A 19) A 20) A 3 Compute and interpret the linear correlation coefficient. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) The linear correlation coefficient remains unchanged. 4 Determine whether a linear relation exists between two variables. 1) A 2) A 3) A Page 68
4) A 5) A 6) A 7) A 8) A 9) A 10) A 5 Explain the difference between correlation and causation. 1) A 2) A positive correlation exists between the number of schools in a city and the number of robberies but this is an example of correlation not causation. Building more schools is unlikely to lead to an increase in robberies. A likely lurking variable is the population of the city and this lurking variable accounts for the positive correlation. Larger cities tend to have both more schools and more robberies. 3) A negative correlation exists between the infant mortality rate and the number of cars per capita but this is an example of correlation not causation. If people buy more cars, this is unlikely to lead to a decrease in the infant mortality rate. A likely lurking variable is wealth and this lurking variable accounts for the negative correlation. More affluent countries tend to have both more cars per capita and lower infant mortality rates. 4) A positive correlation exists between income and blood pressure but this is an example of correlation not causation. An increase in salary is unlikely to lead to an increase in blood pressure. Age and level of job stress are possible lurking variables and these lurking variables account for the positive correlation. Older men tend to have both higher blood pressures and higher incomes. Also men in high stress jobs tend to have both higher blood pressures and higher incomes.
4.2 Least-Squares Regression 1 Find the least-squares regression line and use the line to make predictions. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) The regression lines are not necessarily the same.
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15) SSxx = ∑x2 -
SSxy = ∑xy -
∑x
2
n
= 0.948622 -
∑x ∑y = 295.54 - (3.642)(1,134) = 0.538 n
y=
∑y = 1,134 = 81
x=
∑x = 3.642 = 0.26014
n
(3.642)2 = 0.00118171 14
14
14
n
14
SSxy ^ 0.538 b1 = = = 455.27 SSxx 0.00118171 ^
^
b0 = y - b1 x = 81 - 455.27(0.26014) = -37.434 ^
The least squares equation is y= -37.434 + 455.27x. SSxy 3031.7125 ^ 16) b1 = = = 2.4804 SSxx 1222.2771 ^
^
b0 = y - b1x = 95.0625 - 2.4804(21.2675) = 42.3106 ^
The least squares prediction equation is y = 42.3106 + 2.4804x. 17) A 18) A 19) A 2 Interpret the slope and the y-intercept of the least-squares regression line. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) b1 = -0.08365. For every 1 mile per hour increase in the maximum attained speed of a new car, we estimate the elapsed 0 to 60 acceleration time to decrease by .08365 seconds. 12) a. E(y) = b0 + b1 x ^
^
^
b. y= b0 + b1 x = 168.43 - 0.8195x c. We would expect approximately 168 grunts after feeding a warthog that was just born. However, since the value 0 in 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 0.8195. 13) A 14) A Page 70
15) A 16) A 17) A 18) A 19) A 20) a) y 100 90 80
Calculus II
70 60 50 50
60
70
80
90
100 x
Calculus I ^
b) y= 1.044x - 5.990 ^
c) When x = 80, y = 78. 21) A 3 Compute the sum of squared residuals. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A
4.3 Diagnostics on the Least-Squares Regression Line 1 Compute and interpret the coefficient of determination. 1) A 2) A 3) A 4) A 5) A 6) The coefficient of determination, R 2, = 0.701. That is, 70.1% of the variation is explained and 29.9% of the variation is unexplained. 7) The coefficient of determination, R 2, = 0.391. That is, 39.1% of the variation is explained and 60.9% of the variation is unexplained. 8) The coefficient of determination, R 2, = 1. That is, 100% of the variation is explained and there is no variation that is unexplained. 9) R2 = 0.627; Approximately 62.7% of the variation in the number of grunts is explained by age. Page 71
10) A 11) A 12) A 13) A 14) A 15) A 16) R2 = 92.96% of the variation in the sample monthly sales values about their mean can be explained by using months on the job in a linear model. 17) R2 = 0.5315 18) A 19) The model explains 83% of sample variation in cost. 20) A 21) A 22) A 2 Perform residual analysis on a regression model. 1) A 2) A 3) A 4) A 5) A 6) A 3 Identify influential observations. 1) A 2) A 3) A 4) A
4.4 Contingency Tables and Association 1 Compute the marginal distribution of a variable. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 2 Use the conditional distribution to identify association among categorical data. 1) The proportion who prefers country music increases as age increases. The proportion who prefers rock is roughly constant as age increases. The proportion who prefers pop decreases as age increases. The proportion who prefers classical increases as age increases. 2) The proportion of home owners increases as the household income increases. The proportions of renters and of those living with family decrease as household income increases. 3 Explain Simpson's Paradox. 1) (a) Treatment A appears to be more effective: treatment A was effective in 76.4% of cases treatment B was effective in 75.7% of cases. (b) Mild cases: treatment A was effective in 79.5% of cases treatment B was effective in 82.6% of cases. (c) Advanced cases: treatment A was effective in 68.8% of cases treatment B was effective in 73.9% of cases. (d) Within each category of disease, B has a higher rate of effectiveness and yet overall it has a lower rate of effectiveness. The initial analysis failed to take into account the lurking variable - seriousness of the case. Treatment B is used more often in the more serious cases where success rates are lower for both methods. Page 72
2) (a) The acceptance rate appears to be higher for those not belonging to minority groups: 13.2% of applications from people belonging to minority groups were accepted 13.6% of applications from people not belonging to minority groups were accepted (b) Department A: minority applications: 9.4% accepted non-minority applications: 8.3% accepted (c) Department B: minority applications: 21.6% accepted non-minority applications: 18.5% accepted Department C: minority applications: 14.9% accepted non-minority applications: 12.7% accepted. (d) Within each department, the rate of acceptance is higher for people belonging to minority groups and yet overall the acceptance rate is higher for people not belonging to minority groups. The initial analysis failed to take into account the lurking variable - the department of the company. In department A, there are more applications from minorities and this department has the lowest acceptance rates. Department B has the fewest applications from minorities and this department has the highest acceptance rates.
4.5 Nonlinear Regression: Transformations (online) 1 Convert between exponential and logarithmic expressions. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 2 Simplify logarithmic expressions. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 3 Use logarithmic transformations to linearize exponential relations. 1) Day, x Population, y Log y 0 1,929 3.285,3 1 5,257 3.720,7 2 14,240 4.153,5 3 38,816 4.589 4 105,586 5.023,6 5 287,189 5.458,2 2) A
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4 Use logarithmic transformations to linearize power relations. 1) Length, x Log x Period, y Log y 1 0 1.1 0.041,4 2 0.3010 1.6 0.204,1 3 0.4771 1.9 0.278,8 4 0.6021 2.2 0.342,4 5 0.6990 2.5 0.397,9 6 0.7782 2.7 0.431,4 2) A 3) A 4) A 5) A
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Ch. 5 Probability 5.1 Probability Rules 1 Understand random processes and the Law of Large Numbers. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Identify the sample space of the probability experiment: tossing a coin. Answer: 2) Identify the sample space of the probability experiment: answering a true or false question. Answer: 3) Identify the sample space of the probability experiment: tossing four coins and recording the number of heads Answer: 4) Identify the sample space of the probability experiment: answering a multiple choice question with A, B, C, D and E as the possible answers. Answer: 5) Identify the sample space of the probability experiment: determining the puppies’ gender for a litter of three puppies (Use M for male and F for female.). Answer: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6) A(n) of a probability experiment is the collection of all outcomes possible. A) Sample space B) Event set C) Bernoulli space
D) Prediction set
Answer: A 2 Apply the rules of probabilities. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Which of the following probabilities for the sample points A, B, and C could be true if A, B, and C are the only sample points in an experiment? A) P(A) = 0, P(B) = 1/3, P(C) = 2/3 B) P(A) = 1/8, P(B) = 1/9, P(C) = 1/5 C) P(A) = -1/4, P(B) = 1/2, P(C) = 3/4 D) P(A) = 1/4, P(B) = 1/4, P(C) = 1/4 Answer: A 2) If A, B, C, and D, are the only possible outcomes of an experiment, find the probability of D using the table below. Outcome A B C D . Probability 1/13 1/13 1/13 A) 10/13 B) 1/13 C) 1/4 D) 3/13 Answer: A
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3) In a 1-pound bag of skittles the possible colors were red, green, yellow, orange, and purple. The probability of drawing a particular color from that bag is given below. Is this a probability model? Answer Yes or No. Color Red Green Orange Yellow Purple
Probability 0.2299 0.1908 0.2168 0.1889 0.1816
A) Yes
B) No
Answer: A 4) A bag contains 25 wooden beads. The colors of the beads are red, blue, white, green, black, brown, and grey. The probability of randomly selecting a bead of a particular color from the bag is given below. Is this a probability model? Answer yes or No. Color Red Probability 0.28
Blue 0.24
White 0.20
Green Black 0.16 0.12
A) No
Brown Grey 0.08 0.03 B) Yes
Answer: A 5) Which of the following cannot be the probability of an event? A) -23
B) 0
C) 0.001
D)
5 3
Answer: A 6) The probability that event A will occur is P(A) = A) False
Number of successful outcomes Number of unsuccessful outcomes B) True
Answer: A 7) The probability that event A will occur is P(A) = A) True
Number of successful outcomes Total number of all possible outcomes B) False
Answer: A 8) In terms of probability, a(n) ___________________ is any process with uncertain results that can be repeated. A) Experiment B) Sample space C) Event D) Outcome Answer: A 9) True or False: An event is any collection of outcomes from a probability experiment. A) True B) False Answer: A 10) An unusual event is an event that has a A) Low probability of occurring C) Probability which exceeds 1 Answer: A
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B) Probability of 1 D) A negative probability
3 Compute and interpret probabilities using the empirical method. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The table below represents a random sample of the number of deaths per 100 cases for a certain illness over time. If a person infected with this illness is randomly selected from all infected people, find the probability that the person dies 3-4 years after diagnosis. Express your answer as a simplified fraction and as a decimal rounded to three decimal places. Years after Diagnosis Number deaths 1-2 15 3-4 35 5-6 16 7-8 9 9-10 6 11-12 4 13-14 2 15+ 13 7 1 7 7 A) ; 0.35 B) ; 0.029 C) ; 0.538 D) ; 0.058 20 35 13 120 Answer: A 2) At a particular point in time, the stock market took big swings up and down. A survey of 1,014 adult investors asked how often they tracked their portfolio. The table shows the investor responses. What is the probability that an adult investor tracks his or her portfolio daily? Express your answer as a simplified fraction and as a decimal rounded to three decimal places. How frequently? Response Daily 235 Weekly 275 Monthly 292 Couple times a year 148 Don't track 64 235 275 292 148 A) ; 0.232 B) ; 0.271 C) ; 0.288 D) ; 0.146 1,014 1,014 1,014 1,014 Answer: A The chart below shows the percentage of people in a questionnaire who bought or leased the listed car models and were very satisfied with the experience. Model A 81% Model B 79% Model C 73% Model D 61% Model E 59% Model F 57% 3) With which model was the greatest percentage satisfied? Estimate the empirical probability that a person with this model is very satisfied with the experience. Express the answer as a fraction with a denominator of 100. 81 0.81 57 0.57 A) Model A; B) Model A: C) Model F; D) Model F; 100 100 100 100 Answer: A
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4) The empirical probability that a person with a model shown is very satisfied with the experience is is the model? A) F
B) D
C) E
57 . What 100
D) B
Answer: A Provide an appropriate response. 5) True or False: The probability of an event E in an empirical experiment may change from experiment to experiment. A) True B) False Answer: A 4 Compute and interpret probabilities using the classical method. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Use the spinner below to answer the question. Assume that it is equally probable that the pointer will land on any one of the five numbered spaces. If the pointer lands on a borderline, spin again.
Find the probability that the arrow will land on 2 or 5. 2 A) B) 2 5
C)
5 3
D)
3 2
Answer: A 2) Use the spinner below to answer the question. Assume that it is equally probable that the pointer will land on any one of the five numbered spaces. If the pointer lands on a borderline, spin again.
Find the probability that the arrow will land on an odd number. 3 2 A) B) C) 1 5 5
D) 0
Answer: A 3) You are dealt one card from a standard 52-card deck. Find the probability of being dealt an ace or a 7. 2 4 13 A) B) C) D) 8 13 13 2 Answer: A 4) A die is rolled. The set of equally likely outcomes is {1, 2, 3, 4, 5, 6}. Find the probability of getting a 5. 1 5 A) B) C) 5 D) 0 6 6 Answer: A
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5) A die is rolled. The set of equally likely outcomes is {1, 2, 3, 4, 5, 6}. Find the probability of getting a 9. 9 A) 0 B) 1 C) 9 D) 6 Answer: A 6) You are dealt one card from a standard 52-card deck. Find the probability of being dealt a picture card. 3 1 3 3 A) B) C) D) 13 13 26 52 Answer: A 7) A fair coin is tossed two times in succession. The set of equally likely outcomes is {HH, HT, TH, TT}. Find the probability of getting the same outcome on each toss. 1 1 3 A) B) C) D) 1 2 4 4 Answer: A 8) A single die is rolled twice. The set of 36 equally likely outcomes is {(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)}. Find the probability of getting two numbers whose sum is greater than 9. 1 1 1 A) B) C) D) 6 6 4 12 Answer: A 9) A single die is rolled twice. The set of 36 equally likely outcomes is {(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)}. Find the probability of getting two numbers whose sum is less than 13. 1 1 A) 1 B) 0 C) D) 2 4 Answer: A 10) A single die is rolled twice. The set of 36 equally likely outcomes is {(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)}. Find the probability of getting two numbers whose sum is greater than 9 and less than 13. 1 5 7 A) B) 0 C) D) 6 36 36 Answer: A 11) This problem deals with eye color, an inherited trait. For purposes of this problem, assume that only two eye colors are possible, brown and blue. We use "b" to represent a blue eye gene and "B" a brown eye gene. If any B genes are present, the person will have brown eyes. The table shows the four possibilities for the children of two Bb (brown-eyed) parents, where each parent has one of each eye color gene. Second Parent B b B BB Bb First Parent b Bb bb Find the probability that these parents give birth to a child who has blue eyes. 1 1 A) B) C) 1 D) 0 4 2 Answer: A Page 5
12) Three fair coins are tossed in the air and land on a table. The up side of each coin is noted. How many elements are there in the sample space? A) 8 B) 3 C) 6 D) 4 Answer: A 13) The sample space for tossing three fair coins is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. What is the probability of exactly two heads? 3 1 5 A) B) 3 C) D) 8 2 8 Answer: A 14) In the game of roulette in the United States a wheel has 38 slots: 18 slots are black, 18 slots are red, and 2 slots are green. You watched a friend play roulette for two hours. In that time you noted that the wheel was spun 22 50 times and that out of those 50 spins black came up 22 times. Based on this data, the P(black ) = = 0.44. 50 This is an example of what type of probability? A) Empirical B) Classical
C) Subjective
D) Observational
Answer: A 15) In the game of roulette in the United States a wheel has 38 slots: 18 slots are black, 18 slots are red, and 2 slots 18 ≈ 0.47. This is an example of what type of probability? are green. The P(Red) = 38 A) Classical
B) Empirical
C) Simulated
D) Subjective
Answer: A 5 Recognize and interpret subjective probabilities. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Classify the statement as an example of classical probability, empirical probability, or subjective probability. The probability that it will snow tomorrow is 62%. A) subjective probability B) classical probability C) empirical probability Answer: A 2) Classify the statement as an example of classical probability, empirical probability, or subjective probability. It is known that the probability of hitting a pothole while driving on a certain road is 1%. A) empirical probability B) classical probability C) subjective probability Answer: A 3) Classify the statement as an example of classical probability, empirical probability, or subjective probability. The probability that Uber fares will rise during the winter is 0.05. A) subjective probability B) classical probability C) empirical probability Answer: A 4) Classify the statement as an example of classical probability, empirical probability, or subjective probability. In 1 one state lottery, a person selects a 4-digit number. The probability of winning this state's lottery is . 10,000 A) classical probability Answer: A
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B) empirical probability
C) subjective probability
5) Classify the statement as an example of classical probability, empirical probability, or subjective probability. 1 The probability that a newborn kitten is a male is . 2 A) classical probability
B) empirical probability
C) subjective probability
Answer: A 6) The ______________ probability of an outcome is a probability based on personal judgment. A) Subjective B) Classical C) Empirical D) Conditional Answer: A 7) The ______________ probability of an outcome is obtained by dividing the frequency of occurrence of an event by the number of trials of the experiment. A) Empirical B) Subjective C) Classical D) Conditional Answer: A 8) The ______________ probability of an outcome is obtained by dividing the number of ways an event can occur by the number of possible outcomes. A) Classical B) Subjective C) Empirical D) Conditional Answer: A
5.2 The Addition Rule and Complements 1 Use the Addition Rule for Disjoint Events. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A probability experiment is conducted in which the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Let event A = {4, 5, 6, 7} and event B = {10, 11, 12}. Assume that each outcome is equally likely. List the outcomes in A and B. Are A and B mutually exclusive? A) { }; yes B) { }; no C) {4, 5, 6, 7, 10, 11, 12}; no D) {4, 5, 6, 7, 10, 11, 12}; yes Answer: A 2) The events A and B are mutually exclusive. If P(A) = 0.1 and P(B) = 0.2, what is P(A or B)? A) 0.3 B) 0 C) 0.02 D) 0.1 Answer: A 3) The table lists the drinking habits of a group of college students. If a student is chosen at random, find the probability of getting someone who is a regular or heavy drinker. Round your answer to three decimal places. Sex Non-drinker Regular Drinker Heavy Drinker Total Man 135 41 5 181 Woman 187 21 15 223 Total 322 62 20 404 A) 0.203 B) 0.683 C) 0.236 D) 0.139 Answer: A
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4) The table lists the drinking habits of a group of college students. If a student is chosen at random, find the probability of getting someone who is a man or a woman. Round your answer to three decimal places. Sex Non-drinker Regular Drinker Heavy Drinker Total Man 135 64 5 204 Woman 187 21 14 222 Total 322 85 19 426 A) 1 B) 0.918 C) 0.756 D) 0.244 Answer: A 5) The table lists the drinking habits of a group of college students. If a student is chosen at random, find the probability of getting someone who is a non-drinker. Round your answer to three decimal places. Sex Non-drinker Regular Drinker Heavy Drinker Total Man 135 68 5 208 Woman 187 21 6 214 Total 322 89 11 422 A) 0.763 B) 0.936 C) 1 D) 0.237 Answer: A 6) The distribution of Bachelor's degrees conferred by a university is listed in the table. Assume that a student majors in only one subject. What is the probability that a randomly selected student with a Bachelor's degree majored in Physics or Philosophy? Round your answer to three decimal places. Major Frequency Physics 229 Philosophy 204 Engineering 86 Business 176 Chemistry 222 A) 0.472 B) 0.528 C) 0.250 D) 0.222 Answer: A 7) The distribution of Bachelor's degrees conferred by a university is listed in the table. Assume that a student majors in only one subject. What is the probability that a randomly selected student with a Bachelor's degree majored in Business, Chemistry or Engineering? Round your answer to three decimal places. Major Frequency Physics 216 Philosophy 207 Engineering 88 Business 174 Chemistry 225 A) 0.535 B) 0.465 C) 0.288 D) 0.344 Answer: A 8) A card is drawn from a standard deck of 52 playing cards. Find the probability that the card is an ace or a king. 2 1 4 8 A) B) C) D) 13 13 13 13 Answer: A 9) If two events have no outcomes in common they are said to be A) Disjoint B) Independent C) Conditional Answer: A 10) True or False: Mutually exclusive events are not disjoint events. A) False B) True Answer: A Page 8
D) At odds
11) The table below shows the probabilities generated by rolling one die 50 times and recording the number rolled. Are the events A = {roll an odd number } and B = {roll a number less than or equal to two} disjoint? Roll 1 Probability 0.22
2 0.10
3 0.18
4 0.12
5 0.18
6 0.20
A) No
B) Yes
Answer: A 12) In the game of craps, two dice are tossed and the up faces are totaled. Is the event getting a total of 9 and one of the dice showing a 6 mutually exclusive? Answer Yes or No. A) No B) Yes Answer: A 13) Using a standard deck of 52 playing cards are the events of getting an ace and getting a jack on the card drawn mutually exclusive? Answer Yes or No. A) Yes B) No Answer: A 14) The below table shows the probabilities generated by rolling one die 50 times and noting the up face. What is the probability of getting an odd up face? Roll 1 Probability 0.22
2 0.10
A) 0.58
3 0.18
4 0.12
B) 0.42
5 0.18
6 0.20 C) 0.50
D) 0.55
Answer: A 15) In the game of craps two dice are rolled and the up faces are totaled. If the person rolling the dice on the first roll rolls a 7 or an 11 total they win. If they roll a 2, 3, or 12 on the first roll they lose. If they roll any other total then on subsequent rolls they must roll that total before rolling a 7 to win. What is the probability of winning on the first roll? A) 0.22 B) 0.17 C) 0.06 D) 0.50 Answer: A 2 Use the General Addition Rule. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A probability experiment is conducted in which the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Let event A = {5, 6, 7, 8} and event B = {7, 8, 9, 10, 11}. Assume that each outcome is equally likely. List the outcomes in A and B. Are A and B mutually exclusive? A) {7, 8}; no B) {7, 8}; yes C) {5, 6, 7, 8, 9, 10, 11}; no D) {5, 6, 7, 8, 9, 10, 11}; yes Answer: A
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2) A probability experiment is conducted in which the sample space of the experiment is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Let event A = {7, 8, 9, 10} and event B = {9, 10, 11, 12, 13}. Assume that each outcome is equally likely. List the outcomes in A or B. Find P(A or B). 7 2 A) {7, 8, 9, 10, 11, 12, 13}; B) {9, 10}; 15 15 C) {7, 8, 9, 10, 11, 11, 12, 13};
3 5
D) {7, 8, 9, 10, 12, 13};
2 5
Answer: A 3) The events A and B are mutually exclusive. If P(A) = 0.7 and P(B) = 0.1, what is P(A and B)? A) 0 B) 0.07 C) 0.5 D) 0.8 Answer: A 4) Given that P(A or B) = fraction. 17 A) 72
1 1 1 , P(A) = , and P(A and B) = , find P(B). Express the probability as a simplified 4 8 9
B)
7 96
C)
35 72
D)
19 72
Answer: A 5) The table lists the drinking habits of a group of college students. If a student is chosen at random, find the probability of getting someone who is a man or a non-drinker. Round your answer to three decimal places. Sex Non-drinker Regular Drinker Heavy Drinker Total Man 135 39 5 179 Woman 187 21 14 222 Total 322 60 19 401 A) 0.913 B) 0.946 C) 0.942 D) 0.850 Answer: A 6) The table lists the drinking habits of a group of college students. If a student is chosen at random, find the probability of getting someone who is a woman or a heavy drinker. Round your answer to three decimal places. Sex Non-drinker Regular Drinker Heavy Drinker Total Man 135 59 5 199 Woman 187 21 14 222 Total 322 80 19 421 A) 0.539 B) 0.917 C) 0.810 D) 0.173 Answer: A 7) A card is drawn from a standard deck of 52 playing cards. Find the probability that the card is a queen or a club. Express the probability as a simplified fraction. 4 7 2 3 A) B) C) D) 13 52 13 13 Answer: A 8) One hundred people were asked, "Do you favor stronger laws on gun control?" Of the 33 that answered "yes" to the question, 14 were male. Of the 67 that answered "no" to the question, six were male. If one person is selected at random, what is the probability that this person answered "yes" or was a male? Round the the nearest hundredth. A) 0.39 B) 0.53 C) 0.67 D) 0.13 Answer: A
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9) The below table shows the probabilities generated by rolling one die 50 times and noting the up face. What is the probability of getting an odd up face and a two or less? Round to the the nearest hundredth. Roll 1 Probability 0.22 A) 0.68
2 0.10
3 0.18
4 0.12
B) 0.90
5 0.18
6 0.20 C) 0.66
D) 0.32
Answer: A 10) You roll two dice and total the up faces. What is the probability of getting a total of 8 or two up faces that are the same? Round to the the nearest hundredth. A) 0.28 B) 0.31 C) 0.33 D) 0.50 Answer: A 11) Consider the data in the table shown which represents the marital status of males and females 18 years or older in the United States in 2003. Determine the probability that a randomly selected U.S. resident 18 years or older is divorced or a male? Round to the nearest hundredth. Males Females Total (in millions) (in millions) (in millions) Never married 28.6 23.3 51.9 Married 62.1 62.8 124.9 Widowed 2.7 11.3 14.0 Divorced 9.0 12.7 21.7 Total (in millions) 102.4 110.1 212.5 Source: U.S. Census Bureau, Current Population reports A) 0.54 B) 0.58 C) 0.50
D) 0.04
Answer: A 12) If one card is drawn from a standard 52 card playing deck, determine the probability of getting a ten, a king or a diamond. Round to the nearest hundredth. A) 0.37 B) 0.40 C) 0.31 D) 0.29 Answer: A 13) If one card is drawn from a standard 52 card playing deck, determine the probability of getting a jack, a three, a club or a diamond. Round to the nearest hundredth. A) 0.58 B) 0.65 C) 0.50 D) 0.15 Answer: A 14) Two dice are rolled. What is the probability of having both faces the same (doubles) or a total of 4 or 10? Round to the nearest hundredth. A) 0.28 B) 0.33 C) 0.06 D) 0.15 Answer: A
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3 Compute the probability of an event using the Complement Rule. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A probability experiment is conducted in which the sample space of the experiment is S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Let event A = {5, 6, 7, 8, 9}. Assume that each outcome is equally likely. List the outcomes in Ac. Find P(Ac). A) {2, 3, 4, 10, 11, 12}; C) {10, 11, 12};
3 11
6 11
B) {5, 6, 7, 8, 9};
5 11
D) {2, 3, 4, 9, 10, 11, 12};
7 11
Answer: A 2) You are dealt one card from a 52-card deck. Find the probability that you are not dealt a 10. Express the probability as a simplified fraction. 12 1 9 1 A) B) C) D) 13 13 10 10 Answer: A 3) You are dealt one card from a 52-card deck. Find the probability that you are not dealt a heart. Express the probability as a simplified fraction. 3 1 4 2 A) B) C) D) 4 4 13 5 Answer: A 4) In 5-card poker, played with a standard 52-card deck, 2,598,960 different hands are possible. If there are 624 different ways a "four-of-a-kind" can be dealt, find the probability of not being dealt a "four-of-a-kind". Express the probability as a fraction, but do not simplify. 2,598,336 624 625 1248 A) B) C) D) 2,598,960 2,598,960 2,598,960 2,598,960 Answer: A 5) A certain disease only affects men 20 years of age or older. The chart shows the probability that a man with the disease falls in the given age group. What is the probability that a randomly selected man with the disease is not between the ages of 55 and 64? Age Group Probability 20-24 0.004 25-34 0.006 35-44 0.14 45-54 0.29 55-64 0.32 65-74 0.17 75+ 0.07 A) 0.68 B) 0.32 C) 0.29 D) 0.24 Answer: A
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6) A certain disease only affects men 20 years of age or older. The chart shows the probability that a man with the disease falls in the given age group. What is the probability that a randomly selected man with the disease is between the ages of 35 and 64? Age Group Probability 20-24 0.004 25-34 0.006 35-44 0.14 45-54 0.29 55-64 0.32 65-74 0.17 75+ 0.07 A) 0.75 B) 0.14 C) 0.32 D) 0.29 Answer: A 7) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a company's success has been proven time and again to be customer service. A study was conducted to study 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. What is the probability that a respondent did not have a low level of satisfaction with the company? Round to the the nearest hundredth. Satisfaction level Frequency of Use High Medium Low TOTAL 2 per month 250 140 10 400 < 2 - 5 per month 140 55 5 200 70 25 5 100 > 5 per month TOTAL 460 220 20 700 A) 0.97 B) 0.03 C) 0.14 D) 0.86 Answer: A 8) A sample of 255 shoppers at a large suburban mall were asked two questions: (1) Did you see a television ad for the sale at department store X during the past 2 weeks? (2) Did you shop at department store X during the past 2 weeks? The responses to the questions are summarized in the table. What is the probability that a randomly selected shopper from the 255 questioned did not shop at department store X? Round to the the nearest thousandth. Shopped at X Did Not Shop at X Saw ad 135 20 Did not see ad 20 80 A) 0.392 B) 0.078 C) 0.314 D) 0.608 Answer: A 9) The breakdown of workers in a particular state according to their political affiliation and type of job held is shown here. Suppose a worker is selected at random within the state and the worker's political affiliation and type of job are noted. Find the probability the worker is not an Independent. Round the the nearest hundredth. Political Affiliation Republican Democrat Independent White collar 7% 20% 14% Type of job Blue Collar 6% 18% 35% A) 0.51 B) 0.49 C) 0.27 D) 0.24 Answer: A
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10) 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 62% regularly use the golf course, 47% regularly use the tennis courts, and 3% use neither of these facilities regularly. What percentage of the 600 use at least one of the golf or tennis facilities? A) 97% B) 3% C) 106% D) 12% Answer: A 11) Fill in the blank. The A) complement
of an event A is the event that A does not occur. B) intersection
C) union
D) Venn diagram
Answer: A 12) The following Venn diagram is for the six sample points possible when rolling a fair die. Let A be the event rolling an even number and let B be the event rolling a number greater than 1. Which of the following events describes the event rolling a 1?
A) Bc
B) Ac
C) B
D) A ∪ B
Answer: A 13) True or False: P(E) + P(Ec) > 1 A) False
B) True
Answer: A 14) The complement of 4 heads in the toss of 4 coins is A) At least one tail B) All tails
C) Exactly one tail
D) Three heads
Answer: A 15) A game has three outcomes. The probability of a win is 0.4, the probability of tie is 0.5, and the probability of a loss is 0.1. What is the probability of not winning in a single play of the game. A) 0.6 B) 0.5 C) 0.1 D) 0.33 Answer: A
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5.3 Independence and the Multiplication Rule 1 Identify independent events. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) There are 30 chocolates in a box, all identically shaped. There are 11 filled with nuts, 10 filled with caramel, and 9 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Is this an example of independence? A) No B) Yes Answer: A 2) Numbered disks are placed in a box and one disk is selected at random. There are 6 red disks numbered 1 through 6, and 7 yellow disks numbered 7 through 13. In an experiment a disk is selected, the number and color noted, replaced, and then a second disk is selected. Is this an example of independence? A) Yes B) No Answer: A 3) Two events are __________________ if the occurrence if the occurrence of event E in a probability experiment does not affect the probability of event F in the same experiment. A) independent B) mutually exclusive C) dependent D) disjoint Answer: A 4) Two events are ________________ if the occurrence of event E in a probability experiment changes the probability of event F in the same experiment. A) dependent B) mutually exclusive C) independent D) disjoint Answer: A 5) True or False: Mutually exclusive events are always independent. A) False B) True Answer: A 2 Use the Multiplication Rule for independent events. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Suppose that events E and F are independent, P(E) = 0.3 and P(F ) = 0.8. What is the P(E and F )? A) 0.24 B) 1.1 C) 0.024 D) 0.86 Answer: A 2) A single die is rolled twice. Find the probability of getting a 6 the first time and a 3 the second time. Express the probability as a simplified fraction. 1 1 1 1 A) B) C) D) 36 6 12 3 Answer: A
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3) You are dealt one card from a 52 card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of getting a picture card the first time and a heart the second time. Express the probability as a simplified fraction. 3 1 3 1 A) B) C) D) 52 13 13 4 Answer: A 4) If you toss a fair coin 11 times, what is the probability of getting all heads? Express the probability as a simplified fraction. 1 1 1 1 A) B) C) D) 2048 1024 4096 2 Answer: A 5) A human gene carries a certain disease from the mother to the child with a probability rate of 44%. That is, there is a 44% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has five children. Assume that the infections of the five children are independent of one another. Find the probability that all five of the children get the disease from their mother. Round to the nearest thousandth. A) 0.016 B) 0.984 C) 0.055 D) 0.043 Answer: A 6) 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) = 0.97, P(C) = 0.99, and P(D) = 0.93. Find the probability that the machine works properly. Round to the nearest ten-thousandth. A) 0.866,3 B) 0.893,1 C) 0.931,5 D) 0.133,7 Answer: A 7) Suppose a basketball player is an excellent free throw shooter and makes 92% of his free throws (i.e., he has a 92% chance of making a single free throw). Assume that free throw shots are independent of one another. Suppose this player gets to shoot three free throws. Find the probability that he misses all three consecutive free throws. Round to the nearest ten-thousandth. A) 0.000,5 B) 0.221,3 C) 0.778,7 D) 0.999,5 Answer: A 8) What is the probability that in three consecutive rolls of two fair dice, a person gets a total of 7, followed by a total of 11, followed by a total of 7? Round to the nearest ten-thousandth. A) 0.0015 B) 0.1667 C) 0.2876 D) 0.0012 Answer: A 9) A bag contains 10 white, 12 blue, 13 red, 7 yellow, and 8 green wooded balls. A ball is selected from the bag, its color noted, then replaced. You then draw a second ball, note its color and then replace the ball. What is the probability of selecting 2 red balls? Round to the nearest ten-thousandth. A) 0.0676 B) 0.5200 C) 0.2600 D) 0.0624 Answer: A 10) A bag contains 10 white, 12 blue, 13 red, 7 yellow, and 8 green wooded balls. A ball is selected from the bag, its color noted, then replaced. You then draw a second ball, note its color and then replace the ball. What is the probability of selecting one white ball and one blue ball? Round to the nearest ten-thousandth. A) 0.0480 B) 0.4400 C) 0.2200 D) 0.0088 Answer: A
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3 Compute at-least probabilities. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A human gene carries a certain disease from the mother to the child with a probability rate of 42%. That is, there is a 42% chance that the child becomes infected with the disease. Suppose a female carrier of the gene has three children. Assume that the infections of the three children are independent of one another. Find the probability that at least one of the children get the disease from their mother. Round to the nearest thousandth. A) 0.805 B) 0.141 C) 0.424 D) 0.195 Answer: A 2) 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) = 0.96, P(C) = 0.9, and P(D) = 0.97. Find the probability that at least one of the four parts will work. Round to six decimal places. A) 0.999,995 B) 0.804,557 C) 0.000,005 D) 0.195,443 Answer: A 3) Investing is a game of chance. Suppose there is a 38% 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 four independent risky stocks. Find the probability that at least one of your four investments becomes a total loss. Round to the nearest ten-thousandth when necessary. A) 0.852,2 B) 0.362,4 C) 0.090,6 D) 0.020,9 Answer: A 4) Find the probability that of 25 randomly selected students, at least two share the same birthday. Round to the nearest thousandth. A) 0.569 B) 0.068 C) 0.432 D) 0.995 Answer: A 5) Two companies, A and B, package and market a chemical substance and claim 0.15 of the total weight of the substance is sodium. However, a careful survey of 4,000 packages (half from each company) indicates that the proportion varies around 0.15, with the results shown below. Find the percentage of all chemical B packages that contain a sodium total weight proportion above 0.150. Proportion of Sodium 0.100 - 0.149 0.150 - 0.199 < 0.100 > 0.200 A 25% 10% 10% 5% Chemcal Brand B 5% 5% 10% 30% A) 80% B) 40% C) 50% D) 55% Answer: A 6) Find the probability that of 25 randomly selected students, no two share the same birthday. A) 0.431 B) 0.995 C) 0.569 D) 0.068 Answer: A 7) The probability that a region prone to hurricanes will be hit by a hurricane in any single year is probability of a hurricane at least once in the next 5 years? A) 0.672,32 B) 1 C) 0.999,68 Answer: A
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1 . What is the 5
D) 0.000,32
8) Investment in new issues (the stock of newly formed companies) can be both suicidal and rewarding. Suppose that of 300 newly formed companies in 2010, only 17 appeared to have outstanding prospects. Suppose that you had selected two of these 300 companies back in 2010. Find the probability that at least one of your companies had outstanding prospects. Round to seven decimal places. A) 0.110,301 B) 0.334,693,4 C) 0.889,699 D) 0.053,634,3 Answer: A 9) You toss a fair coin 5 times. What is the probability of at least one head? Round to the nearest ten-thousandth. A) 0.9688 B) 0.7500 C) 0.5000 D) 0.0313 Answer: A 10) You are playing roulette at a casino in the United States. The wheel has 18 red slots, 18 black slots, and two green slots. In 4 spins of the wheel what is the probability of at least one red? Round to the nearest ten-thousandth. A) 0.9048 B) 0.9375 C) 0.0625 D) 0.0953 Answer: A
5.4 Conditional Probability and the General Multiplication Rule 1 Compute conditional probabilities. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. If necessary, round to three decimal places. 1) Suppose that E and F are two events and that P(E and F) = 0.21 and P(E) = 0.8. What is P(F E)? A) 0.262,5 B) 3.81 C) 0.168 D) 1.01 Answer: A 2) Suppose that E and F are two events and that N(E and F) = 330 and N(E) = 790. What is P(F E)? A) 0.418 B) 2.394 C) 0.295 D) 0.042 Answer: A 3) Suppose that E and F are two events and that P(E) = 0.8 and P(F E) = 0.9. What is P(E and F)? A) 0.72 B) 1.7 C) 0.889 D) 0.072 Answer: A Find the indicated probability. Give your answer as a simplified fraction. 4) The overnight shipping business has skyrocketed in the last ten years. The single greatest predictor of a company's success has been proven time and again to be customer service. A study was conducted to study 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. A customer is chosen at random. Given that the customer uses the company less than two times per month, what is the probability that they expressed medium satisfaction with the company? Satisfaction level Frequency of Use High Medium Low TOTAL 250 140 10 400 < 2 per month 2 - 5 per month 140 55 5 200 70 25 5 100 > 5 per month TOTAL 460 220 20 700 7 7 1 24 A) B) C) D) 20 11 5 35 Answer: A
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5) The managers of a corporation were surveyed to determine the background that leads to a successful manager. Each manager 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. Given that a manager is only a fair manager, what is the probability that this manager has no college background? Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals Good 3 4 26 6 39 Fair 6 13 48 20 87 Poor 2 5 9 18 34 Totals 11 22 83 44 160 2 6 3 23 A) B) C) D) 29 11 80 40 Answer: A 6) The managers of a corporation were surveyed to determine the background that leads to a successful manager. Each manager 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. Given that a manager is only a fair manager, what is the probability that this manager's highest level of education is a college degree? Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals Good 1 5 25 8 39 Fair 7 17 46 17 87 Poor 6 3 4 21 34 Totals 14 25 75 46 160 46 23 15 25 A) B) C) D) 87 80 32 29 Answer: A 7) The managers of a corporation were surveyed to determine the background that leads to a successful manager. Each manager 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. Given that a manager is a good manager, what is the probability that this manager has some college background? Educational Background Manager Rating H. S. Degree Some College College Degree Master's or Ph.D. Totals Good 4 5 25 5 39 Fair 6 15 44 22 87 Poor 2 8 9 15 34 Totals 12 28 78 42 160 5 25 1 5 A) B) C) D) 39 39 32 28 Answer: A
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8) A study was recently done that emphasized the problem we all face with drinking and driving. Four hundred accidents that occurred on a Saturday night were analyzed. Two items noted were the number of vehicles involved and whether alcohol played a role in the accident. The numbers are shown below. Given that an accident involved multiple vehicles, what is the probability that it involved alcohol? Number of Vehicles Involved Did Alcohol Play a Role? 1 2 3 or more Totals Yes 57 92 21 170 No 21 172 37 230 Totals 78 264 58 400 113 113 21 21 A) B) C) D) 322 400 58 400 Answer: A 9) A researcher at a large university wanted to investigate if a student's seat preference was related in any way to the gender of the student. The researcher divided the 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 his students sat on a particular day of the class. The researcher's summary table is provided below. Suppose a person sitting in the front, middle portion Area (1) of the class is randomly selected to answer a question. Find the probability the person selected is a female. Area (1) Area (2) Area (3) Total Males 20 10 3 33 Females 12 14 13 39 Total 32 24 16 72 3 4 32 1 A) B) C) D) 8 13 39 6 Answer: A 10) 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 his car and its make (foreign or domestic). 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? Age of Car (in years) Make 0-2 3-5 6 - 10 over 10 Total Foreign 38 28 12 22 100 Domestic 43 26 14 17 100 Total 81 54 26 39 200 31 19 62 38 A) B) C) D) 50 50 119 119 Answer: A 11) 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 his car and its make (foreign or domestic). A car was randomly selected from the lot. Given that the car selected was a domestic car, what is the probability that it was older than 2 years? Age of Car (in years) Make 0-2 3-5 6 - 10 over 10 Total Foreign 44 23 10 23 100 Domestic 39 29 11 21 100 Total 83 52 21 44 200 61 39 39 83 A) B) C) D) 100 83 200 200 Answer: A
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12) 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 his car and its make (foreign or domestic). Age of Car (in years) Make 0-2 3-5 6 - 10 over 10 Total Foreign 40 21 12 27 100 Domestic 41 26 11 22 100 Total 81 47 23 49 200 A car was randomly selected from the lot. Given that the car selected is older than two years old, find the probability that it is not a foreign car. 59 60 59 3 A) B) C) D) 119 119 100 5 Answer: A Find the indicated probability. Give your answer as a decimal rounded to the nearest thousandth. 13) A fast-food restaurant chain with 700 outlets in the United States describes the geographic location of its restaurants with the accompanying table of percentages. A restaurant is to be chosen at random from the 700 to test market a new style of chicken. Given that the restaurant is located in the eastern United States, what is the probability it is located in a city with a population of at least 10,000? Region NE SE SW NW 1% 6% 3% 0% <10,000 Population of City 10,000 - 100,000 15% 4% 12% 5% 20% 4% 5% 25% >100,000 A) 0.86 B) 0.478 C) 0.43 D) 0.14 Answer: A 14) The breakdown of workers in a particular state according to their political affiliation and type of job held is shown here. Suppose a worker is selected at random within the state and the worker's political affiliation and type of job are noted. Given the worker is a Democrat, what is the probability that the worker is in a white collar job. Political Affiliation Republican Democrat Independent White collar 18% 19% 20% Type of job Blue Collar 12% 13% 18% A) 0.594 B) 0.333 C) 0.271 D) 0.457 Answer: A 15) 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 67% regularly use the golf course, 43% regularly use the tennis courts, and 5% use neither 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) 0.349 B) 0.224 C) 0.136 D) 0.158 Answer: A Provide an appropriate response. 16) The conditional probability of event G, given the knowledge that event H has occurred, would be written as . A) P(G|H) Answer: A
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B) P(G)
C) P(H|G)
D) P(H)
17) Computing the probability of the event "drawing a second red ball from a bag of colored balls after having kept the red ball that was drawn first from the bag" is an example of A) conditional probability. B) independence of events. C) mutual exclusiveness. D) disjoint events. Answer: A 18) True or False: Conditional probabilities leave the sample space the same when considering sequential events. A) False B) True Answer: A 19) Numbered disks are placed in a box and one disk is selected at random. If there are 6 red disks numbered 1 through 6, and 4 yellow disks numbered 7 through 10, find the probability of selecting a yellow disk, given that the number selected is less than or equal to 3 or greater than or equal to 8. 1 3 3 3 A) B) C) D) 2 4 5 10 Answer: A 2 Compute probabilities using the General Multiplication Rule. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. Express your answer as a simplified fraction unless otherwise noted. 1) There are 35 chocolates in a box, all identically shaped. There 8 are filled with nuts, 11 with caramel, and 16 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting 2 solid chocolates in a row. 24 256 8 48 A) B) C) D) 119 1225 595 245 Answer: A 2) There are 36 chocolates in a box, all identically shaped. There 10 are filled with nuts, 12 with caramel, and 14 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting 2 nut candies. 1 25 1 5 A) B) C) D) 14 324 90 72 Answer: A 3) There are 39 chocolates in a box, all identically shaped. There 16 are filled with nuts, 10 with caramel, and 13 are solid chocolate. You randomly select one piece, eat it, and then select a second piece. Find the probability of selecting a solid chocolate candy followed by a nut candy. 8 5 1 16 A) B) C) D) 57 39 114 117 Answer: A 4) Consider a political discussion group consisting of 6 Democrats, 3 Republicans, and 7 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting an Independent and then a Democrat. 7 21 1 7 A) B) C) D) 40 128 40 240 Answer: A
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5) Consider a political discussion group consisting of 7 Democrats, 6 Republicans, and 4 Independents. Suppose that two group members are randomly selected, in succession, to attend a political convention. Find the probability of selecting an Independent and then a Republican. 3 24 7 1 A) B) C) D) 34 289 272 68 Answer: A 6) An ice chest contains 7 cans of apple juice, 6 cans of grape juice, 5 cans of orange juice, and 2 cans of pineapple juice. Suppose that you reach into the container and randomly select three cans in succession. Find the probability of selecting no grape juice. 91 343 273 1 A) B) C) D) 285 855 1000 57 Answer: A 7) Numbered disks are placed in a box and one disk is selected at random. If there are 6 red disks numbered 1 through 6, and 5 yellow disks numbered 7 through 11, find the probability of selecting a disk numbered 3, given that a red disk is selected. 1 6 1 5 A) B) C) D) 6 11 11 11 Answer: A 8) Numbered disks are placed in a box and one disk is selected at random. If there are 6 red disks numbered 1 through 6, and 8 yellow disks numbered 7 through 14, find the probability of selecting a red disk, given that an odd-numbered disk is selected. 3 4 3 2 A) B) C) D) 7 7 14 7 Answer: A 9) A group of students were asked if they carry a n ATM card The responses are listed in the table. If a student is selected at random, find the probability that he or she owns an ATM card given that the student is a freshman. Round your answer to three decimal places. ATM Card Not an ATM Card Class Carrier Carrier Total Freshman 21 39 60 Sophomore 27 13 40 Total 48 52 100 A) 0.350 B) 0.650 C) 0.438 D) 0.210 Answer: A 10) Four employees drive to work in the same car. The workers claim they were late to work because of a flat tire. Their managers ask the workers to identify the tire that went flat; front driver's side, front passenger's side, rear driver's side, or rear passenger's side. If the workers didn't really have a flat tire and each randomly selects a tire, what is the probability that all four workers select the same tire? 1 1 1 1 A) B) C) D) 64 4 256 8 Answer: A 11) Find the probability that of 25 randomly selected housewives, no two share the same birthday. Round your answer to the nearest thousandth. A) 0.431 B) 0.995 C) 0.569 D) 0.068 Answer: A
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12) A fast-food restaurant chain with 700 outlets in the United States describes the geographic location of its restaurants with the accompanying table of percentages. A restaurant is to be chosen at random from the 700 to test market a new style of chicken. Given that the restaurant is located in the eastern United States, what is the probability it is located in a city with a population of at least 10,000? Round your answer to the nearest thousandth. Region NE SE SW NW 4% 6% 3% 0% <10,000 Population of City 10,000 - 100,000 15% 3% 12% 5% 20% 4% 5% 23% >100,000 A) 0.808 B) 0.483 C) 0.42 D) 0.192 Answer: A 13) A bag contains 10 white, 12 blue, 13 red, 7 yellow, and 8 green wooded balls. A ball is selected from the bag and kept. You then draw a second ball and keep it also. What is the probability of selecting one white ball and one blue ball? Round your answer to four decimal places. A) 0.0490 B) 0.0480 C) 0.0090 D) 0.0088 Answer: A 14) A bag contains 10 white, 12 blue, 13 red, 7 yellow, and 8 green wooded balls. A ball is selected from the bag and kept. You then draw a second ball and keep it also. What is the probability of selecting two blue balls? Round your answer to four decimal places. A) 0.0539 B) 0.0588 C) 0.0528 D) 0.0576 Answer: A 15) Five cards are randomly selected without replacement from a standard deck of 52 playing cards. What is the probability of getting 5 hearts? Round your answer to four decimal places. A) 0.0005 B) 0.0012 C) 0.0010 D) 0.0004 Answer: A 16) CampusFest is a student festival where local businesses come on campus to sell their goods to students at vastly reduced prices. As part of a give-away promotion, a local cellular phone company gave away 400 cellular phones to students who signed up for their calling service. Unbeknownst to the company is that 70 of these cellular phones were faulty and will cause a small explosion when dialed outside the local calling area. Suppose you and your roommate each received one of the giveaway phones. Find the probability that both of you received faulty phones. Round to five decimal places when necessary. A) 0.030,26 B) 0.030,63 C) 0.35 D) 0.144,74 Answer: A 17) A county welfare agency employs 32 welfare workers who interview prospective food stamp recipients. Periodically, the supervisor selects, at random, the forms completed by two workers to audit for illegal deductions. Unknown to the supervisor, five of the workers have regularly been giving illegal deductions to applicants. Given that the first worker chosen has not been giving illegal deductions, what is the probability that the second worker chosen has been giving illegal deductions? Round to the nearest thousandth. A) 0.161 B) 0.129 C) 0.125 D) 0.156 Answer: A 18) A county welfare agency employs 36 welfare workers who interview prospective food stamp recipients. Periodically, the supervisor selects, at random, the forms completed by two workers to audit for illegal deductions. Unknown to the supervisor, nine of the workers have regularly been giving illegal deductions to applicants. What is the probability both workers chosen have been giving illegal deductions? Round to the nearest thousandth. A) 0.057 B) 0.063 C) 0.064 D) 0.056 Answer: A
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19) True or False: If A and B are independent events, then A and B are mutually exclusive also. A) False B) True Answer: A 20) True or False: Two events, A and B, are independent if P(A and B) = P(A) ∙ P(B). A) True B) False Answer: A 21) Assume that P(A) = 0.7 and P(B) = 0.2. If A and B are independent, find P(A and B). A) 0.14 B) 0.76 C) 0.90
D) 1.00
Answer: A 22) If P(A) = 0.45, P(B) = 0.25, and P(B|A) = 0.45, are A and B independent? A) no B) yes
C) cannot determine
Answer: A 23) If P(A) = 0.72, P(B) = 0.11, and A and B are independent, find P(A|B). A) 0.72 B) 0.0792 C) 0.83
D) 0.11
Answer: A 24) Assume that P(E) = 0.15 and P(F) = 0.48. If E and F are independent, find P(E and F). A) 0.072 B) 0.15 C) 0.558
D) 0.630
Answer: A 25) If two events A and B are __________, then P(A and B) = P(A) P(B). A) independent B) simple events C) mutually exclusive D) complements Answer: A 26) True or False: For two events A and B, suppose P(A) = 0.35, P(B) = 0.65, and P(B|A) = 0.35. Then A and B are independent. A) False B) True Answer: A 27) True or False: For two events A and B, suppose P(A) = 0.1, P(B) = 0.8, and P(A|B) = 0.1. Then A and B are independent. A) True B) False Answer: A 28) Given that events A and B are mutually exclusive and P(A) = 0.5 and P(B) = 0.7, are A and B independent? A) no B) yes C) cannot be determined Answer: A 29) Given that events C and D are independent, P(C) = 0.3, and P(D) = 0.6, are C and D mutually exclusive? A) no B) yes C) cannot be determined Answer: A 30) Given events A and B with probabilities P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, are A and B independent? A) yes B) no C) cannot be determined Answer: A
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31) Given events C and D with probabilities P(C) = 0.3, P(D) = 0.2, and P(C and D) = 0.1, are C and D independent? A) no B) yes C) cannot be determined Answer: A 32) Given events A and B with probabilities P(A) = 0.75 and P(B) = 0.15, are A and B mutually exclusive? A) cannot be determined B) no C) yes Answer: A
5.5 Counting Techniques 1 Solve counting problems using the Multiplication Rule. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the factorial expression. 8! 1) 6! A) 56
8 6
B) 2!
C)
B) 89,700
C) 1
D) 8
Answer: A 2)
300! 299! A) 300
D) 299
Answer: A Provide an appropriate response. 3) A person can order a new car with a choice of 6 possible colors, with or without air conditioning, with or without heated seats, with or without anti-lock brakes, with or without power windows, and with or without a CD player. In how many different ways can a new car be ordered in terms of these options? A) 192 B) 96 C) 384 D) 12 Answer: A 4) You are taking a multiple-choice test that has 11 questions. Each of the questions has 4 choices, with one correct choice per question. If you select one of these options per question and leave nothing blank, in how many ways can you answer the questions? A) 4,194,304 B) 44 C) 14,641 D) 15 Answer: A 5) License plates in a particular state display 2 letters followed by 3 numbers. How many different license plates can be manufactured? (Repetitions are allowed.) A) 676,000 B) 6 C) 36 D) 260 Answer: A 6) How many different four-letter secret codes can be formed if the first letter must be an S or a T? A) 35,152 B) 456,976 C) 72 D) 421,824 Answer: A 7) There are 9 performers who are to present their acts at a variety show. How many different ways are there to schedule their appearances? A) 362,880 B) 9 C) 81 D) 72 Answer: A
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8) There are 7 performers who are to present their acts at a variety show. One of them insists on being the first act of the evening. If this request is granted, how many different ways are there to schedule the appearances? A) 720 B) 5,040 C) 49 D) 42 Answer: A 9) You want to arrange 8 of your favorite CD's along a shelf. How many different ways can you arrange the CD's assuming that the order of the CD's makes a difference to you? A) 40,320 B) 5,040 C) 64 D) 56 Answer: A 10) True or False: 0! = 1! A) True
B) False
Answer: A 11) How many different breakfasts can you have at the local diner if you can select 3 different egg dishes (scrambled, fried, poached), 4 choices of meat (steak, ham, bacon, sausage), 5 breads (white, wheat, rye, muffin, bagel), 6 juices (tomato, grape, apple, orange, mixed berry, vegetable cocktail), and 4 beverages (water, milk, coffee, tea)? A) 1440 B) 5 C) 22 D) 360 Answer: A 12) A medical salesperson is to visit the various members of the staff at a clinic. He must see 8 doctors, 6 physicians assistants, 12 nurses, 3 medical technologists, and 3 receptionists. How many different ways can these people be visited by the salesperson if the order is not important? A) 5184 B) 32 C) 25,920 D) 6,220,800 Answer: A 2 Solve counting problems using permutations. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the value of the permutation. 1) 10P4 A) 5,040
B) 151,200
C) 907,200
D) 3,628,800
B) 70
C) 0
D) 5,040
B) 1
C) 3
D) 2
Answer: A 2) 7P0 A) 1 Answer: A 3) 3P3 A) 6 Answer: A Provide an appropriate response. 4) A church has 8 bells in its bell tower. Before each church service 5 bells are rung in sequence. No bell is rung more than once. How many sequences are there? A) 6,720 B) 336 C) 56 D) 672 Answer: A
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5) A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are possible if there are 14 members and any member can be elected to each position? No person can hold more than one office. A) 2,184 B) 1,092 C) 728 D) 24,024 Answer: A 6) In a contest in which 7 contestants are entered, in how many ways can the 5 distinct prizes be awarded? A) 2,520 B) 42 C) 21 D) 84 Answer: A 7) How many arrangements can be made using 4 letters of the word HYPERBOLAS if no letter is to be used more than once? A) 5,040 B) 151,200 C) 210 D) 302,400 Answer: A 8) The Environmental Protection Agency must inspect nine factories for complaints of water pollution. In how many different ways can a representative visit five of these to investigate this week? A) 15,120 B) 362,880 C) 5 D) 45 Answer: A 9) How many ways can five people, A, B, C, D, and E, sit in a row at a concert hall if A and B must sit together? A) 48 B) 120 C) 12 D) 24 Answer: A 10) How many ways can five people, A, B, C, D, and E, sit in a row at a concert hall if C must sit to the right of but not necessarily next to B? A) 60 B) 24 C) 20 D) 48 Answer: A 11) How many ways can five people, A, B, C, D, and E, sit in a row at a concert hall if D and E will not sit next to each other? A) 72
B) 24
C) 48
D) 60
Answer: A 12) In how many different ways can a ski club consisting of 20 people select a person for its officers? The positions available are president, vice president, treasurer, and secretary. No person can hold more than one positions and the each office is filled in order. A) 116,280 B) 4845 C) 1440 D) 74 Answer: A 3 Solve counting problems using combinations. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the value of the combination. 1) 7C6 A) 7
B) 6
C) 42
D) 5,040
B) 90
C) 907,200
D) 4
Answer: A 2) 10C8 A) 45 Answer: A
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3) 8C0 A) 1
B) 8
C) 10,080
D) 20,160
B) 3,628,800
C) 5
D) 725,760
B) 120
C) 30
D) 0.5
B) 2
C) 4
D) 40,320
Answer: A 4) 10C1 A) 10 Answer: A 5) 5C5 A) 1 Answer: A 6)
10C3 6 C4 A) 8 Answer: A
Provide an appropriate response. 7) From 8 names on a ballot, a committee of 3 will be elected to attend a political national convention. How many different committees are possible? A) 56 B) 336 C) 6,720 D) 168 Answer: A 8) To win at LOTTO in a certain state, one must correctly select 6 numbers from a collection of 50 numbers (one through 50.) The order in which the selections is made does not matter. How many different selections are possible? A) 15,890,700 B) 13,983,816 C) 300 D) 720 Answer: A 9) In how many ways can a committee of three men and four women be formed from a group of 11 men and 11 women? A) 54,450 B) 7,840,800 C) 554,400 D) 110 Answer: A 10) A physics exam consists of 9 multiple-choice questions and 6 open-ended problems in which all work must be shown. If an examinee must answer 5 of the multiple-choice questions and 3 of the open-ended problems, in how many ways can the questions and problems be chosen? A) 2520 B) 810 C) 261,273,600 D) 1,814,400 Answer: A 11) A professor wants to arrange his books on a shelf. He has 30 books and only space on the shelf for 20 of them. How many different 20-book arrangements can he make using the 30 books? This is an example of a problem that can be solved using which method? A) Permutations B) Combinations C) Conditional probability D) Randomness Answer: A
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12) Professor Al Whet teaches French and has a class of 24 students. Part of his grading system includes an observation of groups of 3 students engaged in a conversation in French. This is an example of a problem that can be solved using which method? A) Combinations B) Permutations C) Conditional probability D) Randomness Answer: A 4 Solve counting problems involving permutations with nondistinct items. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) How many distinct arrangements can be formed from all the letters of "students"? A) 10,080 B) 1680 C) 720
D) 40,320
Answer: A 2) How many distinct arrangements can be formed from all the letters of "statistics"? A) 50,400 B) 201,600 C) 259,200 D) 72
E) 3,628,800
Answer: A 3) Given the sets of digits {1, 3, 4, 6, 8}, how many different numbers between 40,000 and 80,000 can be written using these digits if repetition of digits is allowed? A) 1250 B) 3125 C) 120 D) 1875 E) 22 Answer: A 4) How many distinct arrangements of the letters in the word Mississippi are possible? A) 34,650 B) 39,916,800 C) 1152
D) 7920
Answer: A 5) How many distinct arrangements of the letters in the word football are possible? A) 10,080 B) 1680 C) 720
D) 40,320
Answer: A 6) A man has 12 coins that consist of 3 pennies, 4 nickels, and 5 quarters. How many distinct arrangements of the coins can he make if he lays them in a row one at a time? A) 27,720 B) 479,001,600 C) 9240 D) 83,160 Answer: A 5 Compute probabilities involving permutations and combinations. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Amy, Jean, Keith, Tom, Susan, and Dave have all been invited to a birthday party. They arrive randomly and each person arrives at a different time. In how many ways can they arrive? In how many ways can Jean arrive first and Keith last? Find the probability that Jean will arrive first and Keith will arrive last. 1 1 1 1 A) 720; 24; B) 720; 15; C) 120; 6; D) 120; 10; 30 48 20 12 Answer: A
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2) Six students, A, B, C, D, E, F, are to give speeches to the class. The order of speaking is determined by random selection. Find the probability that (a) E will speak first (b) that C will speak fifth and B will speak last (c) that the students will speak in the following order: DECABF (d) that A or B will speak first. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A) ; ; ; B) ; ; ; C) ; ; ; D) ; ; ; 6 24 720 3 6 12 720 3 6 36 720 12 6 36 360 3 Answer: A 3) A group consists of 6 men and 5 women. Four people are selected to attend a conference. In how many ways can 4 people be selected from this group of 11? In how many ways can 4 men be selected from the 6 men? Find the probability that the selected group will consist of all men. 1 1 1 1 A) 330; 15; B) 7,920; 360; C) 330; 15; D) 330; 15; 22 22 15,840 1,814,400 Answer: A 4) To play the lottery in a certain state, a person has to correctly select 5 out of 45 numbers, paying $1 for each five-number selection. If the five numbers picked are the same as the ones drawn by the lottery, an enormous sum of money is bestowed. What is the probability that a person with one combination of five numbers will win? What is the probability of winning if 100 different lottery tickets are purchased? 1 100 1 10 A) ; B) ; 1,221,759 1,221,759 146,611,080 14,661,108 C)
1 1 ; 8,145,060 814,506
D)
1 1 ; 5,864,443,200 58,644,432
Answer: A 5) A box contains 22 widgets, 4 of which are defective. If 4 are sold at random, find the probability that (a) all are defective (b) none are defective. 1 612 2 9 1 1 1 2 A) ; B) ; C) ; D) ; 7,315 1,463 11 11 175,560 43,890 22 11 Answer: A 6) A committee consisting of 6 people is to be selected from eight parents and four teachers. Find the probability of selecting three parents and three teachers. 8 2 100 10 A) B) C) D) 33 33 231 11 Answer: A 7) If you are dealt 5 cards from a shuffled deck of 52 cards, find the probability that all 5 cards are picture cards. 33 3 1 1 A) B) C) D) 108,290 13 2,598,960 216,580 Answer: A 8) If you are dealt 5 cards from a shuffled deck of 52 cards, find the probability that none of the 5 cards are picture cards. 108,257 3 33 1 A) B) C) D) 108,290 13 108,290 216,580 Answer: A 9) If you are dealt 6 cards from a shuffled deck of 52 cards, find the probability of getting 3 jacks and 3 aces. 2 2 1 3 A) B) C) D) 2,544,815 13 1,017,926 26 Answer: A
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10) A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are possible if there are 12 members and any member can be elected to each position? No person can hold more than one office. A) 1,320 B) 660 C) 440 D) 11,880 Answer: A 11) From 10 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible? A) 210 B) 5,040 C) 151,200 D) 2,520 Answer: A 12) To win at LOTTO in a certain state, one must correctly select 6 numbers from a collection of 52 numbers (one through 52.) The order in which the selections is made does not matter. How many different selections are possible? A) 20,358,520 B) 18,009,460 C) 312 D) 720 Answer: A 13) In how many ways can a committee of three men and four women be formed from a group of 12 men and 12 women? A) 108900 B) 15,681,600 C) 6652800 D) 165 Answer: A 14) A physics exam consists of 9 multiple-choice questions and 6 open-ended problems in which all work must be shown. If an examinee must answer 7 of the multiple-choice questions and 2 of the open-ended problems, in how many ways can the questions and problems be chosen? A) 540 B) 756 C) 261,273,600 D) 5,443,200 Answer: A
5.6 Simulating Probability Experiments 1 Know Concepts: Probability Rules. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) (a) Roll a pair of dice 40 times, recording the sum each time. Use your results to approximate the probability of getting a sum of 8. (b) Roll a pair of dice 100 times, recording the sum each time. Use your results to approximate the probability of getting a sum of 8. Compare the results of (a) and (b) to the probability that would be obtained using the classical method. Which answer was closer to the probability that would be obtained using the classical method? Is this what you would expect? Answer: Answers to parts (a) and (b) will vary. Using the classical method, the probability of getting a sum of 8 is 5 . 36 In general, as the number of repetitions of a probability experiment increases, the closer the empirical probability should get to the classical probability.
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2) (a) Simulate the experiment of sampling 100 four-child families to estimate the probability that a four-child family has three girls. Assume that the outcomes "have a girl" and "have a boy" are equally likely. (b) Simulate the experiment of sampling 1000 four-child families to estimate the probability that a four-child family has three girls. Assume that the outcomes "have a girl" and "have a boy" are equally likely. 1 The classical probability that a four-child family has three girls is . 4 Compare the results of (a) and (b) to the probability that would be obtained using the classical method. Which answer was closer to the probability that would be obtained using the classical method? Is this what you would expect? Answer: Answers to parts (a) and (b) will vary. The answer in part (b) is likely to be closer to the classical probability of
1 . 4
In general, as the number of repetitions of a probability experiment increases, the closer the empirical probability should get to the classical probability.
5.7 Putting It Together: Which Method Do I Use? 1 Determine the appropriate probability rule to use. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) Find P(A or B) given that P(A) = 0.6, P(B) = 0.1, and A and B are mutually exclusive. A) 0.7 B) 0 C) 0.06
D) 0.5
Answer: A 2) Find P(A and B) given that P(A) = 0.2, P(B) = 0.5, and A and B are independent. A) 0.1 B) 0 C) 0.7
D) 0.3
Answer: A 3) Suppose that the sample space is S = {a, b, c, d, e, f, g, h} and that outcomes are equally likely. Find the probability of the event E = {b, f, h}. A) 0.375 B) 0.3 C) 3 D) 0.33 Answer: A 4) Suppose that the sample space is S = {a, b, c, d, e, f, g, h, i, j} and that outcomes are equally likely. Find the probability of the event F = "a vowel". A) 0.3 B) 0.2 C) 3 D) 0.33 Answer: A 5) If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.1, find P(A or B). A) 1 B) 0.9 C) 1.2
D) 1.1
Answer: A 6) If P(A) = 0.6, P(B) = 0.6, and P( A or B) = 0.7, find P(A and B). A) 0.5 B) 0.4 C) 0.7
D) 1.2
Answer: A 7) If P(B) = 0.6, P(A or B) = 0.3, and P(A and B) = 0.7, find P(A). A) 0.4 B) 0.6 C) 0.7 Answer: A
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D) 0.9
8) Suppose that S and T are two events, P(S) = 0.62 and P(T S) = 0.16. What is P(S and T)? A) 0.099,2 B) 0.78 C) 0.680,8
D) 0.520,8
Answer: A 9) Suppose that M and N are two events, P(M) = 0.20, P(N) = 0.09, and P(M and N) = 0.05. What is P(M N)? A) 0.556 B) 0.250 C) 0.040 D) 0.240 Answer: A 10) Of 1,639 people who came into a blood bank to give blood, 361 were ineligible to give blood. Estimate the probability that the next person who comes in to give blood will be ineligible to give blood. A) 0.22 B) 0.271 C) 0.188 D) 0.139 Answer: A 11) A study conducted at a certain college shows that 61% of the school's graduates move to a different state after graduating. Find the probability that among 9 randomly selected graduates, at least one moves to a different state after graduating. A) 1.00 B) 0.988 C) 0.111 D) 0.610 Answer: A 12) In one town, 54% of adults are employed in the tourism industry. What is the probability that four adults selected at random from the town are all employed in the tourism industry? A) 0.085 B) 0.915 C) 0.045 D) 0.053 Answer: A 13) The age distribution of members of a gymnastics association is shown in the table. A member of the association is selected at random. Find the probability that the person selected is between 26 and 35 inclusive. Round your answer to three decimal places. Age (years) Frequency Under 21 403 21-25 411 26-30 206 31-35 50 Over 35 30 A) 0.233
1,100 B) 0.187
C) 256
D) 0.045
Answer: A 14) The age distribution of members of a gymnastics association is shown in the table. A member of the association is selected at random. Find the probability that the person selected is at least 31. Round your answer to three decimal places. Age (years) Frequency Under 21 412 21-25 418 26-30 210 31-35 54 Over 35 20 A) 0.066
1,114 B) 0.048
C) 74
D) 0.934
Answer: A 15) At Bill's community college, 40.4% of students are Caucasian and 2.8% of students are Caucasian math majors. What percentage of Caucasian students are math majors? A) 6.9% B) 14.4% C) 37.6% D) 43.2% Answer: A Page 34
16) 58% of students at one college drink alcohol regularly, and 12% of those who drink alcohol regularly suffer from depression. What is the probability that a randomly selected student drinks alcohol regularly and suffers from depression? A) 0.069,6 B) 0.7 C) 0.630,4 D) 0.510,4 Answer: A 17) At a certain college, 20% of students speak Spanish, 7% speak Italian, and 2% speak both languages. A student is chosen at random from the college. What is the probability that the student speaks Spanish given that he or she speaks Italian? A) 0.286 B) 0.100 C) 0.050 D) 0.250 Answer: A 18) Suppose that the probability that Sue will pass her statistics test is 0.64, the probability that she will pass her physics test is 0.50, and the probability that she will pass both tests is 0.310. What is the probability that she passes her physics test given that she passed her statistics test? A) 0.484 B) 0.620 C) 0.190 D) 0.830 Answer: A 19) The table below describes the exercise habits of a group of people suffering from high blood pressure. If a woman is selected at random from the group, find the probability that she does not exercise. No Occasional Regular exercise exercise exercise Total Men 366 84 87 537 Women 369 73 75 517 Total 735 157 162 1,054 A) 0.714 B) 0.35 C) 0.502 D) 0.491 Answer: A 20) The table below describes the exercise habits of a group of people suffering from high blood pressure. If one of the 1,025 subjects is randomly selected, find the probability that the person selected is female given that they exercise occasionally. No Occasional Regular exercise exercise exercise Total Men 320 89 82 491 Women 388 85 61 534 Total 708 174 143 1,025 A) 0.489 B) 0.083 C) 0.159 D) 0.17 Answer: A 21) 390 voters are classified by income and political party. The results are shown in the table. If a person is selected at random from the sample, find the probability that the person has high income. Democrat Republican Total Low Income 107 75 182 Medium Income 91 69 160 High Income 18 16 34 Super High Income 9 5 14 Total 225 165 390 A) 0.087 B) 0.046 C) 0.080 D) 0.529 Answer: A
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22) 390 voters are classified by income and political party. The results are shown in the table. If a person is selected at random from the sample, find the probability that the person has medium income and votes Democrat. Democrat Republican Total Low Income 102 72 174 Medium Income 94 70 164 High Income 19 15 34 Super High Income 10 8 18 Total 225 165 390 A) 0.241 B) 0.418 C) 0.573 D) 0.756 Answer: A 23) 390 voters are classified by income and political party. The results are shown in the table. If a person is selected at random from the sample, find the probability that the person has medium income or votes Democrat. Democrat Republican Total Low Income 106 68 174 Medium Income 87 74 161 High Income 22 18 40 Super High Income 10 5 15 Total 225 165 390 A) 0.767 B) 0.387 C) 0.540 D) 0.990 Answer: A 2 Determine the appropriate counting technique to use. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate. 1) 10P4 A) 5,040
B) 151,200
C) 907,200
D) 3,628,800
B) 56
C) 10,080
D) 4
B) 2!
C)
8 6
D) 8
Answer: A 2) 8C6 A) 28 Answer: A 3)
8! 6! A) 56 Answer: A
Solve the problem. 4) In how many ways can Iris choose 6 of 12 books to bring on vacation? A) 924 B) 665,280 C) 720
D) 2,985,984
Answer: A 5) In how many ways can a board of supervisors choose a president, a treasurer, and a secretary from its 11 members ? A) 990 B) 1,331 C) 39,916,800 D) 165 Answer: A
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6) Karen is completing a fitness circuit. There are 9 fitness stations. At each station she can choose from 3 different activities. If she chooses one activity at each fitness station, in how many ways can she complete the circuit? A) 19,683 B) 27 C) 729 D) 12 Answer: A 7) A company makes skirts in 7 different styles. Each style comes in two different fabrics and 3 different colors. How many skirts are available from this company? A) 42 B) 21 C) 10 D) 12 Answer: A 8) License plates are made using 2 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed? A) 676,000 B) 100,000 C) 11,881,376 D) 67,600 Answer: A 9) If a coin is tossed 7 times, how many head-tail sequences are possible? A) 128 B) 14 C) 5,040
D) 49
Answer: A 10) A pool of possible candidates for a student council consists of 13 freshmen and 9 sophomores. How many different councils consisting of 5 freshmen and 7 sophomores are possible? A) 46,332 B) 646,646 C) 932,990,400 D) 18,532,800 Answer: A 11) How many arrangement are there of the letters DISAPPOINT? A) 907,200 B) 3,628,800 C) 1,814,400
D) 1024
Answer: A 12) How many different arrangements are possible using 6 letters from the word PAYMENT? A) 5,040 B) 2,520 C) 7 D) 42 Answer: A 13) There are 8 runners in a race. In how many ways can the first, second, and third place finishes occur? (Assume there are no ties.) A) 336 B) 56 C) 340 D) 54 Answer: A 14) A poet will read 3 of her poems at an award ceremony. How many ways can she choose the 3 poems from 10 poems given that the sequence is important? A) 720 B) 604,800 C) 120 D) 1,209,600 Answer: A 15) A license plate is to consist of 3 letters followed by 4 digits. Determine the number of different license plates possible if the first letter must be an K , L , or M and repetition of letters and numbers is not permitted. A) 9,072,000 B) 1,296,000 C) 54,432,000 D) 9,162,000 Answer: A 16) In how many ways can 7 women and 8 men be seated in a row of 15 seats at a movie theater assuming that all the women must sit together and all the men must sit together? A) 406,425,600 B) 203,212,800 C) 1.307674368e+12 D) 32,768 Answer: A
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17) In how many ways can a club choose a president, a treasurer, a secretary, and three other committee members (with identical duties) from a group of 14 candidates? A) 360,360 B) 2,162,160 C) 7,529,536 D) 3,003 Answer: A
5.8 Bayes's Rule (online) 1 Use the rule of total probability. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) One of the conditions for a sample space S to be portioned into n subsets is that A) Each of the subsets is mutually exclusive. B) Each of the subsets is independent of the other subsets. C) At least one subset must be empty. D) Each subset is orthogonal to the other subsets. Answer: A 2) One of the conditions for a sample space S to be portioned into n subsets is that A) Each of the subsets must contain at least one element of S. B) The subsets are independent of each other. C) Each subset is orthogonal to the other subsets. D) The elements in the subsets are randomly selected for inclusion. Answer: A Use the rule of total probability to find the indicated probability. 3) Use the tree diagram below to find P(E). Round to the nearest thousandth when necessary. P(E|A) = 0.3 P(A) = 0.35 P(EC|A) = 0.7 P(E|B) = 0.6 P(B) = 0.4 P(EC|B) = 0.4 P(E|C) = 0.2 P(C) = 0.25 P(EC|C) = 0.8 A) 0.395
B) 0.105
C) 0.363
D) 0.345
Answer: A 4) Suppose that events A1 and A2 form a partition of the sample space S with P(A1 ) = 0.65 and P(A2 ) = 0.35. If B is an event that is a subset of S and P(B A1 ) = 0.08 and P(B A2 ) = 0.25, find P(B). Round to the nearest ten-thousandth when necessary. A) 0.139,5 B) 0.052 Answer: A
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C) 0.165
D) 0.087,5
5) Suppose that events A1 , A2, and A3 form a partition of the sample space S with P(A1 ) = 0.45, P(A2) = 0.2, and P(A3 ) = 0.35. If B is an event that is a subset of S and P(B A1 ) = 0.25, P(B A2 ) = 0.29, and P(B A3 ) = 0.25, find P(B). Round to the nearest ten-thousandth when necessary. A) 0.258 B) 0.112,5 C) 0.260,7 D) 0.170,5 Answer: A 6) Suppose that there are two buckets. Bucket 1 contains 3 tennis balls and 5 ping-pong balls. Bucket 2 contains 4 tennis balls, 3 ping-pong balls, and 5 baseballs. An unfair coin will decide from which bucket we draw. Heads 1 implies we draw from Bucket 1 and tails implies we draw from Bucket 2. The probability of heads is and the 3 probability of tails is when necessary. A) 0.375
2 . What is the probability of drawing a ping-pong ball? Round to the nearest thousandth 3 B) 0.208
C) 0.167
D) 0.277
Answer: A 7) Two stores sell a certain product. Store A has 45% of the sales, 3% of which are of defective items, and store B has 55% of the sales, 4% of which are of defective items. The difference in defective rates is due to different levels of pre-sale checking of the product. A person receives one of this product as a gift. What is the probability it is defective? Round to the nearest thousandth when necessary. A) 0.036 B) 0.035 C) 0.018 D) 0.48 Answer: A 8) A teacher designs a test so a student who studies will pass 84% of the time, but a student who does not study will pass 9% of the time. A certain student studies for 85% of the tests taken. On a given test, what is the probability that student passes? Round to the nearest thousandth when necessary. A) 0.728 B) 0.714 C) 0.465 D) 0.135 Answer: A 9) In one town, 7% of 18-29 year olds own a house, as do 35% of 30-50 year olds and 57% of those over 50. According to a recent census taken in the town, 25.2% of adults in the town are 18-29 years old, 36.1% are 30-50 years old, and 38.7% are over 50. What is the probability that a randomly selected adult owns a house? Round to the nearest thousandth when necessary. A) 0.365 B) 0.33 C) 0.036 D) 0.238 Answer: A 10) A company manufactures shoes in three different factories. Factory Omaha Produces 25% of the company's shoes, Factory Chicago produces 60%, and factory Seattle produces 15%. One percent of the shoes produced in Omaha are mislabeled, 0.5 % of the Chicago shoes are mislabeled, and 2% of the Seattle shoes are mislabeled. If you purchase one pair of shoes manufactured by this company what is the probability that the shoes are mislabeled? Round to the nearest thousandth. A) 0.009 B) 0.036 C) 0.043 D) 0.020 Answer: A
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11) A survey conducted in one U.S. city together with information from the census bureau yielded the following table. The first two columns give a percentage distribution of adults in the city by ethnic group. The third column gives the percentage of people in each ethnic group who have health insurance. Round to the nearest thousandth. Ethnic Group Caucasian African American Hispanic Asian Other
Percentage Percentage with of adults health insurance 45.3 77 12.0 42 18.2 52 12.2 68 12.3 46
Determine the probability that a randomly selected adult has health insurance. A) 0.633 B) 0.570 C) 0.063
D) 0.577
Answer: A 2 Use Bayes's Rule to compute probabilities. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Suppose that there are two buckets. Bucket 1 contains 3 tennis balls and 5 ping-pong balls. Bucket 2 contains 4 tennis balls, 3 ping-pong balls, and 5 baseballs. An unfair coin will decide from which bucket we draw. Heads 1 implies we draw from Bucket 1 and tails implies we draw from Bucket 2. The probability of a heads is and 3 the probability of a tails is Bucket 2? A) 0.444
2 . Given that a ping-pong ball was selected what is the probability that it came from 3 B) 0.555
C) 0.625
D) 0.250
Answer: A 2) A company manufactures shoes in three different factories. Factory Omaha Produces 25% of the company's shoes, Factory Chicago produces 60%, and factory Seattle produces 15%. One percent of the shoes produced in Omaha are mislabeled, 0.5 % of the Chicago shoes are mislabeled, and 2% of the Seattle shoes are mislabeled. If you purchase one pair of shoes manufactured by this company and you determine they are mislabeled what is the probability they were made in Omaha? A) 0.294 B) 0.353 C) 0.333 D) 0.070 Answer: A 3) A company manufactures shoes in three different factories. Factory Omaha Produces 25% of the company's shoes, Factory Chicago produces 60%, and factory Seattle produces 15%. One percent of the shoes produced in Omaha are mislabeled, 0.5 % of the Chicago shoes are mislabeled, and 2% of the Seattle shoes are mislabeled. If you purchase one pair of shoes manufactured by this company what is the probability that it was labeled correctly? A) 0.991 B) 0.035 C) 0.9 D) 0.08 Answer: A
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Ch. 5 Probability Answer Key 5.1 Probability Rules 1 Understand random processes and the Law of Large Numbers. 1) 2) 3) 4) 5) 6) A 2 Apply the rules of probabilities. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 3 Compute and interpret probabilities using the empirical method. 1) A 2) A 3) A 4) A 5) A 4 Compute and interpret probabilities using the classical method. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 5 Recognize and interpret subjective probabilities. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A
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5.2 The Addition Rule and Complements 1 Use the Addition Rule for Disjoint Events. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 2 Use the General Addition Rule. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 3 Compute the probability of an event using the Complement Rule. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A
5.3 Independence and the Multiplication Rule 1 Identify independent events. 1) A 2) A 3) A 4) A Page 42
5) A 2 Use the Multiplication Rule for independent events. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 3 Compute at-least probabilities. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A
5.4 Conditional Probability and the General Multiplication Rule 1 Compute conditional probabilities. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A 2 Compute probabilities using the General Multiplication Rule. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A Page 43
10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A 20) A 21) A 22) A 23) A 24) A 25) A 26) A 27) A 28) A 29) A 30) A 31) A 32) A
5.5 Counting Techniques 1 Solve counting problems using the Multiplication Rule. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 2 Solve counting problems using permutations. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 3 Solve counting problems using combinations. 1) A 2) A 3) A Page 44
4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 4 Solve counting problems involving permutations with nondistinct items. 1) A 2) A 3) A 4) A 5) A 6) A 5 Compute probabilities involving permutations and combinations. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A
5.6 Simulating Probability Experiments 1 Know Concepts: Probability Rules. 1) Answers to parts (a) and (b) will vary. Using the classical method, the probability of getting a sum of 8 is In general, as the number of repetitions of a probability experiment increases, the closer the empirical probability should get to the classical probability. 2) Answers to parts (a) and (b) will vary. 1 The answer in part (b) is likely to be closer to the classical probability of . 4 In general, as the number of repetitions of a probability experiment increases, the closer the empirical probability should get to the classical probability.
5.7 Putting It Together: Which Method Do I Use? 1 Determine the appropriate probability rule to use. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A Page 45
5 . 36
10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A 20) A 21) A 22) A 23) A 2 Determine the appropriate counting technique to use. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A
5.8 Bayes's Rule (online) 1 Use the rule of total probability. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 2 Use Bayes's Rule to compute probabilities. 1) A 2) A 3) A
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Ch. 6 Discrete Probability Distributions 6.1 Discrete Random Variables 1 Distinguish between discrete and continuous random variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Classify the following random variable according to whether it is discrete or continuous. 1) the number of bottles of juice sold in a cafeteria during lunch A) discrete B) continuous Answer: A 2) the heights of the bookcases in a school library A) continuous
B) discrete
Answer: A 3) the cost of a road t-shirt A) discrete
B) continuous
Answer: A 4) the pressure of water coming out of a fire hose measured in pounds per square inch (psi) A) continuous B) discrete Answer: A 5) the temperature in degrees Celsius on January 1st in Fargo, North Dakota A) continuous B) discrete Answer: A 6) the number of goals scored in a hockey game A) discrete
B) continuous
Answer: A 7) the speed of a car on a New York tollway during rush hour traffic A) continuous B) discrete Answer: A 8) the number of emails received on any given day A) discrete
B) continuous
Answer: A 9) the age of the oldest dog in a kennel A) continuous
B) discrete
Answer: A 10) the number of pills in an aspirin bottle A) discrete Answer: A
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B) continuous
Provide an appropriate response. 11) The peak shopping time at home improvement store is between 8:00am-11:00 am on Saturday mornings. Management at the home improvement store randomly selected 55 customers last Saturday morning and decided to observe their shopping habits. They recorded the number of items that 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 home improvement store. A) number of items - discrete; total time - continuous B) number of items - continuous; total time - continuous C) number of items - continuous; total time - discrete D) number of items - discrete; total time - discrete Answer: A 12) The number of violent crimes committed in a day possesses a distribution with a mean of 2 crimes per day and a standard deviation of four crimes per day. A random sample of 50 days was observed, and the sample mean number of crimes for the sample was calculated. The data that was collected in this experiment could be measured with a __________ random variable. A) discrete B) continuous Answer: A 13) A random variable is A) a numerical measure of the outcome of a probability experiment. B) generated by a random number table. C) the variable for which an algebraic equation is solved. D) a qualitative attribute of a population. Answer: A 2 Identify discrete probability distributions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer Yes or No. x P(x)
5 0.10
10 -0.30
15 0.50
20 0.70
A) No
B) Yes
Answer: A 2) Given the table of probabilities for the random variable x, does this form a probability distribution? Answer Yes or No. x P(x)
0 0.02
A) Yes Answer: A
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1 0.07
2 0.22
3 0.27
4 0.42 B) No
3) Consider the discrete probability distribution given when answering the following question. Find the probability that x equals 5. x 3 5 7 8 P(x) 0.07 ? 0.16 0.12 A) 0.65 B) 0.35 C) 3.25 D) 1.75 Answer: A 4) Consider the discrete probability distribution given when answering the following question. Find the probability that x exceeds 4. x 2 4 7 8 P(x) 0.01 ? 0.11 0.02 A) 0.13 B) 0.99 C) 0.87 D) 0.86 Answer: A 5) An Apple Pie Company knows that the number of pies sold each day 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 $1.50 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies for $4 each, find the probability distribution for her daily profit. A) B) C) D) Profit P(profit) Profit P(profit) Profit P(profit) Profit P(profit) $100 0.5 $400 0.5 $200 0.5 $250 0.5 $300 0.25 $600 0.25 $400 0.25 $375 0.25 $500 0.25 $800 0.25 $600 0.25 $500 0.25 Answer: A 6) The sum of the probabilities of a discrete probability distribution must be A) equal to one. B) between zero and one. C) greater than one. D) less than or equal to zero. Answer: A
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3 Graph discrete probability distributions. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The random variable x represents the number of boys in a family of three children. Assuming that boys and girls are equally likely, (a) construct a probability distribution, and (b) graph the probability distribution. Answer: (a) x P(x) 1 0 8
(b)
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1
3 8
2
3 8
3
1 8
2) The random variable x represents the number of tests that a pet entering an animal shelter will have along with the corresponding probabilities. Graph the probability distribution. x P(x) 3 0 17 1
5 17
2
6 17
3
2 17
4
1 17
Answer:
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3) The random variable x represents the number of credit cards that students have along with the corresponding probabilities. Graph the probability distribution. x P(x) 0 0.49 1 0.05 2 0.32 3 0.07 4 0.07 Answer:
4) In an Italian cafe, the following probability distribution was obtained. The random variable x represents the number of toppings for a large pizza. Graph the probability distribution. x P(x) 0 0.30 1 0.40 2 0.20 3 0.06 4 0.04 Answer:
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5) Use the frequency distribution to (a) construct a probability distribution for the random variable x which represents the number of cars per family in a town of 1000 families, and (b) graph the probability distribution. Cars Families 0 125 1 428 2 256 3 108 4 83 Answer: (a) x P(x) 0 0.125 1 0.428 2 0.256 3 0.108 4 0.083 (b)
4 Compute and interpret the mean and standard deviation of a discrete random variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Calculate the mean for the discrete probability distribution shown here. x 2 6 8 11 P(x) 0.23 0.14 0.31 0.32 A) 7.3 B) 6.75 C) 27 Answer: A
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D) 1.825
2) A lab orders a shipment of 100 rats per week, 52 weeks per year, from a rat supplier for experiments that the lab conducts. Prices for each weekly shipment of rats follow the distribution below: Price $10.00 $12.50 $15.00 Probability 0.25 0.35 0.4 Suppose the mean cost of the rats turned out to be $12.88 per week. Interpret this value. A) The average cost for all weekly rat purchases is $12.88. B) Most of the weeks resulted in rat costs of $12.88. C) The median cost for the distribution of rat costs is $12.88. D) The rat cost that occurs more often than any other is $12.88. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) Calculate the mean for the discrete probability distribution shown here. x 2 5 10 14 P(x) 0.2 0.3 0.3 0.2 Answer: μ = ∑x ∙ p(x) = 2(0.2) + 5(0.3) + 10(0.3) + 14(0.2) = 7.7 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) A baseball player is asked to swing at pitches in sets of four. The player swings at 100 sets of 4 pitches. The probability distribution for making a particular number of hits is given below. Determine the mean for this discrete probability distribution. x P(x)
0 0.02
1 0.07
2 0.22
A) 3
3 0.27
4 0.42
B) 4
C) 2
D) 3.5
Answer: A 5) The produce manager at a farmer's market was interested in determining how many oranges a person buys when they buy oranges. He asked the cashiers over a weekend to count how many oranges a person bought when they bought oranges and record this number for analysis at a later time. The data is given below in the table. The random variable x represents the number of oranges purchased and P(x) represents the probability that a customer will buy x apples. Determine the mean number of oranges purchased by a customer. x P(x)
1 0.05
2 0.19
3 0.20
A) 3.97
4 0.25
5 0.12
6 0.10
7 0
B) 5.50
8 0.08
9 0
10 0.01
C) 3
D) 4
Answer: A 6) A random number generator is set to generate single digits between 0 and 9. One hundred fifty random numbers are generated. The probability distribution for this random number generator is given below. What is the mean of this distribution? x P(x)
0 0.09
A) 4.5 Answer: A Page 8
1 0.12
2 0.11
3 0.08
4 0.09
B) 6.6
5 0.13
6 0.10
7 0.07
8 0.10
C) 5
9 0.11 D) 7
7) A seed company has a test plot in which it is testing the germination of a hybrid seed. They plant 50 rows of 40 seeds per row. After a two-week period, the researchers count how many seed per row have sprouted. They noted that least number of seeds to germinate was 33 and some rows had all 40 germinate. The germination data is given below in the table. The random variable x represents the number of seed in a row that germinated and P(x) represents the probability of selecting a row with that number of seed germinating. Determine the mean number of seeds per row that germinated. 33 0.02
x P(x)
34 0.06
35 0.10
A) 36.9
36 0.20
37 0.24
38 0.26
39 0.10
B) 36.5
40 0.02 C) 36
D) 0.13
Answer: A 8) A manager asked her employees how many times they had given blood in the last year. The results of the survey are given below. The random variable x represents the number of times a person gave blood and P(x) represents the probability of selecting an employee who had given blood that percent of the time. What is the mean number of times a person gave blood based on this survey? x P(x)
0 0.30
A) 1.6
1 0.25
2 0.20
3 0.12
4 0.07
B) 3.0
5 0.04
6 0.02 C) 2.0
D) 0.14
Answer: A 9) The random variable x represents the number of girls in a family of three children. Assuming that boys and girls are equally likely, find the mean and standard deviation for the random variable x. A) mean: 1.50; standard deviation: 0.87 B) mean: 2.25; standard deviation: 0.87 C) mean: 1.50; standard deviation: 0.76 D) mean: 2.25; standard deviation: 0.76 Answer: A 10) The random variable x represents the number of tests that a patient entering a clinic will have along with the corresponding probabilities. Find the mean and standard deviation for the random variable x. x P(x) 3 0 17 1
5 17
2
6 17
3
2 17
4
1 17
A) mean: 1.59; standard deviation: 1.09 C) mean: 2.52; standard deviation: 1.93 Answer: A
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B) mean: 1.59; standard deviation: 3.72 D) mean: 3.72; standard deviation: 2.52
11) The random variable x represents the number of computers that families have along with the corresponding probabilities. Find the mean and standard deviation for the random variable x. x P(x) 0 0.49 1 0.05 2 0.32 3 0.07 4 0.07 A) mean: 1.18; standard deviation: 1.30 B) mean: 1.39; standard deviation: 0.64 C) mean: 1.39; standard deviation: 0.80 D) mean: 1.18; standard deviation: 0.64 Answer: A 12) In a sandwich shop, the following probability distribution was obtained. The random variable x represents the number of condiments used for a hamburger. Find the mean and standard deviation for the random variable x. x P(x) 0 0.30 1 0.40 2 0.20 3 0.06 4 0.04 A) mean: 1.14; standard deviation: 1.04 B) mean: 1.54; standard deviation: 1.30 C) mean: 1.30; standard deviation: 2.38 D) mean: 1.30; standard deviation: 1.54 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 13) From the probability distribution, find the mean and standard deviation for the random variable x, which represents the number of bicycles per household in a town of 1000 households. x P(x) 0 0.125 1 0.428 2 0.256 3 0.108 4 0.083 Answer: μ = 1.596; σ = 1.098 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 14) A baseball player is asked to swing at pitches in sets of four. The player swings at 100 sets of 4 pitches. The probability distribution for hitting a particular number of pitches is given below. Determine the standard deviation for this discrete probability distribution. x P(x)
0 0.02
A) 1.05 Answer: A
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1 0.07
2 0.22
3 0.27 B) 1.10
4 0.42 C) 1.21
D) 0.28
15) The owner of a farmer's market was interested in determining how many oranges a person buys when they buy oranges. He asked the cashiers over a weekend to count how many oranges a person bought when they bought oranges and record this number for analysis at a later time. The data is given below in the table. The random variable x represents the number of oranges purchased and P(x) represents the probability that a customer will buy x oranges. Determine the variance of the number of oranges purchased by a customer. x P(x)
1 0.05
2 0.19
3 0.20
A) 3.57
4 0.25
5 0.12
6 0.10
7 0
B) 1.95
8 0.08
9 0
C) 3.97
10 0.01 D) 0.56
Answer: A 16) A manager at a local company asked his employees how many times they had given blood in the last year. The results of the survey are given below. The random variable x represents the number of times a person gave blood and P(x) represents the probability of selecting an employee who had given blood that percent of the time. What is the standard deviation for the number of times a person gave blood based on this survey? x P(x)
0 0.30
1 0.25
2 0.20
A) 1.54
3 0.12
4 0.07
5 0.04
6 0.02
B) 1.82
C) 1.16
D) 2.23
Answer: A 17) A seed company has a test plot in which it is testing the germination of a hybrid seed. They plant 50 rows of 40 seeds per row. After a two-week period, the researchers count how many seed per row have sprouted. They noted that least number of seeds to germinate was 33 and some rows had all 40 germinate. The germination data is given below in the table. The random variable x represents the number of seed in a row that germinated and P(x) represents the probability of selecting a row with that number of seed germinating. Determine the standard deviation of the number of seeds per row that germinated. x P(x)
33 0.02
A) 1.51
34 0.06
35 0.10
36 0.20
37 0.24
B) 7.13
38 0.26
39 0.10
40 0.02 C) 36.86
D) 6.07
Answer: A 5 Interpret the mean of a discrete random variable as an expected value. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A lab orders a shipment of 100 rats a week, 52 weeks a year, from a rat supplier for experiments that the lab conducts. Prices for each weekly shipment of rats follow the distribution below: Price $10.00 $12.50 $15.00 Probability 0.15 0.25 0.6 How much should the lab budget for next year's rat orders assuming this distribution does not change. (Hint: find the expected price.) A) $708.50 B) $13.63 C) $1,363.00 D) $3,684,200.00 Answer: A
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2) Giselle bakes six pies a day that cost $2 each to produce. On 35% of the days she sells only two pies. On 38% of the days, she sells 4 pies, and on the remaining 27% of the days, she sells all six pies. If Giselle sells her pies for $5 each, what is her expected profit for a day's worth of pies? [Assume that any leftover pies are given away.] A) $7.20 B) $19.20 C) -$7.00 D) -$8.16 Answer: A 3) A local bakery has determined a probability distribution for the number of cheesecakes that they sell in a given day. The distribution is as follows: Number sold in a day 0 5 10 15 20 Prob (Number sold) 0.18 0.05 0.07 0.17 0.53 Find the number of cheesecakes that this local bakery expects to sell in a day. A) 14.1 B) 14.28 C) 15 D) 10 Answer: A 4) A dice game involves throwing three dice and betting on one of the six numbers that are on the dice. The game costs $6 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 Profit Probability of Observing 0 125/216 -$6 1 $6 75/216 2 $8 15/216 3 $18 1/216 Find your expected profit from playing this game. A) -$0.77 B) $0.50 C) $6.18 D) $3.39 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) A legendary football coach was known for his winning seasons. He consistently won nine or more games per season. Suppose x equals the number of games won up to the halfway mark (six games) in a 12-game season. If this coach and his team had a probability p = 0.65 of winning any one game (and the winning or losing of one game was independent of another), then the probability distribution of the number x of winning games in a series of six games is: x P(x) 0 0.001,838 1 0.020,484 2 0.095,102 3 0.235,491 4 0.328,005 5 0.243,661 6 0.075,419 Find the expected number of winning games in the first half of the season for this coach's football teams. Answer: μ = ∑xp(x) ≈ 3.9
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6) On one busy holiday weekend, a national airline has many requests for standby flights at half of the usual one-way air fare. However, past experience has shown that these passengers have only about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, their 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 $192 and the cost of flying first class is $320. 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.] Answer: Let x = cost of fare paid by passenger. The probability distribution for x is: x $96 $320 x(p) 1/5 4/5 1 4 The expected cost is E(x) = μ = ∑x ∙ p(x) = $96 + $320 = $275.20 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. 7) An automobile insurance company estimates the following loss probabilities for the next year on a $25,000 sports car: Total loss: 0.001 50% loss: 0.01 25% loss: 0.05 10% loss: 0.10 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 $660 profit per policy sold? Answer: 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 0.001
$12,000 0.01
$5,750 0.05
$2,000 0.10
-$500 0.839
Note: These losses paid have already considered the $500 deductible paid by the owner. The expected loss paid is: μ = ∑x ∙ p(x) = $24,500(0.001) + $12,000(0.01) + $5,750(0.05) + $2,000(0.10) - $500(0.839) = $212.50 In order to average $660 profit per policy sold, the insurance company must charge an annual premium of $212.50 + $660 = $872.50. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 8) True or False: The expected value of a discrete probability distribution may be negative. A) True B) False Answer: A 9) In a carnival game, a person wagers $2 on the roll of two dice. If the total of the two dice is 2, 3, 4, 5, or 6 then the person gets $4 (the $2 wager and $2 winnings). If the total of the two dice is 8, 9, 10, 11, or 12 then the person gets nothing (loses $2). If the total of the two dice is 7, the person gets $0.75 back (loses $0.25). What is the expected value of playing the game once? A) -$0.04 B) -$0.42 C) $2.00 D) $0.00 Answer: A
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10) In the American version of the Game Roulette, a wheel has 18 black slots, 8 red slots and 2 green slots. All slots are the same size. In a carnival game, a person wagers $2 on the roll of two dice. A person can wager on either red or black. Green is reserved for the house. If a player wagers $5 on either red or black and that color comes up, they win $10 otherwise they lose their wager. What is the expected value of playing the game once? A) -$0.26 B) -$0.50 C) $0.26 D) $0.50 Answer: A 6 Compute the Standard Deviation of a Discrete Random Variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A lab orders a shipment of 100 rats a week, 52 weeks a year, from a rat supplier for experiments that the lab conducts. Prices for each weekly shipment of rats follow the distribution below: Price $10.00 $12.50 $15.00 Probability 0.3 0.25 0.45 Find the standard deviation for the lab budget for next year's rat orders assuming this distribution does not change. A) $2.13 B) $12.88 C) $12.50 D) $2.55 Answer: A 2) A local bakery has determined a probability distribution for the number of cheesecakes that they sell in a given day. The distribution is as follows: Number sold in a day 0 5 10 15 20 Prob (Number sold) 0.22 0.24 0.13 0.25 0.16 Find the standard deviation of cheesecakes that this local bakery expects to sell in a day. A) 7.06 B) 0.59 C) 9.45 D) 10 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) A legendary football coach was known for his winning seasons. He consistently won nine or more games per season. Suppose x equals the number of games won up to the halfway mark (six games) in a 12-game season. If this coach and his team had a probability p = 0.6 of winning any one game (and the winning or losing of one game was independent of another), then the probability distribution of the number x of winning games in a series of six games is: x P(x) 0 0.004096 1 0.036864 2 0.138240 3 0.276480 4 0.311040 5 0.186624 6 0.046656 Find the standard deviation of winning games in the first half of the season for this coach's football teams Answer: 1.2 4) On one busy holiday weekend, a national airline has many requests for standby flights at half of the usual one-way air fare. However, past experience has shown that these passengers have only about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, their 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 $100 and the cost of flying first class is $490. Find the standard deviation of cost of the flight. Answer: 176
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6.2 The Binomial Probability Distribution 1 Determine whether a probability experiment is a binomial experiment. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Decide whether the experiment is a binomial experiment. If it is not, explain why. You observe the gender of the next 200 babies born at a local hospital. The random variable represents the number of boys. Answer: binomial experiment 2) Decide whether the experiment is a binomial experiment. If it is not, explain why. You draw a marble 850 times from a bag with three colors of marbles. The random variable represents the color of marble that is drawn. Answer: Not a binomial experiment. There are more than two outcomes. 3) Decide whether the experiment is a binomial experiment. If it is not, explain why. In a game you spin a wheel that has 10 different letters 1,000 times. The random variable represents the selected letter on each spin of the wheel. Answer: Not a binomial experiment. There are more than two outcomes. 4) Decide whether the experiment is a binomial experiment. If it is not, explain why. Testing a cough suppressant using 500 people to determine if it is effective. The random variable represents the number of people who find the cough suppressant to be effective. Answer: binomial experiment. 5) Decide whether the experiment is a binomial experiment. If it is not, explain why. Survey 150 investors to see how many different stocks they own. The random variable represents the number of different stocks owned by each investor. Answer: Not a binomial experiment. There are more than two outcomes. 6) Decide whether the experiment is a binomial experiment. If it is not, explain why. Survey 50 college students see whether they are enrolled as a new student. The random variable represents the number of students enrolled as new students. Answer: binomial experiment. 7) Decide whether the experiment is a binomial experiment. If it is not, explain why. Each week, a man attends a club meeting in which he has a 38% chance of meeting a new member. The random variable is the number of times he meets a new member in 49 weeks. Answer: binomial experiment. 8) Decide whether the experiment is a binomial experiment. If it is not, explain why. You test four flu medicines. The random variable represents the flue medicine that is most effective. Answer: Not a binomial experiment. There are more than two outcomes. 9) Decide whether the experiment is a binomial experiment. If it is not, explain why. Each week, a gambler plays blackjack at the local casino. The random variable is the number of times per week the player wins. Answer: Not a binomial experiment. There are more than two outcomes. 10) Decide whether the experiment is a binomial experiment. If it is not, explain why. Selecting five cards, one at a time without replacement, from a standard deck of cards. The random variable is the number of picture cards obtained. Answer: Not a binomial experiment. The probability of success is not the same for each trial.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 11) True or False: The trials of a binomial experiment must be mutually exclusive of each other. A) False B) True Answer: A 12) Which of the below is not a requirement for binomial experiment? A) The trials are mutually exclusive. B) The experiment is performed a fixed number of times. C) For each trial there are two mutually exclusive outcomes. D) The probability of success is fixed for each trial of the experiment. Answer: A 13) If p is the probability of success of a binomial experiment, then the probability of failure is x n A) 1 - p B) -p C) D) n x Answer: A 2 Compute probabilities of binomial experiments. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of exactly three girls in ten births. A) 0.117 B) 0.3 C) 15.000 D) 0.03 Answer: A 2) In a recent survey, 60% of the community favored building a health center in their neighborhood. If 14 citizens are chosen, find the probability that exactly 8 of them favor the building of the health center. A) 0.207 B) 0.092 C) 0.571 D) 0.600 Answer: A 3) The probability that an individual has 20-20 vision is 0.11. In a class of 89 students, what is the probability of finding five people with 20-20 vision? A) 0.037 B) 0.056 C) 0.000 D) 0.11 Answer: A 4) According to insurance records a car with a certain protection system will be recovered 86% of the time. Find the probability that 5 of 9 stolen cars will be recovered. A) 0.023 B) 0.556 C) 0.86 D) 0.14 Answer: A 5) The probability that a football game will go into overtime is 20%. What is the probability that two of three football games will go to into overtime? A) 0.096 B) 0.2 C) 0.384 D) 0.04 Answer: A 6) Fifty percent of the people that use the Internet buy their clothes exclusively online. Find the probability that only five of 8 Internet users buy their clothes exclusively online. A) 0.219 B) 0.625 C) 0.004 D) 1.750 Answer: A
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7) The probability that a house in an urban area will develop a leak is 5%. If 59 houses are randomly selected, what is the probability that none of the houses will develop a leak? A) 0.048 B) 0.050 C) 0.000 D) 0.001 Answer: A 8) Sixty-five percent of men consider themselves knowledgeable soccer fans. If 12 men are randomly selected, find the probability that exactly two of them will consider themselves knowledgeable fans. A) 0.001 B) 0.109 C) 0.65 D) 0.167 Answer: A 9) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three girls in ten births. A) 0.172 B) 0.300 C) 0.003 D) 0.333 Answer: A 10) A quiz consists of 10 true or false questions. To pass the quiz a student must answer at least eight questions correctly. If the student guesses on each question, what is the probability that the student will pass the quiz? A) 0.055 B) 0.8 C) 0.20 D) 0.08 Answer: A 11) A quiz consists of 10 multiple choice questions, each with five possible answers, one of which is correct. To pass the quiz a student must get 60% or better on the quiz. If a student randomly guesses, what is the probability that the student will pass the quiz? A) 0.006 B) 0.060 C) 0.377 D) 0.205 Answer: A 12) A recent survey found that 70% of all adults over 50 wear sunglasses for driving. In a random sample of 10 adults over 50, what is the probability that at least six wear sunglasses? A) 0.850 B) 0.700 C) 0.200 D) 0.006 Answer: A 13) According to government data, the probability that an adult was never in a museum is 15%. In a random survey of 10 adults, what is the probability that two or fewer were never in a museum? A) 0.820 B) 0.002 C) 0.800 D) 0.200 Answer: A 14) According to government data, the probability that an adult was never in a museum is 15%. In a random survey of 10 adults, what is the probability that at least eight were in a museum? A) 0.820 B) 0.200 C) 0.002 D) 0.800 Answer: A 15) According to the Federal Communications Commission, 70% of all U.S. households have DVD players. In a random sample of 15 households, what is the probability that at least 13 have DVD players? A) 0.1268 B) 0.7 C) 0.8732 D) 0.5 Answer: A 16) According to the Federal Communications Commission, 70% of all U.S. households have DVD players. In a random sample of 15 households, what is the probability that the number of households with DVD players is between 10 and 12, inclusive? A) 0.5947 B) 0.4053 C) 0.7 D) 0.2061 Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 17) A motel has a policy of booking as many as 150 guests in a building that holds 140. Past studies indicate that only 85% of booked guests show up for their room. Find the probability that if the motel books 150 guests, not enough seats will be available. Answer: 0.0005 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) We believe that 78% of the population of all Calculus I students consider calculus an exciting subject. Suppose we randomly and independently selected 29 students from the population. If the true percentage is really 78%, find the probability of observing 28 or more of the students who consider calculus to be an exciting subject in our sample of 29. A) 0.006,817 B) 0.000,743 C) 0.006,074 D) 0.993,183 Answer: A 19) A psychic network received telephone calls last year from over 1.5 million people. A recent article attempts to shed some light onto the credibility of the psychic network. One of the psychic network's psychics agreed to take part in the following experiment. Five different cards are shuffled, and one is chosen at random. The psychic will then try to identify which card was drawn without seeing it. Assume that the experiment was repeated 50 times and that the results of any two experiments are independent of one another. If we assume that the psychic is a fake (i.e., they are merely guessing at the cards and have no psychic powers), find the probability that they guess at least three correctly. A) 0.998,715 B) 0.001,257 C) 0.004,371 D) 0.001,093 Answer: A 20) A history 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 0.10. What score should be set as the lowest passing grade? A) 11 B) 9 C) 12 D) 10 Answer: A 21) A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample, the paper claims that 39% of all constituents believe their state representative possesses low ethical standards. Assume that responses were randomly and independently collected. A representative of a district with 1,000 people does not believe the paper's claim applies to her. If the claim is true, how many of the representative's constituents believe the state representative possesses low ethical standards? A) 390 B) 39 C) 610 D) 961 Answer: A 22) A recent article in the paper claims that government ethics are at an all-time low. Reporting on a recent sample, the paper claims that 31% of all constituents believe their state representative possesses low ethical standards. Suppose 20 of a representative's constituents 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 constituents sampled believe their state representative possesses low ethical standards. A) 0.127,128 B) 0.090,085 C) 0.257,174 D) 0.176,107 Answer: A
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3 Compute the mean and standard deviation of a binomial random variable. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Suppose that 600 couples each have a baby; find the mean and standard deviation for the number of boys in the 600 babies. Answer: μ = np = 600(0.5) = 300; σ =
npq =
600(0.5)(0.5) = 12.25
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) A quiz consists of 660 true or false questions. If the student guesses on each question, what is the mean number of correct answers? A) 330 B) 0 C) 660 D) 132 Answer: A 3) A quiz consists of 50 true or false questions. If the student guesses on each question, what is the standard deviation of the number of correct answers? A) 3.535,533,91 B) 0 C) 2 D) 5 Answer: A 4) A quiz consists of 30 multiple choice questions, each with five possible answers, only one of which is correct. If a student guesses on each question, what is the mean and standard deviation of the number of correct answers? A) mean: 6; standard deviation: 2.190,890,23 B) mean: 6; standard deviation: 2.449,489,74 C) mean: 15; standard deviation: 2.190,890,23 D) mean: 15; standard deviation: 3.872,983,35 Answer: A 5) The probability that an individual has 20-20 vision is 0.15. In a class of 20 students, what is the mean and standard deviation of the number with 20-20 vision in the class? A) mean: 3; standard deviation: 1.596,871,94 B) mean: 20; standard deviation: 1.596,871,94 C) mean: 3; standard deviation: 1.732,050,81 D) mean: 20; standard deviation: 1.732,050,81 Answer: A 6) A recent survey found that 71% of all adults over 50 wear sunglasses for driving. In a random sample of 80 adults over 50, what is the mean and standard deviation of those that wear sunglasses? A) mean: 56.8; standard deviation: 4.058,571,18 B) mean: 56.8; standard deviation: 7.536,577,47 C) mean: 23.2; standard deviation: 4.058,571,18 D) mean: 23.2; standard deviation: 7.536,577,47 Answer: A 7) According to government data, the probability that an adult was never in a museum is 17%. In a random survey of 50 adults, what is the mean and standard deviation of the number that were never in a museum? A) mean: 8.5; standard deviation: 2.656,125 B) mean: 41.5 standard deviation: 2.915,475,95 C) mean: 41.5; standard deviation: 2.656,125 D) mean: 8.5; standard deviation: 2.915,475,95 Answer: A 8) According to insurance records, a car with a certain protection system will be recovered 89% of the time. If 400 stolen cars are randomly selected, what is the mean and standard deviation of the number of cars recovered after being stolen? A) mean: 356; standard deviation: 6.257,795,14 B) mean: 356; standard deviation: 39.16 C) mean: -1,244: standard deviation: 6.257,795,14 D) mean: -1,244: standard deviation: 39.16 Answer: A
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9) The probability that a football game will go into overtime is 16%. In 140 randomly selected football games, what is the mean and the standard deviation of the number that went into overtime? A) mean: 22.4; standard deviation: 4.337,741,35 B) mean: 22.4; standard deviation: 4.732,863,83 C) mean: 21; standard deviation: 4.337,741,35 D) mean: 21; standard deviation: 4.732,863,83 Answer: A 10) In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizens are chosen, what is the mean number favoring the health center? A) 12 B) 15 C) 8 D) 10 Answer: A 11) In a recent survey, 80% of the community favored building a health center in their neighborhood. If 15 citizens are chosen, what is the standard deviation of the number favoring the health center? A) 1.55 B) 2.40 C) 0.98 D) 0.55 Answer: A 12) The probability that a house in an urban area will develop a leak is 5%. If 20 houses are randomly selected, what is the mean of the number of houses that developed leaks? A) 1 B) 2 C) 0.5 D) 1.5 Answer: A 13) A psychic network received telephone calls last year from over 1.5 million people. A recent article attempts to shed some light onto the credibility of the psychic network. One of the psychic network's psychics agreed to take part in the following experiment. Five different cards are shuffled, and one is chosen at random. The psychic will then try to identify which card was drawn without seeing it. Assume that the experiment was repeated 15 times and that the results of any two experiments are independent of one another. If we assume that the psychic is a fake (i.e., they are merely guessing at the cards and have no psychic powers), how many of the 15 cards do we expect the psychic to guess correctly? A) 3 B) 2 C) 0 D) 5 Answer: A
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4 Graph a binomial probability distribution. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Draw the probability graph and label the mean for n = 6 and p = 0.4 Answer:
2) Draw the probability graph and label the mean for n = 7 and p = 0.5 Answer:
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3) Draw the probability graph and label the mean for n = 8 and p = 0.3 Answer:
4) Draw the probability graph and label the mean for n = 9 and p = 0.7 Answer:
6.3 The Poisson Probability Distribution 1 Determine if a probability experiment follows a Poisson process. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Given that a random variable x, the number of successes, follows a Poisson process, the probability of 2 or more successes in any sufficiently small subinterval is A) zero. B) one. C) any number between zero and one. D) none of these Answer: A
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2) Given that a random variable x, the number of successes, follows a Poisson process, then the probability of success for any two intervals of the same size A) is the same. B) are complementary. C) are reciprocals. D) none of these Answer: A 3) Given that a random variable x, the number of successes, follows a Poisson process, then the number of successes in any interval is independent of the number of successes in any other interval provided the intervals A) are disjoint. B) overlap. C) have at least one element in common. D) are the same size and are independent. Answer: A 2 Compute probabilities of a Poisson random variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A history professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in the 10:30 a.m. time slot exactly two students will arrive. A) 0.224,1 B) 0.609,1 C) 0.199,2 D) 0.541,5 Answer: A 2) A help desk receives an average of four calls per hour on its toll-free number. For any given hour, find the probability that it will receive exactly nine calls. Use the Poisson distribution. A) 0.013,2 B) 0.000,1 C) 146.370,0 D) 0.000,3 Answer: A 3) The local police department receives an average of two calls per hour. Use the Poisson distribution to determine the probability that in a randomly selected hour the number of calls is five. A) 0.036,1 B) 0.001,8 C) 0.028,2 D) 0.001,4 Answer: A 4) A dictionary contains 500 pages. If there are 200 typing errors randomly distributed throughout the book, use the Poisson distribution to determine the probability that a page contains exactly two errors. A) 0.053,6 B) 0.010,8 C) 0.442,3 D) 0.089,3 Answer: A 5) A customer service firm receives an average of three calls per hour on its toll-free number. For any given hour, find the probability that it will receive at least three calls. Use the Poisson distribution. A) 0.5768 B) 0.1891 C) 0.4232 D) 0.6138 Answer: A 6) An online retailer receives an average of five orders per 500 hits on its website. If it gets 100 hits on its website, find the probability of receiving at least two orders. Use the Poisson distribution. A) 0.2642 B) 0.1839 C) 0.9048 D) 0.9596 Answer: A 7) A local animal rescue organization receives an average of 0.55 rescue calls per hour. Use the Poisson distribution to find the probability that during a randomly selected hour, the organization will receive fewer than two calls. A) 0.894 B) 0.106 C) 0.317 D) 0.087 Answer: A
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8) 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 6. Find the probability that exactly two road construction projects are currently taking place in this city. A) 0.044,618 B) 0.006,767 C) 0.079,320 D) 0.012,030 Answer: A 9) 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 7. Find the probability that more than four road construction projects are currently taking place in the city. A) 0.827,008 B) 0.918,235 C) 0.172,992 D) 0.081,765 Answer: A 10) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.7. Find the probability that less than three accidents will occur next month on this stretch of road. A) 0.007,920 B) 0.026,203 C) 0.992,080 D) 0.973,797 Answer: A 11) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.1. Find the probability of observing exactly two accidents on this stretch of road next month. A) 0.009,958 B) 4.439,674 C) 0.041,642 D) 18.566,242 Answer: A 12) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distribution with a mean of 6.8. Find the probability that the next two months will both result in three accidents each occurring on this stretch of road. A) 0.003,407 B) 0.058,368 C) 0.116,736 D) 0.000,520 Answer: A 13) Suppose the number of babies born during an 8-hour shift at a hospital's maternity wing follows a Poisson distribution with a mean of 7 an hour. Find the probability that three babies are born during a particular 1-hour period in this maternity wing. A) 0.052,129 B) 0.046,544 C) 0.006,516 D) 0.000,315 Answer: A 14) Suppose the number of babies born during an 8-hour shift at a hospital's maternity wing follows a Poisson distribution with a mean of 7 an hour. 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, County Hospital experienced eight consecutive hours with more than eight births. Based on this fact, comment on the belief that the full moon increases the number of births. A) The belief is supported as the probability of observing this many births would be 0.0000291. B) The belief is not supported as the probability of observing this many births is 0.271. C) The belief is supported as the probability of observing this many births would be 0.271. D) The belief is not supported as the probability of observing this many births is 0.0000291. Answer: A 15) 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 7.2 tickets per day. Find the probability that less than six tickets are written on a randomly selected day from this distribution. A) 0.275,897 B) 0.420,356 C) 0.724,103 D) 0.579,644 Answer: A
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16) 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 6.2 tickets per day. Find the probability that exactly four tickets are written on a randomly selected day from this distribution. A) 0.124,948 B) 0.457,020 C) 1.127,659 D) 4.124,609 Answer: A 17) The number of goals scored at State College hockey games follows a Poisson distribution with a mean of 5 goals per game. Find the probability that a randomly selected State College hockey game would have more than three goals scored. A) 0.734,974 B) 0.265,026 C) 0.875,348 D) 0.124,652 Answer: A 18) The number of goals scored at State College hockey games follows a Poisson distribution with a mean of 5 goals per game. Find the probability that each of four randomly selected State College hockey games resulted in three goals being scored. A) 0.000,388,28 B) 0.561,495,61 C) 0.005,402,43 D) 0.438,504,39 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 19) A small life insurance company has determined that on the average it receives 3 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day. Answer: Let x = the number of death claims received per day. Then x is a Poisson random variable with λ = 3. P(x ≥ 7) = 1 - P(x ≤ 6) = 0.033,509 20) The number of traffic accidents that occurs on a particular stretch of road during a month follows a Poisson distribution with a mean of 8.5. Find the probability that less than two accidents will occur on this stretch of road during a randomly selected month. Answer: Let x = the number of accidents that occur on the stretch of road during a month. Then x is a Poisson random variable with λ = 8.5. P(x < 2) = P(x = 0) + P(x = 1) = 0.001,933 21) Suppose the number of babies born during an eight-hour shift at a hospital's maternity wing follows a Poisson distribution with a mean of 3 an hour. Find the probability that exactly two babies are born during a randomly selected hour. Answer: Let x = the number of babies born during a one hour period at this hospital's maternity wing. Then x is a Poisson random variable with λ = 3. P(x = 2) = 0.224,042
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3 Find the mean and standard deviation of a Poisson random variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) 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 tickets per day. Interpret the value of the mean. A) If we sampled all days, the arithmetic average number of tickets written would be 9.5 tickets per day. B) The number of tickets that is written most often is 9.5 tickets per day. C) Half of the days have less than 9.5 tickets written and half of the days have more than 9.5 tickets written. D) The mean has no interpretation since 0.5 ticket can never be written. Answer: A 2) Suppose x is a random variable for which a Poisson probability distribution with λ = 7.9 provides a good characterization. Find μ for x. A) 7.9 B) 2.81 C) 4 D) 62.41 Answer: A 3) Suppose x is a random variable for which a Poisson probability distribution with λ = 3.6 provides a good characterization. Find σ for x. A) 1.9 B) 3.6 C) 1.8 D) 12.96 Answer: A 4) The number of goals scored at State College soccer games follows a Poisson process with a goal scored approximately every 18 minutes (a soccer game consists of 2 45-minute halves). What is the mean number of goals scored during a game A) 5 B) 2.5 C) 0.11 D) 1.25 Answer: A 5) The number of goals scored at State College soccer games follows a Poisson process with a goal scored approximately every 18 minutes (a soccer game consists of 2 45-minute halves). What is the standard deviation of the mean number of goals scored during a game? A) 2.34 B) 1.58 C) 1.12 D) 1.27 Answer: A 6) The university police department keeps track of the number of tickets it write in a year. Last year the campus police wrote 1460 tickets. Ticket writing on campus follows a Poisson process. What is the mean number of tickets written per day by the campus police? A) 4 B) 2 C) 6.08 D) 5 Answer: A 7) The university police department keeps track of the number of tickets it write in a year. Last year the campus police wrote 1460 tickets. Ticket writing on campus follows a Poisson process. What is the standard deviation of the number of tickets written per day by the campus police? A) 2 B) 1.4 C) 2.47 D) 1.73 Answer: A
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6.4 The Hypergeometric Probability Distribution (Online) 1 Determine whether a probability experiment is a hypergeometric experiment. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the probability experiment represents a hypergeometric probability experiment. If it does, determine the values of N, n, and k and list the possible values of the random variable X. 1) A jury is to be selected from a pool of 38 potential jurors. The defendant faces the death penalty if convicted. Of the potential jurors, 7 are opposed to the death penalty. The jury consists of 12 randomly selected jurors. The random variable X represents the number of jurors who oppose the death penalty. A) hypergeometric; N = 38, n = 12, k = 7, x = 0, 1, 2, . . . , 7 B) hypergeometric; N = 38, n = 7, k = 12, x = 0, 1, 2, . . . , 7 C) hypergeometric; N = 38, n = 12, k = 7, x = 0, 1, 2, . . . ,12 D) not hypergeometric Answer: A 2) An electronics store receives a shipment of 55 flat screen TVs of which 12 are defective. During the quality control inspection, 4 TVs are selected at random from the shipment for testing. The random variable X represents the number of defective computers in the sample A) hypergeometric; N = 55, n = 4, k = 12, x = 0, 1, 2, 3, 4 B) hypergeometric; N = 55, n = 12, k = 4, x = 0, 1, 2, 3, 4 C) hypergeometric; N = 55, n = 4, k = 12, x = 0, 1, 2, . . . , 12 D) not hypergeometric Answer: A 3) A university must choose a team of 5 students to participate in a TV quiz show. The students will be chosen at random from a pool of 50 potential participants of whom 29 are women. The random variable X represents the number of women on the team. A) hypergeometric; N = 50, n = 5, k = 29, x = 0, 1, 2, 3, 4, 5 B) hypergeometric; N = 50, n = 29, k = 5, x = 0, 1, 2, 3, 4, 5 C) hypergeometric; N = 50, n = 5, k = 29, x = 0, 1, 2, . . . , 29 D) not hypergeometric Answer: A 4) Of the 3800 students at a college, 280 are mature students. 50 students are selected at random from the 3800 students at the college and asked to participate in a survey. The questions in the survey relate to the student's need for financial assistance. The random variable X represents the number of mature students in the sample. A) hypergeometric; N = 3800, n = 50, k = 280, x = 0, 1, 2, . . . , 50 B) hypergeometric; N = 3800, n = 280, k = 50, x = 0, 1, 2, . . . , 50 C) hypergeometric; N = 3800, n = 50, k = 280, x = 0, 1, 2, . . . , 280 D) not hypergeometric Answer: D 2 Compute the probabilities of hypergeometric experiments. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A hypergeometric probability experiment is conducted with the given parameters. Compute the probability of obtaining x successes. N = 130, n = 10, k = 25, x = 3 A) 0.196,5 B) 0.191,4 C) 0.186,7 D) 0.172,2 Answer: A
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2) A manufacturer receives a shipment of 180 laptop computers of which 12 are defective. To test the shipment, the quality control engineer randomly selects 15 computers from the shipment and tests them. The random variable X represents the number of defective computers in the sample. What is the probability of obtaining 4 defective computers? A) 0.009,4 B) 0.012,6 C) 0.013,3 D) 0.009,9 Answer: A 3) A university must choose a team of 6 students to participate in a TV quiz show. The students will be chosen from a pool of 32 potential participants of whom 14 are women. If the 6 students are chosen at random, what is the probability that the team will contain 2 women? A) 0.307,3 B) 0.276,6 C) 0.287,4 D) 0.258,7 Answer: A Solve the problem. 4) In a lottery, a player selects six numbers between 1 and 30 inclusive. The six winning numbers (all different) are selected at random from the numbers 1-30. To win a prize, the player must match three or more of the winning numbers. What is the probability that the player matches exactly 3 numbers? A) 0.068,2 B) 0.056,8 C) 0.079,5 D) 0.085,2 Answer: A 5) An electronics store receives a shipment of 50 flat screen TVs of which 9 are defective. During the quality control inspection, 3 TVs are selected at random from the shipment for testing. The shipment will only be accepted if all 3 TVs pass the inspection. What is the probability that the shipment will be accepted? A) 0.543,9 B) 0.551,4 C) 0.005,8 D) 0.004,3 Answer: A 6) A IRS auditor randomly selects 3 tax returns from 59 returns of which 10 contain errors. What is the probability that none of the returns she selects contains an error? A) 0.566,7 B) 0.572,8 C) 0.004,9 D) 0.003,7 Answer: A 7) Among the contestants in a competition are 42 women and 24 men. If 5 winners are randomly selected, what is the probability that they are all men? A) 0.004,76 B) 0.049,97 C) 0.104,36 D) 0.060,93 Answer: A Provide an appropriate response. 8) A jury is to be selected from a pool of 36 potential jurors. The defendant faces the death penalty if convicted. Of the potential jurors, 7 are opposed to the death penalty and would not convict regardless of the evidence. The prosecutor knows that if even one juror opposes the death penalty, they will have no chance of getting a conviction. If none of the jurors opposes the death penalty they will have a chance of getting a conviction. What is the probability that none of the jurors opposes the death penalty, if the jury consists of 12 randomly selected jurors? A) 0.041,5 B) 0.039,4 C) 0.074,7 D) 0.078,4 Answer: A
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3 Compute the mean and standard deviation of a hypergeometric random variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute the mean and standard deviation of the hypergeometric random variable X. 1) N = 70, n = 15, k = 25 A) μ X = 5.36, σX = 1.66 B) μ X = 5.36, σX = 1.86 C) μ X = 0.36, σX = 1.66
D) μ X = 5.89, σX = 1.86
Answer: A 2) In a lottery, a player must choose 6 numbers between 1 and 43 inclusive. Six balls are then randomly selected from an urn containing 43 balls numbered from 1 to 43. The random variable X represents the number of matching numbers. What are the mean and standard deviation of the random variable X? A) μ X = 0.837, σX = 33.459 B) μ X = 0.837, σX = 0.849 C) μ X = 0.140, σX = 33.459
D) μ X = 0.140, σX = 0.849
Answer: A 3) A manufacturer receives a shipment of 200 laptop computers of which 7 are defective. To test the shipment, the quality control engineer randomly selects 15 computers from the shipment and tests them. The random variable X represents the number of defective computers in the sample. What are the mean and standard deviation of the random variable X? A) μ X = 0.525, σX = 0.686 B) μ X = 0.525, σX = 0.712 C) μ X = 0.035, σX = 0.686
D) μ X = 0.525, σX = 0.471
Answer: A
6.5 Combining Random Variables 1 Compute the mean of the sum of random variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) If E(X) = 1046 and σx =57, compute E(2X). A) 2092 B) 523
C) 1103
D) 2206
C) 25000
D) 200
Answer: A 2) If E(X) = 250 and σx=50, compute E(X-100). A) 150 B) 350 Answer: A 3) Stephen's average commute time to work is 47minutes with standard deviation of 12 minutes. What is Stephen ’s expected commute time total for a five day work week? A) 235 B) 47 C) 52 D) 94 Answer: A 2 Compute the variance and standard deviation of the sum of random variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) If E(X) = 1046 and σx=57, compute σ X-10 A) 57 Answer: A
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B) 47
C) 3259
D) 58
2) If E(X) = 250 and σx=50, compute σ 2X. A) 10
B) 100
C) 500
D) 350
Answer: A 3) Stephen's average commute time to work is 47 minutes with standard deviation of 12 minutes. What is Stephen ’s standard deviation in a five day work week? A) 60 B) 50 C) 235 D) 564 Answer: A
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Ch. 6 Discrete Probability Distributions Answer Key 6.1 Discrete Random Variables 1 Distinguish between discrete and continuous random variables. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 2 Identify discrete probability distributions. 1) A 2) A 3) A 4) A 5) A 6) A
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3 Graph discrete probability distributions. 1) (a) x P(x) 1 0 8
(b)
2)
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1
3 8
2
3 8
3
1 8
3)
4)
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5) (a) x P(x) 0 0.125 1 0.428 2 0.256 3 0.108 4 0.083 (b)
4 Compute and interpret the mean and standard deviation of a discrete random variable. 1) A 2) A 3) μ = ∑x ∙ p(x) = 2(0.2) + 5(0.3) + 10(0.3) + 14(0.2) = 7.7 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) μ = 1.596; σ = 1.098 14) A 15) A 16) A 17) A 5 Interpret the mean of a discrete random variable as an expected value. 1) A 2) A 3) A 4) A 5) μ = ∑xp(x) ≈ 3.9
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6) Let x = cost of fare paid by passenger. The probability distribution for x is: x $96 $320 x(p) 1/5 4/5 1 4 The expected cost is E(x) = μ = ∑x ∙ p(x) = $96 + $320 = $275.20 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. 7) 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 0.001
$12,000 0.01
$5,750 0.05
$2,000 0.10
-$500 0.839
Note: These losses paid have already considered the $500 deductible paid by the owner. The expected loss paid is: μ = ∑x ∙ p(x) = $24,500(0.001) + $12,000(0.01) + $5,750(0.05) + $2,000(0.10) - $500(0.839) = $212.50 In order to average $660 profit per policy sold, the insurance company must charge an annual premium of $212.50 + $660 = $872.50. 8) A 9) A 10) A 6 Compute the Standard Deviation of a Discrete Random Variable. 1) A 2) A 3) 1.2 4) 176
6.2 The Binomial Probability Distribution 1 Determine whether a probability experiment is a binomial experiment. 1) binomial experiment 2) Not a binomial experiment. There are more than two outcomes. 3) Not a binomial experiment. There are more than two outcomes. 4) binomial experiment. 5) Not a binomial experiment. There are more than two outcomes. 6) binomial experiment. 7) binomial experiment. 8) Not a binomial experiment. There are more than two outcomes. 9) Not a binomial experiment. There are more than two outcomes. 10) Not a binomial experiment. The probability of success is not the same for each trial. 11) A 12) A 13) A 2 Compute probabilities of binomial experiments. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A Page 35
11) A 12) A 13) A 14) A 15) A 16) A 17) 0.0005 18) A 19) A 20) A 21) A 22) A 3 Compute the mean and standard deviation of a binomial random variable. 1) μ = np = 600(0.5) = 300; σ = npq = 600(0.5)(0.5) = 12.25 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 4 Graph a binomial probability distribution. 1)
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2)
3)
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4)
6.3 The Poisson Probability Distribution 1 Determine if a probability experiment follows a Poisson process. 1) A 2) A 3) A 2 Compute probabilities of a Poisson random variable. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) Let x = the number of death claims received per day. Then x is a Poisson random variable with λ = 3. P(x ≥ 7) = 1 - P(x ≤ 6) = 0.033,509 20) Let x = the number of accidents that occur on the stretch of road during a month. Then x is a Poisson random variable with λ = 8.5. P(x < 2) = P(x = 0) + P(x = 1) = 0.001,933 21) Let x = the number of babies born during a one hour period at this hospital's maternity wing. Then x is a Poisson random variable with λ = 3. P(x = 2) = 0.224,042 Page 38
3 Find the mean and standard deviation of a Poisson random variable. 1) A 2) A 3) A 4) A 5) A 6) A 7) A
6.4 The Hypergeometric Probability Distribution (Online) 1 Determine whether a probability experiment is a hypergeometric experiment. 1) A 2) A 3) A 4) D 2 Compute the probabilities of hypergeometric experiments. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 3 Compute the mean and standard deviation of a hypergeometric random variable. 1) A 2) A 3) A
6.5 Combining Random Variables 1 Compute the mean of the sum of random variables. 1) A 2) A 3) A 2 Compute the variance and standard deviation of the sum of random variables. 1) A 2) A 3) A
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Ch. 7 The Normal Probability Distribution 7.1 Properties of the Normal Distribution 1 Use the uniform probability distribution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) High temperatures in a certain city for the month of August follow a uniform distribution over the interval 62°F to 88°F. What is the probability that a randomly selected August day has a high temperature that exceeded 67°F? A) 0.807,7 B) 0.192,3 C) 0.446,7 D) 0.038,5 Answer: A 2) High temperatures in a certain city for the month of August follow a uniform distribution over the interval 76°F to 106°F. Find the high temperature which 90% of the August days exceed. A) 79°F B) 103°F C) 86°F D) 106°F Answer: A 3) The diameter of ball bearings produced in a manufacturing process can be explained using a uniform distribution over the interval 5.5 to 7.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 6.1 millimeters? A) 0.7 B) 0.813,3 C) 3 D) 0.469,2 Answer: A 4) Suppose x is a uniform random variable which could take on values between 40 and 80. Find the probability that a randomly selected observation exceeds 64. A) 0.4 B) 0.6 C) 0.1 D) 0.9 Answer: A 5) Suppose x is a uniform random variable which could take on values between 20 and 90. Find the probability that a randomly selected observation is between 23 and 85. A) 0.89 B) 0.11 C) 0.5 D) 0.8 Answer: A 6) A machine is set to pump cleanser into a process at the rate of 7 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 6.5 to 9.5 gallons per minute. Find the probability that between 7.0 gallons and 8.0 gallons are pumped during a randomly selected minute. A) 0.33 B) 0.67 C) 0 D) 1 Answer: A 7) Suppose a uniform random variable can be used to describe the outcome of an experiment with the outcomes ranging from 10 to 90. What is the probability that this experiment results in an outcome less than 20? A) 0.13 B) 0.1 C) 0.18 D) 1 Answer: A 8) A random number generator is set to generate integer random numbers between 1 and 10 inclusive following a uniform distribution. What is the probability of the random number generator generating a 7 or less? A) 0.7 B) 0 C) 0.07 D) 0.5 Answer: A
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9) True or False: In a uniform probability distribution, any random variable is just as likely as any other random variable to occur, provided the random variables belong to the distribution. A) True B) False Answer: A 2 Graph a normal curve. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 4 and standard deviation of 1. The graphs would A) Have the same height but one would be shifted 4 units to the right. B) Have the same height but one would be shifter 4 units to the left. C) Have no horizontal shift but one would be steeper that the other. D) Have no horizontal shift but one would be flatter than the other. Answer: A 2) Compare a graph of the normal density function with mean of 0 and standard deviation of 1 with a graph of a normal density function with mean equal to 0 and standard deviation of 0.5. The graphs would A) Have no horizontal shift but one would be steeper that the other. B) Have no horizontal shift but one would be flatter than the other. C) Have the same height but one would be shifted 4 units to the right. D) Have the same height but one would be shifter 4 units to the left. Answer: A 3) Draw a normal curve with μ = 150 and σ = 20. Label the mean and the inflection points. A)
130
150
170
B)
110
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150
190
C)
140
150
160
D)
130 Answer: A
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150
170
3 State the properties of the normal curve. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) You are performing an obesity study about the effects of school lunch choices on preschoolers and randomly sample one all-day class for the preschoolers’ weights in pounds. A previous study found the weights to be normally distributed with a mean of 30 and a standard deviation of 4. In your sample of 30 preschool children their weights are as follows. 25 25 26 26.5 27 27 27.5 28 28 28.5 29 29 30 30 30.5 31 31 32 32.5 32.5 33 33 34 34.5 35 35 37 37 38 38 a) Draw a histogram to display the data. Is it reasonable to assume that the weights are normally distributed? Why? b) Find the mean and standard deviation of your sample. c) Is there a high probability that the mean and standard deviation of your sample are consistent with those found in previous studies? Explain your reasoning. Answer: (a)
It is not reasonable to assume that the heights are normally distributed since the histogram is skewed. (b) μ = 31, σ = 3.86 (c) Yes. The mean and standard deviation are close.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) The graph of a normal curve is given. Use the graph to identify the value of μ and σ.
-12 -10 A) μ = -6, σ = 2
-8
-6
-4 -2 B) μ = 2, σ = -6
0 C) μ = -6, σ = 6
D) μ = 6, σ = -6
Answer: A 3) The normal density curve is symmetric about A) Its mean B) The horizontal axis C) An inflection point D) A point located one standard deviation from the mean Answer: A 4) The highest point on the graph of the normal density curve is located at A) its mean B) an inflection point C) μ + σ
D) μ + 3σ
Answer: A 5) Approximately ____% of the area under the normal curve is between μ - σ and μ + σ. A) 68 B) 95 C) 99.7 Answer: A
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D) 50
Determine whether the graph can represent a normal curve. If it cannot, explain why. 6)
A) The graph cannot represent a normal density function because it increases as x becomes very large or very small. B) The graph cannot represent a normal density function because it takes negative values for some values of x. C) The graph cannot represent a normal density function because the area under the graph is greater than 1. D) The graph can represent a normal density function. Answer: A 7)
A) The graph can represent a normal density function. B) The graph cannot represent a normal density function because it has no inflection points. C) The graph cannot represent a normal density function because as x increases without bound, the graph takes negative values. D) The graph cannot represent a normal density function because the area under the graph is greater than 1. Answer: A
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8)
A) The graph cannot represent a normal density function because the graph takes negative values for some values of x. B) The graph cannot represent a normal density function because the area under the graph is less than 1. C) The graph cannot represent a normal density function because it is not symmetric. D) The graph can represent a normal density function. Answer: A 9)
A) The graph cannot represent a normal density function because it does not approach the horizontal axis as x increases or decreases without bound. B) The graph cannot represent a normal density function because it is not bell shaped. C) The graph can represent a normal density function. D) The graph cannot represent a normal density function because it has no inflection points. Answer: A
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10)
A) The graph can represent a normal density function. B) The graph cannot represent a normal density function because it has no inflection points. C) The graph cannot represent a normal density function because its maximum value is too small. D) The graph cannot represent a normal density function because the area under the graph is less than 1. Answer: A 11)
A) The graph cannot represent a normal density function because it is not symmetric. B) The graph cannot represent a normal density function because as x increases without bound, the graph takes negative values. C) The graph cannot represent a normal density function because it is bimodal. D) The graph can represent a normal density function. Answer: A
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12)
A) The graph cannot represent a normal density function because it is bimodal. B) The graph cannot represent a normal density function because a normal density curve should approach but not reach the horizontal axis as x increases and decreases without bound. C) The graph can represent a normal density function. D) A and B are both true. Answer: D 4 Explain the role of area in the normal density function. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The analytic scores on a standardized aptitude test are known to be normally distributed with mean μ = 610 and standard deviation σ = 115. (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of test takers who scored less than 725. (c) Suppose the area under the normal curve to the left of X = 725 is 0.8413. Provide two interpretations of this result. Answer: (a), (b)
(c) The two interpretations are: (1) the proportion of test takers who scored less than 725 is 0.8413 and (2) the probability that a randomly selected test taker has a score less than 725 is 0.8413.
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2) The weight of 2-year old lions is known to be normally distributed with mean μ = 2200 grams and standard deviation σ = 365 grams (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of lions who weighed more than 2930 grams. (c) Suppose the area under the normal curve to the left of X = 2930 is 0.0228. Provide two interpretations of this result. Answer: (a), (b)
(c) The two interpretations are: (1) the proportion of hyraxes who weighed more than 2930 is 0.0228 and (2) the probability that a randomly selected hyrax weighs more than 2930 is 0.0228. 3) The average mpg (miles per gallon) of a new model of motorcycle is known to be normally distributed with mean μ = 27.4 mpg and standard deviation σ = 2.9 mpg. (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of mpgs between 29.3 and 23.7. (c) Suppose the area under the normal curve to between X = 29.3 and X = 23.7 is 0.6419. Provide two interpretations of this result. Answer: (a), (b)
(c) The two interpretations are: (1) the proportion of mpg between 29.3 and 23.7 is 0.6419 and (2) the probability that a randomly selected mpg is between 29.3 and 23.7 is 0.6419. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) True or False: The area under the normal curve drawn with regard to the population parameters is the same as the proportion of the population that has these characteristics. A) True B) False Answer: A
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5) True or False: The area under the normal curve drawn with regard to the population parameters is the same as the probability that a randomly selected individual of a population has these characteristics. A) True B) False Answer: A 6) True or False: The proportion of the population that has certain characteristics is the same as the probability that a randomly selected individual of the population has these same characteristics. A) True B) False Answer: A
7.2 Applications of the Normal Distribution 1 Find and interpret the area under a normal curve. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Find the area under the standard normal curve to the left of z = 1.5. A) 0.9332 B) 0.0668 C) 0.5199
D) 0.7612
Answer: A 2) Find the area under the standard normal curve to the left of z = 1.25. A) 0.8944 B) 0.1056 C) 0.2318
D) 0.7682
Answer: A 3) Find the area under the standard normal curve to the right of z = 1. A) 0.1587 B) 0.8413 C) 0.1397
D) 0.5398
Answer: A 4) Find the area under the standard normal curve to the right of z = -1.25. A) 0.8944 B) 0.5843 C) 0.6978
D) 0.7193
Answer: A 5) Find the area under the standard normal curve between z = 0 and z = 3. A) 0.4987 B) 0.9987 C) 0.0010
D) 0.4641
Answer: A 6) Find the area under the standard normal curve between z = 1 and z = 2. A) 0.1359 B) 0.8413 C) 0.5398
D) 0.2139
Answer: A 7) Find the area under the standard normal curve between z = -1.5 and z = 2.5. A) 0.9270 B) 0.7182 C) 0.6312
D) 0.9831
Answer: A 8) Find the area under the standard normal curve between z = 1.5 and z = 2.5. A) 0.0606 B) 0.9938 C) 0.9332
D) 0.9816
Answer: A 9) Find the area under the standard normal curve between z = -1.25 and z = 1.25. A) 0.7888 B) 0.8817 C) 0.6412 Answer: A
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D) 0.2112
10) Find the sum of the areas under the standard normal curve to the left of z = -1.25 and to the right of z = 1.25. A) 0.2112 B) 0.7888 C) 0.1056 D) 0.3944 Answer: A Determine the area under the standard normal curve that lies between: 11) z = 1 and z = 3 A) 0.157,4 B) 0.842,6 C) 0.0004
D) 0.0007
Answer: A 12) z = 0.5 and z = 1.4 A) 0.227,7
B) 0.691,5
C) 0.9192
D) 0.308,5
B) 0.420,7
C) 0.579,3
D) 0.5
B) 0.242
C) 0.758
D) 0.0228
Answer: A 13) z = -0.2 and z = 0.2 A) 0.158,6 Answer: A 14) z = -2 and z = -0.7 A) 0.219,2 Answer: A Find the indicated probability. 15) Assume that the random variable X is normally distributed, with mean μ = 40 and standard deviation σ = 8. Compute the probability P(X < 50). A) 0.8944 B) 0.8849 C) 0.1056 D) 0.9015 Answer: A 16) Assume that the random variable X is normally distributed, with mean μ = 60 and standard deviation σ = 5. Compute the probability P(X > 64). A) 0.2119 B) 0.1977 C) 0.7881 D) 0.2420 Answer: A 17) Assume that the random variable X is normally distributed, with mean μ = 80 and standard deviation σ = 12. Compute the probability P(47 < X < 95). A) 0.8914 B) 0.8819 C) 0.8944 D) 0.7888 Answer: A Provide an appropriate response. 18) A physical fitness association is including the mile run in its high school fitness test. The time for this event is known to possess a normal distribution with a mean of 470 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected high school student can run the mile in less than 332 seconds. A) 0.0107 B) 0.4893 C) 0.9893 D) 0.5107 Answer: A 19) A physical fitness association is including the mile run in its high school fitness test. The time for this event is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected high school student will take longer than 312 seconds to run the mile. A) 0.9893 B) 0.4893 C) 0.0107 D) 0.5107 Answer: A
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20) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.19 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains fewer than 12.09 ounces of beer. A) 0.0062 B) 0.4938 C) 0.9938 D) 0.5062 Answer: A 21) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 11.14 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains more than 11.14 ounces of beer. A) 0.5 B) 1 C) 0 D) 0.4 Answer: A 22) Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.13 ounces and a standard deviation of 0.04 ounce. Find the probability that the bottle contains between 12.03 and 12.09 ounces. A) 0.1525 B) 0.8351 C) 0.1649 D) 0.8475 Answer: A 23) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 5.5 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 5.0 minutes. A) 0.3085 B) 0.1915 C) 0.3551 D) 0.2674 Answer: A 24) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will take between 2.0 and 4.5 minutes to find a parking spot in the library lot. A) 0.7745 B) 0.4938 C) 0.0919 D) 0.2255 Answer: A 25) The amount of soda a dispensing machine pours into a 12 ounce can of soda follows a normal distribution with a mean of 12.03 ounces and a standard deviation of 0.02 ounce. The cans only hold 12.05 ounces of soda. Every can that has more than 12.05 ounces of soda poured into it causes a spill and the can needs to go through a special cleaning process before it can be sold. What is the probability a randomly selected can will need to go through this process? A) 0.1587 B) 0.3413 C) 0.8413 D) 0.6587 Answer: A 26) A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls followed a normal distribution with an average of $700 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 a randomly selected month had a PCE of between $575.00 and $790.00? A) 0.9579 B) 0.0421 C) 0.9999 D) 0.0001 Answer: A 27) A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $600 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 a PCE that falls below $450. A) 0.0013 B) 0.750,0 C) 0.250,0 D) 0.9987 Answer: A Page 13
28) 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 3,000 miles. What is the probability a particular tire of this brand will last longer than 57,000 miles? A) 0.8413 B) 0.1587 C) 0.2266 D) 0.7266 Answer: A 29) 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 1,700 miles. What is the probability a certain tire of this brand will last between 56,430 miles and 56,940 miles? A) 0.0180 B) 0.9813 C) 0.4920 D) 0.4649 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 30) A firm believes the internal rate of return for its proposed investment can best be described by a normal distribution with mean 22% and standard deviation 3%. What is the probability that the internal rate of return for the investment will be at least 17.5%? Answer: Let x be the internal rate of return. Then x is a normal random variable with μ = 22% and σ = 3%. To determine the probability that x is at least 17.5%, we need to find the z-value for x = 17.5%. x - μ 17.5 - 22 z= = = -1.5 σ 3 P(x ≥ 17.5%) = P(z ≥ -1.5) = 1 - P(z ≤ -1.5) = 1 - 0.0668 = 0.9332 31) A firm believes the internal rate of return for its proposed investment can best be described by a normal distribution with mean 44% and standard deviation 3%. What is the probability that the internal rate of return for the investment exceeds 50%? Answer: Let x be the internal rate of return. Then x is a normal random variable with μ = 44% and σ = 3%. To determine the probability that x exceeds 50%, we need to find the z-value for x = 50%. z=
x - μ 50 - 44 = =2 σ 3
P(x > 50%) = P(z ≥ 2) = 1 - P(z ≤ 2) = 1 - 0.9772 = 0.0228 32) 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 equal to 449 tomatoes per day and a standard deviation equal to 30 tomatoes per day. If there are 407 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? Answer: Let x be the number of tomatoes sold per day. Then x is a normal random variable with μ = 449 and σ = 30. To determine if all 83 tomatoes will be sold, we need to find the z-value for x = 407. z=
x - μ 407 - 449 = = -1.4 σ 30
P(x ≥ 407) = P(z ≥ -1.4) = 1 - P(z ≤ -1.4) = 1 - 0.0808 = 0.9192
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 33) Given a distribution that follows a standard normal curve, what does the graph of the curve do as z increases in the positive direction? A) The graph of the curve approaches zero. B) The graph of the curve approaches 1. C) The graph of the curve approaches an inflection point. D) The graph of the curve eventually intersects the horizontal axis. Answer: A 2 Find the value of a normal random variable. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated z-score. 1) Find the z-score for which the area under the standard normal curve to its left is 0.96 A) 1.75 B) 1.82 C) 1.03
D) -1.38
Answer: A 2) Find the z-score for which the area under the standard normal curve to its left is 0.40 A) -0.25 B) 0.25 C) 0.57
D) -0.57
Answer: A 3) Find the z-score for which the area under the standard normal curve to its left is 0.09. A) -1.34 B) -1.39 C) -1.26
D) -1.45
Answer: A 4) Find the z-score for which the area under the standard normal curve to its left is 0.04. A) -1.75 B) -1.89 C) -1.48
D) -1.63
Answer: A 5) Find the z-score for which the area under the standard normal curve to its left is 0.70. A) 0.53 B) 0.98 C) 0.81
D) 0.47
Answer: A 6) Find the z-score for which the area under the standard normal curve to its right is 0.07. A) 1.48 B) 1.39 C) 1.26 D) 1.45 Answer: A 7) Find the z-score for which the area under the standard normal curve to its right is 0.70. A) -0.53 B) -0.98 C) -0.81 D) -0.47 Answer: A 8) Find the z-score for which the area under the standard normal curve to its right is 0.09. A) 1.34 B) 1.39 C) 1.26 D) 1.45 Answer: A 9) Find the z-score having area 0.86 to its right under the standard normal curve; that is, find z 0.86 . A) -1.08
B) 1.08
C) 0.8051
D) 0.5557
Answer: A 10) For a standard normal curve, find the z-score that separates the bottom 90% from the top 10%. A) 1.28 B) 0.28 C) 1.52 D) 2.81 Answer: A
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11) For a standard normal curve, find the z-score that separates the bottom 30% from the top 70%. A) -0.53 B) -0.98 C) -0.47 D) -0.12 Answer: A 12) For a standard normal curve, find the z-score that separates the bottom 70% from the top 30%. A) 0.53 B) 0.98 C) 0.47 D) 0.12 Answer: A 13) Determine the two z-scores that separate the middle 87.4% of the distribution from the area in the tails of the standard normal distribution. A) -1.53, 1.53 B) -1.39, 1.39 C) -1.46, 1.46 D) -1.45, 1.45 Answer: A 14) Determine the two z-scores that separate the middle 96% of the distribution from the area in the tails of the standard normal distribution. A) -2.05 and 2.05 B) -1.75 and 1.75 C) 0 and 2.05 D) -2.33 and 2.33 Answer: A 15) Find the z-scores for which 90% of the distribution's area lies between -z and z. A) (-1.645, 1.645) B) (-2.33, 2.33) C) (-1.96, 1.96)
D) (-0.99, 0.99)
Answer: A 16) Find the z-scores for which 98% of the distribution's area lies between -z and z. A) (-2.33, 2.33) B) (-1.645, 1.645) C) (-1.96, 1.96)
D) (-0.99, 0.99)
Answer: A Find the value of z α. 17) z0.13 A) 1.13
B) -1.13
C) 0.55
D) 1.08
Answer: A Find the indicated percentile. 18) Assume that the random variable X is normally distributed with mean μ = 70 and standard deviation σ = 14. Find the 11th percentile for X. A) 52.78 B) 87.22 C) 62.44 D) 50.68 Answer: A Provide an appropriate response. 19) A physical fitness association is including the mile run in its hight school fitness test. The time for this event is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 40 seconds. The fitness association wants to recognize students whose times are among the top (or fastest) 10% with certificates of recognition. What time would a student need to beat in order to earn a certificate of recognition from the fitness association? A) 398.8 sec B) 384.2 sec C) 501.2 sec D) 515.8 sec Answer: A 20) A physical fitness association is including the mile run in its high school fitness test. The time for this event is known to possess a normal distribution with a mean of 440 seconds and a standard deviation of 60 seconds. Between what times do we expect most (approximately 95%) of the students to run the mile? A) between 322.4 and 557.6 sec B) between 341.3 and 538.736 sec C) between 345 and 535 sec D) between 0 and 538.736 sec Answer: A
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21) The amount of corn chips dispensed into a 48-ounce bag by a dispensing machine has been identified as possessing a normal distribution with a mean of 48.5 ounces and a standard deviation of 0.2 ounce. What chip amount represents the 67th percentile for the bag weight distribution? Round to the nearest hundredth. A) 48.59 oz B) 48.09 oz C) 48.63 oz D) 48.13 oz Answer: A 22) Suppose a brewery has a filling machine that fills 12-ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.19 ounces and a standard deviation of 0.04 ounce. The company is interested in reducing the amount of extra beer that is poured into the 12 ounce bottles. The company is seeking to identify the highest 1.5% of the fill amounts poured by this machine. For what fill amount are they searching? Round to the nearest thousandth. A) 12.277 oz B) 11.913 oz C) 12.103 oz D) 12.087 oz Answer: A 23) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 6.5 minutes and a standard deviation of 1 minute. Find the cut-off time which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot. A) 7.2 min B) 7.0 min C) 6.8 min D) 7.3 min Answer: A 24) The amount of soda a dispensing machine pours into a 12 ounce can of soda follows a normal distribution with a standard deviation of 0.02 ounce. Every can that has more than 12.05 ounces of soda poured into it causes a spill and the can needs to 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 need to be cleaned because of spillage to 3%? A) 12.012,4 oz B) 12.087,6 oz C) 12.006,6 oz D) 12.093,4 oz Answer: A 25) A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows 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 point in the distribution below which 2.5% of the PCE's fell. A) $402.00 B) $598.00 C) $12.50 D) $487.50 Answer: A 26) A brewery has a beer dispensing machine that dispenses beer into the company's 12 ounce bottles. The distribution for the amount of beer dispensed by the machine follows a normal distribution with a standard deviation of 0.17 ounce. The company can control the mean amount of beer dispensed by the machine. What value of the mean should the company use if it wants to guarantee that 98.5% of the bottles contain at least 12 ounces (the amount on the label)? Round to the nearest thousandth. A) 12.369 oz B) 12.413 oz C) 12.003 oz D) 12.001 oz Answer: A 27) 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 1,400 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty? A) 57,550 mi B) 62,450 mi C) 58,600 mi D) 61,400 mi Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 28) 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 562 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. Answer: Let x be a score on this exam. Then x is a normally distributed random variable with μ = 562 and σ = 72. We want to find the value of x0 , such that P(x > x0 ) = 0.10. The z-score for the value x = x0 is z=
x0 - μ σ
=
x0 - 562 72
x0 - 562
P(x > x0 ) = P z > We find
.
72
x0 - 562 72
= 0.10
≈ 1.28.
x0 - 562 = 1.28(72) ⇒ x0 = 562 + 1.28(72) = 654.16 29) 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 554 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. Answer: Let x be a score on this exam. Then x is a normally distributed random variable with μ = 554 and σ = 72. We want to find the value of x0 , such that P(x > x0 ) = 0.80. The z-score for the value x = x0 is z=
x0 - μ σ
=
x0 - 554 72
P(x > x0 ) = P z > We find
x0 - 554 72
.
x2 - 554 72
= 0.80
≈ -0.84.
x0 - 554 = -0.84(72) ⇒ x0 = 554 - 0.84(72) = 493.52 30) 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 equal to 138 tomatoes per day and a standard deviation equal to 30 tomatoes per day. 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? Answer: Let x be the number of tomatoes sold per day. Then x is a normal random variable with μ = 138 and σ = 30. We want to find the value x0 , such that P(x > x0 ) = .015. The z-value for the point x = x0 is z=
x - μ x0 - 138 . = σ 30
P(x > x0 ) = P(z > We find
x0 - 138 30
x0 - 138 30
)= 0.015
= 2.17
x0 - 138 = 2.17(30) ⇒ x0 = 138 + 2.17(30) = 203
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7.3 Assessing Normality 1 Use normal probability plots to assess normality. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. 1) An industrial psychologist conducted an experiment in which 40 employees that were identified as "chronically tardy" by their managers were divided into two groups of size 20. Group 1 participated in the new "It's Great to be Awake!" program, while Group 2 had their pay docked. The following data represent the number of minutes that employees in Group 1 were late for work after participating in the program. 77 85 100 104 100 116 101 136 91 94 106 82 91 106 111 112 112 113 84 106 A) normally distributed B) not normally distributed Answer: A 2) The following data represent a random sample of the number of shares of a pharmaceutical company's stock traded for 20 days in 2000. 4.93 9.85 11.35 8.47 14.79 6.24 10.64 9.89 13.05 14.2 5.19 9.49 12.54 7.7 28.49 8.66 11.09 5.23 8.19 11.83 A) not normally distributed B) normally distributed Answer: A Provide an appropriate response. 3) A normal probability plot is a graph that plots _____________ versus _____________. A) observed data, normal scores B) normal score, observed data C) normal data, observed scores D) observed scores, normal data Answer: A 4) If sample data are taken from a population that is normally distributed, a normal probability plot of the observed data values versus the expected z scores will A) be approximately linear. B) have no discernable pattern. C) look exponential in nature. D) have a correlation coefficient near 0. Answer: A
7.4 The Normal Approximation to the Binomial Probability Distribution 1 Approximate binomial probabilities using the normal distribution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Compute P(x) using the binomial probability formula. Then determine whether the normal distribution can be used as an approximation for the binomial distribution. If so, approximate P(x) and compare the result to the exact probability. 1) n = 80, p = 0.6, x = 42 (Round the standard deviation to three decimal places to work the problem.) A) Exact: 0.035,4; Approximate: 0.034,4 B) Exact: 0.035,4; Approximate: 0.033,7 C) Exact: 0.034,7; Approximate: 0.033,7 D) Exact: 0.035,4; Approximate: 0.035,1 Answer: A
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Provide an appropriate response. 2) A student answers all 48 questions on a multiple-choice test by guessing. Each question has four possible answers, only one of which is correct. Find the probability that the student gets exactly 15 correct answers. Use the normal distribution to approximate the binomial distribution. A) 0.0823 B) 0.8577 C) 0.7967 D) 0.0606 Answer: A 3) If the probability of a newborn kitten being underweight is 0.5, find the probability that in 100 births, 55 or more will be underweight. Use the normal distribution to approximate the binomial distribution. A) 0.1841 B) 0.7967 C) 0.8159 D) 0.0606 Answer: A 4) A local concert center reports that it has been experiencing a 15% rate of no-shows on advanced reservations. Among 150 advanced reservations, find the probability that there will be fewer than 20 no-shows. Round the standard deviation to three decimal places to work the problem. A) 0.2451 B) 0.7549 C) 0.7967 D) 0.3187 Answer: A 5) Find the probability that in 200 tosses of a fair six-sided die, a three will be obtained at least 40 times. A) 0.1190 B) 0.8810 C) 0.0871 D) 0.3875 Answer: A 6) Find the probability that in 200 tosses of a fair six-sided die, a three will be obtained at most 40 times. A) 0.9147 B) 0.1190 C) 0.8810 D) 0.0853 Answer: A 7) A salesperson found that there was a 1% chance of a sale from her phone solicitations. Find the probability of getting 5 or more sales for 1000 telephone calls. Round the standard deviation to three decimal places to work the problem. A) 0.9599 B) 0.0401 C) 0.8810 D) 0.0871 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8) The author of an economics book has trouble deciding whether to use new or historical data in the book's examples. To solve the problem, the author flips a coin each time the problem arises. If a head shows, the author uses historical data and if a tail shows, the author uses new data from within the last 6 months. If this problem occurs 100 times in the book, what is the probability that historical data will be used 58 times? Answer: P(57.5 < x < 58.5) = P(1.50 < z < 1.70) = 0.9554 - 0.9332 = 0.0222 9) In a recent survey, 84% of the community favored building more parks in their neighborhood. You randomly select 16 citizens and ask each if they think the community needs more parks. Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the mean and standard deviation. If not, explain why. Answer: cannot use normal distribution, nq = (16)(0.16) = 2.56 < 5 10) A recent survey found that 73% of all teenagers own cell phones. You randomly select 35 teenagers, and ask if they own a cell phone. Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the mean and standard deviation, If not, explain why. Round to the nearest hundredth when necessary. Answer: use normal distribution, μ = 25.55 and σ = 2.63.
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11) According to government data, the probability than an adult never had the flu is 14%. You randomly select 45 adults and ask if they ever had the flu. Decide whether you can use the normal distribution to approximate the binomial distribution, If so, find the mean and standard deviation, If not, explain why. Round to the nearest hundredth when necessary. Answer: use normal distribution, μ = 6.3 and σ = 2.33. 12) For the following conditions, determine if it is appropriate to use the normal distribution to approximate a binomial distribution with n = 11 and p = 0.2. Answer: cannot use normal distribution 13) For the following conditions, determine if it is appropriate to use the normal distribution to approximate a binomial distribution with n = 44 and p = 0.7. Answer: can use normal distribution 14) The failure rate in a German class is 30%. In a class of 50 students, find the probability that exactly five students will fail. Use the normal distribution to approximate the binomial distribution. Round the standard deviation to three decimal places to work the problem. Answer: P(4.5 < X < 5.5) = P(-3.24 < z < -2.93) = 0.0017 - 0.0006 = 0.0011 15) A local rental car agency has 200 cars. The rental rate for the winter months is 60%. Find the probability that in a given winter month at least 140 cars will be rented. Use the normal distribution to approximate the binomial distribution. Round the standard deviation to three decimal places to work the problem. Answer: P(x ≥ 139.5) = 0.002,5 16) A local rental car agency has 100 cars. The rental rate for the winter months is 60%. Find the probability that in a given winter month fewer than 70 cars will be rented. Use the normal distribution to approximate the binomial distribution. Answer: P(x ≤ 69.5) = 0.973,8 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 17) True or False: In order to use a normal approximation to the binomial probability distribution, np(1 - p) ≥ 10. A) True B) False Answer: A 18) Assuming that all conditions are met to approximate a binomial probability distribution with the standard normal distribution, then to compute P(x ≥ 19) from the binomial distribution we must compute as the normal approximation. A) P(x ≥ 18.5)
B) P(x ≤ 18.5)
C) P(x ≥ 19.1)
D) P(x ≤ 18.9)
Answer: A 19) Assuming that all conditions are met to approximate a binomial probability distribution with the standard normal distribution, then to compute P(x ≤ 23) from the binomial distribution we must compute as the normal approximation. A) P(x ≤ 23.5)
B) P(x ≥ 23.5)
C) P(x ≤ 22.9)
D) P(x ≤ 23.1)
Answer: A 20) Assuming that all conditions are met to approximate a binomial probability distribution with the standard normal distribution, then to compute P(12 ≤ x ≤ 15) from the binomial distribution we must compute as the normal approximation. A) P(11.5 < x < 15.5) C) P(x > 15.5) and P(x < 11.5) Answer: A Page 21
B) P(12.5 < x < 14.5) D) P(x > 12.5) and P(x < 14.5)
Ch. 7 The Normal Probability Distribution Answer Key 7.1 Properties of the Normal Distribution 1 Use the uniform probability distribution. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 2 Graph a normal curve. 1) A 2) A 3) A 3 State the properties of the normal curve. 1) (a)
It is not reasonable to assume that the heights are normally distributed since the histogram is skewed. (b) μ = 31, σ = 3.86 (c) Yes. The mean and standard deviation are close. 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) D
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4 Explain the role of area in the normal density function. 1) (a), (b)
(c) The two interpretations are: (1) the proportion of test takers who scored less than 725 is 0.8413 and (2) the probability that a randomly selected test taker has a score less than 725 is 0.8413. 2) (a), (b)
(c) The two interpretations are: (1) the proportion of hyraxes who weighed more than 2930 is 0.0228 and (2) the probability that a randomly selected hyrax weighs more than 2930 is 0.0228. 3) (a), (b)
(c) The two interpretations are: (1) the proportion of mpg between 29.3 and 23.7 is 0.6419 and (2) the probability that a randomly selected mpg is between 29.3 and 23.7 is 0.6419. 4) A 5) A 6) A
7.2 Applications of the Normal Distribution 1 Find and interpret the area under a normal curve. 1) A 2) A 3) A Page 23
4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A 20) A 21) A 22) A 23) A 24) A 25) A 26) A 27) A 28) A 29) A 30) Let x be the internal rate of return. Then x is a normal random variable with μ = 22% and σ = 3%. To determine the probability that x is at least 17.5%, we need to find the z-value for x = 17.5%. x - μ 17.5 - 22 z= = = -1.5 σ 3 P(x ≥ 17.5%) = P(z ≥ -1.5) = 1 - P(z ≤ -1.5) = 1 - 0.0668 = 0.9332 31) Let x be the internal rate of return. Then x is a normal random variable with μ = 44% and σ = 3%. To determine the probability that x exceeds 50%, we need to find the z-value for x = 50%. z=
x - μ 50 - 44 = =2 σ 3
P(x > 50%) = P(z ≥ 2) = 1 - P(z ≤ 2) = 1 - 0.9772 = 0.0228 32) Let x be the number of tomatoes sold per day. Then x is a normal random variable with μ = 449 and σ = 30. To determine if all 83 tomatoes will be sold, we need to find the z-value for x = 407. z=
x - μ 407 - 449 = = -1.4 σ 30
P(x ≥ 407) = P(z ≥ -1.4) = 1 - P(z ≤ -1.4) = 1 - 0.0808 = 0.9192 33) A 2 Find the value of a normal random variable. 1) A 2) A 3) A 4) A 5) A 6) A Page 24
7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A 20) A 21) A 22) A 23) A 24) A 25) A 26) A 27) A 28) Let x be a score on this exam. Then x is a normally distributed random variable with μ = 562 and σ = 72. We want to find the value of x0 , such that P(x > x0 ) = 0.10. The z-score for the value x = x0 is z=
x0 - μ σ
=
x0 - 562 72
x0 - 562
P(x > x0) = P z > We find
.
72
x0 - 562 72
= 0.10
≈ 1.28.
x0 - 562 = 1.28(72) ⇒ x0 = 562 + 1.28(72) = 654.16 29) Let x be a score on this exam. Then x is a normally distributed random variable with μ = 554 and σ = 72. We want to find the value of x0 , such that P(x > x0 ) = 0.80. The z-score for the value x = x0 is z=
x0 - μ σ
=
x0 - 554 72
P(x > x0) = P z > We find
x0 - 554 72
.
x2 - 554 72
= 0.80
≈ -0.84.
x0 - 554 = -0.84(72) ⇒ x0 = 554 - 0.84(72) = 493.52 30) Let x be the number of tomatoes sold per day. Then x is a normal random variable with μ = 138 and σ = 30. We want to find the value x0, such that P(x > x0 ) = .015. The z-value for the point x = x0 is z=
x - μ x0 - 138 . = σ 30
P(x > x0) = P(z > We find
x0 - 138 30
x0 - 138 30
)= 0.015
= 2.17
x0 - 138 = 2.17(30) ⇒ x0 = 138 + 2.17(30) = 203 Page 25
7.3 Assessing Normality 1 Use normal probability plots to assess normality. 1) A 2) A 3) A 4) A
7.4 The Normal Approximation to the Binomial Probability Distribution 1 Approximate binomial probabilities using the normal distribution. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) P(57.5 < x < 58.5) = P(1.50 < z < 1.70) = 0.9554 - 0.9332 = 0.0222 9) cannot use normal distribution, nq = (16)(0.16) = 2.56 < 5 10) use normal distribution, μ = 25.55 and σ = 2.63. 11) use normal distribution, μ = 6.3 and σ = 2.33. 12) cannot use normal distribution 13) can use normal distribution 14) P(4.5 < X < 5.5) = P(-3.24 < z < -2.93) = 0.0017 - 0.0006 = 0.0011 15) P(x ≥ 139.5) = 0.002,5 16) P(x ≤ 69.5) = 0.973,8 17) A 18) A 19) A 20) A
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Ch. 8 Sampling Distributions 8.1 Distribution of the Sample Mean 1 Describe the distribution of the sample mean: normal population. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine μ
and σ from the given parameters of the population and the sample size. Round the answer to the x x nearest thousandth where appropriate. 1) μ = 38, σ = 24, n = 16 A) μ = 38, σ = 6 B) μ = 38, σ = 24 x x x x C) μ = 38, σ = 1.5 D) μ = 9.5, σ = 6 x x x x Answer: A 2) μ = 56, σ = 11, n = 19 A) μ = 56, σ = 2.524 x x C) μ = 56, σ = 11 x x
= 0.579 x D) μ = 32.332, σ = 2.524 x x B) μ
x
= 56, σ
Answer: A Provide an appropriate response. 3) What are the values of μ
A) μ C) μ
x x
= 380, σ = 380, σ
Answer: A
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x x
x
360 = 20 = 60
and σ
380
x
for the sampling distribution of the sample mean shown?
400 B) μ D) μ
x x
= 20, σ
x = 380, σ
= 380 x
= 40
4) What are the values of μ
A) μ C) μ
x
= 7, σ
x =
7, σ
x
x
6.95 0.05 =
x =
and σ
7
x
for the sampling distribution of the sample mean shown?
7.05 =7 x D) μ = 7, σ = 0.1 x x B) μ
0.15
x
= 0.05, σ
Answer: A 5) The sampling distribution of the sample mean is shown. If the sample size is n = 16, what is the standard deviation of the population from which the sample was drawn? Round to the nearest thousandth where appropriate.
310 A) σ = 40
320 330 B) σ = 160
C) σ = 2.5
D) σ = 0.625
Answer: A 6) The sampling distribution of the sample mean is shown. If the sample size is n = 16, what is the standard deviation of the population from which the sample was drawn? Round to the nearest thousandth where appropriate.
7.97 A) σ = 0.12 Answer: A Page 2
8
8.03 B) σ = 0.48
C) σ = 0.008
D) σ = 0.002
7) Suppose a population has a mean of 7 for some characteristic of interest and a standard deviation of 9.6. A sample is drawn from this population of size 64. What is the standard error of the mean? A) 1.2 B) 0.7 C) 0.15 D) 3.3 Answer: A 8) Suppose a population has a mean weight of 180 pounds and a standard deviation of 25 pounds. A sample of 100 items is drawn from this population. What is the standard error of the mean? A) 2.5 B) 1.8 C) 18.0 D) 0.25 Answer: A 9) The number of violent crimes committed in a day possesses a distribution with a mean of 2.8 crimes per day and a standard deviation of 6 crimes per day. A random sample of 100 days was observed, and the mean number of crimes for the sample was calculated. Describe the sampling distribution of the sample mean. A) approximately normal with mean = 2.8 and standard deviation = 0.6 B) approximately normal with mean = 2.8 and standard deviation = 6 C) shape unknown with mean = 2.8 and standard deviation = 6 D) shape unknown with mean = 2.8 and standard deviation = 0.6 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) A random sample of size n is to be drawn from a population with μ = 600 and σ = 200. What size sample would be necessary in order to ensure a standard error of 25? Answer: The standard error is σx = 25 =
σ 200 ⇒ = n n
σ . If the standard error is desired to be 25, we get: n
n ∙ 25 = 200 ⇒
n=
200 = 8 ⇒ n = 64 25
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 11) The amount 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 0.2 ounce. Suppose 100 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 100 bags exceeded 10.6 ounces. A) approximately 0 B) 0.1915 C) 0.3085 D) 0.6915 Answer: A 12) The average score of all golfers for a particular course has a mean of 80 and a standard deviation of 4.5. Suppose 81 golfers played the course today. Find the probability that the average score of the 81 golfers exceeded 81. A) 0.0228 B) 0.1293 C) 0.4772 D) 0.3707 Answer: A 13) One year, professional sports players salaries averaged $1.7 million with a standard deviation of $0.8 million. Suppose a sample of 100 major league players was taken. Find the approximate probability that the average salary of the 100 players exceeded $1.1 million. A) approximately 1 B) approximately 0 C) 0.2357 D) 0.7357 Answer: A 14) Furnace repair bills are normally distributed with a mean of 267 dollars and a standard deviation of 25 dollars. If 100 of these repair bills are randomly selected, find the probability that they have a mean cost between 267 dollars and 269 dollars. A) 0.2881 B) 0.7881 C) 0.2119 D) 0.5517 Answer: A Page 3
15) Assume that the heights of men are normally distributed with a mean of 71.2 inches and a standard deviation of 2.1 inches. If 36 men are randomly selected, find the probability that they have a mean height greater than 72.2 inches. A) 0.0021 B) 0.0210 C) 0.9005 D) 0.9979 Answer: A 16) Assume that blood pressure readings are normally distributed with a mean of 117 and a standard deviation of 8. If 100 people are randomly selected, find the probability that their mean blood pressure will be less than 119. A) 0.9938 B) 0.0062 C) 0.9998 D) 0.8615 Answer: A 17) According to the law of large numbers, as more observations are added to the sample, the difference between the sample mean and the population mean A) Tends to become smaller B) Tends to become larger C) Remains about the same D) Is inversely affected by the data added Answer: A 18) The standard error of the mean is given by σ A) B) μ - x n
C) μ - x
D) μ ± σ
Answer: A 2 Describe the distribution of the sample mean: nonnormal population. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The amount of money collected by a snack bar at a large university has been recorded daily for the past five years. Records indicate that the mean daily amount collected is $2,400 and the standard deviation is $550. The distribution is skewed to the right due to several high volume days (including football game days). Suppose that 100 days were randomly selected from the five years and the average amount collected from those days was recorded. Which of the following describes the sampling distribution of the sample mean? A) normally distributed with a mean of $2,400 and a standard deviation of $55 B) normally distributed with a mean of $2,400 and a standard deviation of $550 C) normally distributed with a mean of $240 and a standard deviation of $55 D) skewed to the right with a mean of $2,400 and a standard deviation of $550 Answer: A 2) A farmer was interested in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 60 or more flights on average in the next 40 rows down which he drives his tractor? A) 0.0571 B) 0.4429 C) 0.9429 D) 0.5710 Answer: A 3) A farmer was interest in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 52 or fewer flights on average in the next 40 rows down which he drives his tractor? A) 0.0041 B) 0.4959 C) 0.9959 D) 0.0410 Answer: A
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4) A farmer was interested in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12. What is the probability of the farmer will count 52 or fewer flights or 60 or more flights on average in the next 40 rows down which he drives his tractor? A) 0.0612 B) 0.4959 C) 0.9388 D) 0.0530 Answer: A 5) The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will exceed $7500? A) 0.0853 B) 0.9131 C) 0.0869 D) 0.9147 Answer: A 6) The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will be less than $7000? A) 0.1814 B) 0.8186 C) 0.5675 D) 0.4325 Answer: A 7) The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will be between $7000 and $7500? A) 0.7333 B) 0.2667 C) 0.9147 D) 0.8186 Answer: A 8) The ages of five randomly chosen cars in a parking garage are determined to be 7, 9, 3, 4, and 6 years old. If we consider this sample of 5 in groups of 3, how many groups can be formed? A) 10 B) 5 C) 30 D) 60 Answer: A 9) The ages of five randomly chosen cars in a parking garage are determined to be 7, 9, 3, 4, and 6 years old. If we consider this sample of 5 in groups of 3, what is the probability of the population mean falling between 5.5 and 6.5 years? A) 0.5 B) 0.4 C) 0.55 D) 0.6 Answer: A 10) The ages of five randomly chosen cars in a parking garage are determined to be 7, 9, 3, 4, and 6 years old. If we consider this sample of 5 in groups of 3, what is the probability of the population mean being more than 6 years? A) 0.4 B) 0.5 C) 0.1 D) 0.6 Answer: A 11) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time how many, how many groups can be formed? A) 20 B) 10 C) 30 D) 15 Answer: A 12) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time what is the probability of selecting a group that has an average value of at least $26? A) 0.55 B) 0.50 C) 0.60 D) 0.45 Answer: A
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13) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time what is the probability of selecting a group that has an average value of between $40 and $45? A) 0.15 B) 0.05 C) 0.25 D) 0.20 Answer: A 14) John has six bills of paper money in the following denominations: $1, $5, $10, $20, $50, and $100 If he selects 3 bills at a time what is the probability of selecting a group that has an average value of equal to or less than $25? A) 0.45 B) 0.86 C) 0.50 D) 0.35 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 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 10 minutes and a standard deviation of 1.3 minutes. If data were collected from 60 randomly selected walks, describe the sampling distribution of x, the sample mean time. Answer: By the Central Limit Theorem, the sampling distribution of x is approximately normal with μ x = μ = 10 minutes and σx =
σ 1.3 = = 0.167,8 minutes. n 60
8.2 Distribution of the Sample Proportion 1 Describe the sampling distribution of a sample proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ^
Describe the sampling distribution of p. 1) N = 22,000, n = 550, p = 0.8 A) Approximately normal; μ p = 0.8, σp = 0.017 C) Approximately normal; μ p = 0.8, σp = 0.085,3
B) Exactly normal; μ p = 0.8, σp = 0.017 D) Binomial; μ p = 440, σp = 9.38
Answer: A Provide an appropriate response. 2) Professor Zuro surveyed a random sample of 420 statistics students. One of the questions was "Will you take another mathematics class?" The results showed that 252 of the students said yes. What is the sample ^
proportion, p of students who say they will take another math class? A) 0.6 B) 0.42 C) 0.252
D) 0.775
Answer: A 3) To assess attitudes towards issues that affect the residents of a village, the village randomly chose 800 families to participate in a survey of life attitudes. The village received 628 completed surveys. What is the sample proportion of completed surveys? A) 0.785 B) 0.628 C) 1.274 D) 0.886 Answer: A 4) A national caterer determined that 87% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of 5000. The 144 people are asked to sample the ^
caterer's food. Will the distribution of p, the sample proportion saying that the food is delicious, be approximately normal? Answer Yes or No. A) Yes B) No Answer: A
Page 6
5) A national caterer determined that 87% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of 5000. The 144 people are asked to sample the ^
caterer's food. If p is the sample proportion saying that the food is delicious, what is the mean of the sampling ^
distribution of p? A) 0.87
B) 1.25
C) 0.19
D) 0.42
Answer: A 6) A national caterer determined that 37% of the people who sampled their food said that it was delicious. A random sample of 144 people is obtained from a population of 5000. The 144 people are asked to sample the ^
caterer's food. If p is the sample proportion saying that the food is delicious, what is the standard deviation of ^
the sampling distribution of p? A) 0.04 B) 0.002
C) 0.48
D) 0.23
Answer: A 7) A greenhouse in a tri-county area has kept track of its customers for the last several years and has determined that it has about 10,000 regular customers. Of those customers, 28% of them plant a vegetable garden in the spring. The greenhouse obtains a random sample of 800 of its customers. Is it safe to assume that the sampling ^
distribution of p, the sample proportion of customers that plant a vegetable garden, is approximately normal? Answer Yes or No. A) No B) Yes Answer: A 8) A greenhouse in a tri-county area has kept track of its customers for the last several years and has determined that 28% of its customers plant a vegetable garden in the spring. The greenhouse obtains a random sample of ^
1000 of its customers. What is the mean of the sampling distribution of p, the sample proportion of customers that plant a vegetable garden in the spring? A) 0.28 B) 0.72 C) 2800 D) 0.002 Answer: A 9) If a population proportion is believed to be 0.6, how many items must be sampled to ensure that the sampling ^
distribution of p will be approximately normal? Assume that the size of the population is N = 10,000. A) 42 B) 30 C) 13 D) 60 Answer: A 10) A simple random sample of size n = 1,300 is obtained from a population whose size is N = 1,300,000 and whose ^
population proportion with a specified characteristic is p = 0.36. Describe the sampling distribution of p. A) Approximately normal; μ p = 0.36, σp = 0.013 B) Exactly normal; μ p = 0.36, σp = 0.013 C) Approximately normal; μ p = 0.36, σp = 0.133,1
D) Exactly normal; μ p = 0.36, σp = 0.133,1
Answer: A 11) According to a study conducted in one city, 25% of adults in the city have credit card debts of more than $2000. ^
A simple random sample of n = 100 adults is obtained from the city. Describe the sampling distribution of p, the sample proportion of adults who have credit card debts of more than $2000. A) Approximately normal; μ p = 0.25, σp = 0.043 B) Exactly normal; μ p = 0.25, σp = 0.043 C) Approximately normal; μ p = 0.25, σp = 0.001,9 Answer: A
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D) Binomial; μ p = 25, σp = 4.33
2 Compute probabilities of a sample proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find less than 20 substandard welds? A) 0.9066 B) 0.4066 C) 0.0934 D) 0.5934 Answer: A 2) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find more than 25 substandard welds? A) 0.0040 B) 0.4960 C) 0.5040 D) 0.9960 Answer: A 3) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, what is the probability that he will find between 10 and 20 substandard welds? A) 0.8132 B) 0.4066 C) 0.2033 D) 0.6377 Answer: A 4) Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the afternoon shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, would it be unusual for Smith to find 30 or more substandard welds? A) Yes B) No Answer: A 5) The U.S. housing market estimates that 31% of all homes purchased in 2019 were considered investment properties. If a sample of 800 homes sold in 2019 is obtained what is the probability that at most 250 homes are going to be used as investment property? A) 0.5608 B) 0.4392 C) 0.0934 D) 0.5935 Answer: A 6) The U.S. housing market estimates that 31% of all homes purchased in 2019 were considered investment properties. If a sample of 800 homes sold in 2019 is obtained what is the probability that at least 175 homes are going to be used as investment property? A) 0.9999 B) 0.0195 C) 0.2236 D) 0.5189 Answer: A 7) The U.S. housing market estimates that 31% of all homes purchased in 2019 were considered investment properties. If a sample of 800 homes sold in 2019 is obtained and it was noted that 248 homes were to be used as investment property, would this be unusual? Answer Yes or No. A) No B) Yes Answer: A 8) The U.S. housing market estimates that 31% of all homes purchased in 2019 were considered investment properties. If a sample of 800 homes sold in 2019 is obtained what is the probability that between 250 and 300 homes are going to be used as investment property? A) 0.4392 B) 0.9099 C) 0.7764 D) 0.2236 Answer: A
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Ch. 8 Sampling Distributions Answer Key 8.1 Distribution of the Sample Mean 1 Describe the distribution of the sample mean: normal population. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A σ 10) The standard error is σx = . If the standard error is desired to be 25, we get: n 25 =
σ 200 ⇒ = n n
n ∙ 25 = 200 ⇒
n=
200 = 8 ⇒ n = 64 25
11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 2 Describe the distribution of the sample mean: nonnormal population. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) By the Central Limit Theorem, the sampling distribution of x is approximately normal with μ x = μ = 10 minutes and σx =
σ 1.3 = = 0.167,8 minutes. n 60
8.2 Distribution of the Sample Proportion 1 Describe the sampling distribution of a sample proportion. 1) A 2) A 3) A 4) A 5) A 6) A Page 9
7) A 8) A 9) A 10) A 11) A 2 Compute probabilities of a sample proportion. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A
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Ch. 9 Estimating the Value of a Parameter 9.1 Estimating a Population Proportion 1 Obtain a point estimate for the population proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) When 425 junior college students were surveyed,175 said that they have previously owned a motorcycle. Find a point estimate for p, the population proportion of students who have previously owned a motorcycle. A) 0.412 B) 0.588 C) 0.700 D) 0.292 Answer: A 2) A survey of 100 fatal accidents showed that in 22 cases the driver at fault was inadequately insured. Find a point estimate for p, the population proportion of accidents where the driver at fault was inadequately insured A) 0.22 B) 0.78 C) 0.282 D) 0.180 Answer: A 3) A survey of 700 non-fatal accidents showed that 171 involved faulty equipment. Find a point estimate for p, the population proportion of accidents that involved faulty equipment. A) 0.244 B) 0.756 C) 0.323 D) 0.196 Answer: A 4) A survey of 250 households showed 32 owned at least one snow blower. Find a point estimate for p, the population proportion of households that own at least one snow blower. A) 0.128 B) 0.872 C) 0.147 D) 0.113 Answer: A 5) A survey of 2,470 musicians showed that 342 of them are left-handed. Find a point estimate for p, the population proportion of musicians that are left-handed. A) 0.138 B) 0.862 C) 0.161 D) 0.122 Answer: A 6) A marketing research company wanted to estimate which of two evacuation plans its employees prefer. A random sample of n employees produced the following 90% confidence interval for the proportion of employees who prefer plan A: (0.367, 0.527). Identify the point estimate for estimating the true proportion of employees who prefer plan A. A) 0.447 B) 0.08 C) 0.367 D) 0.527 Answer: A 7) It is thought that not as many Americans buy presents to celebrate Valentine’s Day anymore. A random sample of 4000 Americans yielded 2,120 who bought their significant other a present and celebrated Valentine ’s Day. Find the point estimate for estimating the proportion of all Americans who celebrate Valentine’s Day. A) 0.5,300 B) 2,120 C) 4000 D) 0.4,700 Answer: A 2 Construct and interpret a confidence interval for the population proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine the critical value z α/2 that corresponds to the given level of confidence. 1) 86% A) 1.48 Answer: A Page 1
B) 1.08
C) 0.81
D) 1.16
2) 97% A) 2.17
B) 1.88
C) 0.83
D) 1.92
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 3) When 495 junior college students were surveyed, 145 said they have a passport. Construct a 95% confidence interval for the proportion of junior college students that have a passport. Round to the nearest thousandth. Answer: (0.253, 0.333) 4) A survey of 300 non-fatal accidents showed that 108 involved uninsured drivers. Construct a 99% confidence interval for the proportion of fatal accidents that involved uninsured drivers. Round to the nearest thousandth. Answer: (0.289, 0.431) 5) In a survey of 10 musicians, 2 were found to be left-handed. Is it practical to construct the 90% confidence interval for the population proportion, p? Explain. ^
^
Answer: It is not practical to find the confidence interval. It is necessary that np(1 - p) ≥ 10 to insure that the ^
^
^
distribution of p be normal. (np(1 - p) = 1.6) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6) An article from a Florida newspaper reported on the topics that teenagers want to discuss more with their parents. The results of the 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 519 teenagers. Estimate the proportion of all teenagers who want more family discussions about ^
school. Use a 98% confidence level. Express the answer in the form p ± E and round to the nearest thousandth. A) 0.37 ± 0.049 B) 0.37 ± 0.002 C) 0.63 ± 0.049 D) 0.63 ± 0.002 Answer: A 7) It is thought that not as many Americans buy presents to celebrate Valentine’s Day anymore. A random sample of 4000 Americans yielded 2200 who bought their significant other a present and celebrated Valentine’s Day. Estimate the true proportion of all Americans who celebrate Valentine’sDay using a 90% confidence ^
interval. Express the answer in the form p ± E and round to the nearest ten-thousandth. A) 0.5625 ± 0.0129 B) 0.5625 ± 0.4,048 C) 0.4375 ± 0.0129 D) 0.4375 ± 0.4,048 Answer: A 8) 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 95% confidence interval to estimate the true proportion of students on ^
financial aid. Express the answer in the form p ± E and round to the nearest thousandth. A) 0.59 ± 0.068 B) 0.59 ± 0.002 C) 0.59 ± 0.474 D) 0.59 ± 0.005 Answer: A 9) True or False: The general form of a large-sample (1 - α) 100% confidence interval for a population proportion ^
p is p ± z α/2 interest. A) True Answer: A
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^
^
^ x p(1 - p) , where p = is the sample proportion of observations with the characteristic of n n
B) False
10) What is the best point estimate for p in order to construct a confidence interval for p? ^
A) p
B) μ p
C) p
~
D) p
Answer: A 11) A confidence interval for p can be constructed using ^
A) p ± z α/2
^
^
p(1 - p) n
B) p ± z
σ n
^
C) p ± z
σ n
D) p ± zα/2
p(1 - p) n
Answer: A 3 Determine the sample size necessary for estimating a population proportion within a specified margin of error. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A researcher at a major clinic wishes to estimate the proportion of the adult population of the United States that has sleep deprivation. What size sample should be obtained in order to be 95% confident that the sample proportion will not differ from the true proportion by more than 5%? A) 385 B) 10 C) 769 D) 271 Answer: A 2) A senator wishes to estimate the proportion of United States voters who favor new road construction. What size sample should be obtained in order to be 90% confident that the sample proportion will not differ from the true proportion by more than 4%? A) 423 B) 256 C) 11 D) 846 Answer: A 3) A private opinion poll is conducted for a politician to determine what proportion of the population favors adding more national parks. What size sample should be obtained in order to be 99% confident that the sample proportion will not differ from the true proportion by more than 4%? A) 1,037 B) 849 C) 2,073 D) 17 Answer: A 4) A pollster wishes to estimate the number of left-handed scientists. What size sample should be obtained in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 4%? A previous study indicates that the proportion of left-handed scientists is 11%. A) 333 B) 236 C) 374 D) 18 Answer: A 5) A researcher wishes to estimate the number of households with two tablets. What size sample should be obtained in order to be 99% confident that the sample proportion will not differ from the true proportion by more than 6%? A previous study indicates that the proportion of households with two tablets is 23%. A) 327 B) 268 C) 424 D) 8 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A state highway patrol official wishes to estimate the number of legally intoxicated drivers on a certain road. a) What size sample should be obtained in order to be 98% confident that the sample proportion will not differ from the true proportion by more than 5%? b) Repeat part (a) assuming previous studies found that 65% of the drivers on this road are legally intoxicated. Answer: a) 543 b) 495
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) A local outdoor equipment store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will randomly sample among its 100,000 items in order to determine the proportion of merchandise that is outdated. The current owners have never determined their outdated percentage and can not help the buyers. Approximately What size sample should the buyers obtain in order to insure that they are 95% confident that the margin of error is within 3%? A) 1,068 B) 2,135 C) 4,269 D) 545 Answer: A 8) A confidence interval was used to estimate the proportion of math majors that double-major in another subject area. A random sample of 72 math majors generated the following confidence interval: (0.438, 0.642). Using the information above, what size sample would be necessary if we wanted to estimate the true proportion to within 1% using 90% confidence? A) 6,722 B) 6,766 C) 7,036 D) 6,495 Answer: A 9) It is thought that not as many Americans buy presents to celebrate Valentine’s Day anymore. A random sample of 4000 Americans yielded 2200 who bought their significant other a present and celebrated Valentine’s Day. What size sample should be obtained if a 90% confidence interval was desired to estimate the true proportion within 4%? A) 3,341 B) 3,529 C) 3,394 D) 3,122 Answer: A 10) 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 4% with 90% reliability, how many students would need to be sampled? A) 410 B) 99 C) 249 D) 17 Answer: A 11) In a college student poll, it is of interest to estimate the proportion p of students in favor of changing from a quarter-system to a semester-system. How many students should be polled so that we can estimate p to within 0.09 using a 99% confidence interval? A) 205 B) 182 C) 261 D) 114 Answer: A 12) True or False? When choosing the sample size for estimating a population proportion p to within E units with confidence (1 - α)100%, if you take p ≈ 0.5 as the approximation to p, you will always obtain a sample size that is at least as large as required. A) True B) False Answer: A 13) True or False? If no estimate of p exists when determining the sample size, we can use 0.5 in the formula to get a value for n. A) True B) False Answer: A
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9.2 Estimating a Population Mean 1 Obtain a point estimate for the population mean. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Determine the point estimate of the population mean and margin of error for the confidence interval with lower bound 14 and upper bound: 26. A) x = 20, E = 6
B) x = 20, E = 12
C) x = 14, E = 12
D) x = 26, E = 6
Answer: A 2 State properties of Student's t-distribution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A simple random sample of size n < 30 for a quantitative variable has been obtained. Using the normal probability plot, the correlation between the variable and expected z-score, and the boxplot, judge whether a t-interval should be constructed. 1) n = 14; Correlation = 0.956
A) Yes Answer: A
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B) No
2) n = 10; Correlation = 0.896
A) No
B) Yes
Answer: A A simple random sample of size n < 30 has been obtained. From the boxplot, judge whether a t-interval should be constructed. 3)
A) No, though there are no outliers, the data are not normally distributed but right skewed B) No, there are outliers and the data are not normally distributed but right skewed C) Yes; the data are normally distributed and there are no outliers D) No; the data are normally distributed, but there are outliers Answer: A
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4)
A) No; the data appear roughly normally distributed but there are outliers B) Yes; the data appear normally distributed and there are no outliers C) No, there are outliers and the data are not normally distributed D) No, there are no outliers but the data are not normally distributed Answer: A 5)
A) Yes; the data appear roughly normally distributed and there are no outliers B) No; the data are not normally distributed and there are outliers C) No; there are no outliers, but the data are not normally distributed D) No; the data appear roughly normally distributed, but there are outliers Answer: A Provide an appropriate response. 6) Suppose a 98% confidence interval for μ turns out have a lower bound of 1000 and an upper bound of 2100. If this interval was based on a sample of size n = 18, explain what assumptions are necessary for this interval to be valid. A) The population must have an approximately normal distribution. B) The sampling distribution of the sample mean must have a normal distribution. C) The population of salaries must have an approximate t distribution. D) The sampling distribution must be biased with 17 degrees of freedom. Answer: A 7) A computer package was used to generate the following printout for estimating the sale price of condominiums in a particular neighborhood. X = sale_price SAMPLE MEAN OF X =46,500 SAMPLE STANDARD DEV 1=3,747 SAMPLE SIZE OF X = 15 CONFIDENCE = 98 UPPER BOUND = 55,813.80 SAMPLE MEAN OF X =46,500 LOWER BOUND = 37,186.20 What assumptions are necessary for any inferences derived from this printout to be valid? A) The sample was randomly selected from an approximately normal population. B) The sample variance equals the population variance. C) The population mean has an approximate normal distribution. D) All of these are necessary. Answer: A Page 7
8) A computer package was used to generate the following printout for estimating the sale price of condominiums in a particular neighborhood. X = sale_price SAMPLE MEAN OF X =46,300 SAMPLE STANDARD DEV 1=3,747 SAMPLE SIZE OF X = 15 CONFIDENCE = 98 UPPER BOUND = 55,613.80 SAMPLE MEAN OF X =46,300 LOWER BOUND = 36,986.20 A friend suggests that the mean sale price of homes in this neighborhood is $47,000. Comment on your friend's suggestion. A) Based on this printout, all you can say is that the mean sale price might be $47,000. B) Your friend is wrong, and you are 98% certain. C) Your friend is correct, and you are 98% certain. D) Your friend is correct, and you are 100% certain. Answer: A 9) To select the correct Student's t-distribution requires knowing the degrees of freedom. How many degrees of freedom are there for a sample of size n? A) n - 1
B) n
C) n + 1
D)
x-μ s/ n
Answer: A 10) True or False: Every Student's t-distribution with n < N, n the number in the sample and N the number in the population, will be less peaked and have thinner tails. A) False B) True Answer: A 11) The area under the graph of every Student's t-distribution is A) 1
B) Less than the standard normal distribution
C) Greater than the standard normal distribution
D)
Area of the standard normal distribution s/ n
Answer: A 12) Which of the following is not a characteristic of Student's t-distribution? A) mean of 1 B) symmetric distribution C) depends on degrees of freedom. D) For large samples, the t and z distributions are nearly equivalent. Answer: A
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3 Determine t-values. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the t-value. 1) Let t0 be a specific value of t. Find t0 such that the statement is true: P(t ≥ t0 ) = 0.01 where df = 20. A) 2.528
B) -2.528
C) 2.539
D) -2.539
Answer: A 2) Find the t-value such that the area in the right tail is 0.005 with 28 degrees of freedom. A) 2.763 B) 1.701 C) -2.763
D) 2.771
Answer: A 3) Find the t-value such that the area left of the t-value is 0.005 with 29 degrees of freedom. A) -2.756 B) -1.699 C) 2.756 D) 2.763 Answer: A 4) Find the critical t-value that corresponds to 99% confidence and n = 10. A) 3.250 B) 2.821 C) 2.262
D) 1.833
Answer: A 5) Find the critical t-value that corresponds to 95% confidence and n = 16. A) 2.131 B) 2.947 C) 2.602
D) 1.753
Answer: A 6) Find the critical t-value that corresponds to 90% confidence and n = 15. A) 1.761 B) 1.345 C) 2.145
D) 2.624
Answer: A 4 Construct and interpret a confidence interval for a population mean. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) How much money does the average professional hockey fan spend on food at a single hockey game? That question was posed to 10 randomly selected hockey fans. The sampled results show that sample mean and standard deviation were $12.00 and $3.45, respectively. Use this information to create a 90% confidence interval for the mean. Express the answer in the form x ± tα/2(s/ n). A) 12 ± 1.833(3.45/ 10) C) 12 ± 1.796(3.45/ 10)
B) 12 ± 1.812(3.45/ 10) D) 12 ± 1.383(3.45/ 10)
Answer: A 2) Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A sample of 20 part-time workers had mean annual earnings of $3120 with a standard deviation of $677. Round to the nearest dollar. A) ($2803, $3437) B) ($1324, $1567) C) ($2135, $2567) D) ($2657, $2891) Answer: A 3) Construct a 90% confidence interval for the population mean, μ. Assume the population has a normal distribution. A sample of 15 randomly selected math majors has a grade point average of 2.86 with a standard deviation of 0.78. Round to the nearest hundredth. A) (2.51, 3.21) B) (2.41, 3.42) C) (2.37, 3.56) D) (2.28, 3.66) Answer: A
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4) Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A sample of 25 randomly English majors has a mean test score of 81.5 with a standard deviation of 10.2. Round to the nearest hundredth. A) (77.29, 85.71) B) (56.12, 78.34) C) (66.35, 69.89) D) (87.12, 98.32) Answer: A 5) Construct a 95% confidence interval for the population mean, μ. Assume the population has a normal distribution. A random sample of 16 lithium batteries has a mean life of 645 hours with a standard deviation of 31 hours. Round to the nearest tenth. A) (628.5, 661.5) B) (876.2, 981.5) C) (531.2, 612.9) D) (321.7, 365.8) Answer: A 6) Construct a 99% confidence interval for the population mean, μ. Assume the population has a normal distribution. A group of 19 randomly selected employees has a mean age of 22.4 years with a standard deviation of 3.8 years. Round to the nearest tenth. A) (19.9, 24.9) B) (16.3, 26.9) C) (17.2, 23.6) D) (18.7, 24.1) Answer: A 7) Construct a 98% confidence interval for the population mean, μ. Assume the population has a normal distribution. A study of 14 car owners showed that their average repair bill was $192 with a standard deviation of $8. Round to the nearest cent. A) ($186.33, $197.67) B) ($222.33, $256.10) C) ($328.33, $386.99) D) ($115.40, $158.80) Answer: A 8) Construct a 90% confidence interval for the population mean, μ. Assume the population has a normal distribution. In a recent study of 22 eighth graders, the mean number of hours per week that they played video games was 19.6 with a standard deviation of 5.8 hours. Round to the nearest hundredth. A) (17.47, 21.73) B) (18.63, 20.89) C) (5.87, 7.98) D) (19.62, 23.12) Answer: A 9) A random sample of 10 parking meters in a resort community showed the following incomes for a day. Assume the incomes are normally distributed. Find the 95% confidence interval for the true mean. Round to the nearest cent. $3.60 $4.50 $2.80 $6.30 $2.60 $5.20 $6.75 $4.25 $8.00 $3.00 A) ($3.39, $6.01) B) ($2.11, $5.34) C) ($4.81, $6.31) D) ($1.35, $2.85) Answer: A 10) The grade point averages for 10 randomly selected junior college students are listed below. Assume the grade point averages are normally distributed. Find a 98% confidence interval for the true mean. Round to the nearest hundredth. 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 A) (1.55, 3.53) B) (0.67, 1.81) C) (2.12, 3.14) D) (3.11, 4.35) Answer: A 11) A local bank needs information concerning the savings account balances of its customers. A random sample of 15 accounts was checked. The mean balance was $686.75 with a standard deviation of $256.20. Find a 98% confidence interval for the true mean. Assume that the account balances are normally distributed. Round to the nearest cent. A) ($513.17, $860.33) B) ($238.23, $326.41) C) ($326.21, $437.90) D) ($487.31, $563.80) Answer: A
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12) 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 gave a mean nicotine content of 28.4 milligrams and standard deviation of 2.9 milligrams for a sample of n = 9 cigarettes. The FDA claims that the mean nicotine content exceeds 31.2 milligrams for this brand of cigarette, and their stated reliability is 90%. Do you agree? A) No, since the value 31.2 does not fall in the 90% confidence interval. B) Yes, since the value 31.2 does fall in the 90% confidence interval. C) Yes, since the value 31.2 does not fall in the 90% confidence interval. D) No, since the value 31.2 does fall in the 90% confidence interval. Answer: A 13) What effect will an outlier have on a confidence interval that is based on a small sample size? A) The confidence interval will be wider than an interval without the outlier. B) The interval will be smaller than an interval without the outlier. C) The interval will be the same with or without the outlier. D) The interval will reveal exclusionary data. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 14) In a random sample of 26 laptop computers, the mean repair cost was $146 with a standard deviation of $31. Assume the population has a normal distribution. Construct a 95% confidence interval for the population mean, μ. Suppose you did some research on repair costs for laptop computers and found that the standard deviation is σ = 31. Use the normal distribution to construct a 95% confidence interval for the population mean, μ. Compare the results. Round to the nearest cent. Answer: ($133.48, $158.52); ($134.08, $157.92), The t-confidence interval is wider. 5 Determine the sample size needed to estimate a population mean within a specified margin of error. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Determine the sample size required to estimate the mean score on a standardized test within 4 points of the true mean with 90% confidence. Assume that s = 15 based on earlier studies. A) 39 B) 7 C) 141 D) 1 Answer: A 2) A doctor at a local hospital is interested in estimating the birth weight of infants. How large a sample must she select if she desires to be 90% confident that her estimate is within 2 ounces of the true mean? Assume that s = 7 ounces based on earlier studies. A) 34 B) 33 C) 6 D) 5 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) In order to set rates, an insurance company is trying to estimate the number of sick days that full time workers at a local bank take per year. Based on earlier studies, they will assumed that s = 2.2 days. a) How large a sample must be selected if the company wants to be 90% confident that their estimate is within 1 day of the true mean? b) Repeat part (a) using a 95% confidence interval. Which level of confidence requires a larger sample size? Explain. Answer: a) 14 b) 19; A 95% confidence interval requires a larger sample than a 90% confidence interval because more information is needed from the population to be 95% confident.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) The grade point averages for 10 randomly selected students in an algebra class with 125 students are listed below. What is the effect on the width of the confidence interval if the sample size is increased to 20? 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 A) The width decreases. B) The width increases. C) The width remains the same. D) It is impossible to tell without more information. Answer: A 5) The principal at Riverside High School would like to estimate the mean length of time each day that it takes all the buses to arrive and unload the students. How large a sample is needed if the principal would like to assert with 90% confidence that the sample mean is off by, at most, 7 minutes. Assume that s = 14 minutes based on previous studies. A) 11 B) 10 C) 12 D) 13 Answer: A 6) True or False: As the level of confidence increases the number of items to be included in a sample will decrease when the error and the standard deviation are held constant. A) False B) True Answer: A
9.3 Estimating a Population Standard Deviation 1 Find critical values for the chi-square distribution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 2 2 1) Find the critical values, χ 1 - α/2 and χ α/2 , for 95% confidence and n = 12. A) 3.816 and 21.920
B) 3.053 and 24.725
C) 4.575 and 26.757
D) 2.603 and 19.675
Answer: A 2 2 2) Find the critical values, χ 1 - α/2 and χ α/2 , for 90% confidence and n = 15. A) 6.571 and 23.685
B) 4.075 and 31.319
C) 4.660 and 29.131
D) 5.629 and 26.119
Answer: A 2 2 3) Find the critical values, χ 1 - α/2 and χ α/2 , for 98% confidence and n = 20. A) 7.633 and 36.191
B) 6.844 and 27.204
C) 8.907 and 38.582
D) 10.117 and 32.852
Answer: A 2 2 4) Find the critical values, χ 1 - α/2 and χ α/2 , for 99% confidence and n = 10. A) 1.735 and 23.587 Answer: A
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B) 2.156 and 25.188
C) 2.088 and 21.666
D) 2.558 and 23.209
5) True or False: The chi-square distribution is a symmetric distribution for all degrees of freedom. A) False B) True Answer: A 6) True or False: The chi-square distribution is a symmetric distribution is negative when the degrees of freedom become large. A) False B) True Answer: A 7) True or False: As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric. A) True B) False Answer: A 2 Construct and interpret confidence intervals for the population variance and standard deviation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Construct a 95% confidence interval for the population standard deviation σ of a random sample of 15 crates which have a mean weight of 165.2 pounds and a standard deviation of 12.2 pounds. Assume the population is normally distributed. A) (8.9, 19.2) B) (79.8, 370.2) C) (2.6, 5.5) D) (9.4, 17.8) Answer: A 2) Assume that the heights of bookcases are normally distributed. A random sample of 16 bookcases in one company have a mean height of 67.5 inches and a standard deviation of 2.4 inches. Construct a 99% confidence interval for the population standard deviation, σ. A) (1.6, 4.3) B) (1.7, 4.5) C) (1.0, 2.8) D) (1.7, 4.1) Answer: A 3) Assume that the heights of female executives are normally distributed. A random sample of 20 executives have a mean height of 62.5 inches and a standard deviation of 3.7 inches. Construct a 98% confidence interval for the population variance, σ2 . A) (7.2, 34.1)
B) (2.7, 5.8)
C) (1.9, 9.2)
D) (7.6, 35.9)
Answer: A 4) The mean replacement time for a random sample of 12 cd players is 8.6 years with a standard deviation of 3.8 years. Construct the 98% confidence interval for the population variance, σ2 . Assume the data are normally distributed A) (6.4, 52.0)
B) (2.5, 7.2)
C) (1.7, 13.7)
D) (6.1, 44.5)
Answer: A 5) A student randomly selects 10 paperbacks at a store. The mean price is $8.75 with a standard deviation of $1.50. Construct a 95% confidence interval for the population standard deviation, σ. Assume the data are normally distributed. A) ($1.03, $2.74) B) ($0.43, $1.32) C) ($1.43, $2.70) D) ($1.76, $3.10) Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) The July utility bills (in dollars) of 20 randomly selected homeowners in one city are listed below. Construct a 99% confidence interval for the variance, σ2 . Assume the population is normally distributed. 70 67
72 71
71 70
70 74
69 69
73 68
69 71
68 71
70 71
71 72
Answer: s = $1.73; ($1.47, $8.31) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) The June precipitation (in inches) for 10 randomly selected cities are listed below. Construct a 90% confidence interval for the population standard deviation, σ. Assume the data are normally distributed. 2.0 3.2 1.8 2.9 0.9 4.0 3.3 2.9 3.6 0.8 A) (0.81, 1.83) B) (0.32, 0.85) C) (0.53, 1.01) D) (1.10, 2.01) Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8) A container of soda is supposed to contain 1000 milliliters of soda. A quality control manager wants to be sure that the standard deviation of the soda containers is less than 20 milliliters. He randomly selects 10 cans of soda with a mean of 997 milliliters and a standard deviation of 32 milliliters. Use these sample results to construct a 95% confidence interval for the true value of σ. Does this confidence interval suggest that the variation in the soda containers is at an acceptable level? Answer: The 95% confidence interval is (22.01, 58.42). Because this interval does not contain 20, the standard deviation is not at an acceptable level. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 9) True or False? When constructing a (1 - α) 100% confidence interval for a population variance σ2 , the population from which the random sample is selected can have any distribution. A) False B) True Answer: A 10) The best point estimate for the standard deviation of a population is A) The standard deviation of the sample.
B) The variance of the population.
C) The variance of the sample.
D)
Answer: A
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(n - 1)s2 . σ2
9.4 Putting It Together: Which Method Do I Use? 1 Determine the appropriate confidence interval to construct. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) In a random sample of 60 dog owners enrolled in obedience training, it was determined that the mean amount of money spent per owner was $109.33 per class. Assuming the population standard deviation of the amount spent per owner is $12, construct and interpret a 95% confidence interval for the mean amount spent per owner for an obedience class. A) ($106.29, $112.37); we are 95% confident that the mean amount spent per dog owner for a single obedience class is between $106.29 and $112.37. B) ($106.78, $111.88); we are 95% confident that the mean amount spent per dog owner for a single obedience class is between $106.78 and $111.88. C) ($106.23, $112.43); we are 95% confident that the mean amount spent per dog owner for a single obedience class is between $106.23 and $112.43. D) ($106.74, $111.92); we are 95% confident that the mean amount spent per dog owner for a single obedience class is between $106.74 and $111.92. Answer: A 2) A survey of 1010 college seniors working towards an undergraduate degree was conducted. Each student was asked, "Are you planning or not planning to pursue a graduate degree?" Of the 1010 surveyed, 658 stated that they were planning to pursue a graduate degree. Construct and interpret a 98% confidence interval for the proportion of college seniors who are planning to pursue a graduate degree. A) (0.616, 0.686); we are 98% confident that the proportion of college seniors who are planning to pursue a graduate degree is between 0.616 and 0.686. B) (0.620, 0.682); we are 98% confident that the proportion of college seniors who are planning to pursue a graduate degree is between 0.620 and 0.682. C) (0.621, 0.680); we are 98% confident that the proportion of college seniors who are planning to pursue a graduate degree is between 0.621 and 0.680. D) (0.612, 0.690); we are 98% confident that the proportion of college seniors who are planning to pursue a graduate degree is between 0.612 and 0.690. Answer: A Construct a 95% Z-interval or a 95% t-interval about the population mean. Assume the data come from a population that is approximately normal with no outliers. 3) The heights of 20- to 29-year-old females are known to have a population standard deviation σ = 2.7 inches. A simple random sample of n = 15 females 20 to 29 years old results in the following data: 63.1 63.3 68.4
67.9 66.2 69.9
64.8 68.2 67.3
62.2 69.7 64.5
65.4 64.1 70.2
A) (64.98, 67.72); we are 95% confident that the mean height of 20- to 29-year-old females is between 64.98 and 67.72 inches. B) (64.85, 67.85); we are 95% confident that the mean height of 20- to 29-year-old females is between 64.85 and 67.85 inches. C) (65.12, 67.58); we are 95% confident that the mean height of 20- to 29-year-old females is between 65.12 and 67.58 inches. D) (65.20, 67.50); we are 95% confident that the mean height of 20- to 29-year-old females is between 65.20 and 67.50 inches. Answer: A
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4) Fifteen randomly selected men were asked to run on a treadmill for 6 minutes. After the 6 minutes, their pulses were measured and the following data were obtained: 105 101 122
94 99 114
98 85 97
88 84 101
104 124 90
A) (93.7, 107.1); we are 95% confident that the mean pulse rate of men after 6 minutes of exercise is between 93.7 and 107.1 beats per minute. B) (94.9, 105.9); we are 95% confident that the mean pulse rate of men after 6 minutes of exercise is between 94.9 and 105.9 beats per minute. C) (94.2, 106.6); we are 95% confident that the mean pulse rate of men after 6 minutes of exercise is between 94.2 and 106.6 beats per minute. D) (95.2, 105.6); we are 95% confident that the mean pulse rate of men after 6 minutes of exercise is between 95.2 and 105.6 beats per minute. Answer: A
9.5 Estimating with Bootstrapping 1 Estimate a parameter using the bootstrap method. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) If we wish to obtain a 95% confidence interval of a parameter using the bootstrap method, which percentiles of the resampled distribution will form the lower and upper bounds of the interval? A) 2.5, 97.5 B) 5, 95 C) 5, 97.5 D) 2.5, 95 Answer: A 2) A random sample of 20 electricians is obtained and the monthly income is recorded for each one. A researcher plans to use the bootstrap method with 1000 resamples to obtain a 90% confidence interval for the mean monthly income of all electricians. Which of the following is not true of the resamples? A) Each resample will be selected from the population. B) Each resample will be selected with replacement. C) Each resample will be selected from the original sample. D) Each resample will be of the same size as the original sample. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) The table shows the monthly rents (in dollars) for 10 studio apartments selected randomly from all studio apartments in one city. 950 880
840 990
1330 760
1020 1250
1180 825
Explain the algorithm in using the bootstrap method with 1000 resamples to obtain a 90% confidence interval for the mean monthly rent of all studio apartments in the city. Answer: The sample data is treated as the population. A computer is used to obtain 1000 independent resamples of size n = 10 with replacement from the sample data. For each resample, the sample mean is obtained. The lower bound of the confidence interval is the 5th percentile of the 1000 sample means and the upper bound is the 95th percentile of the 1000 sample means.
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4) A college nurse obtained a random sample of 20 students from the college. Each student was asked if they were taking antidepressants. In the data, a 1 indicates the student is taking antidepressants and a 0 indicates they are not taking antidepressants. 0 1 0 0
0 0 1 0
1 0 0 1
0 0 0 0
1 0 0 1
Treat these data as a simple random sample of all students at the college. Explain the algorithm in using the bootstrap method with 1000 resamples to obtain a 99% confidence interval for the proportion of all students at the college who are taking antidepressants. Answer: The sample data is treated as the population. A computer is used to obtain 1000 independent resamples of size n = 20 with replacement from the sample data. For each resample, the sample proportion of students taking antidepressants is obtained. The lower bound of the confidence interval is the 0.5th percentile of the 1000 sample proportions and the upper bound is the 99.5th percentile of the 1000 sample proportions.
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Ch. 9 Estimating the Value of a Parameter Answer Key 9.1 Estimating a Population Proportion 1 Obtain a point estimate for the population proportion. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 2 Construct and interpret a confidence interval for the population proportion. 1) A 2) A 3) (0.253, 0.333) 4) (0.289, 0.431) ^
^
^
5) It is not practical to find the confidence interval. It is necessary that np(1 - p) ≥ 10 to insure that the distribution of p be ^
^
normal. (np(1 - p) = 1.6) 6) A 7) A 8) A 9) A 10) A 11) A 3 Determine the sample size necessary for estimating a population proportion within a specified margin of error. 1) A 2) A 3) A 4) A 5) A 6) a) 543 b) 495 7) A 8) A 9) A 10) A 11) A 12) A 13) A
9.2 Estimating a Population Mean 1 Obtain a point estimate for the population mean. 1) A 2 State properties of Student's t-distribution. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A Page 18
10) A 11) A 12) A 3 Determine t-values. 1) A 2) A 3) A 4) A 5) A 6) A 4 Construct and interpret a confidence interval for a population mean. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) ($133.48, $158.52); ($134.08, $157.92), The t-confidence interval is wider. 5 Determine the sample size needed to estimate a population mean within a specified margin of error. 1) A 2) A 3) a) 14 b) 19; A 95% confidence interval requires a larger sample than a 90% confidence interval because more information is needed from the population to be 95% confident. 4) A 5) A 6) A
9.3 Estimating a Population Standard Deviation 1 Find critical values for the chi-square distribution. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 2 Construct and interpret confidence intervals for the population variance and standard deviation. 1) A 2) A 3) A 4) A 5) A 6) s = $1.73; ($1.47, $8.31) 7) A 8) The 95% confidence interval is (22.01, 58.42). Because this interval does not contain 20, the standard deviation is not at an acceptable level. 9) A Page 19
10) A
9.4 Putting It Together: Which Method Do I Use? 1 Determine the appropriate confidence interval to construct. 1) A 2) A 3) A 4) A
9.5 Estimating with Bootstrapping 1 Estimate a parameter using the bootstrap method. 1) A 2) A 3) The sample data is treated as the population. A computer is used to obtain 1000 independent resamples of size n = 10 with replacement from the sample data. For each resample, the sample mean is obtained. The lower bound of the confidence interval is the 5th percentile of the 1000 sample means and the upper bound is the 95th percentile of the 1000 sample means. 4) The sample data is treated as the population. A computer is used to obtain 1000 independent resamples of size n = 20 with replacement from the sample data. For each resample, the sample proportion of students taking antidepressants is obtained. The lower bound of the confidence interval is the 0.5th percentile of the 1000 sample proportions and the upper bound is the 99.5th percentile of the 1000 sample proportions.
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Ch. 10 Hypothesis Tests Regarding a Parameter 10.1 The Language of Hypothesis Testing 1 Determine the null and alternative hypotheses. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The null and alternative hypotheses are given. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed and the parameter that is being tested. 1) H 0 : μ = 9.3 H 1 : μ ≠ 9.3 A) Two-tailed, μ
B) Two-tailed, x
C) Right-tailed, μ
D) Left-tailed, x
B) Left-tailed, p
C) Right-tailed, p
D) Left-tailed, p
B) Right-tailed, μ
C) Right-tailed, σ
D) Left-tailed, s
Answer: A 2) H 0 : p = 0.93 H 1 : p > 0.93 A) Right-tailed, p
^
^
Answer: A 3) H 0 : σ = 8.1 H 1 : σ < 8.1 A) Left-tailed, σ Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 4) The mean annual return for an employee's IRA is less than 5.6 percent. Write the null and alternative hypotheses to test the statement. Answer: H 0 : μ = 5.6, H1 : μ < 5.6 5) The mean age of lawyers in New York is 46.6 years. Write the null and alternative hypotheses to test the statement. Answer: H 0 : μ = 46.6, H1 : μ ≠ 46.6 6) The mean repair bill of cars is greater than $110. Write the null and alternative hypotheses to test the statement.. Answer: H 0 : μ = $110, H 1 : μ > $110 7) The mean utility bill in one city during the summer was less than $100. Write the null and alternative hypotheses to test the statement. Answer: H 0 : μ = $100, H 1 : μ < $100 8) The mean annual return for an employee's IRA is more than 3.6 percent. Write the null and alternative hypotheses to test the statement. Answer: H 0 : μ = 3.6, H1 : μ < 3.6 9) A popular referendum on the ballot is favored by more than half of the voters. Write the null and alternative hypotheses to test the statement. Answer: H 0 : p = 0.5, H1 : p > 0.5 Page 1
10) The owner of an outdoor store recommends against buying the new model of one brand of GPS receivers because they vary more than the old model, which had a standard deviation of 50 meters. Write the null and alternative hypotheses to test the statement. Answer: H 0 : σ = 50, H1 : σ > 50 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 11) A ______________ is a statement or claim regarding a characteristic of one or more populations. A) hypothesis B) conclusion C) conjecture D) fact Answer: A 12) The ______________ hypothesis contains the "=" sign. A) null B) alternative
C) explanatory
D) conditional
Answer: A 13) A hypothesis test is a "two-tailed" if the alternative hypothesis contains a _______ sign. A) ≠ B) + C) < D) > Answer: A 2 Explain Type I and Type II errors. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The mean age of judges in Los Angeles is 53.6 years. Identify the type I and type II errors for the hypothesis test of this claim. Answer: type I: rejecting H0 : μ = 53.6 when μ = 53.6 type II: failing to reject H0 : μ = 53.6 when μ ≠ 53.6 2) The mean cost of textbooks for one class is greater than $110. Identify the type I and type II errors for the hypothesis test of this claim. Answer: type I: rejecting H0 : μ = $110 when μ ≤ $110 type II: failing to reject H0 : μ = $110 when μ > $110 3) The mean monthly cell phone bill for one household was less than $92. Identify the type I and type II errors for the hypothesis test of this claim. Answer: type I: rejecting H0 : μ = $92 when μ ≥ $92 type II: failing to reject H0 : μ = $92 when μ < $92 4) A referendum for an upcoming election is favored by more than half of the voters. Identify the type I and type II errors for the hypothesis test of this claim. Answer: type I: rejecting H0 : p = 0.5 when p ≤ 0.5 type II: failing to reject H0 : p = 0.5 when p > 0.5
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) A local tennis pro-shop strings tennis rackets at the tension (pounds per square inch) requested by the customer. Recently a customer made a claim that the pro-shop consistently strings rackets at lower tensions, on average, than requested. To support this claim, the customer asked the pro shop to string 15 new rackets at 53 psi. Upon receiving the rackets, the customer measured the tension of each and calculated the following summary statistics: x = 51 psi, s = 2.2 psi. In order to conduct the test, the customer selected a significance level of α = .01. Interpret this value. A) The probability of concluding that the true mean is less than 53 psi when in fact it is equal to 53 psi is only .01. B) The smallest value of α that you can use and still reject H 0 is 0.01. C) The probability of making a Type II error is 0.99. D) There is a 1% chance that the sample will be biased. Answer: A 6) True or False: If I specify β to be equal to 0.34, then the value of α must be 0.66. A) False B) True Answer: A 7) What is the probability associated with not making a Type II error? A) (1 - β) B) α C) β
D) (1 - α)
Answer: A 8) We never conclude "Accept H0 " in a test of hypothesis. This is because: A) β = p(Type II error) is not known. C) The rejection region is not known.
B) α is the probability of a Type I error. D) The p-value is not small enough.
Answer: A 9) If we reject the null hypothesis when the null hypothesis is true, then we have made a A) Type I error B) Type II error C) Correct decision
D) Type α error
Answer: A 10) If we do not reject the null hypothesis when the null hypothesis is in error, then we have made a A) Type II error B) Type I error C) Correct decision D) Type β error Answer: A 11) The level of significance, α, is the probability of making a A) Type I error B) Type II error
C) Correct decision
Answer: A 12) True or False: Type I and Type II errors are independent events. A) False B) True Answer: A
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D) Type β error
3 State conclusions to hypothesis tests. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The mean age of principals in a local school district is 53.4 years. If a hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to reject the claim μ = 53.4. B) There is not sufficient evidence to reject the claim μ = 53.4. C) There is sufficient evidence to support the claim μ = 53.4. D) There is not sufficient evidence to support the claim μ = 53.4. Answer: A 2) The mean age of professors at a university is 53.9 years. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is not sufficient evidence to reject the claim μ = 53.9. B) There is sufficient evidence to reject the claim μ = 53.9. C) There is sufficient evidence to support the claim μ = 53.9. D) There is not sufficient evidence to support the claim μ = 53.9. Answer: A 3) The mean age of professors at a university is greater than 57.5 years. If a hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to support the claim μ > 57.5. B) There is sufficient evidence to reject the claim μ > 57.5. C) There is not sufficient evidence to reject the claim μ > 57.5. D) There is not sufficient evidence to support the claim μ > 57.5. Answer: A 4) The mean age of judges in Dallas is greater than 58.7 years. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is not sufficient evidence to support the claim μ > 58.7. B) There is sufficient evidence to reject the claim μ > 58.7. C) There is not sufficient evidence to reject the claim μ > 58.7. D) There is sufficient evidence to support the claim μ > 58.7. Answer: A 5) The mean monthly gasoline bill for one household is greater than $140. If a hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to support the claim μ > $140. B) There is sufficient evidence to reject the claim μ > $140. C) There is not sufficient evidence to reject the claim μ > $140. D) There is not sufficient evidence to support the claim μ > $140. Answer: A 6) The mean monthly gasoline bill for one household is greater than $140. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is not sufficient evidence to support the claim μ > $140. B) There is sufficient evidence to reject the claim μ > $140. C) There is not sufficient evidence to reject the claim μ > $140. D) There is sufficient evidence to support the claim μ > $140. Answer: A
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7) The mean number of rushing yards for one NFL team was less than 99 yards per game. If a hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to support the claim μ < 99. B) There is sufficient evidence to reject the claim μ < 99. C) There is not sufficient evidence to reject the claim μ < 99. D) There is not sufficient evidence to support the claim μ < 99. Answer: A 8) The mean number of rushing yards for one NFL team was less than 94 yards per game. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is not sufficient evidence to support the claim μ < 94. B) There is sufficient evidence to reject the claim μ < 94. C) There is not sufficient evidence to reject the claim μ < 94. D) There is sufficient evidence to support the claim μ < 94. Answer: A 9) The dean of a major university claims that the mean number of hours students study at her University (per day) is less than 4.9 hours. If a hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to reject the claim μ = 4.9. B) There is not sufficient evidence to reject the claim μ = 4.9. C) There is sufficient evidence to support the claim μ = 4.9. D) There is not sufficient evidence to support the claim μ = 4.9. Answer: A 10) The dean of a major university claims that the mean number of hours students study at her University (per day) is less than 5.2 hours. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is not sufficient evidence to reject the claim μ = 5.2. B) There is sufficient evidence to reject the claim μ = 5.2. C) There is sufficient evidence to support the claim μ = 5.2. D) There is not sufficient evidence to support the claim μ = 5.2. Answer: A 11) A candidate for state representative of a certain state claims to be favored by more than half of the voters. If a hypothesis test is performed, how should you interpret a decision that rejects the null hypothesis? A) There is sufficient evidence to reject the claim p =0.5. B) There is not sufficient evidence to reject the claim p = 0.5. C) There is sufficient evidence to support the claim p = 0.5. D) There is not sufficient evidence to support the claim p = 0.5. Answer: A 12) A candidate for state representative of a certain state claims to be favored by more than half of the voters. If a hypothesis test is performed, how should you interpret a decision that fails to reject the null hypothesis? A) There is not sufficient evidence to reject the claim p = 0.5. B) There is sufficient evidence to reject the claim p = 0.5. C) There is sufficient evidence to support the claim p = 0.5. D) There is not sufficient evidence to support the claim p =0.5. Answer: A
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10.2 Hypothesis Tests for a Population Proportion 1 Explain the logic of hypothesis testing. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Find the critical value for a left-tailed test with α = 0.05. A) -1.645 B) -1.96
C) -1.28
D) -2.33
C) ±1.645
D) ±1.96
Answer: A 2) Find the critical value for a two-tailed test with α = 0.01. A) ±2.575 B) ±2.33 Answer: A 3) Suppose you want to test the claim that μ = 3.5. Given a sample size of n = 36 and a level of significance of α = 0.05, when should you reject H0 ? A) Reject H 0 if the standardized test statistic is greater than 1.96 or less than -1.96. B) Reject H 0 if the standardized test statistic is greater than 2.33 or less than -2.33. C) Reject H 0 if the standardized test statistic is greater than 1.645 or less than -1.645 D) Reject H 0 if the standardized test statistic is greater than 2.575 or less than -2.575 Answer: A 4) Suppose you want to test the claim that μ > 25.6. Given a sample size of n = 47 and a level of significance of α = 0.01, when should you reject H 0? A) Reject H 0 if the standardized test statistic is greater than 2.33. B) Reject H 0 if the standardized test statistic is greater than 1.28. C) Reject H 0 if the standardized test statistic is greater than 1.96. D) Reject H 0 if the standardized test statistic is greater than 2.575. Answer: A 5) Suppose you want to test the claim that μ < 65.4. Given a sample size of n = 35 and a level of significance of α = 0.05, when should you reject H 0? A) Reject H 0 if the standardized test statistic is less than -1.645. B) Reject H 0 if the standardized test is less than -2.33. C) Reject H 0 if the standardized test statistic is less than -1.28. D) Reject H 0 if the standardized test statistic is less than -1.96. Answer: A 6) When the results of a hypothesis test are determined to be statistically significant, then we _______________ the null hypothesis. A) reject B) fail to reject C) polarize D) compartmentalize Answer: A
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2 Test hypotheses about a population proportion. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The business college wants to determine the proportion of business students who have extended time between classes. If the proportion differs from 30%, then the lab will modify a proposed enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is 2.5. Find the P-value for a two-tailed test of hypothesis. A) 0.01<p<0.05 B) p<0.01 C) 0.05<p<0.10 D) p>0.10 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) Increasing numbers of businesses are offering child-care benefits for their workers. However, one union claims that more than 80% of firms in the manufacturing sector still do not offer any child-care benefits to their workers. A random sample of 390 manufacturing firms is selected and asked if they offer child-care benefits. Suppose the P-value for this test was reported to be p = 0.1,235. State the conclusion of interest to the union. Use α = 0.05. Answer: At α = 0.05, α < P-value = 0.1,235, so H 0 cannot be rejected. There is insufficient evidence to indicate that more than 80% of the firms do not offer any child-care benefits. ^
3) Test the claim about the population proportion p > 0.015 given n = 150 and p = 0.027. Use α = 0.01. Answer: test statistic ≈ 1.21; P-value = 2.33; fail to reject H0 ; There is not sufficient evidence to support the claim. 4) Fifty percent of registered voters in a congressional district are registered Democrats. The Republican candidate takes a poll to assess his chances in a two-candidate race. He polls 1200 potential voters and finds that 621 plan to vote for the Democratic candidate. Does the Republican candidate have a chance to win? Use α = 0.05. Answer: P-value = 0.1131; test statistic ≈ 1.21; fail to reject H0 ; There is not sufficient evidence to support the claim p > 0.5. The Republican candidate has no chance. 5) An airline claims that the no-show rate for passengers is less than 5%. In a sample of 420 randomly selected ^
reservations, 19 were no-shows. At α = 0.01, test the airline's claim. Round p to the nearest thousandth when calculating the test statistic. Answer: P-value = 0.3264; test statistic ≈ -0.45; fail to reject H0 ; There is not sufficient evidence to support the airline's claim. 6) A recent study claimed that at least 15% of junior high students are overweight. In a sample of 160 students, 18 were found to be overweight. At α = 0.05, test the claim. Answer: P-value = 0.9082; test statistic ≈ -1.33; fail to reject H0 ; There is not sufficient evidence to reject the claim. 7) The engineering school at a major university claims that 20% of its graduates are women. In a graduating class ^
of 210 students, 58 were females. Does this suggest that the school is believable? Use α = 0.05. Round p to the nearest ten-thousandth when calculating the test statistic. Answer: P-value = 0.0058; test statistic ≈ 2.76; reject H 0; There is sufficient evidence to reject the university's claim. 8) A coin is tossed 1000 times and 540 heads appear. At α = 0.05, test the claim that this is not a biased coin. Answer: P-value = 0.0114; test statistic ≈ 2.53; reject H 0; There is sufficient evidence to reject the claim that this is not a biased coin.
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3 Test hypotheses about a population proportion using the binomial probability distribution. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) According to a national statistics bureau, 3.7% of people living in the Southwest were retired exterminators. A researcher believes that the percentage has increased since then. She randomly selects 250 people in the Southwest and finds that 4 of them are retired exterminators. Test this researcher's claim at the α = 0.1 level of significance. Answer: H 0 : p = 0.037 and H 1 : p > 0.037 Since 250(0.037)(1 - 0.037) = 8.91 < 10, we calculate the P-value: P(X ≥ 4) = 1 - P(X < 4) = 1 - P(X ≤ 3) = 1 - 0.016 = 0.984. Since 0.984 > 0.1, we do not reject H0 . There is not significant evidence to conclude that the percentage has increased. 2) According to a prestigious historical society, in 1999, 7.2% of recent high school graduates believe that the Romans invented mayonnaise. A classics scholar believes that the percentage has increased since then. He randomly selects 125 recent high school graduates and finds that 17 of them believe in the Roman invention of mayonnaise. Test this researcher's claim at the α = 0.01 level of significance. Answer: H 0 : p = 0.072 and H 1 : p > 0.072 Since 125(0.072)(1 - 0.072) = 8.35 < 10, we calculate the P-value: P(X ≥ 17) = 1 - P(X < 17) = 1 - P(X ≤ 16) = 1 - 0.992 = 0.008. Since 0.008 < 0.01, we reject H0 . There is evidence to conclude that the percentage has increased. 3) According to a local chamber of commerce, in 1993, 5.9% of local area residents owned more than five cars. A local car dealer claims that the percentage has increased. He randomly selects 180 local area residents and finds that 12 of them own more than five cars. Test this car dealer's claim at the α = 0.05 level of significance. Answer: H 0 : p = 0.059 and H 1 : p > 0.059 Since 180(0.059)(1 - 0.059) = 9.99 < 10, we calculate the P-value: P(X ≥ 12) = 1 - P(X < 12) = 1 - P(X ≤ 11) = 1 - 0.626 = 0.374. Since 0.374 > 0.05, we do not reject H0 . There is not significant evidence to conclude that the percentage has increased. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) An event is considered unusual if the probability of observing the event is A) less than 0.05 B) less than 0.025 C) less than 0.10
D) greater than 0.95
Answer: A
10.3 Hypothesis Tests for a Population Mean 1 Test hypotheses about a mean. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the critical value. 1) Determine the critical value for a right-tailed test of a population mean at the α = 0.005 level of significance with 28 degrees of freedom. A) 2.763 B) 1.701 C) -2.763 D) 2.771 Answer: A
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2) Determine the critical value for a left-tailed test of a population mean at the α = 0.025 level of significance based on a sample size of n = 18. A) -2.11 B) -3.222 C) 2.110 D) 2.101 Answer: A 3) Determine the critical values for a two-tailed test of a population mean at the α = 0.1 level of significance based on a sample size of n = 8. A) ±1.895 B) ±1.86 C) ±1.415 D) ±1.397 Answer: A Provide an appropriate response. 4) A relative frequency histogram for the sale prices of homes sold in one city during 2010 is shown below. Based on the histogram, is a large sample necessary to conduct a hypothesis test about the mean sale price? If so, why?
A) Yes; data do not appear to be normally distributed but skewed right. B) Yes; data do not appear to be normally distributed but skewed left. C) No; data appear to be normally distributed. D) Yes; data do not appear to be normally distributed but bimodal. Answer: A
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5) The ages of a group of patients being treated at one hospital for osteoporosis are summarized in the frequency histogram below. Based on the histogram, is a large sample necessary to conduct a hypothesis test about the mean age? If so, why?
A) Yes; data do not appear to be normally distributed but skewed left. B) Yes; data do not appear to be normally distributed but skewed right. C) Yes; data do not appear to be normally distributed but bimodal. D) No; data appear to be normally distributed with no outliers. Answer: A 6) The weekly salaries (in dollars) of randomly selected employees of a company are summarized in the boxplot below. Based on the boxplot, is a large sample necessary to conduct a hypothesis test about the mean salary? If so, why?
A) Yes; data do not appear to be normally distributed but skewed right. B) Yes; data do not appear to be normally distributed but skewed left. C) No; data appear to be normally distributed. D) Yes; data contain outliers. Answer: A 7) The weights (in ounces) of a sample of tomatoes of a particular variety are summarized in the boxplot below. Based on the boxplot, is a large sample necessary to conduct a hypothesis test about the mean weight? If so, why?
A) Yes; data contain outliers. B) Yes; data do not appear to be normally distributed but skewed left. C) Yes; data do not appear to be normally distributed but skewed right. D) No; data appear to be normally distributed with no outliers. Answer: A
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8) Find the standardized test statistic t for a sample with n = 12, x = 13.2, s = 2.2, and α = 0.01 if H 0: μ = 12. Round your answer to three decimal places. A) 1.890 B) 1.991
C) 2.132
D) 2.001
Answer: A 9) Find the standardized test statistic t for a sample with n = 10, x = 16.2, s = 1.3, and α = 0.05 if H 0: μ ≤ 17.1. Round your answer to three decimal places. A) -2.189 B) -3.186
C) -3.010
D) -2.617
Answer: A 10) Find the standardized test statistic t for a sample with n = 15, x = 6.7, s = 0.8, and α = 0.05 if H 0 : μ ≥ 6.4. Round your answer to three decimal places. A) 1.452 B) 1.728
C) 1.631
D) 1.312
Answer: A 11) Find the standardized test statistic t for a sample with n = 20, x = 11.9, s = 2.0, and α = 0.05 if H 1: μ < 12.3. Round your answer to three decimal places. A) -0.894 B) -0.872
C) -1.265
D) -1.233
Answer: A 12) Find the standardized test statistic t for a sample with n = 25, x = 14, s = 3, and α = 0.005 if H1 : μ > 13. Round your answer to three decimal places. A) 1.667 B) 1.997
C) 1.452
D) 1.239
Answer: A 13) Find the standardized test statistic t for a sample with n = 12, x = 19.9, s = 2.1, and α = 0.01 if H 1: μ ≠ 20.4. Round your answer to three decimal places. A) -0.825 B) -0.008
C) -0.037
D) -0.381
Answer: A 2 Test hypotheses about a mean. (P-Value Approach). SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Use a t-test to test the claim μ = 26 at α = 0.01, given the sample statistics n = 12, x = 27.2, and s = 2.2. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ 1.890, 0.05 < P-value < 0.10, fail to reject H 0 ; There is not sufficient evidence to reject the claim. 2) Use a t-test to test the claim μ ≥ 10.1 at α = 0.05, given the sample statistics n = 10, x = 9.2, and s = 1.3. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ -2.189, 0.025 < P-value < 0.05, reject H0 ; There is sufficient evidence to reject the claim 3) Use a t-test to test the claim μ ≤ 9.4 at α = 0.05, given the sample statistics n = 15, x = 9.7, and s = 0.8. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ 1.452, 0.05 < P-value < 0.10, fail to reject H 0 ; There is not sufficient evidence to reject the claim
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4) Use a t-test to test the claim μ < 12.9 at α = 0.10, given the sample statistics n = 20, x = 12.5, and s = 2.0. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ -0.894, 0.15 < P-value < 0.20, fail to reject H 0 ; There is not sufficient evidence to support the claim 5) Use a t-test to test the claim μ > 15 at α = 0.005, given the sample statistics n = 25, x = 16, and s = 3. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ 1.667, 0.05 < P-value < 0.10, fail to reject H 0 ; There is not sufficient evidence to support the claim 6) Use a t-test to test the claim μ = 21.3 at α = 0.01, given the sample statistics n = 12, x = 20.8, and s = 2.1. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ -0.825, 0.40 < P-value < 0.50, fail to reject H 0 ; There is not sufficient evidence to support the claim 7) A local retailer claims that the mean waiting time is less than 5 minutes. A random sample of 20 waiting times has a mean of 3.4 minutes with a standard deviation of 2.1 minutes. At α = 0.01, test the retailer's claim. Assume the distribution is normally distributed. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ -3.407; 0.001 < P-value < 0.0025; reject H 0 ; There is sufficient evidence to support the retailer's claim. 8) A manufacturer claims that the mean lifetime of its lithium batteries is 1,300 hours. A homeowner selects 25 of these batteries and finds the mean lifetime to be 1,280 hours with a standard deviation of 80 hours. Test the manufacturer's claim. Use α = 0.05. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ -1.25; 0.20 < P-value < 0.30; fail to reject H 0 ; There is not sufficient evidence to reject the manufacturer's claim. 9) A shipping firm suspects that the mean life of a certain brand of tire used by its trucks is less than 38,000 miles. To check the claim, the firm randomly selects and tests 18 of these tires and gets a mean lifetime of 37,350 miles with a standard deviation of 1200 miles. At α = 0.05, test the shipping firm's claim. Round the test statistic to the nearest thousandth. Answer: test statistic -2.298; 0.01 < P-value < 0.02; reject H0 ; There is sufficient evidence to support the shipping firm's claim. 10) A local juice manufacturer distributes juice in bottles labeled 12 ounces. A government agency thinks that the company is cheating its customers. The agency selects 20 of these bottles, measures their contents, and obtains a sample mean of 11.7 ounces with a standard deviation of 0.7 ounce. Use a 0.01 significance level to test the agency's claim that the company is cheating its customers. Round the test statistic to the nearest thousandth. Answer: test statistic ≈ -1.917; 0.025 < P-value < 0.05; fail to reject H 0; There is not sufficient evidence to support the government agency's claim. 11) A local group claims that the police issue at least 56 parking tickets a day in their area. To prove their point, they randomly select two weeks. Their research yields the number of tickets issued for each day. The data are listed below. At α = 0.01, test the group's claim. Round the test statistic to the nearest thousandth. 70 48 41 68 69 55 70 57 60 83 32 60 72 58 Answer: x = 60.21, s = 13.43; test statistic ≈ 1.173; 0.10 < P-value < 0.15; fail to reject H0 ; There is not sufficient evidence to reject the claim.
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12) A bank claims that the mean waiting time in line is less than 4.4 minutes. A random sample of 20 customers has a mean of 4.2 minutes with a standard deviation of 0.8 minute. If α = 0.05, test the bank's claim using P-values. Answer: Standardized test statistic ≈ -1.118; Therefore, at 19 degrees of freedom, the P-value must lie between 0.10 and 0.25. Since the P-value > α, fail to reject H0 . There is not sufficient evidence to support the bank's claim. 13) A local hardware store claims that the mean waiting time in line is less than 3.5 minutes. A random sample of 20 customers has a mean of 3.7 minutes with a standard deviation of 0.8 minute. If α = 0.05, test the store's claim using P-values. Answer: Standardized test statistic ≈ 1.118; Therefore, at a degree of freedom of 19, the P-value must lie between 0.10 and 0.25. Since the P-value > α, fail to reject H0 . There is not sufficient evidence to support the store's claim. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 14) A local tennis pro-shop strings tennis rackets at the tension (pounds per square inch) requested by the customer. Recently a customer made a claim that the pro-shop consistently strings rackets at lower tensions, on average, than requested. To support this claim, the customer asked the pro shop to string 7 new rackets at 42 psi. Suppose the two-tailed P-value for the test described above (obtained from a computer printout) is 0.08. Give the proper conclusion for the test. Use α = 0.10. A) There is sufficient evidence to conclude that μ, the true mean tension of the rackets, is less than 42 psi. B) Accept H0 and conclude that μ, the true mean tension of the rackets, equals 42 psi. C) There is insufficient evidence to conclude that μ, the true mean tension of the rackets, is less than 42 psi. D) Reject H 0 and conclude that μ, the true mean tension of the rackets, equals 42 psi. Answer: A
10.4 Hypothesis Tests for a Population Standard Deviation 1 Test hypotheses about a population standard deviation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the critical value(s). 1) Determine the critical value for a right-tailed test of a population standard deviation with 16 degrees of freedom at the α = 0.05 level of significance. A) 26.296 B) 24.996 C) 34.267 D) 7.962 Answer: A 2) Determine the critical value for a left-tailed test of a population standard deviation for a sample of size n = 21 at the α = 0.05 level of significance. A) 10.851 B) 11.591 C) 31.41 D) 32.671 Answer: A 3) Determine the critical values for a two-tailed test of a population standard deviation for a sample of size n = 24 at the α = 0.01 level of significance. A) 9.260, 44.181 B) 9.886, 45.559 C) 10.196, 41.638 D) 10.856, 42.980 Answer: A Provide an appropriate response. Round the test statistic to the nearest thousandth. 4) Compute the standardized test statistic, χ 2 , to test the claim σ2 = 25.8 if n = 12, s2 = 21.6, and α = 0.05. A) 9.209 Answer: A
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B) 12.961
C) 18.490
D) 0.492
5) Compute the standardized test statistic, χ 2 , to test the claim σ2 ≥ 16.2 if n = 15, s2 = 13.5, and α = 0.05. A) 11.667 B) 8.713 C) 12.823 D) 23.891 Answer: A 6) Compute the standardized test statistic, χ 2 , to test the claim σ2 ≤ 16 if n = 20, s2 = 31, and α = 0.01. A) 36.813 B) 9.322 C) 12.82 D) 33.41 Answer: A 7) Compute the standardized test statistic, χ 2 , to test the claim σ2 > 7.6 if n = 18, s2 = 10.8, and α = 0.01. A) 24.158 B) 28.175 C) 33.233 D) 43.156 Answer: A 8) Compute the standardized test statistic, χ 2 , to test the claim σ2 < 5.6 if n = 28, s2 = 3.5, and α = 0.10. A) 16.875 B) 14.324 C) 18.132 D) 21.478 Answer: A 9) Compute the standardized test statistic, χ 2 to test the claim σ2 ≠ 54.4 if n = 10, s2 = 60, and α = 0.01. A) 9.926 B) 3.276 C) 4.919 D) 12.008 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) Test the claim that σ2 = 38.7 if n = 12, s2 = 32.4 and α = 0.05. Assume that the population is normally distributed. 2 2 Answer: critical values χ L = 3.816 and χ R = 21.920; standardized test statistic χ 2 = 9.209; fail to reject H0 ; There is not sufficient evidence to reject the claim. 11) Test the claim that σ2 ≤ 7.2 if n = 15, s2 = 6, and α = 0.05. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 6.571; standardized test statistic χ 2 ≈ 11.667; fail to reject H 0; There is not sufficient evidence to reject the claim. 12) Test the claim that σ2 ≥ 22.4 if n = 20, s2 = 43.4, and α = 0.01. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 36.191; standardized test statistic χ 2 ≈ 36.813; reject H0 ; There is sufficient evidence to reject the claim. 13) Test the claim that σ2 > 11.4 if n = 18, s2 = 16.2, and α = 0.01. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 33.409; standardized test statistic χ 2 ≈ 24.158; fail to reject H0 ; There is not sufficient evidence to reject the claim. 14) Test the claim that σ2 < 39.2 if n = 28, s2 = 24.5, and α = 0.10. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 18.114; standardized test statistic χ 2 ≈ 16.875; reject H0 ; There is sufficient evidence to support the claim. Page 14
15) Test the claim that σ2 ≠ 61.2 if n = 10, s2 = 67.5, and α = 0.01. Assume that the population is normally distributed. 2 2 Answer: critical values χ L = 1.735 and χ R = 23.589; standardized test statistic χ 2 ≈ 9.926; fail to reject H 0 ; There is not sufficient evidence to support the claim. 16) Test the claim that σ = 8.28 if n = 12, s = 7.6, and α = 0.05. Assume that the population is normally distributed. 2 2 Answer: critical values χ L = 3.816 and χ R = 21.920; standardized test statistic χ 2 ≈ 9.267; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 17) Test the claim that σ ≤ 9.38 if n = 15, s = 8.54, and α = 0.05. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 6.571; standardized test statistic χ 2 ≈ 11.605; fail to reject H 0; There is not sufficient evidence to reject the claim. 18) Test the claim that σ ≥ 14.32 if n = 20, s = 19.92, and α = 0.01. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 36.191; standardized test statistic χ 2 ≈ 36.766; reject H0 ; There is sufficient evidence to reject the claim. 19) Test the claim that σ > 4.14 if n = 18, s = 4.92, and α = 0.01. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 33.409; standardized test statistic χ 2 ≈ 24.009; fail to reject H0 ; There is not sufficient evidence to support the claim. 20) Test the claim that σ < 11.85 if n = 28, s = 9.35 and α = 0.10. Assume that the population is normally distributed. 2 Answer: critical value χ 0 = 18.114; standardized test statistic χ 2 ≈ 16.809; reject H0 ; There is sufficient evidence to support the claim. 21) Test the claim that σ ≠ 10.44 if n = 10, s = 10.96, and α = 0.01. Assume that the population is normally distributed. 2 2 Answer: critical values χ L = 1.735 and χ R = 23.589; standardized test statistic χ 2 ≈ 9.919; fail to reject H 0 ; There is not sufficient evidence to support the claim. 22) Listed below are the April utility bills (in dollars) for one neighborhood. Assuming that the data is normally distributed, test the claim that the standard deviation for the data is $15. Use α = 0.01. 70 48 41 68 69 55 70 57 60 83 32 60 72 58 2 2 Answer: critical values χ L = 3.565 and χ R = 29.819; standardized test statistic χ 2 ≈ 10.42; fail to reject H 0 ; There is not sufficient evidence to reject the claim.
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23) The commute times (in minutes) of 20 randomly selected adult males are listed below. Test the claim that the variance is less than 6.25. Use α = 0.05. Assume the population is normally distributed. 70 72 71 70 69 73 69 68 70 71 67 71 70 74 69 68 71 71 71 72 2 Answer: critical value χ 0 = 10.117; standardized test statistic χ 2 ≈ 9.098; reject H0 ; There is sufficient evidence to support the claim. 24) A shipping firm suspects that the variance for a certain brand of tire used by its trucks is greater than 1,000,000. To check the claim, the firm puts 101 of these tires on its trucks and gets a standard deviation of 1200 miles. At α = 0.05, test the shipping firm's claim. 2 Answer: critical value χ 0 = 124.342; standardized test statistic χ 2 = 144; reject H0 ; There is sufficient evidence to support the claim. 25) A brokerage firm needs information concerning the standard deviation of the account balances of its customers. From previous information it was assumed to be $250. A random sample of 61 accounts was checked. The standard deviation was $286.20. At α = 0.01, test the firm's assumption. Assume that the account balances are normally distributed. 2 2 Answer: critical values χ L = 35.534 and χ R = 91.952; standardized test statistic χ 2 ≈ 78.634; fail to reject H0 ; There is not sufficient evidence to reject the claim. 26) In one area, monthly incomes of technology related workers have a standard deviation of $650. It is believed that the standard deviation of monthly incomes of non-technology workers is higher. A sample of 71 non-technology workers are randomly selected and found to have a standard deviation of $950. Test the claim that non-technology workers have a higher standard deviation. Use α = 0.05. 2 Answer: critical value χ 0 = 90.531; standardized test statistics χ 2 = 149.527; reject H 0 ; There is sufficient evidence to support the claim. 27) A statistics professor at the college level determined that the standard deviation of men's heights is 2.5 inches. The professor then randomly selected 41 female students and found the standard deviation to be 2.9 inches. Test the professor's claim that the standard deviation of female heights is greater than 2.5 inches. Use α = 0.01. 2 Answer: critical value χ 0 = 63.691; standardized test statistic χ 2 = 53.824; fail to reject H 0 ; There is not sufficient evidence to support the claim. 28) A new gun-like 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 be less than 0.05 to ensure proper inoculation. A random sample of 25 injections was measured. Suppose the P-value for the test is p = 0.0024. State the proper conclusion using α = 0.01. Answer: Since α = 0.01 > p = 0.0024, H 0 can be rejected. There is sufficient evidence to indicate that the variance in the amount of serum injected exceeds 0.05.
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10.5 Putting It Together: Which Method Do I Use? 1 Determine the appropriate hypothesis test to perform. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) In 2020, 36% of adults in a certain country were morbidly obese. A health practitioner suspects that the percent has changed since then. She obtains a random sample of 1042 adults and finds that 393 are morbidly obese. Is ^
this sufficient evidence to support the practitioner's claim at the α = 0.1 level of significance? Round p to five decimal places when calculating the test statistic. Answer: H 0 : p = 0.36, H1 : p ≠ 0.36, Z = 1.15, P-value = 0.2485; do not reject H0 . There is not sufficient evidence at the α = 0.1 level of significance to support the practitioner's claim that the percentage of adults who are morbidly obese has change since 2010. 2) The mean expenditure for auto insurance in a certain state was $806. An insurance salesperson in this state believes that the mean expenditure for auto insurance is less today. She obtains a simple random sample of 32 auto insurance policies and determines the mean expenditure to be $781 with a standard deviation of $39.13. Is there enough evidence to support the claim that the mean expenditure for auto insurance is less than the 2010 amount at the α = 0.05 level of significance? Answer: H 0 : μ = $806, H 1 :μ < $806, T0 = -3.61, P-value = 0.0005; reject H0 . There is sufficient evidence at the α = 0.05 level of significance to support the salesperson's claim that the mean expenditure for auto insurance is less than the 2010 amount. 3) A pharmaceutical company manufactures a 81-mg pain reliever. Company specifications include that the standard deviation of the amount of the active ingredient must be less than 2 mg. The quality-control manager selects a random sample of 30 tablets from a certain batch and finds that the standard deviation is 2.4 mg. Assume that the amount of the active ingredient is normally distributed. Test the claim that the standard deviation of the amount of the active ingredient is greater than 2 mg at the α = 0.05 level of significance. Answer: H 0 : σ = 2 mg, H1 :σ > 2 mg, χ 2 = 41.76, P-value = 0.0590; do not to reject H0 . There is not sufficient evidence at the α = 0.05 level of significance to support the claim that the standard deviation of the amount of the active ingredient is greater than 2 mg. 4) The mean for the number of pets owned per household was 1.9. A poll of 1023 households conducted this year reported the mean for the number of pets owned per household to be 1.8. Assuming σ = 1.1, is there sufficient evidence to support the claim that the mean number of pets owned has changed since at the α = 0.1 level of significance? Answer: H 0 : μ = 1.9, H1 :μ ≠ 1.9, Z0 = -2.91, P-value = 0.0036; reject H0 . There is sufficient evidence at the α = 0.10 level of significance to support the claim that the mean number of pets owned per household has changed since 2010.
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10.6 The Probability of a Type II Error and the Power of the Test 1 Determine the probability of making a Type II error and compute the power of the test. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) It is desired to test H0 : μ = 45 against H1 : μ < 45 using α = 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 40, what is the probability that the hypothesis test would lead the investigator to commit a Type II error? A) 0.1469 B) 0.8531 C) 0.3531 D) 0.2938 Answer: A 2) It is desired to test H0 : μ = 10 against H1 : μ ≠ 10 using α = 0.05. The population in question is uniformly distributed with a standard deviation of 1.5. A random sample of 100 will be drawn from this population. If μ is really equal to 9.9, what is the value of β associated with this test? A) 0.1028 B) 0.8972 C) 0.3972 D) 0.0514 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) 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 estimate. The following data resulted: x = 31.2 miles per gallon, s = 7 miles per gallon. Calculate the value of β if the true value of the mean is really 32 miles per gallon. Use α = 0.025. Answer: Since the alternative hypothesis is H1 : μ > 30, the test is one-tailed. Thus, α = 0.025 is required in the upper tail of the z distribution, and we have z 0.025 = 1.96. The value of x on the border between the rejection region and the acceptance region is found using z=
x - 30 σ ⇒x = z + 30 ⇒ x = 1.96 σ/ n n
β = P(x < 31.96, when μ a = 32) = P z <
7 + 30 ⇒ x = 31.96 49 31.96 - 32 = P(z < -0.04) = 0.5 - 0.0160 = 0.4840 7/ 49
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) The probability of making a Type II error is indicated by the letter A) β B) α C) μ
D) σ
Answer: A 5) A Type II error is an error that A) fails to reject H 0, given that H1 is true.
B) rejects H 0 , given that H1 is false.
C) fails to reject H 0, given that H1 is false.
D) rejects H 0 , given that H1 is true.
Answer: A 6) In a hypothesis test as the population mean gets closer to the hypothesized mean, β A) increases. B) decreases. C) does not change. D) becomes the reciprocal of α. Answer: A Page 18
7) The power of the test is A) 1 - β.
B) α.
C) β.
D) 1 - α.
Answer: A 8) The greater the power of the test the more likely the test will A) reject H0 when H1 is true. B) reject H0 when H1 is false. C) fail to reject H 0 when H1 is true.
D) fail to reject H0 when H 1 is false.
Answer: A 9) If β is computed to be 0.763, then the power of the test is A) 0.237. B) 0.763.
C) 0.263.
Answer: A 10) A power curve is a graphic that plots A) the power of the test against values of the population mean that make H 0 false. B) the power of the test against values of the population mean that make H 0 true. C) the power of the test against values of the hypothesized mean that make H 0 false. D) the power of the test against values of the hypothesized mean that make H 0 true. Answer: A
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D) 0.737.
Ch. 10 Hypothesis Tests Regarding a Parameter Answer Key 10.1 The Language of Hypothesis Testing 1 Determine the null and alternative hypotheses. 1) A 2) A 3) A 4) H0 : μ = 5.6, H 1 : μ < 5.6 5) H0 : μ = 46.6, H1 : μ ≠ 46.6 6) H0 : μ = $110, H1 : μ > $110 7) H0 : μ = $100, H1 : μ < $100 8) H0 : μ = 3.6, H 1 : μ < 3.6 9) H0 : p = 0.5, H 1: p > 0.5 10) H0 : σ = 50, H1 : σ > 50 11) A 12) A 13) A 2 Explain Type I and Type II errors. 1) type I: rejecting H0 : μ = 53.6 when μ = 53.6 type II: failing to reject H0 : μ = 53.6 when μ ≠ 53.6 2) type I: rejecting H0 : μ = $110 when μ ≤ $110 type II: failing to reject H0 : μ = $110 when μ > $110 3) type I: rejecting H0 : μ = $92 when μ ≥ $92 type II: failing to reject H0 : μ = $92 when μ < $92 4) type I: rejecting H0 : p = 0.5 when p ≤ 0.5 type II: failing to reject H0 : p = 0.5 when p > 0.5 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 3 State conclusions to hypothesis tests. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A
10.2 Hypothesis Tests for a Population Proportion 1 Explain the logic of hypothesis testing. 1) A Page 20
2) A 3) A 4) A 5) A 6) A 2 Test hypotheses about a population proportion. 1) A 2) At α = 0.05, α < P-value = 0.1,235, so H 0 cannot be rejected. There is insufficient evidence to indicate that more than 80% of the firms do not offer any child-care benefits. 3) test statistic ≈ 1.21; P-value = 2.33; fail to reject H0 ; There is not sufficient evidence to support the claim. 4) P-value = 0.1131; test statistic ≈ 1.21; fail to reject H0 ; There is not sufficient evidence to support the claim p > 0.5. The Republican candidate has no chance. 5) P-value = 0.3264; test statistic ≈ -0.45; fail to reject H0 ; There is not sufficient evidence to support the airline's claim. 6) P-value = 0.9082; test statistic ≈ -1.33; fail to reject H0 ; There is not sufficient evidence to reject the claim. 7) P-value = 0.0058; test statistic ≈ 2.76; reject H 0 ; There is sufficient evidence to reject the university's claim. 8) P-value = 0.0114; test statistic ≈ 2.53; reject H 0 ; There is sufficient evidence to reject the claim that this is not a biased coin. 3 Test hypotheses about a population proportion using the binomial probability distribution. 1) H0 : p = 0.037 and H1 : p > 0.037 Since 250(0.037)(1 - 0.037) = 8.91 < 10, we calculate the P-value: P(X ≥ 4) = 1 - P(X < 4) = 1 - P(X ≤ 3) = 1 - 0.016 = 0.984. Since 0.984 > 0.1, we do not reject H0 . There is not significant evidence to conclude that the percentage has increased. 2) H0 : p = 0.072 and H1 : p > 0.072 Since 125(0.072)(1 - 0.072) = 8.35 < 10, we calculate the P-value: P(X ≥ 17) = 1 - P(X < 17) = 1 - P(X ≤ 16) = 1 - 0.992 = 0.008. Since 0.008 < 0.01, we reject H0 . There is evidence to conclude that the percentage has increased. 3) H0 : p = 0.059 and H1 : p > 0.059 Since 180(0.059)(1 - 0.059) = 9.99 < 10, we calculate the P-value: P(X ≥ 12) = 1 - P(X < 12) = 1 - P(X ≤ 11) = 1 - 0.626 = 0.374. Since 0.374 > 0.05, we do not reject H0 . There is not significant evidence to conclude that the percentage has increased. 4) A
10.3 Hypothesis Tests for a Population Mean 1 Test hypotheses about a mean. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 2 Test hypotheses about a mean. (P-Value Approach). 1) test statistic ≈ 1.890, 0.05 < P-value < 0.10, fail to reject H0 ; There is not sufficient evidence to reject the claim. Page 21
2) test statistic ≈ -2.189, 0.025 < P-value < 0.05, reject H0 ; There is sufficient evidence to reject the claim 3) test statistic ≈ 1.452, 0.05 < P-value < 0.10, fail to reject H0 ; There is not sufficient evidence to reject the claim 4) test statistic ≈ -0.894, 0.15 < P-value < 0.20, fail to reject H0 ; There is not sufficient evidence to support the claim 5) test statistic ≈ 1.667, 0.05 < P-value < 0.10, fail to reject H0 ; There is not sufficient evidence to support the claim 6) test statistic ≈ -0.825, 0.40 < P-value < 0.50, fail to reject H0 ; There is not sufficient evidence to support the claim 7) test statistic ≈ -3.407; 0.001 < P-value < 0.0025; reject H0 ; There is sufficient evidence to support the retailer's claim. 8) test statistic ≈ -1.25; 0.20 < P-value < 0.30; fail to reject H0 ; There is not sufficient evidence to reject the manufacturer's claim. 9) test statistic -2.298; 0.01 < P-value < 0.02; reject H0 ; There is sufficient evidence to support the shipping firm's claim. 10) test statistic ≈ -1.917; 0.025 < P-value < 0.05; fail to reject H 0 ; There is not sufficient evidence to support the government agency's claim. 11) x = 60.21, s = 13.43; test statistic ≈ 1.173; 0.10 < P-value < 0.15; fail to reject H0 ; There is not sufficient evidence to reject the claim. 12) Standardized test statistic ≈ -1.118; Therefore, at 19 degrees of freedom, the P-value must lie between 0.10 and 0.25. Since the P-value > α, fail to reject H0 . There is not sufficient evidence to support the bank's claim. 13) Standardized test statistic ≈ 1.118; Therefore, at a degree of freedom of 19, the P-value must lie between 0.10 and 0.25. Since the P-value > α, fail to reject H0 . There is not sufficient evidence to support the store's claim. 14) A
10.4 Hypothesis Tests for a Population Standard Deviation 1 Test hypotheses about a population standard deviation. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 2 2 10) critical values χ L = 3.816 and χ R = 21.920; standardized test statistic χ 2 = 9.209; fail to reject H0 ; There is not sufficient evidence to reject the claim. 2 11) critical value χ 0 = 6.571; standardized test statistic χ 2 ≈ 11.667; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 2 12) critical value χ 0 = 36.191; standardized test statistic χ 2 ≈ 36.813; reject H0 ; There is sufficient evidence to reject the claim. 2 13) critical value χ 0 = 33.409; standardized test statistic χ 2 ≈ 24.158; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 2 14) critical value χ 0 = 18.114; standardized test statistic χ 2 ≈ 16.875; reject H0 ; There is sufficient evidence to support the claim. 2 2 15) critical values χ L = 1.735 and χ R = 23.589; standardized test statistic χ 2 ≈ 9.926; fail to reject H0 ; There is not sufficient evidence to support the claim.
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2 2 16) critical values χ L = 3.816 and χ R = 21.920; standardized test statistic χ 2 ≈ 9.267; fail to reject H0 ; There is not sufficient evidence to reject the claim. 2 17) critical value χ 0 = 6.571; standardized test statistic χ 2 ≈ 11.605; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 2 18) critical value χ 0 = 36.191; standardized test statistic χ 2 ≈ 36.766; reject H0 ; There is sufficient evidence to reject the claim. 2 19) critical value χ 0 = 33.409; standardized test statistic χ 2 ≈ 24.009; fail to reject H 0 ; There is not sufficient evidence to support the claim. 2 20) critical value χ 0 = 18.114; standardized test statistic χ 2 ≈ 16.809; reject H0 ; There is sufficient evidence to support the claim. 2 2 21) critical values χ L = 1.735 and χ R = 23.589; standardized test statistic χ 2 ≈ 9.919; fail to reject H0 ; There is not sufficient evidence to support the claim. 2 2 22) critical values χ L = 3.565 and χ R = 29.819; standardized test statistic χ 2 ≈ 10.42; fail to reject H0 ; There is not sufficient evidence to reject the claim. 2 23) critical value χ 0 = 10.117; standardized test statistic χ 2 ≈ 9.098; reject H0 ; There is sufficient evidence to support the claim. 2 24) critical value χ 0 = 124.342; standardized test statistic χ 2 = 144; reject H0 ; There is sufficient evidence to support the claim. 2 2 25) critical values χ L = 35.534 and χ R = 91.952; standardized test statistic χ 2 ≈ 78.634; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 2 26) critical value χ 0 = 90.531; standardized test statistics χ 2 = 149.527; reject H0 ; There is sufficient evidence to support the claim. 2 27) critical value χ 0 = 63.691; standardized test statistic χ 2 = 53.824; fail to reject H0 ; There is not sufficient evidence to support the claim. 28) Since α = 0.01 > p = 0.0024, H 0 can be rejected. There is sufficient evidence to indicate that the variance in the amount of serum injected exceeds 0.05.
10.5 Putting It Together: Which Method Do I Use? 1 Determine the appropriate hypothesis test to perform. 1) H0 : p = 0.36, H1 : p ≠ 0.36, Z = 1.15, P-value = 0.2485; do not reject H 0 . There is not sufficient evidence at the α = 0.1 level of significance to support the practitioner's claim that the percentage of adults who are morbidly obese has change since 2010. 2) H0 : μ = $806, H1 :μ < $806, T0 = -3.61, P-value = 0.0005; reject H0 . There is sufficient evidence at the α = 0.05 level of significance to support the salesperson's claim that the mean expenditure for auto insurance is less than the 2010 amount.
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3) H0 : σ = 2 mg, H1 :σ > 2 mg, χ 2 = 41.76, P-value = 0.0590; do not to reject H0 . There is not sufficient evidence at the α = 0.05 level of significance to support the claim that the standard deviation of the amount of the active ingredient is greater than 2 mg. 4) H0 : μ = 1.9, H 1 :μ ≠ 1.9, Z 0 = -2.91, P-value = 0.0036; reject H 0. There is sufficient evidence at the α = 0.10 level of significance to support the claim that the mean number of pets owned per household has changed since 2010.
10.6 The Probability of a Type II Error and the Power of the Test 1 Determine the probability of making a Type II error and compute the power of the test. 1) A 2) A 3) Since the alternative hypothesis is H 1 : μ > 30, the test is one-tailed. Thus, α = 0.025 is required in the upper tail of the z distribution, and we have z 0.025 = 1.96. The value of x on the border between the rejection region and the acceptance region is found using z=
x - 30 σ ⇒x = z + 30 ⇒ x = 1.96 σ/ n n
β = P(x < 31.96, when μ a = 32) = P z < 4) A 5) A 6) A 7) A 8) A 9) A 10) A
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7 + 30 ⇒ x = 31.96 49 31.96 - 32 = P(z < -0.04) = 0.5 - 0.0160 = 0.4840 7/ 49
Ch. 11 Inferences on Two Population Paraeters 11.1 Inference about Two Population Proportions 1 Distinguish between independent and dependent sampling. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Classify the two given samples as independent or dependent. Sample 1: Pre-training blood pressure of 24 people Sample 2: Post-training blood pressure of 24 people A) dependent B) independent Answer: A 2) Classify the two given samples as independent or dependent. Sample 1: The heights in inches of 15 newborn females Sample 2: The heights in inches of 15 newborn males A) independent B) dependent Answer: A 3) Classify the two given samples as independent or dependent. Sample 1: The scores of 19 students who took a statistics final Sample 2: The scores of 19 different students who took a physics final A) independent B) dependent Answer: A 4) If the individuals selected for a sample have no influence upon which individuals are selected for a second sample, then the samples are said to be A) independent B) dependent C) inconsistent D) consistent Answer: A 5) Two samples are said to be dependent if A) the individuals in one sample are used to determine the individuals in a second sample. B) the individuals in one sample have no influence over the selection of the individuals in a second sample. C) some individuals, but not all, in one sample exert influence over who is selected for inclusion in a second ample. D) sampling for inclusion in the two samples is done with replacement. Answer: A Determine whether the sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. 6) A psychologist wants to measure the effect of music on memory. He randomly selects 80 students and measures their scores on a memory test conducted in silence. The next day he measures their scores on a similar test conducted while classical music is playing. The mean score without music is compared to the mean score with music. A) quantitative, dependent B) qualitative, dependent C) quantitative, independent D) qualitative, independent Answer: A 7) A researcher randomly selected 100 adults aged 18-25 and 100 adults aged 50-60. Within each age group, she recorded the number of smokers. A) qualitative, independent B) quantitative, independent C) qualitative, dependent D) quantitative, dependent Answer: A Page 1
8) One hundred men suffering from high cholesterol were randomly assigned to receive placebo or a cholesterol-lowering medication. After three months, the mean cholesterol level of those receiving placebo was compared with the mean cholesterol level of those receiving the medication. A) quantitative, independent B) quantitative, dependent C) qualitative, dependent D) qualitative, independent Answer: A 9) A group of wine tasters rated Chardonnay wines from two different wineries as poor, acceptable, good or excellent. A) qualitative, dependent B) quantitative, independent C) quantitative, dependent D) qualitative, independent Answer: A 10) A university compared the mean salary of its science graduates ten years after graduation with the mean salary of its social science graduates ten years after graduation. A) quantitative, independent B) qualitative, dependent C) quantitative, dependent D) qualitative, independent Answer: A 2 Test hypotheses regarding two proportions from independent samples. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Find the standardized test statistic, z to test the hypothesis that p1 = p2 . Use α = 0.05. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 50, x1 = 35, and n 2 = 60, x2 = 40 A) 0.374
B) 0.982
C) 1.328
D) 2.361
Answer: A 2) Find the standardized test statistic estimate, z, to test the hypothesis that p1 > p2. Use α = 0.01. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 100, x1 = 38, and n 2 = 140, x2 = 50 A) 0.362
B) 2.116
C) 1.324
D) 0.638
Answer: A 3) Find the standardized test statistic, z, to test the hypothesis that p1 < p2 . Use α = 0.10. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 550, x1 = 121, and n 2 = 690, x2 = 195 A) -2.513
B) -2.132
C) -0.985
D) 1.116
Answer: A 4) Find the standardized test statistic, z, to test the hypothesis that p1 ≠ p2 . Use α = 0.02. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 1000, x1 = 250, and n 2 = 1200, x2 = 195 A) 5.087 Answer: A
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B) 2.798
C) 4.761
D) 3.212
5) To perform a hypothesis test of two population proportions, the pooled estimate of p must be determined. The ^
pooled estimate, p, is ^ x1 + x2 A) p = n1 + n2
^
B) p =
x1 n1
+
x2 n2
^
C) p =
n 2x 1 + n 1 x 2 n1 + n2
^
D) p =
x1 + x2 n 1n 2
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) Test the hypothesis that p1 = p2 . Use α = 0.05. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 50, x1 = 35, and n 2 = 60, x2 = 40 Answer: test statistic z 0 ≈ 0.37; P-value = 0.7114; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 7) Test the hypothesis that p1 > p2 . Use α = 0.01. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 100, x1 = 38, and n 2 = 140, x2 = 50 Answer: test statistic z 0 ≈ 0.36; P-value = 0.3594; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 8) Test the hypothesis that p1 < p2 . Use α = 0.10. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 550, x1 = 121, and n 2 = 690, x2 = 195 Answer: test statistic z 0 ≈ -2.51; P-value = 0.0060; reject H 0; There is sufficient evidence to support the hypothesis. 9) Test the hypothesis that p1 ≠ p2 . Use α = 0.02. The sample statistics listed below are from independent samples. Sample statistics: n 1 = 1000, x1 = 250, and n 2 = 1200, x2 = 195. Answer: test statistic z 0 ≈ 5.09; P-value ≈ 0.0000; reject H0 ; There is sufficient evidence to support the hypothesis. 10) In a recent survey of drinking laws, a random sample of 1000 women showed that 65% were in favor of increasing the legal drinking age. In a random sample of 1000 men, 60% favored increasing the legal drinking age. Test the hypothesis that the percentage of men and women favoring a higher legal drinking age is the same. Use α = 0.05. Answer: α = 0.05: p1 = p2 ; test statistic z 0 ≈ 2.31; P-value = 0.0208; reject the null hypothesis; There is sufficient evidence to reject the hypothesis. 11) A recent survey showed that in a sample of 100 elementary school teachers, 15 were single. In a sample of 180 high school teachers, 36 were single. Is the proportion of high school teachers who were single greater than the proportion of elementary teachers who were single? Use α = 0.01. Answer: hypothesis: p1 < p2 ; test statistic z 0 ≈ -1.04; P-value = 0.1515; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 12) To test the effectiveness of a new drug designed to relieve flu symptoms, 200 patients were randomly selected and divided into two equal groups. One group of 100 patients was given a pill containing the drug while the other group of 100 was given a placebo. What can we conclude about the effectiveness of the drug if 62 of those actually taking the drug felt a beneficial effect while 41 of the patients taking the placebo felt a beneficial effect? Use α = 0.05. Answer: hypothesis: p1 > p2 ; test statistic z 0 ≈ 2.97; P-value = 0.0015; reject H 0; The new drug is effective.
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13) A random sample of 100 students at a high school was asked whether they would ask their father or mother for help with a financial problem. A second sample of 100 different students was asked the same question regarding a dating problem. If 43 students in the first sample and 47 students in the second sample replied that they turned to their mother rather than their father for help, test the hypothesis of no difference in the proportions. Use α = 0.02. Answer: hypothesis: p1 = p2 ; test statistic z 0 ≈ -0.57; P-value = 0.5685; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. 14) In the initial test of the Salk vaccine for polio, 400,000 children were selected and divided into two groups of 200,000. One group was vaccinated with the Salk vaccine while the second group was vaccinated with a placebo. Of those vaccinated with the Salk vaccine, 33 later developed polio. Of those receiving the placebo, 115 later developed polio. Test the hypothesis that the Salk vaccine is effective in lowering the polio rate. Use α = 0.01. Answer: hypothesis: p1 < p2 ; test statistic z 0 ≈ -6.74; P-value ≈ 0.0000; reject H0 ; There is sufficient evidence to support the hypothesis. 15) A well-known study of 22,000 male physicians was conducted to determine if taking aspirin daily reduces the chances of a heart attack. Half of the physicians were given a regular dose of aspirin while the other half was given placebos. Six years later, among those who took aspirin, 104 suffered heart attacks while among those who took placebos, 189 suffered heart attacks. Does it appear that the aspirin can reduce the number of heart attacks among the sample group that took aspirin? Use α = 0.01. Answer: hypothesis: p1 < p2 ; test statistic z 0 ≈ -5.00; P-value ≈ 0.0000; reject H0 ; There is sufficient evidence to support the hypothesis. 3 Construct and interpret confidence intervals for the difference between two population proportions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Construct a 95% confidence interval for p1 - p2 . The sample statistics listed below are from independent samples. Sample statistics: n 1 = 50, x1 = 35, and n 2 = 60, x2 = 40 A) (-0.141, 0.208)
B) (-0.871, 0.872)
C) (-1.341, 1.781)
D) (-2.391, 3.112)
Answer: A 2) Construct a 98% confidence interval for p1 - p2 . The sample statistics listed below are from independent samples. Sample statistics: n 1 = 1000, x1 = 250, and n 2 = 1200, x2 = 195 A) (0.047, 0.128)
B) (-0.621, 0.781)
C) (0.581, 1.819)
D) (1.516, 3.021)
Answer: A 3) In a recent survey of drinking laws, a random sample of 1000 women showed that 65% were in favor of increasing the legal drinking age. In a random sample of 1000 men, 60% favored increasing the legal drinking age. Construct a 95% confidence interval for p1 - p2 . A) (0.008, 0.092)
B) (0.587, 0.912)
C) (-1.423, 1.432)
D) (-2.153, 1.679)
Answer: A 4) Construct a 95% confidence interval for p1 - p2 for a survey that finds 30% of 240 males and 41% of 200 females are opposed to the death penalty. A) (-0.200, -0.021) B) (-1.532, 1.342) Answer: A
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C) (-0.561, 0.651)
D) (-1.324, 1.512)
5) A random sample of 100 students at a high school was asked whether they would ask their father or mother for help with a financial problem. A second sample of 100 different students was asked the same question regarding a dating problem. Forty-three students in the first sample and 47 students in the second sample replied that they turned to their mother rather than their father for help. Construct a 98% confidence interval for p1 - p2. A) (-0.204, 0.124)
B) (-1.324, 1.521)
C) (-0.591, 0.762)
D) (-1.113, 1.311)
Answer: A 6) True or False: When constructing a confidence interval for the difference of two population proportions, a pooled estimate of p is not required. A) True B) False Answer: A 7) To construct a confidence interval for the difference of two population proportions the samples must be independently obtained random samples, both must consist of less than 5% of the population, and ^
^
^
^
A) both np1 (1 - p1 ) ≥ 10 and np2 (1 - p2 ) ≥ 10 must be true. ^
^
^
^
B) only one of np1 (1 - p1 ) ≥ 10 or np2 (1 - p2 ) ≥ 10 must be true. ^
^
^
^
^
^
C) np1 (1 - p1 ) + np2 (1 - p2) ≥ 20. ^
^
D) np1 (1 - p1 ) np2 (1 - p2 ) ≥ 100. Answer: A 4 Determine the sample size necessary for estimating the difference between two population proportions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) 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 the two types 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. How large a sample would be necessary in order to estimate the difference in the true success rates to within 0.10 with 95% reliability? n Number of Successes Procedure A 100 88 Procedure B 100 84 A) n 1 = n 2 = 93 B) n 1 = n 2 = 64 C) n 1 = n 2 = 47 D) n 1 = n 2 = 192 Answer: A 2) A controversial bill is being debated in the state legislature. Representative Williams wants to estimate within 2 percentage points and with 90% confidence the difference in the proportion of her male and female constituents who favor the bill. What sample size should she obtain? A) n 1 = n 2 = 3,383 B) n 1 = n 2 = 2,049 C) n 1 = n 2 = 68 D) n 1 = n 2 = 1,692 Answer: A
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11.2 Inference about Two Means: Dependent Samples 1 Test hypotheses for a population mean from matched-pairs data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Data sets A and B are dependent. Find d. A 22 20 39 35 23 B 20 16 17 27 14 Assume that the paired data came from a population that is normally distributed. A) 9.0 B) -5.1 C) 33.1
D) 25.2
Answer: A 2) Data sets A and B are dependent. Find d. A 2.7 3.7 5.6 2.6 2.7 B 5.1 4.0 3.9 3.8 5.2 Assume that the paired data came from a population that is normally distributed. A) -0.94 B) -0.76 C) 0.58
D) 0.89
Answer: A 3) Data sets A and B are dependent. Find sd. A 36 34 53 49 37 B 34 30 31 41 28 Assume that the paired data came from a population that is normally distributed. A) 7.8 B) 5.6 C) 6.8
D) 8.9
Answer: A 4) Data sets A and B are dependent. Find sd. A 6.0 7.0 8.9 5.9 6.0 B 8.4 7.3 7.2 7.1 8.5 Assume that the paired data came from a population that is normally distributed. A) 1.73 B) 1.21 C) 1.32
D) 1.89
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) Data sets A and B are dependent. Test the claim that μ d = 0. Use α = 0.05. A 16 14 33 29 17 B 14 10 11 21 8 Assume that the paired data came from a population that is normally distributed. Answer: critical values t0 = ±2.776; test statistic t ≈ 2.580; fail to reject H 0; There is not sufficient evidence to reject the claim. 6) Data sets A and B are dependent. Test the claim that μ d =0. Use α = 0.01. A 9.0 10.0 11.9 8.9 9.0 B 11.4 10.3 10.2 10.1 11.5 Assume that the paired data came from a population that is normally distributed. Answer: critical values t0 = ±4.604; test statistic t ≈ -1.215; fail to reject H0 ; There is not sufficient evidence to reject the claim.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) When performing a hypothesis test upon two dependent samples, the variable of interest is A) the differences that exist between the matched-pair data. B) all of the combined data. C) the absolute value of the differences that exist between the matched-pair data. D) the data that is the same in both samples. Answer: A 8) Robustness in hypothesis testing means A) departures from normality do not adversely affect the results. B) there are no departures from normality. C) the data is effected by outliers. D) all processes can be exactly duplicated by selecting another pair of samples. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9) Data sets A and B are dependent. Test the claim that μ d = 0. Use α = 0.05. A 23 21 40 36 24 B 21 17 18 28 15 Assume that the paired data came from a population that is normally distributed. Answer: test statistic t ≈ 2.580; 0.05 < P-value < 0.10; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 10) Data sets A and B are dependent. Test the claim that μ d =0. Use α = 0.01. A 9.1 10.1 12.0 9.0 9.1 B 11.5 10.4 10.3 10.2 11.6 Assume that the paired data came from a population that is normally distributed. Answer: test statistic t ≈ -1.215; 0.20 < P-value < 0.30; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 11) Nine students took the SAT. Their scores are listed below. Later on, they read a book on test preparation and retook the SAT. Their new scores are listed below. Test the claim that the book had no effect on their scores. Use α = 0.05. Assume that the distribution is normally distributed. Student 1 2 3 4 5 6 7 8 9 Scores before reading book 720 860 850 880 860 710 850 1200 950 Scores after reading book 740 860 840 920 890 720 840 1240 970 Answer: claim: μ d = 0; test statistic t ≈ -2.401; 0.04 < P-value < 0.05; reject H0 ; There is sufficient evidence to reject the claim. 12) A football coach claims that players can increase their strength by taking a certain supplement. To test the theory, the coach randomly selects 9 athletes and gives them a strength test using a bench press. The results are listed below. Thirty days later, after regular training using the supplement, they are tested again. The new results are listed below. Test the claim that the supplement is effective in increasing the athletes' strength. Use α = 0.05. Assume that the distribution is normally distributed. Athlete 1 2 3 4 5 6 7 8 9 Before 215 240 188 212 275 260 225 200 185 After 225 245 188 210 282 275 230 195 190 Answer: claim: μ d < 0; test statistic t ≈ -2.177; 0.025 < P-value < 0.05; reject H0 ; There is sufficient evidence to support the claim.
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13) A pharmaceutical company wishes to test a new drug with the expectation of lowering cholesterol levels. Ten subjects are randomly selected and pretested. 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.) Test the company's claim that the drug lowers cholesterol levels. Use α = 0.01. Assume that the distribution is normally distributed. Subject 1 2 3 4 5 6 7 8 9 10 Before 195 225 202 195 175 250 235 268 190 240 After 180 220 210 175 170 250 205 250 190 225 Answer: claim: μ d > 0; test statistic t ≈ 2.752; 0.01 < P-value < 0.02 fail to reject H0 ; There is not sufficient evidence to support the claim. 14) In a study of the effectiveness of diet on weight loss, 20 people were randomly selected to participate in a diet program for 30 days. Test the claim that diet had no bearing on weight loss. Use α = 0.02.. Assume that the distribution is normally distributed. Weight before Program178 210 156 188 193 225 190 165 168 200 (in pounds) Weight after program 182 205 156 190 183 220 195 155 165 200 (in pounds) Weight before Program186 172 166 184 225 145 208 214 148 174 (in pounds) Cont. Weight after program 180 173 165 186 240 138 203 203 142 174 (in pounds) Cont. Answer: claim: μ d = 0; test statistic t ≈ 1.451; 0.10 < P-value < 0.20; fail to reject H0 ; There is not sufficient evidence to reject the claim. 15) A local company is concerned about the number of days missed by its employees due to illness. A random sample of 10 employees is selected. The number of days absent in one year is listed below. An incentive program is offered in an attempt to decrease the number of days absent. The number of days absent in one year after the incentive program is listed below. Test the claim that the incentive program cuts down on the number of days missed by employees. Use α = 0.05. Assume that the distribution is normally distributed. Employee A B C D E F G H I J Days absent before 3 8 7 2 9 4 2 0 7 5 incentive Days absent after 1 7 7 0 8 2 0 1 5 5 incentive Answer: claim: μ d > 0; test statistic t ≈ 3.161; 0.005 < P-value < 0.01; reject H0 ; There is sufficient evidence to support the claim. 16) A physician claims that a person's diastolic blood pressure can be lowered if, instead of taking a drug, the person meditates each evening. Ten subjects are randomly selected and pretested. Their blood pressures, measured in millimeters of mercury, are listed below. The 10 patients are instructed in basic meditation and told to practice it each evening for one month. At the end of the month, their blood pressures are taken again. The data are listed below. Test the physician's claim. Use α = 0.01. Patient 1 2 3 4 5 6 7 8 9 10 Before 85 96 92 83 80 91 79 98 93 96 After 82 90 92 75 74 80 82 88 89 80 Answer: claim: μ d > 0; test statistic t ≈ 3.490; 0.0025 < P-value < 0.005; reject H0 ; There is sufficient evidence to support the claim.
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2 Construct and interpret confidence intervals about the population mean difference of matched-pairs data. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Construct a 95% confidence interval for data sets A and B. Data sets A and B are dependent. A 30 28 47 43 31 B 28 24 25 35 22 Assume that the paired data came from a population that is normally distributed. A) (-0.696, 18.700) B) (-1.324, 8.981) C) (-0.113, 12.761) D) (-15.341, 15.431) Answer: A 2) Construct a 99% confidence interval for data sets A and B. Data sets A and B are dependent. A 5.8 6.8 8.7 5.7 5.8 B 8.2 7.1 7.0 6.9 8.3 Assume that the paired data came from a population that is normally distributed. A) (-4.502, 2.622) B) (-25.123, 5.761) C) (-21.342, 18.982) D) (-15.123, 15.123) Answer: A 3) Nine students took the SAT. Their scores are listed below. Later on, they read a book on test preparation and retook the SAT. Their new scores are listed below. Construct a 95% confidence interval for μ d. Assume that the distribution is normally distributed. Student 1 2 3 4 5 6 7 8 9 Scores before reading book 720 860 850 880 860 710 850 1200 950 Scores after reading book 740 860 840 920 890 720 840 1240 970 A) (-30.503, -0.617) B) (-20.341, 4.852) C) (-10.321, 15.436)
D) (1.651, 30.590)
Answer: A 4) We are interested in comparing the average supermarket prices of two leading colas in the Tampa area. Our sample was taken by randomly going to each of eight supermarkets and recording the price of a six-pack of cola of each brand. The data are shown in the following table. Find a 98% confidence interval for the difference in mean price of brand 1 and brand 2. Assume that the paired data came from a population that is normally distributed. Price Supermarket Brand 1 Brand 2 Difference 1 $2.25 $2.30 $-0.05 2 2.47 2.45 0.02 3 2.38 2.44 -0.06 4 2.27 2.29 -0.02 5 2.15 2.25 -0.10 6 2.25 2.25 0.00 7 2.36 2.42 -0.06 8 2.37 2.40 -0.03 x1 = 2.3125 x2 = 2.3500 d = -0.0375 s1 = 0.1007 s2 = 0.0859 sd = 0.0381 A) (-0.0779, 0.0029) B) (-0.1768, 0.1018) C) (-0.0846, 0.0096) Answer: A
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D) (-0.0722, -0.0028)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests before its introduction into the market. A typical test performed is the following: 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 μ 1 be the true mean weight of individuals before starting the diet and let μ 2 be the true mean weight of individuals after 3 weeks on the diet. Calculate a 90% confidence interval for the difference between the mean weights before and after the diet is used. Assume that the paired data came from a population that is normally distributed. Person Weight Before Diet Weight After Diet 1 163 156 2 208 203 3 201 198 4 210 204 5 217 213 Summary information is as follows: d = 5, sd = 1.58. Answer: The matched pairs confidence interval for μ d is xd ± tα/2
sd n
Confidence coefficient .90 ⇒ α = 1 - .90 = .10. α/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 ± 1.51 ⇒ (3.49, 6.51) 5
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6) Seven randomly selected plants that bottle the same beverage implemented a time management program in hopes of improving productivity. The average time, in minutes, that it took the companies to produce the same quantity of bottles before and after the program are listed below. Assume the two population distributions are normal. Construct a 90% confidence interval for μ d . Assume that the paired data came from a population that is normally distributed. Plant 1 2 Before 75 89 After 70 80 A) (0.21, 10.93)
3 31 30
4 90 85
5 6 7 120 50 40 100 49 42 B) (1.60, 9.54)
C) (-0.22, 11.36)
D) (-22, 33.3)
Answer: A 7) When forming a confidence interval for matched-pair data the point estimate is the A) mean of the differences. B) difference of the means. C) standard deviation of the differences. D) differences of the standard deviations. Answer: A
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11.3 Inference about Two Means: Independent Samples 1 Test hypotheses regarding the difference of two independent means. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Find the test statistic, t, to test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected and come from populations that are normal. The sample statistics are given below. n 1 = 25 n 2 = 30 x1 = 23
x2 = 21
s1 = 1.5 A) 4.361
s2 = 1.9 B) 3.287
C) 2.892
D) 1.986
Answer: A 2) Find the test statistic, t, to test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected and come from populations that are normal. The sample statistics are given below. n 1 = 14 n 2 = 12 x1 = 7
x2 = 8
s1 = 2.5 A) -0.954
s2 = 2.8 B) -0.915
C) -1.558
D) -0.909
Answer: A 3) Find the test statistic, t, to test the hypothesis that μ 1 > μ 2 . Two samples are randomly selected and come from populations that are normal. The sample statistics are given below. n 1 = 18 n 2 = 13 x1 = 640 s1 = 40 A) 1.282
x2 = 625 s2 = 25 B) 3.271
C) 2.819
D) 1.865
Answer: A 4) Find the test statistic, t, to test the hypothesis that μ 1 < μ 2 . Two samples are randomly selected and come from populations that are normal. The sample statistics are given below. n 1 = 15 n 2 = 15 x1 = 28.09 s1 = 2.9 A) -2.450
x2 = 30.64 s2 = 2.8 B) -3.165
C) -1.667
D) -0.669
Answer: A 5) Find the test statistic, t, to test the hypothesis that μ 1 ≠ μ 2 . Two samples are randomly selected and come from populations that are normal. The sample statistics are given below. n 1 = 11 n 2 = 18 x1 = 5.1
x2 = 5.5
s1 = 0.76 A) -1.546
s2 = 0.51
Answer: A
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B) -1.821
C) -2.123
D) -1.326
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) Test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 25
n 2 = 30
x1 = 19
x2 = 17
s1 = 1.5
s2 = 1.9
Answer: critical value t0 = ±2.064; test statistic ≈ 4.361; reject H0 ; There is sufficient evidence to reject the hypothesis. 7) Test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 14
n 2 = 12
x1 = 18
x2 = 19
s1 = 2.5
s2 = 2.8
Answer: critical value t0 = ±2.201; test statistic ≈ -0.954; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. 8) Test the hypothesis that μ 1 > μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 18
n 2 = 13
x1 = 650
x2 = 635
s1 = 40
s2 = 25
Answer: critical value t0 = 3.055; test statistic ≈ 1.282; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 9) Test the hypothesis that μ 1 < μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 15
n 2 = 15
x1 = 23.1
x2 = 25.65
s1 = 2.9
s2 = 2.8
Answer: critical value t0 = -1.761; test statistic = -2.450; reject H 0; There is sufficient evidence to support the hypothesis. 10) Test the hypothesis that μ 1 ≠ μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 11
n 2 = 18
x1 = 3.8
x2 = 4.2
s1 = 0.76
s2 = 0.51
Answer: critical value t0 = ±2.764; test statistic ≈ -1.546; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis.
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11) A local supermarket claims that the waiting time for its customers to be served is the lowest in the area. A competitor's supermarket checks the waiting times at both supermarkets. The sample statistics are listed below. Test the local supermarket's hypothesis. Use α = 0.05. Local Supermarket Competitor Supermarket n 1 = 15 n 2 = 16 x1 = 5.3 minutes s1 = 1.1 minutes
x2 = 5.6 minutes s2 = 1.0 minutes
Answer: critical value t0 = -1.761; test statistic ≈ -0.793; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 12) A study was conducted to determine if the salaries of librarians from two neighboring cities were equal. A sample of 15 librarians from each city was randomly selected. The mean from the first city was $28,900 with a standard deviation of $2300. The mean from the second city was $30,300 with a standard deviation of $2100. Test the hypothesis that the salaries from both cities are equal. Answer: critical value t0 = ±2.145; test statistic t ≈ -1.741; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 13) Find the test statistic to test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.05. n 1 = 50 n 2 = 60 x1 = 16
x2 = 14
s1 = 1.5 A) 6.2
s2 = 1.9 B) 8.1
C) 4.2
D) 3.8
Answer: A 14) Find the test statistic to test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.05. n 1 = 40 n 2 = 35 x1 = 18
x2 = 19
s1 = 2.5 A) -1.6
s2 = 2.8 B) -0.8
C) -2.6
D) -1.0
Answer: A 15) Find the test statistic to test the hypothesis that μ 1 > μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.05. n 1 = 100 n 2 = 125 x1 = 755
x2 = 740
s1 = 45 A) 2.98
s2 = 25
Answer: A
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B) 2.81
C) 1.86
D) 0.91
16) Find the test statistic to test the hypothesis that μ 1 < μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.05. n 1 = 35 n 2 = 42 x1 = 27.04 s1 = 2.9 A) -3.90
x2 = 29.59 s2 = 2.8 B) -3.16
C) -2.63
D) -1.66
Answer: A 17) Find the test statistic to test the hypothesis that μ 1 ≠ μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.02. n 1 = 51 n 2 = 38 x1 = 5.3
x2 = 5.7
s1 = 0.76 A) -2.97
s2 = 0.51 B) -1.82
C) -2.12
D) -2.32
Answer: A 18) What is the H0 for testing differences of the means of two independent samples? A) H0 :μ 1 - μ 2 = 0
B) H 0 :μ 1 ≠ μ 2
C) H 0:μ 1 > μ 2
D) H0 :μ 1 < μ 2
Answer: A 19) The degrees of freedom used when testing two independent samples where the population standard deviation is unknown is A) the smaller of n 1 - 1 or n 2 - 1. B) the larger of n 1 - 1 or n 2 - 1. C) n 1 + n 2 - 2.
D) n 1 + n 2 - 1.
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 20) Test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. Use α = 0.05. n 1 = 25 n 2 = 30 x1 = 26
x2 = 24
s1 = 1.5
s2 = 1.9
Answer: test statistic ≈ 4.361; P-value < 0.0010; reject H0 ; There is sufficient evidence to reject the hypothesis. 21) Test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. Use α = 0.05. n 1 = 14 n 2 = 12 x1 = 18
x2 = 19
s1 = 2.5
s2 = 2.8
Answer: test statistic ≈ -0.954; 0.30 < P-value < 0.40; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis.
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22) Test the hypothesis that μ 1 > μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. Use α = 0.005. n 1 = 18 n 2 = 13 x1 = 505
x2 = 490
s1 = 40
s2 = 25
Answer: test statistic ≈ 1.282; 0.10 < P-value < 0.15; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 23) Test the hypothesis that μ 1 < μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. Use α = 0.05. n 1 = 15 n 2 = 15 x1 = 23.55
x2 = 26.1
s1 = 2.9
s2 = 2.8
Answer: test statistic = -2.450; 0.01 < P-value < 0.02; reject H0 ; There is sufficient evidence to support the hypothesis. 24) Test the hypothesis that μ 1 ≠ μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. Use α = 0.01. n 1 = 11 n 2 = 18 x1 = 4.1
x2 = 4.5
s1 = 0.76
s2 = 0.51
Answer: test statistic ≈ -1.546; 0.10 < P-value < 0.20; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 25) A local supermarket claims that the waiting time for its customers to be served is the lowest in the area. A competitor's supermarket checks the waiting times at both supermarkets. The sample statistics are listed below. Test the local supermarket's hypothesis. Use α = 0.05. Local Supermarket Competitor Supermarket n 1 = 15 n 2 = 16 x1 = 5.3 minutes s1 = 1.1 minutes
x2 = 5.6 minutes s2 = 1.0 minutes
Answer: test statistic ≈ -0.793; 0.20 < P-value < 0.25; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 26) A study was conducted to determine if the salaries of librarians from two neighboring cities were equal. A sample of 15 librarians from each city was randomly selected. The mean from the first city was $28,900 with a standard deviation of $2300. The mean from the second city was $30,300 with a standard deviation of $2100. Test the hypothesis that the salaries from both cities are equal. Use α = 0.025. Answer: test statistic t ≈ -1.741; 0.10 < P-value < 0.20; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis.
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27) Test the hypothesis that μ 1 = μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.05. n 1 = 50
n 2 = 60
x1 = 19
x2 = 17
s1 = 1.5
s2 = 1.9
Answer: test statistic ≈ 6.17; P-value ≈ 0.0000; reject H0 ; There is sufficient evidence to reject the hypothesis. 28) Test the hypothesis that μ 1 > μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.01. n 1 = 100 n 2 = 125 x1 = 525
x2 = 510
s1 = 45
s2 = 25
Answer: test statistic ≈ 2.99; 0.001 < P-value < 0.0025; reject H0 ; There is sufficient evidence to support the hypothesis. 29) Test the hypothesis that μ 1 < μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.05. n 1 = 35 n 2 = 42 x1 = 21.45
x2 = 24
s1 = 2.9
s2 = 2.8
Answer: test statistic ≈ -3.90; P-value < 0.0005; reject H0 ; There is sufficient evidence to support the hypothesis. 30) Test the hypothesis that μ 1 ≠ μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. Use α = 0.02. n 1 = 51 n 2 = 38 x1 = 5.3
x2 = 5.7
s1 = 0.76
s2 = 0.51
Answer: test statistic ≈ -2.97; 0.005 < P-Value < 0.01; reject H0 ; There is sufficient evidence to support the hypothesis. 31) A study was conducted to determine if the salaries of librarians from two neighboring states were equal. A sample of 100 librarians from each state was randomly selected. The mean from the first state was $28,400 with a standard deviation of $2300. The mean from the second state was $29,800 with a standard deviation of $2100. Test the hypothesis that the salaries from both states are equal. Use α = 0.05. Answer: test statistic ≈ -4.50; P-value < 0.001; reject H 0; There is sufficient evidence to reject the hypothesis. 32) At a local store, 65 female employees were randomly selected and it was found that their mean monthly income was $620 with a standard deviation of $121.50. Seventy-five male employees were also randomly selected and their mean monthly income was found to be $662 with a standard deviation of $168.70. Test the hypothesis that male employees have a higher monthly income than female employees. Use α = 0.01. Answer: test statistic ≈ -1.71; 0.025 < P-value < 0.05; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis.
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33) A medical researcher suspects that the pulse rate of drinkers is higher than the pulse rate of non-drinkers. Use the sample statistics below to test the researcher's suspicion. Use α = 0.05. Drinkers Nondrinkers n 1 = 100 n 2 = 100 x1 = 88
x2 = 85
s1 = 4.8
s2 = 5.3
Answer: test statistic ≈ 4.20; P-value < 0.0005; reject H 0; There is sufficient evidence to support the hypothesis. 34) A history teacher believes that students in an afternoon class score higher than the students in a morning class. The results of a special exam are shown below. Can the teacher conclude that the afternoon students have a higher score? Use α = 0.01. Morning Students Afternoon Students n 1 = 36 n 2 = 41 x1 = 66
x2 = 69
s1 = 5.8
s2 = 6.3
Answer: test statistic ≈ -2.18; 0.01 < P-value < 0.02; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 35) A local restaurant claims that the waiting time for its patrons to be served is the lowest in the area. A competitor restaurant checks the waiting times at both restaurants. The sample statistics are listed below. Test the local restaurant's hypothesis. Use α = 0.05. Local Restaurant Competitor Restaurant n 1 = 45 n 2 = 50 x1 = 5 minutes
x2 = 5.3 minutes
s1 = 1.1 minutes
s2 = 1.0 minute
Answer: test statistic ≈ -1.39; 0.05 < P-value < 0.10; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 36) A university wanted to see whether there was a significant difference in age between its day staff and evening staff. A random sample of 35 staff members is selected from each group. The data are given below. Test the hypothesis that there is no difference in age between the two groups. Use α = 0.05. Day Staff 22 24 24 23 19 19 23 22 18 21 21 18 18 25 29 24 23 22 22 21 20 20 20 27 17 19 18 21 20 23 26 30 25 21 25 Evening Staff 18 23 25 23 21 21 23 24 27 31 24 20 20 23 19 25 24 27 23 20 20 21 25 24 23 28 20 19 23 24 20 27 21 29 30 Answer: day staff x1 = 22, s1 = 3.13; evening staff x2 = 23.29, s2 = 3.27; test statistic = -1.69; 0.10 < P-value < 0.20; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis.
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2 Construct and interpret confidence intervals regarding the difference of two independent means. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Construct a 95% confidence interval for μ 1 - μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 10 n 2 = 12 x1 = 25
x2 = 23
s1 = 1.5 s2 = 1.9 A) (0.487, 3.513)
B) (0.579, 3.421)
C) (1.413, 3.124)
D) (1.554, 3.651)
Answer: A 2) Construct a 95% confidence interval for μ 1 - μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n 1 = 11 n 2 = 18 x1 = 4.8
x2 = 5.2
s1 = 0.76 s2 = 0.51 A) (-0.950, 0.150)
B) (-0.907, 0.107)
C) (-2.762, 2.762)
D) (-1.762, 1.762)
Answer: A 3) Construct a 95% confidence interval for μ 1 - μ 2 . Two samples are randomly selected from normal populations. The sample statistics are given below. n1 = 8 n2 = 7 x1 = 4.1
x2 = 5.5
s1 = 0.76 s2 = 2.51 A) (-3.734, 0.934)
B) (-1.132, 1.543)
C) (-1.679, 1.987)
D) (2.112, 2.113)
Answer: A 4) A study was conducted to determine if the salaries of librarians from two neighboring cities were equal. A sample of 15 librarians from each city was randomly selected. The mean from the first city was $28,900 with a standard deviation of $2300. The mean from the second city was $30,300 with a standard deviation of $2100. Construct a 95% confidence interval for μ 1 - μ 2 . A) (-3048, 248)
B) (-2976, 176)
C) (-2871, 567)
D) (-4081, 597)
Answer: A 5) Construct a 95% confidence interval for μ 1 - μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. n 1 = 50 n 2 = 60 x1 = 25
x2 = 23
s1 = 1.5 s2 = 1.9 A) (1.357, 2.643) Answer: A
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B) (1.364, 2.636)
C) (1.723, 3.012)
D) (1.919, 3.142)
6) Construct a 95% confidence interval for μ 1 - μ 2 . Two samples are randomly selected from each population. The sample statistics are given below. n 1 = 40 n 2 = 35 x1 = 12
x2 = 13
s1 = 2.5 s2 = 2.8 A) (-2.230, 0.230)
B) (-2.209, 0.209)
C) (-1.968, 1.561)
D) (-1.673, 1.892)
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) A researcher wishes to determine whether people with high blood pressure can lower their blood pressure by performing yoga exercises. A treatment group and a control group are selected. The sample statistics are given below. Construct a 90% confidence interval for the difference between the two population means, μ 1 - μ 2 . Would you recommend using yoga exercises? Explain your reasoning. Treatment GroupControl Group n 1 = 100 n 2 = 100 x1 = 178
x2 = 193
s1 = 35
s2 = 37
Answer: confidence interval: -23.38 < μ 1 - μ 2 < -6.623; Since the interval does not contain zero, we can reject the claim of μ 1 = μ 2 . Since the interval is negative, it appears that the yoga exercises lower blood pressure. 8) A recent study of 100 social workers in a southern state found that their mean salary was $23,600 with a standard deviation of $2100. A similar study of 100 social workers in a western state found that their mean salary was $34,000 with a standard deviation of $3200. Test the claim that the salaries of social workers in the western state is more than $10,000 greater than that of social workers in the southern state. Use α = 0.05. Answer: critical value z 0 = 1.645; standardized test statistic z ≈ 1.05; fail to reject H 0 ; There is not sufficient evidence to support the claim.
11.4 Inference about Two Population Standard Deviations 1 Find critical values of the F-distribution. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Find the right-hand critical value F0 for a two-tailed test using α = 0.05, degrees of freedom in the numerator = 5, and degrees of freedom in the denominator = 10. A) 4.24 B) 4.07 C) 4.47
D) 6.62
Answer: A 2) Find the critical value F0 for a right-tailed test using α = 0.01, degrees of freedom in the numerator = 3, and degrees of freedom in the denominator = 20. A) 4.94 B) 25.58
C) 5.82
D) 3.09
Answer: A 3) Find the critical value F0 for a left-tailed test using α = 0.05, degrees of freedom in the numerator = 6, and degrees of freedom in the denominator = 16. A) 0.255 B) 2.74 Answer: A
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C) 2.66
D) 0.365
4) Find the left-hand critical value F0 for a two-tailed test using α = 0.02, degrees of freedom in the numerator = 5, and degrees of freedom in the denominator = 10. A) 0.099 B) 10.05
C) 5.64
D) 0.177
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) Find the left-tailed and right-tailed critical F-values for a two-tailed test. Let α = 0.02, degrees of freedom in the numerator = 7, and degrees of freedom in the denominator = 5. Answer: (0.134, 10.46) 6) Find the left-tailed and right tailed critical F-values for a two-tailed test. Use the sample statistics below. Let α = 0.05. n1 = 5 n2 = 6 2 s 1 = 5.8
2 s 2 = 2.7
Answer: (0.107, 7.39) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) Two conditions are required to test a claim about two population standard deviations. What are they? A) The samples are independent and the population is normal. B) The samples are mutually exclusive and the population is normal. C) The samples are independent and the populations are randomly selected. D) The samples are mutually exclusive and the population are randomly selected. Answer: A 2 Test hypotheses regarding two population standard deviations. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Test the indicated hypothesis. Assume that the populations are normally distributed. 1) Test the hypothesis that σ1 > σ2 at the α = 0.01 level of significance for the given sample data. Population 1 Population 2 n 61 31 s 34.5 26.7 A) Test statistic: F = 1.67. Critical value = 2.21. Do not reject H 0 . B) Test statistic: F = 1.67. Critical value = 2.03. Reject H 0 . C) Test statistic: F = 44.58. Critical value = 2.03. Reject H 0 . D) Test statistic: F = 1.29. Critical value = 2.21. Do not reject H 0 . Answer: A 2) Test the hypothesis that σ1 > σ2 at the α = 0.01 level of significance for the given sample data. Population 1 Population 2 n 25 17 s 4.82 2.16 A) Test statistic: F = 4.98. Critical value: F = 3.18. Reject H 0. B) Test statistic: F = 4.98. Critical value: F = 3.18. Do not reject H0 . C) Test statistic: F = 2.23. Critical value = 1.87. Reject H 0 . D) Test statistic: F = 2.23. Critical value = 3.18. Do not reject H 0 . Answer: A
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3) Test the hypothesis that σ1 < σ2 at the α = 0.05 level of significance for the given sample data. Population 1 Population 2 n 9 10 s 0.0782 0.188,9 A) Test statistic: F = 0.17 Critical value: 0.295. Reject H0 . B) Test statistic: F = 0.17 Critical value: 0.310. Reject H0 . C) Test statistic: F = 0.17 Critical value: 0.318. Do not reject H0 . D) Test statistic: F = 0.17 Critical value: 0.295. Do not reject H0 . Answer: A 4) Test the hypothesis that σ1 ≠ σ2 at the α = 0.10 level of significance for the given sample data. Population 1 Population 2 n 31 25 s 5.39 4.77 A) Test statistic: F = 1.28. Critical values = 0.529, 1.94. Do not reject H0 . B) Test statistic: F = 1.28. Critical values = 0.515, 1.94. Do not reject H0 . C) Test statistic: F = 1.13. Critical values = 0.529, 1.94. Do not reject H0 . D) Test statistic: F = 1.13. Critical values = 0.515, 1.94. Reject H0 . Answer: A 5) Test the hypothesis that σ1 ≠ σ2 at the α = 0.05 level of significance for the given sample data. Population 1 Population 2 n 16 13 s 20.1 22.9 A) Test statistic: F = 0.77. Critical values = 0.338, 3.18. Do not reject H0 . B) Test statistic: F = 0.77. Critical values = 0.314, 3.18. Do not reject H0 . C) Test statistic: F = 0.77. Critical values = 0.338, 3.18. Reject H0 . D) Test statistic: F = 0.77. Critical values = 0.403, 2.62. Reject H0 . Answer: A 6) Test the hypothesis that σ1 < σ2 at the α = 0.10 level of significance for the given sample data.
n s
Population 1 16 19.6
Population 2 13 22
A) Test statistic: F = 0.79. Critical value = 0.495. Do not reject H0 . B) Test statistic: F = 0.79. Critical value = 0.476. Do not reject H0 . C) Test statistic: F = 0.79. Critical value = 0.495. Reject H0 . D) Test statistic: F = 0.79. Critical value = 0.476. Reject H0 . Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Test the indicated hypothesis. 7) A local supermarket claims that the variance of waiting time for its customers to be served is the lowest in the area. A competitor supermarket checks the waiting time at both supermarkets. The sample statistics are listed below. Test the local bank's hypothesis. Use α = 0.05. Local Supermarket Competitor Supermarket n 1 = 13 n 2 = 16 s1 = 1.65 minutes s2 = 1.95 minutes Assume the samples were randomly selected from normal populations. Answer: critical value F0 = 0.382; test statistic F ≈ 0.716; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. 8) A study was conducted to determine if the variances of librarian salaries from two neighboring cities were equal. A sample of 25 librarians from each city was selected. The first city had a standard deviation of s1 = $6,440, and the second city had a standard deviation s2 = $5,880. Test the hypothesis that the variances of the salaries from both cities are equal. Use α = 0.05 Assume the samples were randomly selected from normal populations. Answer: right-hand critical value F0 = 2.27; test statistic F ≈ 1.200; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 9) A random sample of 21 women had cholesterol levels with a variance of 1,107.2. A random sample of 18 men had cholesterol levels with a variance of 737.28. Test the hypothesis that the cholesterol levels for women have a larger variance than those for men. Use α = 0.01. Assume the samples were randomly selected from normal populations. Answer: critical value F0 = 3.16; test statistic F ≈ 1.502; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 10) The time spent watching television (per day) of 121 women between the ages of 55 and 64 had a standard deviation of 63 minutes. The time spent watching television (per day) of a random sample of 121 women between the ages of 25 and 34 had a standard deviation of 84 minutes. Test the hypothesis that the older women are from a population with a variance less than that for women in the 25 to 34 age group. Use α = 0.05. Assume the samples were randomly selected from normal populations. Answer: critical value F0 = 0.741; test statistic F ≈ 0.563; reject H0 ; There is sufficient evidence to support the hypothesis. 11) At a retail store, 61 female employees were randomly selected and it was found that their monthly income had a standard deviation of $85.05. For 121 male employees, the standard deviation was $118.09. Test the hypothesis that variance of monthly incomes is higher for male employees than it is for female employees. Use α = 0.01. Assume the samples were randomly selected from normal populations. Answer: critical value F0 = 1.73; test statistic F ≈ 1.928; reject H0 ; There is sufficient evidence to support the hypothesis.
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12) A medical researcher suspects that the variance of the pulse rate of drinkers is higher than the variance of the pulse rate of non-drinkers. Use the sample statistics below to test the researcher's suspicion. Use α = 0.05. Drinkers Non-drinkers n 1 = 61 n 2 = 121 s1 = 3.9 s2 = 2.65 Assume the samples were randomly selected from normal populations. Answer: critical value F0 =1.43; test statistic F ≈ 2.166; reject H0 ; There is sufficient evidence to support the hypothesis. 13) An algebra teacher believes that the variances of test scores of students in her evening algebra class are lower than the variances of test scores of students in her morning class. The results of an exam, given to the morning and evening students, are shown below. Can the teacher conclude that her evening students have a lower variance? Use α = 0.01. Evening Students Morning Students n 1 = 41 n 2 = 36 s1 = 18.55 s2 = 34.3 Assume the samples were randomly selected from normal populations. Answer: critical value F0 = 0.466; test statistic F ≈ 0.292; reject H 0 ; There is sufficient evidence to support the hypothesis. 14) A local restaurant claims that the variance of waiting time for its patrons to be served is the lowest in the area. A competitor restaurant checks the waiting time at both restaurants. The sample statistics are listed below. Test the local restaurant's hypothesis. Use α = 0.05. Local Restaurant Competitor Restaurant n 1 = 41 n 2 = 61 s1 = 1.32 minutes s2 = 2.4 minutes Assume the samples were randomly selected from normal populations. Answer: critical value F0 = 0.610; test statistic F ≈ 0.303; reject H0 ; There is sufficient evidence to support the hypothesis. 15) A university wants to see whether there is a significant difference in the variances of the ages between day staff and night staff. A random sample of 31 staff members is selected from each group. The data are given below. Test the hypothesis that there is no difference in age between the two groups. Use α = 0.05. Day Staff 22 24 24 23 19 19 23 22 18 21 21 18 18 25 29 24 23 22 22 21 20 20 20 27 17 19 18 21 20 23 26 Evening Staff 18 23 25 23 21 21 23 24 27 31 34 20 20 23 19 25 24 27 23 20 20 21 25 24 23 28 20 19 23 24 20 Assume the samples were randomly selected from normal populations. Answer: right-hand critical value F0 = 2.07; test statistic F ≈ 1.549; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis.
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11.5 Putting It Together: Which Method Do I Use? 1 Determine the appropriate hypothesis test to perform. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Perform the appropriate hypothesis test. 1) A random sample of n 1 = 40 individuals results in x1 = 10 successes. An independent sample of n 2 = 40 individuals results in x2 = 8 successes. Does this represent sufficient evidence to conclude that p1 > p2 at the α = 0.01 level of significance? Answer: H 0 : p1 = p2 vs H1 : p1 > p2 z 0 = 0.54 < z 0.01= 2.33 Do not reject H0 There is not sufficient evidence at the α = 0.01 level of significance to conclude that p1 > p2 . 2) A random sample of n 1 = 40 individuals results in x1 = 10 successes. An independent sample of n 2 = 40 individuals results in x2 = 8 successes. Does this represent sufficient evidence to conclude that p1 > p2 at the α = 0.01 level of significance? Answer: H 0 : p1 = p2 vs H1 : p1 > p2 z 0 = 0.54. Because the P-value is greater than α = 0.01, do not reject H 0. There is not sufficient evidence at the α = 0.01 level of significance to conclude that p1 > p2 . 3) A random sample of n 1 = 60 individuals results in x1 = 42 successes. An independent sample of n 2 = 40 individuals results in x2 = 24 successes. Does this represent sufficient evidence to conclude that p1 ≠ p2 at the α = 0.05 level of significance? Answer: H 0 : p1 = p2 vs H1 : p1 ≠ p2 z 0 = 1.03. Because the P-value is greater than α = 0.05, do not reject H 0. There is not sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the proportions. 4) A random sample of size n = 50 results in a sample mean of 21 and a sample standard deviation of 1.5. An independent sample of size n = 60 results in a sample mean of 19 and a sample standard deviation of 1.9. Does this constitute sufficient evidence to conclude that the population means differ at the α = 0.05 level of significance? Answer: H 0 : μ 1 = μ 2 vs H1 : μ 1 ≠ μ 2 t0 = 6.17; P-value < 0.001 Reject H0 . There is sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the population means. 5) The table below shows the weights, in pounds, of seven subjects before and after following a particular diet program for three months. Does the sample evidence suggest that the diet program is effective in reducing weight? Use the α = 0.01 level of significance. Assume that the differenced data come from a population that is normally distributed with no outliers. Subject A B C D E F G Before, Xi 192 168 150 157 156 188 177 After, Yi
185
159
148
162
142
190
165
Answer: H 0 : μ d = 0 vs H 1 : μ d < 0, di = Yi - X i t0 = -1.954; 0.025 < P-value < 0.05 Do not reject H0 . At the 1% significance level, the data do not provide sufficient evidence to conclude that the diet program is effective in reducing weight. Page 24
6) The data represent the measure of a variable before and after a treatment. Does the sample evidence suggest that the treatment is effective in increasing the value of the response variable? Use the α = 0.01 level of significance. Assume that the differenced data come from a population that is normally distributed with no outliers. Subject A B C D E F G Before, Xi 9.5 9.6 9.6 9.6 9.7 9.5 9.7 After, Yi
9.6
9.8
9.6
9.5
9.8
9.8
9.5
Answer: H 0 : μ d = 0 vs H 1 : μ d > 0, di = Yi - X i t0 = 0.880; 0.20 < P-value < 0.25 Do not reject H0 . At the 1% significance level, the data do not provide sufficient evidence to conclude that the treatment is effective in increasing the value of the response variable. 7) A random sample of 100 male employees of a retail store results in a mean monthly salary of $430 with a standard deviation of $44. An independent random sample of 125 female employees of the same store results in a mean monthly salary of $415 with a standard deviation of $25. Does this constitute sufficient evidence to conclude that the mean monthly salary for male employees of the store is higher than the mean monthly salary for female employees of the store? Use the α = 0.01 level of significance. Answer: H 0 : μ 1 = μ 2 vs H1 : μ 1 > μ 2 z 0 = 3.039; 0.001 < P-value < 0.0025 Reject H0 . There is sufficient evidence at the α = 0.01 level of significance to conclude that the mean monthly salary for male employees of the store is higher than the mean monthly salary for female employees. 8) In a survey, 46 of 189 women said that they regularly remember their dreams, while 41 of 179 men said that they regularly remember their dreams. Does this constitute sufficient evidence to conclude that women are more likely to remember their dreams than men? Use the α = 0.05 level of significance. Answer: H 0 : p 1 = p 2 vs
H1 : p 1 > p 2
z 0 = 0.320. Because the P-value is greater than α = 0.05, do not reject H0 . At the 5% significance level, the data do not provide sufficient evidence to conclude that women are more likely to remember their dreams than men. 9) A random samples of 8 apples of variety A and an independent random sample of 13 apples of variety B yielded the following weights in ounces. Do the data provide sufficient evidence to conclude that the mean weight of apples of variety A differs from the mean weight of apples of variety B? Use the α = 0.10 level of significance. Assume that the sample data come from normally distributed populations with no outliers. Variety A Variety B 3.7 3.2 3.8 2.8 3.2 3.0 2.5 4.0 3.0 3.6 2.7 3.6 3.9 2.6 3.8 2.8 3.4 4.0 2.5 3.6 3.9 Answer: H 0 : μ 1 = μ 2 vs H1 : μ 1 ≠ μ 2 t0 = -1.506; 0.10 < P-value < 0.20 Do not reject H0 . At the 10% significance level, the data do not provide sufficient evidence to conclude that the mean weight of apples of variety A differs from the mean weight of apples of variety B.
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Ch. 11 Inferences on Two Population Paraeters Answer Key 11.1 Inference about Two Population Proportions 1 Distinguish between independent and dependent sampling. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 2 Test hypotheses regarding two proportions from independent samples. 1) A 2) A 3) A 4) A 5) A 6) test statistic z 0 ≈ 0.37; P-value = 0.7114; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 7) test statistic z 0 ≈ 0.36; P-value = 0.3594; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 8) test statistic z 0 ≈ -2.51; P-value = 0.0060; reject H 0 ; There is sufficient evidence to support the hypothesis. 9) test statistic z 0 ≈ 5.09; P-value ≈ 0.0000; reject H 0 ; There is sufficient evidence to support the hypothesis. 10) α = 0.05: p1 = p2 ; test statistic z 0 ≈ 2.31; P-value = 0.0208; reject the null hypothesis; There is sufficient evidence to reject the hypothesis. 11) hypothesis: p1 < p2 ; test statistic z 0 ≈ -1.04; P-value = 0.1515; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 12) hypothesis: p1 > p2 ; test statistic z 0 ≈ 2.97; P-value = 0.0015; reject H 0 ; The new drug is effective. 13) hypothesis: p1 = p2 ; test statistic z 0 ≈ -0.57; P-value = 0.5685; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 14) hypothesis: p1 < p2 ; test statistic z 0 ≈ -6.74; P-value ≈ 0.0000; reject H0 ; There is sufficient evidence to support the hypothesis. 15) hypothesis: p1 < p2 ; test statistic z 0 ≈ -5.00; P-value ≈ 0.0000; reject H0 ; There is sufficient evidence to support the hypothesis. 3 Construct and interpret confidence intervals for the difference between two population proportions. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 4 Determine the sample size necessary for estimating the difference between two population proportions. 1) A 2) A
11.2 Inference about Two Means: Dependent Samples 1 Test hypotheses for a population mean from matched-pairs data. 1) A 2) A 3) A Page 26
4) A 5) critical values t0 = ±2.776; test statistic t ≈ 2.580; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 6) critical values t0 = ±4.604; test statistic t ≈ -1.215; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 7) A 8) A 9) test statistic t ≈ 2.580; 0.05 < P-value < 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim. 10) test statistic t ≈ -1.215; 0.20 < P-value < 0.30; fail to reject H0 ; There is not sufficient evidence to reject the claim. 11) claim: μ d = 0; test statistic t ≈ -2.401; 0.04 < P-value < 0.05; reject H 0 ; There is sufficient evidence to reject the claim. 12) claim: μ d < 0; test statistic t ≈ -2.177; 0.025 < P-value < 0.05; reject H 0; There is sufficient evidence to support the claim. 13) claim: μ d > 0; test statistic t ≈ 2.752; 0.01 < P-value < 0.02 fail to reject H0 ; There is not sufficient evidence to support the claim. 14) claim: μ d = 0; test statistic t ≈ 1.451; 0.10 < P-value < 0.20; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 15) claim: μ d > 0; test statistic t ≈ 3.161; 0.005 < P-value < 0.01; reject H 0 ; There is sufficient evidence to support the claim. 16) claim: μ d > 0; test statistic t ≈ 3.490; 0.0025 < P-value < 0.005; reject H0 ; There is sufficient evidence to support the claim. 2 Construct and interpret confidence intervals about the population mean difference of matched-pairs data. 1) A 2) A 3) A 4) A sd 5) The matched pairs confidence interval for μ d is xd ± tα/2 n Confidence coefficient .90 ⇒ α = 1 - .90 = .10. α/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 ± 1.51 ⇒ (3.49, 6.51) 5
6) A 7) A
11.3 Inference about Two Means: Independent Samples 1 Test hypotheses regarding the difference of two independent means. 1) A 2) A 3) A 4) A 5) A 6) critical value t0 = ±2.064; test statistic ≈ 4.361; reject H 0 ; There is sufficient evidence to reject the hypothesis. 7) critical value t0 = ±2.201; test statistic ≈ -0.954; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 8) critical value t0 = 3.055; test statistic ≈ 1.282; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 9) critical value t0 = -1.761; test statistic = -2.450; reject H 0 ; There is sufficient evidence to support the hypothesis. 10) critical value t0 = ±2.764; test statistic ≈ -1.546; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 11) critical value t0 = -1.761; test statistic ≈ -0.793; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 12) critical value t0 = ±2.145; test statistic t ≈ -1.741; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. Page 27
13) A 14) A 15) A 16) A 17) A 18) A 19) A 20) test statistic ≈ 4.361; P-value < 0.0010; reject H 0 ; There is sufficient evidence to reject the hypothesis. 21) test statistic ≈ -0.954; 0.30 < P-value < 0.40; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 22) test statistic ≈ 1.282; 0.10 < P-value < 0.15; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 23) test statistic = -2.450; 0.01 < P-value < 0.02; reject H 0 ; There is sufficient evidence to support the hypothesis. 24) test statistic ≈ -1.546; 0.10 < P-value < 0.20; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 25) test statistic ≈ -0.793; 0.20 < P-value < 0.25; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 26) test statistic t ≈ -1.741; 0.10 < P-value < 0.20; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 27) test statistic ≈ 6.17; P-value ≈ 0.0000; reject H 0; There is sufficient evidence to reject the hypothesis. 28) test statistic ≈ 2.99; 0.001 < P-value < 0.0025; reject H0 ; There is sufficient evidence to support the hypothesis. 29) test statistic ≈ -3.90; P-value < 0.0005; reject H 0 ; There is sufficient evidence to support the hypothesis. 30) test statistic ≈ -2.97; 0.005 < P-Value < 0.01; reject H 0 ; There is sufficient evidence to support the hypothesis. 31) test statistic ≈ -4.50; P-value < 0.001; reject H 0 ; There is sufficient evidence to reject the hypothesis. 32) test statistic ≈ -1.71; 0.025 < P-value < 0.05; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 33) test statistic ≈ 4.20; P-value < 0.0005; reject H 0 ; There is sufficient evidence to support the hypothesis. 34) test statistic ≈ -2.18; 0.01 < P-value < 0.02; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 35) test statistic ≈ -1.39; 0.05 < P-value < 0.10; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 36) day staff x1 = 22, s1 = 3.13; evening staff x2 = 23.29, s2 = 3.27; test statistic = -1.69; 0.10 < P-value < 0.20; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 2 Construct and interpret confidence intervals regarding the difference of two independent means. 1) A 2) A 3) A 4) A 5) A 6) A 7) confidence interval: -23.38 < μ 1 - μ 2 < -6.623; Since the interval does not contain zero, we can reject the claim of μ 1 = μ 2 . Since the interval is negative, it appears that the yoga exercises lower blood pressure. 8) critical value z 0 = 1.645; standardized test statistic z ≈ 1.05; fail to reject H0 ; There is not sufficient evidence to support the claim.
11.4 Inference about Two Population Standard Deviations 1 Find critical values of the F-distribution. 1) A 2) A 3) A 4) A 5) (0.134, 10.46) 6) (0.107, 7.39) 7) A 2 Test hypotheses regarding two population standard deviations. 1) A 2) A 3) A Page 28
4) A 5) A 6) A 7) critical value F0 = 0.382; test statistic F ≈ 0.716; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 8) right-hand critical value F0 = 2.27; test statistic F ≈ 1.200; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. 9) critical value F0 = 3.16; test statistic F ≈ 1.502; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 10) critical value F0 = 0.741; test statistic F ≈ 0.563; reject H 0; There is sufficient evidence to support the hypothesis. 11) critical value F0 = 1.73; test statistic F ≈ 1.928; reject H 0 ; There is sufficient evidence to support the hypothesis. 12) critical value F0 =1.43; test statistic F ≈ 2.166; reject H 0 ; There is sufficient evidence to support the hypothesis. 13) critical value F0 = 0.466; test statistic F ≈ 0.292; reject H0 ; There is sufficient evidence to support the hypothesis. 14) critical value F0 = 0.610; test statistic F ≈ 0.303; reject H 0; There is sufficient evidence to support the hypothesis. 15) right-hand critical value F0 = 2.07; test statistic F ≈ 1.549; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis.
11.5 Putting It Together: Which Method Do I Use? 1 Determine the appropriate hypothesis test to perform. 1) H0 : p1 = p2 vs H1 : p1 > p2 z0 = 0.54 < z 0.01= 2.33 Do not reject H 0 There is not sufficient evidence at the α = 0.01 level of significance to conclude that p1 > p2 . 2) H0 : p1 = p2 vs H1 : p1 > p2 z0 = 0.54. Because the P-value is greater than α = 0.01, do not reject H 0 . There is not sufficient evidence at the α = 0.01 level of significance to conclude that p1 > p2. 3) H0 : p1 = p2 vs H1 : p1 ≠ p2 z0 = 1.03. Because the P-value is greater than α = 0.05, do not reject H 0 . There is not sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the proportions. 4) H0 : μ 1 = μ 2 vs H1 : μ 1 ≠ μ 2 t0 = 6.17; P-value < 0.001 Reject H 0 . There is sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the population means. 5) H0 : μ d = 0 vs H1 : μ d < 0, di = Y i - Xi t0 = -1.954; 0.025 < P-value < 0.05 Do not reject H 0. At the 1% significance level, the data do not provide sufficient evidence to conclude that the diet program is effective in reducing weight. 6) H0 : μ d = 0 vs H1 : μ d > 0, di = Y i - Xi t0 = 0.880; 0.20 < P-value < 0.25 Do not reject H 0. At the 1% significance level, the data do not provide sufficient evidence to conclude that the treatment is effective in increasing the value of the response variable. 7) H0 : μ 1 = μ 2 vs H1 : μ 1 > μ 2 z0 = 3.039; 0.001 < P-value < 0.0025 Reject H 0 . There is sufficient evidence at the α = 0.01 level of significance to conclude that the mean monthly salary for male employees of the store is higher than the mean monthly salary for female employees. 8) H0 : p 1 = p 2 vs H1 : p 1 > p 2 z0 = 0.320. Because the P-value is greater than α = 0.05, do not reject H 0 . At the 5% significance level, the data do not provide sufficient evidence to conclude that women are more likely to remember their dreams than men.
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9) H0 : μ 1 = μ 2 vs H1 : μ 1 ≠ μ 2 t0 = -1.506; 0.10 < P-value < 0.20 Do not reject H 0. At the 10% significance level, the data do not provide sufficient evidence to conclude that the mean weight of apples of variety A differs from the mean weight of apples of variety B.
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Ch. 12 Inference on Categorical Data 12.1 Goodness-of-Fit Test 1 Perform a goodness-of-fit test. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Determine the expected counts for each outcome. n = 400 pi Expected Counts A) pi
0.30 0.10 0.40 0.20 B) 0.30
Expected Counts 120
0.10 0.40 0.20
pi
0.30 0.10 0.40 0.20
Expected Counts
30
10
0.10 0.40 0.20
pi
0.30
0.10 0.40
38
Expected Counts 1,200 400 1,600 800
40
160
80
C)
40
20
D) pi
0.30
Expected Counts 124
161
77
0.20
Answer: A 2) Many track hurdlers 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 data lists the number of wins for track hurdlers in the different starting 2 positions. Calculate the chi-square test statistic χ 0 to test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position Number of Wins A) 6.750
1 2 3 4 5 6 44 32 36 45 33 50 B) 9.326
C) 12.592
D) 15.541
Answer: A 3) Many track hurdlers 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 data lists the number of wins for track hurdlers in the different starting 2 positions. Find the critical value χ α to test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position Number of Wins A) 11.070 Answer: A
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1 2 3 4 5 6 32 50 45 36 44 33 B) 9.236
C) 15.086
D) 12.833
4) A company wants to determine if its employees have any preference among 5 different health plans which it 2 offers to them. A sample of 200 employees provided the data below. Calculate the chi-square test statistic χ α to test the claim that the probabilities show no preference. Use α = 0.01. Plan 1 2 3 4 5 Employees 65 55 18 32 30 A) 37.45 B) 45.91
C) 48.91
D) 55.63
Answer: A 5) A company wants to determine if its employees have any preference among 5 different health plans which it 2 offers to them. A sample of 200 employees provided the data below. Find the critical value χ α to test the claim that the probabilities show no preference. Use α = 0.01. Plan 1 2 3 4 5 Employees 30 55 18 32 65 A) 13.277 B) 9.488
C) 11.143
D) 14.860
Answer: A 6) A teacher figures that final grades in the chemistry department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following number of grades were recorded. 2 Calculate the chi-square test statistic χ α to determine if the grade distribution for the department is different than expected. Use α = 0.01. Grade A B C D F Number 42 36 60 14 8 A) 5.25
B) 6.87
C) 0.6375
D) 4.82
Answer: A 7) A teacher figures that final grades in the chemistry department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following number of grades were recorded. Find 2 the critical value χ α to determine if the grade distribution for the department is different than expected. Use α = 0.01. Grade A B C D F Number 36 42 60 8 14 A) 13.277 Answer: A
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B) 15.086
C) 9.488
D) 7.779
8) A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be 1 selected by the spinner approximately of the time. A student suspects that a certain spinner is defective. The 6 2 suspected spinner is spun 90 times. The results are shown below. Calculate the chi-square test statistic χ α to test the student's claim. Use α = 0.10. Color Blue Yellow Red Green White Orange Frequency 11 16 15 17 12 19 A) 3.067 B) 2.143
C) 5.013
D) 4.312
Answer: A 9) A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be 1 selected by the spinner approximately of the time. A student suspects that a certain spinner is defective. The 6 2 suspected spinner is spun 90 times. The results are shown below. Find the critical value χ α to test the student's claim. Use α = 0.10. Color Blue Yellow Red Green White Orange Frequency 17 11 16 12 15 19 A) 9.236 B) 1.610
C) 10.645
D) 11.071
Answer: A 10) A random sample of 160 car purchases are selected and categorized by age. 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% 2 for the 45-65 group, and 12% for the group over 65. Calculate the chi-square test statistic χ α to test the claim that all ages have purchase rates proportional to their driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Purchases 66 39 25 30 A) 75.101 B) 85.123
C) 101.324
D) 95.431
Answer: A 11) A random sample of 160 car purchases are selected and categorized by age. 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% 2 for the 45-65 group, and 12% for the group over 65. Find the critical value χ α to test the claim that all ages have purchase rates proportional to their driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Purchases 66 39 25 30 A) 7.815 B) 6.251 Answer: A
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C) 11.143
D) 9.348
12) As the number of the degrees of freedom increases, the χ 2 distribution A) becomes more symmetric. B) becomes less symmetric. C) does not change shape as the degrees of freedom change. D) becomes exponential. Answer: A 13) A __________________ test is an inferential procedure used to determine whether a frequency distribution follows a defined distribution. A) goodness-of-fit B) χ 2 C) F D) normality Answer: A 14) The degrees of freedom for a χ 2 goodness-of-fit test when there are 6 categories and a sample of size 1200 is A) 5 B) 6 C) 1199 D) 1205 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 15) A multinomial experiment with k = 4 cells and n = 300 produced the data shown in the following table.
ni
Cell 1 2 65 69
3 80
4 86
Do these data provide sufficient evidence to contradict the null hypothesis that p1 = 0.20, p2 = 0.20, p3 = 0.30, and p4 = 0.30? Test using α = 0.05. 2 Answer: Since χ 0 = 3.056 and P-value > 0.10 (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. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 16) Many track hurdlers 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 data lists the number of wins for track hurdlers in the different starting positions. Find the P-value to test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position 1 2 3 4 5 6 Number of Wins 44 45 50 33 36 32 A) P-value > 0.10 C) 0.01 < P-value < 0.025 Answer: A
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B) 0.025 < P-value < 0.05 D) P-value < 0.005
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 17) Many track hurdlers 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 data lists the number of wins for track hurdlers in the different starting positions. Test the claim that the probabilities of winning are the same in the different positions. Use α = 0.05. The results are based on 240 wins. Starting Position Number of Wins
1 2 3 4 5 6 50 33 32 36 45 44
2 Answer: chi-square test statistic χ 0 = 6.750; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim. It seems that the probability of winning in different lanes is the same. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) A company wants to determine if its employees have any preference among 5 different health plans which it offers to them. A sample of 200 employees provided the data below. Find the P-value to test the claim that the probabilities show no preference. Use α = 0.01. Plan 1 2 3 4 5 Employees 30 65 55 18 32 A) P-value < 0.005 C) 0.05 < P-value < 0.10
B) P-value > 0.10 D) 0.025 < P-value < 0.05
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 19) A company wants to determine if its employees have any preference among 5 different health plans which it offers to them. A sample of 200 employees provided the data below. Test the claim that the probabilities show no preference. Use α = 0.01. Plan 1 2 3 4 5 Employees 65 30 18 55 32 2 Answer: chi-square test statistic χ 0 = 37.45; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim that employees show no preferences among the plans. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 20) A teacher figures that final grades in the chemistry department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following number of grades were recorded. Find the P-value to determine if the grade distribution for the department is different than expected. Use α = 0.01. Grade A B C D F Number 36 42 60 14 8 A) P-value > 0.10 C) 0.025 < P-value < 0.05 Answer: A
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B) P-value < 0.005 D) 0.01 < P-value < 0.025
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) A teacher figures that final grades in the chemistry department are distributed as: A, 25%; B, 25%; C, 40%; D, 5%; F, 5%. At the end of a randomly selected semester, the following number of grades were recorded. Determine if the grade distribution for the department is different than expected. Use α = 0.01. Grade A B C D F Number 42 36 60 14 8 2 Answer: chi-square test statistic χ 0 = 5.25; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to support the claim that the grades are different than expected. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 22) A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be 1 selected by the spinner approximately of the time. A student suspects that a certain spinner is defective. The 6 suspected spinner is spun 90 times. The results are shown below. Find the P-value to test the student's claim. Use α = 0.10. Color Blue Yellow Red Green White Orange Frequency 11 17 15 19 12 16 A) P-value > 0.10 C) P-value < 0.005
B) 0.05 < P-value < 0.10 D) 0.025 < P-value < 0.05
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 23) A spinner is mounted on a piece of cardboard divided into six areas of equal size. Each of the six areas is a different color (blue, yellow, red, green, white and orange). When the spinner is spun, each color should be 1 selected by the spinner approximately of the time. A student suspects that a certain spinner is defective. The 6 suspected spinner is spun 90 times. The results are shown below. Test the student's claim. Use α = 0.10. Color Blue Yellow Red Green White Orange Frequency 16 19 15 12 17 11 2 Answer: chi-square test statistic χ 0 = 3.067; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to support the claim of a defective spinner. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 24) A random sample of 160 car purchases are selected and categorized by age. 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 P-value to test the claim that all ages have purchase rates proportional to their driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Purchases 66 39 25 30 A) P-value < 0.005 C) 0.01 < P-value < 0.025 Answer: A Page 6
B) P-value > 0.10 D) 0.005 < P-value < 0.01
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 25) A random sample of 160 car purchases are selected and categorized by age. 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 purchase rates proportional to their driving rates. Use α = 0.05. Age Under 26 26 - 45 46 - 65 Over 65 Purchases 66 39 25 30 2 Answer: chi-square test statistic χ 0 = 75.101; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim that all ages have the same purchase rate. 26) The results of a recent national survey reported that 70% of Americans recycle at least some of the time. As part of their final project in statistics class, Nayla and Roberto survey 5 random students on campus and ask them if they recycle at least some of the time. They then repeat this experiment 1000 times. The results of their research are shown below. X (number who recycle out of 5) 0 1 2 3 4 5 Frequency 2 24 136 306 358 174 Is there evidence to support the belief that the random variable X, the number of students out of 5 who recycle at least some of the time, is a binomial random variable with p = 0.7 at the α = 0.05 level? 2 Answer: Since χ 0 = 1.093, and the P-value is greater than α = 0.05 (with 5 degrees of freedom), we do not reject the null hypothesis. There is evidence at the α = 0.05 level that X is a binomial random variable with n = 5 and p = 0.7.
12.2 Tests for Independence and the Homogeneity of Proportions 1 Perform a test for independence. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Test the null hypothesis of independence of the two classifications, A and B, of the 3 × 3 contingency table shown below. Test using α = 0.01.
A
B1
B B2
B3
A1 A2
19
40
60
55
23
22
A3
31
42
47
2 Answer: Since χ 0 = 42.857 and P-value < 0.005, we reject the null hypothesis. There is sufficient evidence to reject the claim that A and B are independent.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 2) The contingency table below shows the results of a random sample of 200 registered voters that was conducted to see whether their opinions on a bill are related to their party affiliation. Party
Opinion Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Find the P-value to test the claim of independence. A) 0.05 < P-value < 0.10 C) P-value < 0.005
B) P-value > 0.10 D) 0.025 < P-value < 0.05
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) The contingency table below shows the results of a random sample of 200 registered voters that was conducted to see whether their opinions on a bill are related to their party affiliation. Party
Opinion Approve Disapprove No Opinion Republican 42 20 14 Democrat 50 24 18 Independent 10 16 6 Test the claim of independence. Use α = 0.05. 2 Answer: chi-square test statistic χ 0 = 8.030; 0.05 < P-value < 0.10; fail to reject H 0; There is not sufficient evidence to reject the claim of independence. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Find the P-value to determine if there is enough evidence to conclude that the number of minutes spent watching television per day is related to gender. Gender
Minutes spent watching TV per day 0 - 30 30 - 60 60 - 90 90 - over Male 25 35 75 45 Female 30 45 45 15 A) P-value < 0.005 C) P-value > 0.10 Answer: A
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B) 0.05 < P-value < 0.10 D) 0.025 < P-value < 0.05
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Is there enough evidence to conclude that the number of minutes spent watching television per day is related to gender? Use α = 0.05. Gender
Minutes spent watching TV per day 0 - 30 30 - 60 60 - 90 90 - over Male 25 35 75 45 Female 30 45 45 15 2 Answer: chi-square test statistic χ 0 = 18.146; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim of independence. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6) A medical researcher is interested in determining if there is a relationship between adults over 50 who exercise regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and the results are given below. Find the P-value to test the claim that regular exercise and low, moderate, and high blood pressure are independent. Blood Pressure Low Moderate Reg. Exercise 35 62 No Reg. Exercise 21 65 A) P-value > 0.10 C) P-value < 0.10
High 25 28 B) 0.05 < P-value < 0.10 D) P-value = 0.05
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) A medical researcher is interested in determining if there is a relationship between adults over 50 who exercise regularly and low, moderate, and high blood pressure. A random sample of 236 adults over 50 is selected and the results are given below. Test the claim that regular exercise and low, moderate, and high blood pressure are independent. Use α = 0.01. Blood Pressure Low Moderate Reg. Exercise 35 62 No Reg. Exercise 21 65
High 25 28
2 Answer: chi-square test statistic χ 0 = 3.473; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 8) A sports statistician is interested in determining if there is a relationship between the number of home team and visiting team losses and different sports. A random sample of 526 games is selected and the results are given below. Find the P-value to test the claim that the number of home team and visiting team losses is independent of the sport. Football Basketball Soccer Baseball Home team losses 39 156 25 83 Visiting team losses 31 98 19 75 A) P-value > 0.10 B) P-value < 0.005 C) 0.005 < P-value < 0.01 D) 0.01 < P-value < 0.025 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9) A sports statistician is interested in determining if there is a relationship between the number of home team and visiting team losses 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 losses is independent of the sport. Use α = 0.01. Football Basketball Soccer Baseball Home team losses 39 156 25 83 Visiting team losses 31 98 19 75 2 Answer: chi-square test statistic χ 0 = 3.290; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence.
10) The data below show the age and favorite type of reading of 779 randomly selected people. Test the claim that age and preferred reading type are independent. Use α = 0.05. Age Current Events Mystery Science Fiction History 15 - 21 21 45 90 33 21 - 30 68 55 42 48 30 - 40 65 47 31 57 40 - 50 60 39 25 53 2 Answer: chi-square test statistic χ 0 = 91.097; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim of independence.
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2 Perform a test for homogeneity of proportions. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) A random sample of 400 men and 400 women was randomly selected and asked whether they planned to attend a concert in the next month. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion of men who plan to attend a concert in the next month is the same as the proportion of women who plan to attend a concert in the next month. Use α = 0.05.
Plan to attend concert Don't plan to attend concert
Men Women 230 255 170 145
2 Answer: chi-square test statistic χ 0 = 3.273; 0.05 < P-value < 0.10; fail to reject H 0; There is not sufficient evidence to reject the claim. 2) A random sample of 100 employees from 5 different companies was randomly selected, and the number who take public transportation to work was recorded. The results are listed below. Perform a homogeneity of proportions test to test the claim that the proportion who take public transportation to work is the same in all 5 companies. Use α = 0.01. Companies 1 2 3 4 5 Use Public Trans. 18 25 12 33 22 Don't Use Public Trans. 82 75 88 67 78 2 Answer: chi-square test statistic χ 0 = 14.336; 0.005 < P-value < 0.01; reject H 0 ; There is sufficient evidence to reject the claim.
12.3 Inference about Two Population Proportions: Dependent Samples 1 Test hypotheses regarding two proportions from dependent samples. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Perform the indicated hypothesis test. 1) Test whether the population proportions differ at the α = 0.05 level of significance by determining the null and alternative hypotheses, the test statistic, and the P-value. Assume that the samples are dependent and that they were obtained randomly. Treatment A Success Failure Success 86 35 Treatment B Failure 11 55 Answer: H 0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 12.522 P-value < 0.005 Reject H0 . There is sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the proportions.
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2) Test whether the population proportions differ at the α = 0.05 level of significance by determining the null and alternative hypotheses, the test statistic, and the P- value. Assume that the samples are dependent and that they were obtained randomly. Treatment A Success Failure Success 378 125 Treatment B Failure 130 430 Answer: H 0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 0.098 P-value > 0.10 Do not reject H0 . There is not sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the proportions. 3) A researcher wants to determine whether there is a difference between two sunscreen lotions. Participants in a marathon race on a hot, sunny day applied lotion A to one arm and lotion B to the other arm. The results are shown in the table. Lotion A No burn (Success) Burn (Failure) No burn (Success) 714 38 Lotion B Burn (Failure) 50 79 Is there a difference in the effectiveness of the two lotions in preventing sunburn? Use the α = 0.05 level of significance. Answer: H 0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 1.636 P-value > 0.10 Do not reject H0 . There is not sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the effectiveness of the two lotions in preventing sunburn. 4) In a survey, students were selected at random from a certain college and were asked two questions. The first question was "Do you believe it is wrong to hunt ?" The second question was "Do you believe it is wrong to eat meat?" The results are shown in the table. Success for the first question is identifying someone who feels that it is wrong to hunt and for the second question is identifying someone who feels that it is wrong to eat meat. Hunt Wrong (Success) Not wrong (Failure) Wrong (Success) 45 54 Eat Meat Not wrong (Failure) 86 257 Is there a significant difference in the proportion of students at this college who feel it is wrong to hunt and the proportion who feel that it is wrong to eat meat? Use the α = 0.05 level of significance. Answer: H 0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 7.314 0.005 < P-value < 0.01 Reject H0 . There is sufficient evidence at the α = 0.05 level of significance to conclude that there is a significant difference in the proportion of students who feel it is wrong to hunt and the proportion who feel that it is wrong to eat meat.
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Ch. 12 Inference on Categorical Data Answer Key 12.1 Goodness-of-Fit Test 1 Perform a goodness-of-fit test. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 2 15) Since χ 0 = 3.056 and P-value > 0.10 (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. 16) A 2 17) chi-square test statistic χ 0 = 6.750; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim. It seems that the probability of winning in different lanes is the same. 18) A 2 19) chi-square test statistic χ 0 = 37.45; P-value < 0.005; reject H 0; There is sufficient evidence to reject the claim that employees show no preferences among the plans. 20) A 2 21) chi-square test statistic χ 0 = 5.25; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to support the claim that the grades are different than expected. 22) A 2 23) chi-square test statistic χ 0 = 3.067; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to support the claim of a defective spinner. 24) A 2 25) chi-square test statistic χ 0 = 75.101; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim that all ages have the same purchase rate. 2 26) Since χ 0 = 1.093, and the P-value is greater than α = 0.05 (with 5 degrees of freedom), we do not reject the null hypothesis. There is evidence at the α = 0.05 level that X is a binomial random variable with n = 5 and p = 0.7.
12.2 Tests for Independence and the Homogeneity of Proportions 1 Perform a test for independence. 2 1) Since χ 0 = 42.857 and P-value < 0.005, we reject the null hypothesis. There is sufficient evidence to reject the claim that A and B are independent. Page 13
2) A 2 3) chi-square test statistic χ 0 = 8.030; 0.05 < P-value < 0.10; fail to reject H 0 ; There is not sufficient evidence to reject the claim of independence. 4) A 2 5) chi-square test statistic χ 0 = 18.146; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim of independence. 6) A 2 7) chi-square test statistic χ 0 = 3.473; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence. 8) A 2 9) chi-square test statistic χ 0 = 3.290; P-value > 0.10; fail to reject H0 ; There is not sufficient evidence to reject the claim of independence. 2 10) chi-square test statistic χ 0 = 91.097; P-value < 0.005; reject H0 ; There is sufficient evidence to reject the claim of independence. 2 Perform a test for homogeneity of proportions. 2 1) chi-square test statistic χ 0 = 3.273; 0.05 < P-value < 0.10; fail to reject H 0 ; There is not sufficient evidence to reject the claim. 2 2) chi-square test statistic χ 0 = 14.336; 0.005 < P-value < 0.01; reject H0 ; There is sufficient evidence to reject the claim.
12.3 Inference about Two Population Proportions: Dependent Samples 1 Test hypotheses regarding two proportions from dependent samples. 1) H0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 12.522 P-value < 0.005 Reject H 0 . There is sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the proportions. 2) H0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 0.098 P-value > 0.10 Do not reject H 0. There is not sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the proportions. 3) H0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 1.636 P-value > 0.10 Do not reject H 0. There is not sufficient evidence at the α = 0.05 level of significance to conclude that there is a difference in the effectiveness of the two lotions in preventing sunburn.
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4) H0 : p A = p B vs H1 : p A ≠ p B 2 χ 0 = 7.314 0.005 < P-value < 0.01 Reject H 0 . There is sufficient evidence at the α = 0.05 level of significance to conclude that there is a significant difference in the proportion of students who feel it is wrong to hunt and the proportion who feel that it is wrong to eat meat.
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Ch. 13 Comparing Three or More Means 13.1 Comparing Three or More Means (One-Way Analysis of Variance) 1 Verify the requirements to perform a one-way ANOVA. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with high cholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, each subject's cholesterol level is recorded. State the requirements that must be satisfied in order to use the one-way ANOVA procedure. Answer: Each group represents a simple random sample from its respective population. All samples are independent of one another. The population are normally distributed. The population have the same variance. 2) Four different types of insecticides are used on strawberry plants. The number of strawberries on each randomly selected plant is recorded. State the requirements that must be satisfied in order to use the one-way ANOVA procedure. Answer: Each group represents a simple random sample from its respective population. All samples are independent of one another. The population are normally distributed. The population have the same variance. 3) A researcher wishes to determine whether there is a difference in the average age of middle school, high school, and college teachers. Teachers are randomly selected. Their ages are recorded. State the requirements that must be satisfied in order to use the one-way ANOVA procedure. Answer: Each group represents a simple random sample from its respective population. All samples are independent of one another. The population are normally distributed. The population have the same variance. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) The analysis of variance is an inferential method that is used to test the equality of A) three or more population means. B) three or more population variances. C) two population means. D) two population variances. Answer: A 5) How, in general, is the requirement of equal population variances to allow the use of analysis of variance method verified? A) The largest sample standard deviation must be less than or equal to twice the smallest sample standard deviation. B) All three sample standard deviations must be equal. C) The largest sample standard deviation is equal to the sum of the other standard deviations. D) The largest sample standard deviation is less than or equal to the variance of the smallest sample. Answer: A
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2 Test a hypothesis regarding three or more means using one-way ANOVA. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with high cholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, each subject's cholesterol level is recorded. Find the critical value F-value to test the hypothesis that there is no difference among the means. Use α = 0.05. Group 1 Group 2 Group 3 9 8 4 12 3 12 11 2 4 15 5 8 13 4 9 8 0 6 A) 3.68 B) 19.43 C) 4.77 D) 39.43 Answer: A 2) A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with high cholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, each subject's cholesterol level is recorded. Find the test statistic F to test the hypothesis that there is no difference among the means. Use α = 0.05. Group 1 Group 2 Group 3 11 8 6 12 2 12 9 3 4 15 5 8 13 4 9 8 0 4 A) 11.095 B) 9.812 C) 8.369 D) 12.162 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with high cholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, each subject's cholesterol level is recorded. Test the hypothesis that there is no difference among the means. Use α = 0.05. Group 1 Group 2 Group 3 11 8 8 12 2 12 13 5 4 15 3 6 9 4 9 8 0 4 Answer: critical value F0.05,2,15 = 3.68; test statistic F ≈ 11.095; reject H 0 ; There is enough evidence that the sample means are different.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 4) Four different types of insecticides are used on strawberry plants. The number of strawberries on each randomly selected plant is given below. Find the critical value F-value to test the hypothesis that the type of insecticide makes no difference in the mean number of strawberries per plant. Use α = 0.01. Insecticide 1 Insecticide 2 Insecticide 3 Insecticide 4 6 5 6 3 6 8 3 5 5 5 2 3 7 5 4 4 7 5 3 4 6 6 3 5 A) 4.94 B) 4.43 C) 26.69 D) 4.18 Answer: A 5) Four different types of insecticides are used on strawberry plants. The number of strawberries on each randomly selected plant is given below. Find the test statistic F to test the hypothesis that the type of insecticide makes no difference in the mean number of strawberries per plant. Use α = 0.01. Insecticide 1 Insecticide 2 Insecticide 3 Insecticide 4 6 8 6 3 7 5 3 5 5 5 3 3 6 5 4 4 7 5 2 5 6 6 3 4 A) 8.357 B) 8.123 C) 7.123 D) 6.912 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) Four different types of insecticides are used on strawberry plants. The number of strawberries on each randomly selected plant is given below. Test the hypothesis that the type of insecticide makes no difference in the mean number of strawberries per plant. Use α = 0.01. Insecticide 1 Insecticide 2 Insecticide 3 Insecticide 4 6 8 6 3 6 5 3 5 7 5 2 3 5 5 3 4 7 5 4 4 6 6 3 5 Answer: critical value F0.01,3,20 = 4.94; test statistic F ≈ 8.357; reject H 0 ; The data provides ample evidence that the sample means are unequal.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) A researcher wishes to determine whether there is a difference in the average age of middle school, high school, and college teachers. Teachers are randomly selected. Their ages are recorded below. Find the critical value F-value to test the hypothesis that there is no difference in the average age of each group. Use α = 0.01. Middle School Teachers High School Teachers College Teachers 23 41 39 28 38 45 27 36 45 37 47 61 52 42 36 25 31 35 A) 6.36 B) 5.09 C) 9.43 D) 5.42 Answer: A 8) A researcher wishes to determine whether there is a difference in the average age of middle school, high school, and college teachers. Teachers are randomly selected. Their ages are recorded below. Find the test statistic F to test the hypothesis that there is no difference in the average age of each group. Use α = 0.01. Middle School Teachers High School Teachers College Teachers 23 41 45 28 36 45 27 38 36 52 47 61 25 42 39 37 31 35 A) 2.517 B) 2.913 C) 3.189 D) 4.312 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9) A researcher wishes to determine whether there is a difference in the average age of middle school, high school, and college teachers. Teachers are randomly selected. Their ages are recorded below. Test the hypothesis that there is no difference in the average age of each group. Use α = 0.01. Middle School Teachers High School Teachers College Teachers 23 41 36 28 38 45 27 36 39 37 47 61 52 42 45 25 31 35 Answer: critical value F0.01,2,15 = 6.36; test statistic F ≈ 2.517; fail to reject H0 ; There is not enough evidence to indicate that the means are different.
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10) The grade point averages of students participating in sports at a local college are to be compared. The data are listed below. Test the hypothesis that there is no difference in the mean grade point averages of the 3 groups. Use α = 0.05. Hockey Track Basketball 3.2 1.8 2.5 2.6 2.1 2.7 2.5 3.3 2.8 3.5 1.9 3.0 3.1 2.5 2.1 Answer: critical valueF0.05,2,12 = 3.89; test statistic F ≈ 1.606; fail to reject H0 ; The data does not provide enough evidence to indicate that the means are unequal. 11) The times (in minutes) to assemble a component for 3 different cell phones are listed below. Workers are randomly selected. Test the hypothesis that there is no difference in the mean time for each cell phone. Use α = 0.01. Phone 1 Phone 2 Phone 3 32 40 25 30 29 31 32 38 29 31 33 28 33 32 31 35 36 Answer: critical value F0.01,2,14 ≈ 6.36; test statistic F ≈ 7.103; reject H 0; There is enough evidence that the sample means are different. 12) A realtor wishes to compare the square footage of houses in 4 different towns, all of which are priced approximately the same. The data are listed below. Can the realtor conclude that the mean square footage in the four towns are equal? Use α = 0.01. Town 1 Town 2 Town 3 Town 4 2150 1,650 1530 2400 1,980 1540 1,600 2350 2,000 1690 1580 2600 2,210 1,780 1,670 2,150 1900 1500 2000 1,750 2,200 2350 Answer: critical value F0.01,3,18 ≈ 4.94; test statistic F ≈ 31.330; reject H0 ; There is enough evidence that the sample means are different. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 13) Find the critical F0 -value to test the hypothesis that the populations have the same mean. Use α = 0.05. Brand 1 n1 = 8
Brand 2 n2 = 8
Brand 3 n3 = 8
x1 = 3.0
x2 = 2.6
x3 = 2.6
s1 = 0.50 A) 3.49
s2 = 0.60
s3 = 0.55 B) 3.210
Answer: A
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C) 2.819
D) 1.892
14) Find the test statistic F to test the hypothesis that the populations have the same mean. Use α = 0.05. Brand 1 Brand 2 Brand 3 n1 = 8 n2 = 8 n3 = 8 x1 = 3.0
x2 = 2.6
x3 = 2.6
s1 = 0.50 A) 1.403
s2 = 0.60
s3 = 0.55 B) 1.021
C) 1.182
D) 0.832
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 15) Test the hypothesis that the populations have the same mean. Use α = 0.05. Brand 1 Brand 2 Brand 3 n1 = 8 n2 = 8 n3 = 8 x1 = 3.0
x2 = 2.6
x3 = 2.6
s1 = 0.50
s2 = 0.60
s3 = 0.55
Answer: critical value F0.05,2,21 ≈ 3.49; test statistic F ≈ 1.403; fail to reject H0 ; The data does not provide enough evidence to indicate that the means are unequal. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 16) 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. Give the null hypothesis for the ANOVA F test shown on the printout. 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 A) H0 : μ green = μ blue = μ gray = μ red , where the μ's represent mean attitudes for the four rooms B) H0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5 , where the μ i represent attitude means for the ith person in each room C) H0 : x1 = x2 = x3 = x4 , where the x's represent the room colors D) H0 : pgreen = pblue = pgray = pred , where the p's represent the proportion with the corresponding attitude Answer: A
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17) 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. At α = 0.05, what would be the best interpretation? 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 A) At α = 0.05, it can be said that color doesn't matter B) At α = 0.05, light green has a higher mean than gray C) At α = 0.05, it can be said that at least one color has a different mean D) At α = 0.05, red is the best color Answer: A 18) Four different leadership styles (A, B, C, and D) used by Big-Six accountants were investigated. As part of a 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. Interpret the results of the ANOVA test shown on the printout for α = 0.05. ONE-WAY ANOVA FOR SUBSTAND BY STYLE SOURCE DF SS MS F P BETWEEN 3 2,793.61 931.204 5.210 0.003 WITHIN 56 10,009.10 178.734 TOTAL 59 12,802.71 A) At α = 0.05, there is sufficient evidence of differences among the substandard work scale means for the four leadership styles. B) At α = 0.05, there is no evidence of interaction. C) At α =0.05, nothing can be said. D) At α =0.05, there is insufficient evidence of differences among the substandard work scale means for the four leadership styles. Answer: A 19) 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 17 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 17 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 : μ C = μ E = μ I B) H 0 : β 1 = β 2 = β 3 = 0 C) H0 : μ C = μ E = μ I = 0 Answer: A
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D) H 0 : p1 = p2 = p3
20) 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 will be subjected to an analysis of variance. The results of the ANOVA are summarized in the following table. Take α = 0.01. Interpret the P-value of the ANOVA F test. Source df SS MS F Value Prob > F Treatments 2 8,088.846 4,044.423 20.73 0.0001 Error 57 11,120.7 195.1 Total 59 19,209.546 A) The means of the total per member per month charges for the three groups of physicians differ at α = 0.01. B) The model is not statistically useful (at α = 0.01) for prediction purposes. C) The variances of the total per number per month charges for the three groups of physicians differ at α = 0.01. D) The means of the total per member per month charges for the three groups of physicians are equal at α = 0.01. Answer: A 21) Which of the following is not an example of a mean square? A) μ 2
B)
C) σ2
D)
n 1(x1 - x) + n 2 (x2 - x) + . . . + n k(xk - x) k- 1 (n 1 - 1)s12 + (n 2 - 1)s2 2 + . . . + (n k - 1)sk2 n-k
Answer: A
13.2 Post Hoc Tests on One-Way Analysis of Variance 1 Perform the Tukey Test. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the critical value. 1) Find the critical value from the Studentized range distribution for α = 0.05, ν = 9, k = 3. A) 3.949 B) 4.415 C) 4.041
D) 5.428
Answer: A 2) Find the critical value from the Studentized range distribution for H0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5 , with n = 20 at α = 0.05. A) 4.367 Answer: A
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B) 4.076
C) 3.958
D) 4.232
Provide an appropriate response. 3) A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with high cholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, each subject's cholesterol level is recorded. Use α = 0.05. Which of the following is a correct statement about the means? Group 1 Group 2 Group 3 9 8 8 12 3 12 13 5 4 15 2 6 11 4 9 8 0 4 A) The mean of Group 1 is not equal to the mean of Group 2. B) The mean of Group 1 is not equal to the mean of Group 3. C) The mean of Group 2 is not equal to the mean of Group 3. D) All of the means are not equal. Answer: A 4) Four different types of insecticides are used on strawberry plants. The number of strawberries on each randomly selected plant is given below. Use α = 0.01. Which of the following is a correct statement about the means? Insecticide 1 Insecticide 2 Insecticide 3 Insecticide 4 6 5 6 3 7 8 3 5 6 5 3 3 5 5 2 4 7 5 4 5 6 6 3 4 A) The mean of Insecticide 1 is equal to the mean of Insecticide 2. B) The mean of Insecticide 2 is equal to the mean of Insecticide 3. C) The mean of Insecticide 3 is equal to the mean of Insecticide 1. D) The mean of Insecticide 1 is equal to the mean of Insecticide 4. Answer: A 5) A researcher wishes to determine whether there is a difference in the average age of middle school, high school, and college teachers. Teachers are randomly selected. Their ages are recorded below. Use α = 0.01. Which of the following is a correct statement about the means? Middle School Teachers High School Teachers College Teachers 23 38 36 28 41 45 27 36 39 25 47 61 37 42 45 52 31 35 A) There is not enough evidence to suggest that the means are different. B) The mean age of middle school teachers is different from the mean age of high school teachers. C) The mean age of middle school teachers is different from the mean age of college teachers. D) The mean age of high school teachers is different from the mean age of college teachers. Answer: A
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6) The grade point averages of students participating in sports at a local community college are to be compared. The data are listed below. Which of the following is a correct statement about the means? Use α = 0.05. Hockey Track Basketball 2.6 1.8 2.7 3.2 1.9 3.0 2.5 2.1 2.8 3.5 3.3 2.5 3.1 2.5 2.1 A) The mean grade point average of hockey players is the same as the mean grade point average of basketball players. B) The mean grade point average of hockey players is the same as the mean grade point average of track participants. C) The mean grade point average of basketball players is the same as the mean grade point average of track participants. D) The mean grade point average of each sport is the same. Answer: A 7) The times (in minutes) to assemble a component for three different cell phones are listed below. Use α = 0.01. Which of the following is correct? Phone 1 Phone 2 Phone 3 32 40 28 32 29 25 30 38 29 31 33 31 33 32 31 35 36 A) The mean time to assemble Phone 1 is the same as the mean time to assemble Phone 2. B) The mean time to assemble Phone 1 is different from the mean time to assemble Phone 3. C) The mean time to assemble Phone 2 is the same as the mean time to assemble Phone 3. D) There is not enough evidence to suggest that the means are different. Answer: A 8) A realtor wishes to compare the square footage of houses in 4 different towns, all of which are priced approximately the same. The data are listed below. Use α = 0.01. Which of the following is correct? Town 1 Town 2 Town 3 Town 4 2150 1,650 1530 2400 1,980 1540 1,750 2350 2,000 1690 1580 2600 2,210 1,780 1,600 2,150 1900 1500 2000 1,670 2,200 2350 A) The mean square footage of houses in Town 2 is equal to the mean square footage in Town 3. B) The mean square footage of houses in Town 1 is equal to the mean square footage in Town 2. C) The mean square footage of houses in Town 3 is equal to the mean square footage in Town 4. D) The mean square footage of houses in Town 4 is equal to the mean square footage in Town 1. Answer: A
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13.3 The Randomized Complete Block Design 1 Conduct analysis of variance on the randomized complete block design. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A researcher wishes to test H 0: μ 1 = μ 2 = μ 3 against H1 :at least one of the means is different. Based on the ANOVA table below, what is the mean square due to error? SOURCE df SS MS F p ______________________________________________________________ Treatments 2 86.22 43.11 13.15 0.0174 Blocks 2 0.889 0.444 0.136 0.8767 Error 4 13.11 3.28 ______________________________________________________________ Total 8 100.22 A) 3.28
B) 13.11
C) 0.444
D) 0.136
Answer: A 2) A researcher wishes to test H 0: μ 1 = μ 2 = μ 3 against H1 :at least one of the means is different. Based on the ANOVA table below, if the level of significance is α = 0.01, is it necessary to conduct Tukey's test? If the level of significance is α = 0.05, is it necessary to conduct Tukey's test? Explain. SOURCE df SS MS F p ______________________________________________________________ Treatments 2 86.22 43.11 13.15 0.0174 Blocks 2 0.889 0.444 0.136 0.8767 Error 4 13.11 3.28 ______________________________________________________________ Total 8 100.22 A) no, H 0 not rejected; yes, H 0 rejected, at least one mean found to be different B) no, H 0 not rejected; no, H0 not rejected; C) yes, H0 rejected, at least one mean found to be different; yes, H0 rejected, at least one mean found to be different D) yes, H0 rejected, at least one mean found to be different ; no, H0 not rejected Answer: A
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3) A researcher wishes to test H 0: μ 1 = μ 2 = μ 3 against H1 :at least one of the means is different. Based on the ANOVA table, the null hypothesis is rejected and the following output represents the results of Tukey's test. What should the researcher conclude? Tukey Simultaneous Tests Response Variable Response All Pairwise Comparisons among Levels of Treatment Treatment = 1 subtracted from: Treatment Difference SE of T- Value Adjusted of Means Difference P-Value 2 7.1887 1.982 3.627 0.0165 3 0.7488 1.982 0.3778 0.9251 Treatment = 2 subtracted from: Treatment Difference SE of T- Value of Means Difference 3 1.982 -6.4393 -3.2489 A) μ 1 μ 3 μ 2 B) μ 1 μ 2 μ 3
Adjusted P-Value 0.0282 C) μ 1 μ 2 μ 3
D) μ 1 μ 2 μ 3
Answer: A 4) A researcher wishes to test H 0: μ 1 = μ 2 = μ 3 against H1 :at least one of the means is different. Based on the ANOVA table, the null hypothesis is rejected and the following output represents the results of Tukey’s test. What should the researcher conclude? Tukey 95% Simultaneous Confidence Intervals Response Variable Response All Pairwise Comparisons among Levels of Treatment Treatment = 1 subtracted from: Treatment Lower Center Upper -----+-----+-----+-----+-----+ 2 4.606 11.699 18.792 (-------- *--------) 3
6.962
14.055
21.148
(--------*--------) -----+-----+-----+-----+-----+ 0.0 5.0 10.0 15.0 20.0
Treatment = 2 subtracted from: Treatment Lower Center Upper -----+-----+-----+-----+-----+ 3 -4.737 2.356 9.449 (-------* -------) -----+-----+-----+-----+-----+ 0.0 5.0 10.0 15.0 20.0 A) μ 1 μ 2 μ 3 B) μ 1 μ 2 μ 3 C) μ 1 μ 2 μ 3 Answer: A
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D) μ 1 μ 2 μ 3
2 Perform a Tukey test. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A researcher is testing the percent weight gain on mice of varying ages based upon the diet they eat. He has 4 mice in each age category: 6 months, 9 months, 12 months and 15 months. They mice are weighed, fed only one of four diets and weighed after 1 month. The weight gain percent is then recorded. The data is given below. Test at α = 0.05 if there is a difference in weights.
6 months 9 months 12 months 15 months
Diet A Diet B Diet C Diet D 5.2 5.3 5.8 5.6 5.0 5.7 5.6 5.2 5.3 6.1 6.1 5.0 4.9 6.4 5.7 5.1
A) Diet does result in significantly different weight gains. B) Diet does not result in significantly different weight gains. Answer: A 2) A farmer has a 160 acre tract of land that he is planting in corn. He has heard about advances in fertilizer technology that impacts yield. He divides his field into four sections and each section into four parts. He has three type of fertilizer to test and no fertilizer. He randomly selects a part of a section and then randomly selects a type of fertilizer for that part of a section so that each section has all four applications applied to it. The yields in bushel per acres are given below. Test at α = 0.05 if there is a difference in yields.
No fertilizer Traditional Liquid Granular
Field A 5.7 4.5 4.1 2.6
Field B 10.4 8.6 6.3 7.2
Field C 7.3 6.1 5.8 4.6
Field D 9.0 10.6 7.2 8.7
A) Fertilizer does result in significantly different yields. B) Fertilizer does not result in significantly different yields. Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) Assume that the data below come from populations that are normally distributed with the same variance. Let μ 1 be the mean for treatment 1, μ 2 the mean for treatment 2, and μ 3 the mean for treatment 3. In a test of H 0 : μ 1 = μ 2 = μ 3 versus H1 : at least one of the means is different, the null hypothesis is rejected at the α = 0.05 level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for each of the pairwise tests and state your conclusion. Block Treatment 1 Treatment 2 Treatment 3 1 17.3 19.1 16.9 2 17.9 21.4 17.5 3 17.2 23.3 18.6 4 19.1 21.6 20.1 5 20.8 23.7 19.6 Answer: H 0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.0016, reject H 0 H 0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.9909, do not reject H 0 H 0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0019, reject H 0 The Tukey test indicates that the mean of treatment 2 is different from the mean of treatment 1 and from the mean of treatment 3. There is no evidence that the mean of treatment 1 differs from the mean of treatment 3. 4) Assume that the data below come from populations that are normally distributed with the same variance. Let μ 1 be the mean for treatment 1, μ 2 the mean for treatment 2, and μ 3 the mean for treatment 3. In a test of H 0 : μ 1 = μ 2 = μ 3 versus H1 : at least one of the means is different, the null hypothesis is rejected at the α = 0.05 level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for each of the pairwise tests and state your conclusion. Block Treatment 1 Treatment 2 Treatment 2 1 7.2 6.6 6.1 2 6.9 7.5 6.7 3 8.2 8.6 8.3 4 9.7 9.3 8.8 5 9.4 10.2 9.5 6 10.3 9.8 9.2 Answer: H 0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.9676, do not reject H 0 H 0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.0707, do not reject H 0 H 0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0474, reject H 0 The Tukey test indicates that the mean of treatment 2 is different from the mean of treatment 3. There is no evidence that the mean of treatment 1 differs from the mean of treatment 2. At the α = 0.05 level of significance, there is not sufficient evidence that the mean of treatment 1 differs from the mean of treatment 3.
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5) A farmer has five different fields. He wishes to test two different fertilizers. The yields of corn may vary from field to field depending on a number of factors such as soil quality, sunlight etc. He divides each field into three sections. For each field, he randomly assigns one of the sections to receive fertilizer A, one of the sections to receive fertilizer B, and one of the sections to receive no fertilizer. At harvest time, he records the corn yields in bushels per acre for each section of each field. The results are shown in the table. Let μ 1 be the mean for fertilizer A, μ 2 the mean for fertilizer B, and μ 3 the mean for no fertilizer. In a test of H0 : μ 1 = μ 2 = μ 3 versus H1 : at least one of the means is different, the null hypothesis is rejected at the α = 0.05 level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for each of the pairwise tests and state your conclusion. Fertilizer A Fertilizer B No Fertilizer Field 1 132 141 128 Field 2 136 135 130 Field 3 144 147 139 Field 4 126 134 122 Field 5 130 128 125 Answer: H 0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.1819, do not reject H 0 H 0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.0560, do not reject H 0 H 0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0037, reject H 0 The Tukey test indicates that the mean for fertilizer B is different from the mean for no fertilizer. There is no evidence that the mean for fertilizer A differs from the mean for fertilizer B. At the α = 0.05 level of significance, there is not sufficient evidence that the mean for fertilizer A differs from the mean for no fertilizer. 6) A researcher wants to determine whether the type of gasoline used in a car affects the gas mileage. Gas mileage is also affected by car type and driver. The researcher selects five different types of car and assigns a driver to each car. For each car (and driver), the researcher randomly selects the sequence in which the different gasoline types will be used. The mileage for each car/gasoline combination was determined by filling the car with 3 gallons of the specified gasoline, driving the car around a race track until the gas tank was empty, and dividing the number of miles driven by the number of gallons used. The results are shown in the table. The mileages are given in miles per gallon. Let μ 1 be the mean for gasoline A, μ 2 the mean for gasoline B, and μ 3 the mean for gasoline C. In a test of H0 : μ 1 = μ 2 = μ 3 versus H1 : at least one of the means is different, the null hypothesis is rejected at the α = 0.05 level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for each of the pairwise tests and state your conclusion. Gasoline A Gasoline B Gasoline C Car 1 25.1 24.6 26.9 Car 2 26.3 26.2 27.1 Car 3 24.3 24.0 25.8 Car 4 28.1 27.4 28.5 Car 5 27.7 27.8 28.6 Answer: H 0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.4479, do not reject H 0 H 0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.0045, reject H 0 H 0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0010, reject H 0 The Tukey test indicates that the mean for gasoline C is different from the mean for gasoline A and from the mean for gasoline B. There is no evidence that the mean for gasoline A differs from the mean for gasoline B.
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) To conduct an analysis of variance on a randomized complete block design a researcher would use A) a two-way ANOVA. B) a one-way ANOVA. C) a Tukey Test. D) a Scheffe' Test. Answer: A 8) True or False: In the completely randomized block design, a single factor is manipulated and fixed at different levels. A) True B) False Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9) Why is the completely randomized block design not always sufficient? Answer: There may be additional factors that cannot be fixed at a single level throughout the experiment.
13.4 Two-Way Analysis of Variance 1 Analyze a two-way ANOVA design. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A builder has two crews that construct homes. He is interested in determining if there is a difference in the time it takes the crews to build three different styles of homes. The time in days for each crew is given below. Is there any interaction effect? Answer Yes or No. Test at α = 0.05.
Smith Jones
Cottage 27, 34, 30, 35, 39 46, 41, 37, 26,24
A) No
Bi-Level 25, 47, 44, 30, 39 31, 39, 44, 45, 32
2 story 28, 24, 29, 32, 41 43, 45, 44, 36, 31 B) Yes
Answer: A 2) A training program is supposed to develop self confidence in managers. The two-week program is administered to six males and six females. All participants are tested after the end of the first week and after the end of the second week. The scores of those tests are given below. The facilitator is interested in determining if there is an increase in confidence the longer the managers train. Answer Yes or No. Test at α = 0.05.
Men Women A) Yes
Week One 65, 81, 81, 75, 88, 75 71, 76, 82, 88, 87, 75
Week Two 90, 94, 80, 91, 81, 98 81, 97, 86, 91, 99, 83 B) No
Answer: A 3) True or False: It is required for the Two-Way ANOVA that the populations from which the samples are drawn are normal. A) True B) False Answer: A
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4) True or False: It is required for the Two-Way ANOVA that the samples be dependent. A) False B) True Answer: A 5) True or False: It is required for the Two-Way ANOVA that the populations all have the same variance. A) True B) False Answer: A 2 Draw interaction plots. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether the interaction plot suggests that there is no interaction among the factors, some interaction or significant interaction. 1)
Mean Response
B2
B1
A1
A2
A3
A) no interaction
B) significant interaction
C) some interaction
B) no interaction
C) some interaction
Answer: A 2)
Mean Response
B2
B1
A1
A2
A) significant interaction Answer: A
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A3
3)
B2
Mean Response
B1
A1
A2
A) some interaction
A3
B) significant interaction
C) no interaction
Answer: A Provide an appropriate response. 4) A builder has two crews that construct homes. He is interested in determining if there is a difference in the time it takes the crews to build three different styles of homes. The time in days for each crew is given below. What type of interaction is revealed by the interaction plot? Test at α = 0.05.
Smith Jones
Cottage 27, 34, 30, 35, 39 46, 41, 37, 26,24
A) No interaction C) Some interaction
Bi-Level 25, 47, 44, 30, 39 31, 39, 44, 45, 32
2 story 28, 24, 29, 32, 41 43, 45, 44, 36, 31 B) Significant interaction D) Complete interaction
Answer: A 5) A training program is supposed to develop self confidence in managers. The two-week program is administered to six males and six females. All participants are tested after the end of the first week and after the end of the second week. The scores of those tests are given below. What type of interaction is revealed by the interaction plot? Test at α = 0.05.
Men Women
Week One 65, 81, 81, 75, 88, 75 71, 76, 82, 88, 87, 75
A) No interaction C) Some interaction Answer: A
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Week Two 90, 94, 80, 91, 81, 98 81, 97, 86, 91, 99, 83 B) Significant interaction D) Complete interaction
3 Perform the Tukey test. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) A two-way analysis of variance is to be performed using the data below. Assume that the data come from populations that are normally distributed with the same variance. If there is a significant difference in the means for the three levels of factor A, use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. If there is a significant difference in the means for the three levels of factor B, use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Factor B Level 1 Level 2 Level 3 Level 1 67 74 82 71 73 76 62 60 87 Level 2
77 81 65
69 75 77
84 80 90
Level 3
59 67 62
70 58 66
72 68 76
Factor A
Answer: Among levels of factor A: Tukey's test indicates that the mean of level 2 is different from the mean of level 3 (P-value = 0.0017) but that there is not sufficient evidence that the mean of level 1 is different from the mean of level 2 or the mean of level 3. Among levels of factor B: Tukey's test indicates that the mean of level 1 is different from the mean of level 3 (P-value = 0.0012) and that the mean of level 2 is different from the mean of level 3 (P-value = 0.0033) but there is not sufficient evidence that the mean of level 1 is different from the mean of level 2.
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2) A researcher is investigating whether gender and exercise are factors which explain resting pulse rate in her adult patients. She randomly selected three patients from each category and recorded their resting pulse rates. The results are given in the table. Pulse rates are in beats per minute. A two-way analysis of variance is to be performed. Assume that the data come from populations that are normally distributed with the same variance. If there is a significant difference in the means for gender, use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for any significant differences. If there is a significant difference in the means for the three exercise levels, use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for any significant differences. Exercise level 1 level 2 level 3 Female 90, 82, 85 76, 83, 80 64, 71, 75 Male 80, 85, 75 71, 74, 68 60, 65, 70 Levels of the factor exercise are defined as follows: level 1: little or no exercise. level 2: regular moderate exercise level 3: regular vigorous exercise Answer: Gender: Tukey's test indicates that the mean for men is different from the mean for women (P-value = 0.0097). Exercise: Tukey's test indicates that the mean for exercise level 1 is different from the mean for exercise level 2 (P-value 0.0319), the mean for exercise level 1 is different from the mean for exercise level 3 (P-value = 0.0002), and the mean for exercise level 2 is different from the mean for exercise level 3 (P-value = 0.0253). 3) A cell phone company is investigating whether age and ethnicity are factors which explain how much people spend on their cell phones. They randomly picked three people from each category and determined each person's average monthly cell phone bill. The results are shown in the table. Amounts are in dollars. A two-way analysis of variance is to be performed. Assume that the data come from populations that are normally distributed with the same variance. If there is a significant difference in the means for the three age groups, use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for any significant differences. If there is a significant difference in the means for the three ethnic groups, use Tukey's test to determine which pairwise means differ using a familywise error rate of α = 0.05. Give the P-value for any significant differences. Ethnicity Age Asian Hispanic Caucasian 18-34 65 50 85 47 64 49 56 40 59 35-54
70 98 64
47 60 58
73 77 90
55 and older
44 36 59
36 28 42
50 42 65
Answer: Tukey's test indicates that the mean for the 35-54 age group is different from the mean for the 55 and older age group (P-value = 0.0007). Tukey's test indicates that the mean for Hispanics is different from the mean for Caucasians (P-value = 0.0138).
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Ch. 13 Comparing Three or More Means Answer Key 13.1 Comparing Three or More Means (One-Way Analysis of Variance) 1 Verify the requirements to perform a one-way ANOVA. 1) Each group represents a simple random sample from its respective population. All samples are independent of one another. The population are normally distributed. The population have the same variance. 2) Each group represents a simple random sample from its respective population. All samples are independent of one another. The population are normally distributed. The population have the same variance. 3) Each group represents a simple random sample from its respective population. All samples are independent of one another. The population are normally distributed. The population have the same variance. 4) A 5) A 2 Test a hypothesis regarding three or more means using one-way ANOVA. 1) A 2) A 3) critical value F0.05,2,15 = 3.68; test statistic F ≈ 11.095; reject H0 ; There is enough evidence that the sample means are different. 4) A 5) A 6) critical value F0.01,3,20 = 4.94; test statistic F ≈ 8.357; reject H0 ; The data provides ample evidence that the sample means are unequal. 7) A 8) A 9) critical value F0.01,2,15 = 6.36; test statistic F ≈ 2.517; fail to reject H 0; There is not enough evidence to indicate that the means are different. 10) critical valueF0.05,2,12 = 3.89; test statistic F ≈ 1.606; fail to reject H0 ; The data does not provide enough evidence to indicate that the means are unequal. 11) critical value F0.01,2,14 ≈ 6.36; test statistic F ≈ 7.103; reject H 0 ; There is enough evidence that the sample means are different. 12) critical value F0.01,3,18 ≈ 4.94; test statistic F ≈ 31.330; reject H 0 ; There is enough evidence that the sample means are different. 13) A 14) A 15) critical value F0.05,2,21 ≈ 3.49; test statistic F ≈ 1.403; fail to reject H0 ; The data does not provide enough evidence to indicate that the means are unequal. 16) A 17) A 18) A 19) A 20) A 21) A
13.2 Post Hoc Tests on One-Way Analysis of Variance 1 Perform the Tukey Test. 1) A 2) A Page 21
3) A 4) A 5) A 6) A 7) A 8) A
13.3 The Randomized Complete Block Design 1 Conduct analysis of variance on the randomized complete block design. 1) A 2) A 3) A 4) A 2 Perform a Tukey test. 1) A 2) A 3) H0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.0016, reject H 0 H0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.9909, do not reject H 0 H0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0019, reject H 0
The Tukey test indicates that the mean of treatment 2 is different from the mean of treatment 1 and from the mean of treatment 3. There is no evidence that the mean of treatment 1 differs from the mean of treatment 3. 4) H0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.9676, do not reject H 0 H0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.0707, do not reject H 0 H0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0474, reject H 0 The Tukey test indicates that the mean of treatment 2 is different from the mean of treatment 3. There is no evidence that the mean of treatment 1 differs from the mean of treatment 2. At the α = 0.05 level of significance, there is not sufficient evidence that the mean of treatment 1 differs from the mean of treatment 3. 5) H0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.1819, do not reject H 0 H0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.0560, do not reject H 0 H0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0037, reject H 0 The Tukey test indicates that the mean for fertilizer B is different from the mean for no fertilizer. There is no evidence that the mean for fertilizer A differs from the mean for fertilizer B. At the α = 0.05 level of significance, there is not sufficient evidence that the mean for fertilizer A differs from the mean for no fertilizer. 6) H0 : μ 1 = μ 2 versus H1 : μ 1 ≠ μ 2 , P-value = 0.4479, do not reject H 0 H0 : μ 1 = μ 3 versus H1 : μ 1 ≠ μ 3 , P-value = 0.0045, reject H 0 H0 : μ 2 = μ 3 versus H1 : μ 2 ≠ μ 3 , P-value = 0.0010, reject H 0 The Tukey test indicates that the mean for gasoline C is different from the mean for gasoline A and from the mean for gasoline B. There is no evidence that the mean for gasoline A differs from the mean for gasoline B. 7) A 8) A 9) There may be additional factors that cannot be fixed at a single level throughout the experiment.
13.4 Two-Way Analysis of Variance 1 Analyze a two-way ANOVA design. 1) A 2) A 3) A 4) A 5) A Page 22
2 Draw interaction plots. 1) A 2) A 3) A 4) A 5) A 3 Perform the Tukey test. 1) Among levels of factor A: Tukey's test indicates that the mean of level 2 is different from the mean of level 3 (P-value = 0.0017) but that there is not sufficient evidence that the mean of level 1 is different from the mean of level 2 or the mean of level 3. Among levels of factor B: Tukey's test indicates that the mean of level 1 is different from the mean of level 3 (P-value = 0.0012) and that the mean of level 2 is different from the mean of level 3 (P-value = 0.0033) but there is not sufficient evidence that the mean of level 1 is different from the mean of level 2. 2) Gender: Tukey's test indicates that the mean for men is different from the mean for women (P-value = 0.0097). Exercise: Tukey's test indicates that the mean for exercise level 1 is different from the mean for exercise level 2 (P-value = 0.0319), the mean for exercise level 1 is different from the mean for exercise level 3 (P-value = 0.0002), and the mean for exercise level 2 is different from the mean for exercise level 3 (P-value = 0.0253). 3) Tukey's test indicates that the mean for the 35-54 age group is different from the mean for the 55 and older age group (P-value = 0.0007). Tukey's test indicates that the mean for Hispanics is different from the mean for Caucasians (P-value = 0.0138).
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Ch. 14 Inference on the Least-Squares Regression Model and Multiple Regression 14.1 Testing the Significance of the Least-Squares Regression Model 1 State the requirements of the least-squares regression model. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) One of the requirements for conducting inference on the least-squares regression model is that the A) mean of the response variable changes at a constant rate while the standard deviation remains constant. B) mean of the explanatory variable changes at a constant rate while the standard deviation remains constant. C) mean of the response variable remains constant while the standard deviation changes at a constant rate. D) mean of the explanatory variable remains constant while the standard deviation changes at a constant rate. Answer: A 2) The least-squares regression model for one explanatory variable is given by the equation A) yi = β 0 + β 1 xi + εi
B) y = mx + b
C) y - yi = m(x - xi)
D) yi =
Ai
x + Ci Bi i
Answer: A 2 Compute the standard error of the estimate. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by 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. Compute the standard error, the point estimate for σ. Number of Grunts, y Age (days), x 99 127 77 143 48 157 53 162 72 169 49 176 71 185 26 191 31 197 Answer: s = 15.35 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ^
2) Find the standard error of estimate, se, for the data below, given that y = 2x + 1. x 1 y 3 A) 0 Answer: A
Page 1
2 5
3 7
4 9 B) 1
C) 2
D) 3
^
3) Find the standard error of estimate, se, for the data below, given that y = -2.5x. x -1 -2 y 2 6 A) 0.866
-3 7
-4 10 B) 0.675
C) 0.532
D) 0.349
Answer: A ^
4) Find the standard error of estimate, se, for the data below, given that y= 2.097x - 0.552. x -5 y -10 A) 0.976
-3 -8
4 9
-1 -2 0 -2 -6 -1 B) 0.990
1 1
2 3
3 6
-4 -8 C) -0.990
D) 0.980
Answer: A ^
5) Find the standard error of estimate, se, for the data below, given that y = -1.885x + 0.758. x -5 y 11 A) 0.613
-3 6
4 -6
-1 -2 0 3 4 1 B) 0.981
1 -1
2 -4
3 -5
-4 8 C) 0.312
D) 0.011
Answer: A ^
6) Find the standard error of estimate, se, for the data below, given that y = -0.206x + 2.097. x -5 y 11 A) 6.306
-3 -6
4 8
-1 -2 0 -2 1 5 B) 3.203
1 -3
2 -5
3 6
-4 7 C) 5.918
D) 8.214
Answer: A 7) The data below are the final exam scores of 10 randomly selected engineering students and the number of ^
hours they slept the night before the exam. Find the standard error of estimate, se, given that y = 5.044x + 56.11. Hours, x Scores, y A) 6.305
3 65
5 80
2 60
8 2 88 66 B) 7.913
4 78
4 85
5 90
6 3 90 71 C) 8.912
D) 9.875
Answer: A 8) The data below are the one-way commute times (in minutes) for selected students during a summer class and the number of absences they had for that class for the term. Find the standard error of estimate, se, given that ^
y = 0.449x - 30.27. Commute time, x Number of absences, y A) 0.934
72 85 91 3 7 10 B) 1.162
90 10
88 98 8 15
75 100 4 15 C) 1.007
80 5 D) 0.815
Answer: A 9) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly ^
selected adults. Find the standard error of estimate, se, given that y= 1.488x + 60.46. Age, x Pressure, y A) 4.199 Answer: A
Page 2
38 116
41 120
45 48 123 131 B) 6.981
51 142
53 145
57 61 148 150 C) 5.572
65 152 D) 3.099
10) The data below are the number of absences and the final grades of 9 randomly selected students in an ^
engineering class. Find the standard error of estimate, se, given that y = -2.75x + 96.14. Number of absences, x Final grade, y A) 1.798
0 3 6 98 86 80 B) 4.531
4 82
9 71
2 92
15 8 55 76 C) 3.876
5 82 D) 2.160
Answer: A 11) A manager wishes to determine the relationship between the number of years her sales representatives have been with the firm and their average monthly sales (in thousands of dollars). Find the standard error of ^
estimate, se, given that y = 3.53x + 37.92. Years with firm, x Sales, y A) 22.062
2 31
3 10 7 33 78 62 B) 15.951
8 65
15 61
3 48
1 11 55 120 C) 10.569
D) 5.122
Answer: A 12) In order for a company's employees to work at a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Find the ^
standard of estimate, se, given that y = 6.91x + 46.26. Number of years, x Grades on test, y A) 4.578
3 61
4 4 68 75 B) 3.412
5 82
3 73
6 90
2 7 3 58 93 72 C) 5.192
D) 6.713
Answer: A 13) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the ^
yield of wheat (bushels per acre). Find the standard error of estimate, se, given that y = 4.379x + 4.267. Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) 3.529 B) 4.759 C) 2.813
D) 1.332
Answer: A 3 Verify that residuals are normally distributed. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The difference between the observed and predicted value of the response variable is a A) residual C) standard error of the estimate
.
B) variance D) test statistic
Answer: A 2) For the least-squares regression model, yi = β 0 + β 1 xi + εi, the predictor variable must be normally distributed. To show that this is true, the A) residuals C) variability Answer: A
Page 3
must also be normal. B) explanatory variable D) standard error of the estimate
4 Conduct inference on the slope of the least-squares regression model. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) If a hypothesis test of the linear relation between the explanatory and the response variable is of the type where H0 : β 1 = 0, H1 : β 1 > 0, then we are testing the claim that A) the slope of the least square regression model is positive. B) the slope of the least squares regression model is negative. C) a relationship exist without regard to the sign of the slope. D) no linear relationship exists. Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 2) Test the claim, at the α = 0.05 level of significance, that a linear relation exists between the two variables, for the ^
data below, given that y = -2.5x. x -1 -2 -3 -4 y 2 6 7 10 Answer: Since t = -6.455 and 0.02 < P-value < 0.04, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 3) Test the claim, at the α = 0.01 level of significance, that a linear relation exists between the two variables, for the ^
data below, given that y = 2.097x - 0.552. x -5 -3 4 1 -1 -2 0 y -10 -8 9 1 -2 -6 -1
2 3
3 6
-4 -8
Answer: Since t = 19.510 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 4) Test the claim, at the α = 0.10 level of significance, that a linear relation exists between the two variables, for the ^
data below, given that y = -1.885x + 0.758. x 1 -1 -2 0 2 -5 -3 4 y 11 6 -6 -1 3 4 1 -4
3 -5
-4 8
Answer: Since t = -27.929 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 5) The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Test the claim, at the α = 0.05 level of significance, that a ^
linear relation exists between the two variables, given that y = 0.449x - 30.27. Temperature, x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 Answer: Since t = 13.031 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 6) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. Test the claim, at the α = 0.01 level of significance, that a linear relation exists between the two ^
variables, given that y = 1.488x + 60.46. Age, x 38 41 45 48 Pressure, y 116 120 123 131
51 142
53 145
57 148
61 150
65 152
Answer: Since t = 9.034 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables.
Page 4
5 Construct a confidence interval about the slope of the least-squares regression model. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Construct a 95% confidence interval about the slope of the true least-squares regression line, for the data ^
below, given that y = -2.5x. x -1 -2 -3 -4 y 2 6 7 10 A) (-4.165, -0.835)
C) (-3.630, -1.370)
B) (-6.226, 1.226)
D) (-3.731, -1.269)
Answer: A 2) Construct a 99% confidence interval about the slope of the true least-squares regression line, for the data ^
below, given that y = 2.097x - 0.552. x 2 3 -5 -3 4 1 -1 -2 0 y -10 -8 9 1 -2 -6 -1 3 6 A) (1.738, 2.456) B) ( -1.177, 5.371)
-4 -8 C) (1.787, 2.407)
D) (1.749, 2.445)
Answer: A 3) Construct a 90% confidence interval about the slope of the true least-squares regression line, for the data ^
below, for the data below, given that y = -1.885x + 0.758. x 1 -1 -2 0 2 3 -4 -5 -3 4 y 11 6 -6 -1 3 4 1 -4 -5 8 A) (-2.010, -1.760) B) (-3.025, -0.745)
C) (-1.979, -1.791)
D) (-2.008, -1.762)
Answer: A 4) The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Construct a 95% confidence interval about the slope of ^
the true least-squares regression line, for the data below, given that y = 0.449x - 30.27. Temperature, x 72 85 91 90 88 98 75 100 80 Number of absences, y 3 7 10 10 8 15 4 15 5 A) (0.367, 0.530) B) (-1.760, 2.658) C) (0.385, 0.513)
D) (0.371, 0.527)
Answer: A 5) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults.Construct a 95% confidence interval about the slope of the true least-squares regression line, for ^
the data below, given that y = 1.488x + 60.46. Age, x 38 41 45 48 51 53 Pressure, y 116 120 123 131 142 145 A) (1.098, 1.877) B) (-8.443, 11.419) Answer: A
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57 61 65 148 150 152 C) (1.175, 1.801)
D) (1.108, 1.868)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A breeder of Thoroughbred horses wishes to model the relationship between the gestation 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. Construct a 90% confidence interval about the slope of the true least-squares regression line. Horse Gestation Life Horse Gestation Life period Length period Length x (days) y (years) x (days) y (years) 1 416 24 5 356 22 2 279 25.5 6 403 23.5 3 298 20 7 265 21 4 307 21.5 Answer: b1 = 0.01087 sb 1 = 0.0134 t0.05 = 2.015 90% confidence interval = (-0.0161, 0.0379) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) In linear regression, what is the unbiased estimator of σ called? A) standard error of the estimate B) standard deviation C) variance D) sample standard deviation Answer: A
14.2 Confidence and Prediction Intervals 1 Construct confidence intervals for mean responses. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. ^
1) Construct a 95% confidence interval about the mean value of y, given x = -3.5, y = 2.097x - 0.552 and se = 0.976. x -5 -3 y -10 -8 A) (-8.921,-6.862)
4 9
1 1
2 3 -1 -2 0 -2 -6 -1 3 6 B) (-10.367, -5.417)
-4 -8 C) (-4.598 ,-1.986)
D) (-12.142 ,-6.475)
Answer: A 2) The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they slept the night before the exam. Construct a 95% confidence interval about the mean value of y, the score ^
on the final exam, given x = 7 hours, y = 5.044x + 56.11 and se = 6.305. Hours, x 3 5 Scores, y 65 80 A) (82.840, 99.996)
2 60
8 2 4 4 88 66 78 85 B) (74.54, 108.30)
5 90
6 3 90 71 C) (77.21, 110.45)
D) (79.16, 112.34)
Answer: A 3) The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Construct a 95% confidence interval about the mean ^
value of y, the number of days absent, given x = 95 degrees, y = 0.449x - 30.27 and se = 0.934. Temperature, x Number of absences, y A) (11.336, 13.350) Answer: A Page 6
72 85 91 90 3 7 10 10 B) (9.957, 14.813)
88 8
98 15
75 100 80 4 15 5 C) (4.321, 6.913)
D) (6.345, 8.912)
4) In order for a company's employees to work at a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Construct ^
a 95% confidence interval about the mean value of y, given x = 2.5, y= 6.91x + 46.26, and se = 4.578. Number of years, x Grades on test, y A) (58.28, 68.79)
3 61
4 4 5 3 68 75 82 73 B) (51.50, 75.57)
6 90
2 7 3 58 93 72 C) (55.12, 87.34)
D) (47.32, 72.13)
Answer: A 5) In an area of Russia, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Construct a 95% confidence interval about the mean value of y, the yield, given x = 11 ^
inches, y= 4.379x + 4.267 and se = 3.529. Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) (49.41, 55.47) B) (43.56, 61.32) C) (40.54 , 64.15)
D) (39.86, 65.98)
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) 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. Construct a 90% confidence interval about the mean value of y when x = 5 months. Months on Job Monthly Sales y ($ thousands) 2 2.4 4 7.0 8 11.3 12 15.0 1 0.8 5 3.7 9 12.0 ^
Answer: For x = 5, y = -0.25 + 1.315(5) = 6.325 The confidence interval is of the form: ^ 1 (x - x)2 y ± tα/2 s + n SSxx Confidence coefficient 0.90 = 1 - α ⇒ α = 1 - 0.90 = 0.10. α/2 = 0.10/2 = 0.05. From a Student's t table, t0.05 = 2.015 with n - 2 = 7 - 2 = 5 df. The confidence interval is: 6.325 ± 2.015(1.577)
Page 7
1 (5 - 5.8571)2 ⇒ 6.325 ± 1.233 ⇒ (5.092, 7.558) + 7 94.8571
2 Construct prediction intervals for an individual response. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) How does a confidence interval differ from a prediction interval? A) Confidence intervals are used to measure the accuracy of the mean response of all the individuals in the population, while a prediction interval is used to measure the accuracy of a single individual's predicted value. B) Confidence intervals are used to measure the accuracy of a single individual's predicted value, while a prediction interval is used to measure the accuracy of the mean response of all the individuals in the population. C) Confidence intervals are constructed about the predicted values of y while prediction intervals a constructed about a particular value of x D) Confidence intervals are constructed about the predicted values of x while prediction intervals a constructed about a particular value of y Answer: A ^
2) Construct a 95% prediction interval for y given x = -3.5, y= 2.097x - 0.552 and se = 0.976. x -5 -3 4 y -10 -8 9 A) (-10.367, -5.417)
1 1
2 3 -1 -2 0 -2 -6 -1 3 6 B) (-3.187, -2.154)
-4 -8 C) (-4.598, -1.986)
D) (-8.921, -6.862)
Answer: A 3) The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they slept the night before the exam. Construct a 95% prediction interval for y, the score on the final exam, ^
given x = 7 hours, y = 5.044x + 56.11 and se = 6.305. Hours, x 3 Scores, y 65 A) (74.54, 108.30)
5 80
2 60
8 2 4 4 88 66 78 85 B) (55.43, 78.19)
5 90
6 3 90 71 C) (77.21, 110.45)
D) (82.840, 99.996)
Answer: A 4) The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Construct a 95% prediction interval for y, the number of ^
days absent, given x = 95 degrees, y= 0.449x - 30.27 and se = 0.934. Temperature, x Number of absences, y A) (9.916, 14.770)
72 85 91 90 3 7 10 10 B) (3.176, 5.341)
88 8
98 15
75 100 80 4 15 5 C) (4.321, 6.913)
D) (11.378, 13.392)
Answer: A 5) In order for a company's employees to work at a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Construct ^
a 95% prediction interval for y given x = 2.5, y= 6.91x + 46.26, and se = 4.578. Number of years, x Grades on test, y A) (51.50, 75.57) Answer: A
Page 8
3 61
4 4 5 3 68 75 82 73 B) (60.23, 91.42)
6 90
2 7 3 58 93 72 C) (55.12, 87.34)
D) (58.28, 68.79)
6) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Construct a 95% prediction interval for y, the yield, given x = 11 inches, ^
y = 4.379x + 4.267 and se = 3.529. Rainfall (in inches), x 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), y 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8 A) (43.56, 61.32) B) (41.68, 63.21) C) (40.54, 64.15)
D) (49.41, 55.47)
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) A breeder of thoroughbred horses wishes to model the relationship between the gestation 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. Construct a 95% prediction interval about the value of y when x = 300 days. Horse Gestation Life Horse Gestation Life period Length period Length x (days) y (years) x (days) y (years) 1 416 24 5 356 22 2 279 25.5 6 403 23.5 3 298 20 7 265 21 4 307 21.5 Answer: The prediction interval is of the form: ^ 1 (x - x)2 y ± tα/2 s 1 + + n SSxx ^
y = 18.89 + 0.01087(300) = 22.151 Confidence coefficient 0.95 = 1 - α ⇒ α = 1 - 0.95 = 0.05. α/2 = 0.05/2 = 0.025. From a Student's t table, t0.025 = 2.571 with n - 2 = 7 - 2 = 5 df. The 95% prediction interval is: 1 (300 - 332)2 ⇒ 22.151 ± 5.528⇒ (16.623, 27.679) 22.151 ± 2.571(1.971) 1 + + 7 21,752 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 8) Is a confidence interval or a prediction interval the wider interval at the same level of significance? A) Prediction Interval B) Confidence interval Answer: A 9) When constructing a confidence interval about the mean response of y in a linear regression, the t-distribution is used with degrees of freedom. A) n - 2 Answer: A
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B) n - 1
C) n + k - 2
D) n 1 + n 2 - 2
14.3 Introduction to Multiple Regression 1 Obtain the correlation matrix. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A minor league baseball team posted the following data for its 16 best players. What is the correlation between the number of strikeouts and the number of hits? At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
A) 0.576
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
B) 0.830
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
C) 0.932
D) 0.019
Answer: A 2) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. What is the correlation between their state board score and their age? Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
A) 0.881 Answer: A
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State Board 620 568 633 516 585 584 B) -0.240
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0 C) -0.143
State Board 639 558 595 511 579 566 D) 0.453
2 Use technology to find a multiple regression equation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A minor league baseball team posted the following data for its 16 best players. What is the regression equation? At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
A) Hits = -2.64 + 0.989 At Bats - 1.05 Strikeouts C) Hits = -1.05 + 0.989 At Bats - 2.64 Strikeouts
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
B) Hits = 0.989 - 2.64 At Bats - 1.05 Strikeouts D) Hits = -2.64 + 1.05 At Bats - 0.989 Strikeouts
Answer: A 2) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. What is the regression equation? Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
State Board 620 568 633 516 585 584
A) State Board = 255 + 12.7 Age + 4.6 GPA C) State Board = 4.6 + 255 Age + 12.7 GPA Answer: A
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Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0
State Board 639 558 595 511 579 566
B) State Board = 12.7 + 4.6 Age + 255 GPA D) State Board = 255 + 4.6 Age + 12.7 GPA
3) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The data for this class is given below. What is the regression equation? Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
Rope Climb 69 56 46 59 65
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17
Rope Climb 68 49 52 63 74
A) Rope climb = 117 - 1.91 Push-up - 1.06 Pull-ups B) Rope climb = -1.91 - 1.06 Push-up + 117 Pull-ups C) Rope climb = -1.06 + 117 Push-up - 1.91 Pull-ups D) Rope climb = 117 - 1.06 Push-up - 1.91 Pull-ups Answer: A 3 Interpret the coefficients of a multiple regression equation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ^
1) You obtain the multiple regression equation y = -3 + 7x1 - 8x2 from a set of sample data. Interpret the slope coefficients for x1 and x2 . ^
A) The slope coefficient of x1 is 7. This indicates that y will increase 7 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is -8. This indicates that y will decrease 8 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. ^
B) The slope coefficient of x1 is -8. This indicates that y will decrease 8 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is 7. This indicates that y will increase 7 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. ^
C) The slope coefficient of x1 is -7. This indicates that y will decrease 7 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is 8. This indicates that y will increase 8 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. ^
D) The slope coefficient of x1 is 8. This indicates that y will increase 8 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is -7. This indicates that y will decrease 7 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. Answer: A
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^
2) You obtain the multiple regression equation y = -3 - 4x1 + 6x2 from a set of sample data. Interpret the slope coefficients for x1 and x2 . ^
A) The slope coefficient of x1 is -4. This indicates that y will decrease 4 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is 6. This indicates that y will increase 6 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. ^
B) The slope coefficient of x1 is 6. This indicates that y will increase 6 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is -4. This indicates that y will decrease 4 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. ^
C) The slope coefficient of x1 is 4. This indicates that y will increase 4 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is -6. This indicates that y will decrease 6 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. ^
D) The slope coefficient of x1 is -6. This indicates that y will decrease 6 units, on average, for every 1-unit ^
increase in x1 , provided that x2 remains constant. The slope coefficient of x2 is 4. This indicates that y will increase 4 units, on average, for every 1-unit increase in x2 , provided that x1 remains constant. Answer: A 4 Determine R^2 and adjusted R^2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The data for this class is given below. What is the correlation between pull-ups and the time to climb the rope? Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
A) -0.821 Answer: A
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Rope Climb 69 56 46 59 65 B) -0.882
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17 C) 0.709
Rope Climb 68 49 52 63 74 D) 0.003
2) A minor league baseball team posted the following data for its 16 best players. Determine R2 for the multiple ^
regression model y = b0 + b1 x1 + b2 x2 where x1 is At Bats, x2 is Strikeouts, and y is the response variable "Hits". At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
A) 99.2%
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
B) 89.6%
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
C) 98.9%
D) 99.1%
Answer: A 3) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders whether a student's age and GPA would be good predictors of their state board score. She obtains the following data. ^ Determine R2 for the multiple regression model y = b0 + b1 x1 + b2 x2 where x1 is Age, x2 is GPA, and y is the response variable "state board score". Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
A) 78.1%
State Board 620 568 633 516 585 584
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0
B) 73.3%
State Board 639 558 595 511 579 566
C) 88.4%
D) 87.9%
Answer: A 4) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in his class. He records the number of push ups each student is able to do in 2 minutes, the number of pull ups each student is able to do in 2 minutes, and the time (in seconds) for each student to climb a rope. The data are shown in the table. The teacher wonders whether the number of push ups and the number of pull ups would be good predictors of the time to climb the rope. Determine R2 for the multiple regression model ^
y = b0 + b1 x1 + b2x2 where x1 is Push Ups, x2 is Pull Ups, and y is the response variable "Rope Climb". Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
A) 86.2% Answer: A Page 14
Rope Climb 69 56 46 59 65 B) 82.3%
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17 C) 92.8%
Rope Climb 68 49 52 63 74 D) 64.5%
5) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if a student's age and GPA would be good predictors of their state board score. She obtains the following data. ^ Determine the adjusted R2 for the multiple regression model y = b0 + b1x1 + b2 x2 where x1 is Age, x2 is GPA, and y is the response variable "state board score". Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
State Board 620 568 633 516 585 584
A) 73.3%
B) 78.1%
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0
State Board 639 558 595 511 579 566
C) 82.9%
D) 67.9%
Answer: A 6) A minor league baseball team posted the following data for its 16 best players. Determine the adjusted R2 for ^
the multiple regression model y = b0 + b1 x1 + b2 x2 where x1 is At Bats, x2 is Strikeouts, and y is the response variable "Hits". At Bats 82 41 87 95 67 69 32 19 A) 99.1% Answer: A
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Strikeouts 17 11 35 38 29 34 15 9
Hits 62 24 49 51 32 29 14 8 B) 99.2%
At Bats 100 80 70 85 38 53 44 57
Strikeouts 31 25 28 33 15 17 14 12
C) 98.9%
Hits 62 49 37 49 19 33 25 42 D) 99.3%
7) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in his class. He records the number of push ups each student is able to do in 2 minutes, the number of pull ups each student is able to do in 2 minutes, and the time (in seconds) for each student to climb a rope. The data are shown in the table. The teacher wonders whether the number of push ups and the number of pull ups would be good predictors of the time to climb the rope. Determine the adjusted R2 for the multiple regression ^
model y = b0 + b1 x1 + b2x2 where x1 is Push Ups, x2 is Pull Ups, and y is the response variable "Rope Climb". Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
A) 82.3%
Rope Climb 69 56 46 59 65 B) 86.2%
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17
Rope Climb 68 49 52 63 74
C) 92.8%
D) 74.5%
Answer: A 5 Perform an F -test for lack of fit. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A minor league baseball team posted the following data for its 16 best players. Test the null hypothesis that all coefficients are zero at the 95% confidence level. At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
A) Reject the null hypothesis. B) Fail to reject the null hypothesis. C) The probability plot of residuals indicates non-normality of residuals so the hypothesis test is inconclusive. D) The probability plot of residuals indicates normality of residuals so the hypothesis test is inconclusive. Answer: A
Page 16
2) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. Test the null hypothesis that all coefficients are zero at the 95% confidence level. Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
State Board 620 568 633 516 585 584
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0
State Board 639 558 595 511 579 566
A) Reject the null hypothesis. B) Fail to reject the null hypothesis. C) The probability plot of residuals indicates non-normality of residuals so the hypothesis test is inconclusive. D) The probability plot of residuals indicates normality of residuals so the hypothesis test is inconclusive. Answer: A 3) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The data for this class is given below. Test the null hypothesis that all coefficients are zero at the 95% confidence level. Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
Rope Climb 69 56 46 59 65
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17
Rope Climb 68 49 52 63 74
A) Reject the null hypothesis. B) Fail to reject the null hypothesis. C) The probability plot of residuals indicates non-normality of residuals so the hypothesis test is inconclusive. D) The probability plot of residuals indicates normality of residuals so the hypothesis test is inconclusive. Answer: A 4) A multiple linear regression model of the form yi = β 0 + β 1 x1i + β 2 x2i + β 3 x3i + εi has how many parameters to be estimated? A) 4
B) 5
C) 3
D) 1
Answer: A 5) A correlation matrix shows the linear correlation among regression model. A) all Answer: A
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B) several
variables under consideration in a multiple C) two
D) three
6) When performing a multiple linear regression, if we test the null hypothesis that the coefficients are all zero, what is the decision rule? A) If p < α, then reject the null hypothesis. B) If p > α, then reject the null hypothesis. C) If p < α, then fail to reject the null hypothesis. D) If p > α, then fail to reject the alternative hypothesis. Answer: A 7) By saying that we reject the null hypothesis in a hypothesis test of the coefficients of a multiple linear regression we are implying that A) at least one coefficient is different from zero. B) all coefficients are different from zero. C) the relationship is not linear. D) non-parametric methods must be used on this problem. Answer: A 6 Test individual regression coefficients for significance. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A minor league baseball team posted the following data for its 16 best players. Test the significance of the individual predictor variables at α = 0.05. At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
A) Both explanatory variables have a significant linear relationship with the response variable. B) Consider removing the variable "at bats" because it is not a good predictor of "hits". C) Consider removing the variable "strikeouts" because it is not a good predictor of "hits". D) Consider removing both variables because neither is a good predictor of "hits". Answer: A
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2) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. Test the significance of the individual predictor variables at α = 0.05. Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
State Board 620 568 633 516 585 584
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0
State Board 639 558 595 511 579 566
A) Consider removing the variable "GPA" because it is not a good predictor of "State Board Scores". B) Both explanatory variables have a significant linear relationship with the response variable. C) Consider removing the variable "Age" because it is not a good predictor of "State Board Score". D) Consider removing both variables because neither is a good predictor of"State Board Scores". Answer: A 3) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The data for this class is given below. Test the significance of the individual predictor variables at α = 0.05. Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
Rope Climb 69 56 46 59 65
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17
Rope Climb 68 49 52 63 74
A) Consider removing the variable "Pull-ups" because it is not a good predictor of "Rope Climb Time". B) Both explanatory variables have a significant linear relationship with the response variable. C) Consider removing the variable "Push-ups" because it is not a good predictor of "Rope Climb Time". D) Consider removing both variables because neither is a good predictor of "Rope Climb Time". Answer: A
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7 Construct confidence and prediction intervals. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A minor league baseball team posted the following data for its 16 best players. Construct a 95% confidence interval for the number of hits player will get if the player is at bat 53 times. At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
A) (26.17, 33.43)
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
B) (16.12, 43.49)
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
C) (22.49, 39.76)
D) (0.23, 62.03)
Answer: A 2) A minor league baseball team posted the following data for its 16 best players. Construct a 95% prediction interval for the number of hits player will get if the player is at bat 53 times. At Bats 82 41 87 95 67 69 32 19
Strikeouts 17 11 35 38 29 34 15 9
A) (16.12, 43.49)
Hits 62 24 49 51 32 29 14 8
At Bats 100 80 70 85 38 53 44 57
B) (26.17, 33.43)
Strikeouts 31 25 28 33 15 17 14 12
Hits 62 49 37 49 19 33 25 42
C) (22.49, 39.76)
D) (0.23, 62.03)
Answer: A 3) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. Construct a 95% confidence interval for the State Board Score a student will get if the student's age is 24. Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
A) (561.30, 587.30) Answer: A Page 20
State Board 620 568 633 516 585 584 B) (527.93, 620.67)
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0 C) (542.5, 608.6)
State Board 639 558 595 511 579 566 D) (476.6, 674.4)
4) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. Construct a 95% prediction interval for the State Board Score a student will get if the student's age is 24. Age
GPA
26 25 30 20 23 23
3.0 3.1 2.3 2.5 3.9 2.5
State Board 620 568 633 516 585 584
Age
GPA
27 25 26 20 25 23
3.3 3.5 2.0 3.2 3.5 4.0
State Board 639 558 595 511 579 566
A) (527.93, 620.67) B) (561.30, 587.30) C) (542.5, 608.6) D) The result is not reliable because the predictor variable is not a good indicator of the State Board Score. Answer: A 5) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The data for this class is given below. Construct a 95% confidence interval for the Rope Climb Time if the student's Push-ups are 22. Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
A) (48.78, 57.77) Answer: A
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Rope Climb 69 56 46 59 65 B) (41.70, 64.85)
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17 C) (47.09, 58.45)
Rope Climb 68 49 52 63 74 D) (38.89, 66.65)
6) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The data for this class is given below. Construct a 95% prediction interval for the Rope Climb Time if the student's Pull-ups are 22. Push-ups
Pull-ups
16 18 24 22 19
13 20 25 18 15
Rope Climb 69 56 46 59 65
Push-ups
Pull-ups
17 23 21 20 16
19 22 21 16 17
Rope Climb 68 49 52 63 74
A) The result is not reliable because the predictor variable is not a good indicator of the Rope Climb Time. B) (41.70, 64.85) C) (47.09, 58.45) D) (38.89, 66.65) Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) Can happiness be predicted? A researcher is investigating whether job satisfaction, income, and health could be good predictors of happiness. She selects a random sample of working adults and obtains the following information for each person. Each person is asked to rate their happiness and their job satisfaction on a scale of 1 to 10. Each person is asked their annual income. Finally, each person's overall physical health is evaluated by a doctor and rated on a scale of 1 to 20. The results are shown in the table. Annual Income Happiness Job satisfaction (thousands of dollars)Health 8 6 38 14 6 5 47 17 4 2 19 12 7 5 22 11 2 3 36 5 9 7 44 14 4 3 75 8 6 8 24 13 7 4 52 14 3 3 27 5 5 4 105 6 8 9 44 12 5 7 21 7 (a) Construct the correlation matrix. Is there any reason to be concerned with collinearity? ^
(b) Find the least squares regression equation y = b0 + b1 x1 + b2x2 + b3 x3 , where x1 is job satisfaction, x2 is income, x3 is health, and y is the response variable "happiness". (c) Test H 0: β 1 = β 2 = β 3 = 0 versus H 1 : at least one of the β i ≠ 0 at the α = 0.05 level of significance. (d) Test the hypotheses H0 : β 1 = 0 versus H1 : β 1 ≠ 0 , H0 : β 2 = 0 versus H 1: β 2 ≠ 0, and H 0 : β3 = 0 versus H 1 : β 3 ≠ 0 at the α = 0.05 level of significance. Should any of the explanatory variables be removed from the model? If so, which one? Why? (e) Determine the least squares regression equation with the explanatory variable identified in part (d) removed. (f) Are both slope coefficients significantly different from zero? If not, remove the appropriate explanatory variable and compute the new least squares regression equation. (g) How does the P-value for your final regression equation compare with the P-value for the original equation Page 22
in part (b)? What does this imply? Answer: (a)
Happiness Job Sat Income Job Sat 0.699 Income 0.002 -0.183 Health 0.735 0.389 -0.175 There is no concern with multicollinearity. ^
(b) y = - 0.73 + 0.494 x1 + 0.0168 x2 + 0.301 x3 (c) Since the P-value is 0.003, we reject the null hypothesis and conclude that at least one of the explanatory variables is linearly associated with happiness. (d) For the explanatory variable "Job Satisfaction", t0 = 2.99 and the P-value is 0.015. For the explanatory variable "Income", t0 = 1.22 and the P-value is 0.254. For the explanatory variable "Health", t0 = 3.32 and the P-value is 0.009. We reject the null hypothesis for job satisfaction and health and conclude that job satisfaction and health are linearly related to happiness. Remove x2 = income as it does not appear to be linearly related to happiness. ^
(e) y = 0.25 + 0.468 x1 + 0.289 x3 (f) Both slope coefficients are significantly different from zero. (g) The P-value for the final equation (0.001) is lower than the P-value for the original equation (0.003) which indicates that the model has improved. 8) A researcher is investigating whether exercise, age, and percent body fat could be good predictors of resting pulse rate. She selects a random sample of women and for each woman records their resting pulse rate, the amount they exercise (on a scale of 1 to 10), age, and percent body fat. The results are shown in the table. Resting Pulse Rate Amount of Exercise Age Percent Body Fat 76 6 22 23 63 8 38 19 82 5 62 25 90 3 54 31 86 2 44 27 77 4 41 24 80 5 59 26 75 4 27 26 58 8 35 16 76 5 76 24 65 9 31 20 85 3 66 28 (a) Construct the correlation matrix. Is there any reason to be concerned with collinearity? Is this what you would expect? ^
(b) Find the least squares regression equation y = b0 + b1 x1 + b2x2 + b3 x3 , where x1 is Exercise, x2 is Age, x3 is Percent body fat, and y is the response variable "resting pulse rate". (c) Test H 0: β 1 = β 2 = β 3 = 0 versus H 1 : at least one of the β i ≠ 0 at the α = 0.05 level of significance. (d) Test the hypotheses H0 : β 1 = 0 versus H1 : β 1 ≠ 0, H0 : β 2 = 0 versus H1 : β 2 ≠ 0, and H 0 : β3 = 0 versus H 1 : β 3 ≠ 0 at the α = 0.05 level of significance.
Should any of the explanatory variables be removed from the model? If so, which one? Why? (e) Determine the least squares regression equation with the explanatory variable identified in part (d) removed. (f) Are both slope coefficients significantly different from zero? Is this what you would expect? If appropriate, remove an explanatory variable and compute the new least squares regression equation. (g) What is the P-value for your final regression equation? What does this imply?
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Answer: (a)
Pulse Exercise Age Exercise -0.900 Age 0.495 -0.406 Body Fat 0.965 0.447 -0.886 Yes, there is a concern with multicollinearity. There is a strong negative correlation between exercise and percent body fat. This is not surprising - the more a person exercises, the lower their percent body fat is likely to be. Including both of these variables in the model might lead to strange results. ^
(b) y = 36.3 - 0.909 x1 + 0.0442 x2 + 1.76 x3 (c) Since the P-value is < 0.001, we reject the null hypothesis and conclude that at least one of the explanatory variables is linearly associated with resting pulse rate. (d) For the explanatory variable "Exercise", t0 = -1.17 and the P-value is 0.277. For the explanatory variable "Age", t0 = 0.84 and the P-value is 0.426. For the explanatory variable "Percent Body Fat", t0 = 4.15 and the P-value is 0.003. We reject the null hypothesis for percent body fat and conclude that percent body fat is linearly related to resting pulse rate. Remove x2 = "age" as the P-value for the slope coefficient of this variable is the largest and age doesn't appear to be linearly related to resting pulse rate. ^
(e) y = 36.7 - 0.926 x1 + 1.83 x3 (f) The slope coefficient for percent body fat is significantly different from zero, however the slope coefficient for exercise is not significantly different from zero so this variable should be removed. Exercise and resting pulse rate have a strong negative correlation, so a slope coefficient significantly different from zero might have been expected for exercise. However, we might be obtaining strange results because of multicollinearity. ^
Removing x2 : y = 21.4 + 2.27 x3 (g) The P-value for the final equation is < 0.001 which indicates that the model is good.
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14.4 Interaction and Dummy Variables 1 Work with multiple regression models with interaction. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Suppose that the response variable y is related to the explanatory variables x1 and x2 by the regression ^
equation: y = 7 - 0.4x1 + 1.8x2 . Construct a graph showing the relationship between the expected value of y and x1 for x2 = 10, 20, and 30. y 70 60 50 40 30 20 10 10
20
30
40
50
x
A) y
^
70
A: y = 25 - 0.4x1 (x2 = 10)
60
B: y = 43 - 0.4x1 (x2 = 20)
50
C: y = 61 - 0.4x1 (x2 = 30)
^
^
C 40 30
B
20 10
A 10
20
30
40
50
x
B) y
^
70
A: y = 18 - 0.4x1 (x2 = 10)
60
B: y = 36 - 0.4x1 (x2 = 20)
50
C: y = 54 - 0.4x1 (x2 = 30)
^
^
40
C
30 20
B
10 10
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20
30
40
A 50
x
C) y
^
70
A: y = 54 - 0.4x1 (x2 = 10)
60
B: y = 29 - 0.4x1 (x2 = 20)
50
C: y = 47 - 0.4x1 (x2 = 30)
^
^
40 C
30 20
B
10 10
20
30
40
50A
x
D) none of these Answer: A 2) Suppose that the response variable y is related to the explanatory variables x1 and x2 by the regression ^
equation: y = 5 - 0.5x1 + 1.5x2 . Construct a graph showing the relationship between the expected value of y and x2 for x1 = 30, 40, and 50. y 70 60 50 40 30 20 10 10
-10
20
30
40
50
x
-20 -30
A) y 70
^
A: y = -10 + 1.5x2 (x1 = 30)
A
60
B
50
C
^
B: y = -15 + 1.5x2 (x1 = 40) ^
C: y = -20 + 1.5x2 (x1 = 50)
40 30 20 10
-10 -20 -30
Page 26
10
20
30
40
50
x
B) y
^
A: y = -15 + 1.5x2 (x1 = 30)
70 A
60 50 40
^
B
B: y = -20 + 1.5x2 (x1 = 40)
C
C: y = -25 + 1.5x2 (x1 = 50)
^
30 20 10 10
-10
20
30
40
50
x
-20 -30
C) y 70
A
60 50
^
B
A: y = -5 + 1.5x2 (x1 = 30)
C
B: y = -10 + 1.5x2 (x1 = 40)
^
^
C: y = -15 + 1.5x2 (x1 = 50)
40 30 20 10
-10
10
20
30
40
50
x
-20 -30
D) none of these Answer: A 2 Build multiple regression models with indicator variables. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Given the regression equation that relates a student's weight to the number of hours spent on the computer, x1 , the number of hours viewing television, x2, and the number of hours on their cell phone, x3 , as ^
y = 125.5 + 3.26x1 + 4.27x2 + 1.75x3 . What is a student's weight if the student spends 4 hours on the computer, 2 hours viewing television, and 3 hours on their cell phone? A) 152.33 B) 151.83 C) 155.86
D) 154.35
Answer: A 2) A minor league baseball team determined the following to be the regression equation for the number of hits a player would get based upon an expected number of times at bat and how many strikeouts they had. The equation is Hits = -2.64 + 0.989 At Bats - 1.05 Strikeouts. A new player on the team last year in a different league was at bat 80 times and had 32 strikeouts. How many expected hits would he have had in this league? Round your answer to the next whole hit. A) 43 B) 55 C) 49 D) 41 Answer: A
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3) Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students and determines the regression equation to be State Board = 255 + 12.7 Age + 4.6 GPA. What would she expect a 27 year-old student with a GPA of 2.6 to achieve on the state board examination? Round your answer to the nearest whole number. A) 610 B) 412 C) 639 D) 584 Answer: A 4) A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The regression equation for this class is Rope climb = 117 - 1.91 Push-up - 1.06 Pull-ups A new student enters the class and is able to do 20 push-ups and 15 pull-ups, what is this student's expected time to climb the rope? Round your answer to the next whole second. A) 63 seconds B) 68 seconds C) 65 seconds D) 57 seconds Answer: A 5) Given the regression equation that relates a student's weight to the number of hours spent on the computer, x1 , the number of hours viewing television, x2, and the number of hours on their cell phone, x3 , per day as ^
y = 125.5 + 3.26x1 + 4.27x2 + 1.75x3 . What is a student's expected weight change if the amount of television is increased by 1 hour, the number of hours watching television and talking on their cell phone stayed the same? A) 4.27 pounds B) 3.26 pounds C) 1.75 pounds D) 125.5 pounds Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) Why is it important for the explanatory variables to have a low correlation? Answer: Answers will vary. We do not want the effect of x1 on the value of the response variable to depend on the value of x2 .
14.5 Polynomial Regression 1 Find a quadratic regression equation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. ^
1) Find the quadratic regression equation y = b0 + b1 x + b2 x2 for the data below. x 1 1.7 2.1 3.5 4.4 5.8 6.2
y 21.7 16.8 12.4 7.9 7.7 13.1 18.5
^
A) y = 1.8377x2 - 14.1420x + 34.5143 ^ C) y = -1.8377x2 + 14.1420x + 34.5143 Answer: A
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^
B) y = 1.9421x2 - 12.0657x + 32.5923 ^ D) y = -1.9421x2 - 12.0657x + 32.5923
^
2) Find the quadratic regression equation y = b0 + b1 x + b2 x2 for the data below. x 2.1 2.5 3.3 4.2 4.7 5.7 6.3
y 39.7 37.4 34.3 33.1 32.5 41.4 47.3
^
^
A) y = 2.5053x2 - 19.3894x + 69.9652 ^ C) y = 2.1295x2 - 17.8246x + 70.0032
B) y = -2.5053x2 + 19.3894x + 69.9652 ^ D) y = -2.1295x2 - 17.8246x + 70.0032
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 3) An engineer collects the following data showing the speed and miles per gallon of a 4-door sedan. Speed, s Miles per Gallon, M 30 17 35 21 40 24 40 25 45 29 50 29 55 33 60 30 65 25 a) Draw a scatter diagram of the data. What type of relation appears to exist between x and y? ^ b) Find the quadratic regression equation y = b0 + b1x + b2 x2. c) Interpret the coefficient of determination. Answer: a) Miles per Gallon - 4-door sedan y 32 30 28 MPG
26 24 22 20 18 16
Speed 30
40
50
60
x
The relation appears quadratic that opens downward. ^ b) y = -0.0256x2 + 2.7533x - 43.5856 c) 91.2% of the variability in miles per gallon is explained by the least-squares regression model.
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14.6 Building a Regression Model 1 Perform a partial F-test. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) For the data set below, use a partial F-test to determine whether the variables x4 and x5 do not significantly help to predict the response variable, y. Use the α = 0.05 level of significance. x1
x2
x3
x4
x5
y
0.9 3.8 2 5.3 4.8 8.3 12.7 6.3 14.4 9.2
2.7 2.4 2.2 2.4 2.7 2.2 2.1 1.8 2.1 1.2
2.6 5.4 7.7 7.2 6.1 8.7 9.1 1 3.5 5.8
10.8 9.5 10.4 9.1 11.6 10.5 11.3 8.5 10.4 8.9
15.2 4.4 1.8 1.6 5.5 5 2.3 22.6 11.4 6.6
12 10.7 10.9 10.4 10.3 10.4 10.4 9 8.7 8.6
Answer: H 0 : β 4 = β 5 = 0 vs. H 1: at least one of β 4 and β 5 is different from zero. F0 = 0.42 < F0.05,2,4 = 6.94. Do not reject H0 . There is not sufficient evidence to suggest that either x4 or x5 explain the variability in the response variable y. 2) For the data set below, use a partial F-test to determine whether the variables x1 and x2 do not significantly help to predict the response variable, y. Use the α = 0.05 level of significance. x1
x2
x3
x4
y
24.8 66.2 13.7 26.5 100.4 15.8 30.7 77.9 14 40 83.5 8.9 33.4 69.2 10.7 41.9 83.7 9.9 25.5 112.8 10.1 33.8 68.4 6.9 23.5 69.6 7.7 39.8 63.2 6.7
3.6 11.5 15.5 8.9 18.4 21.6 16.4 25.6 15.7 30.7
60 66.4 76.3 77.2 82.2 84.4 87.2 88.4 90.6 93.4
Answer: H 0 : β 1 = β 2 = 0 vs. H 1: at least one of β 1 and β 2 is different from zero. F0 = 1.76 < F0.05,2,5 = 5.79. Do not reject H0 . There is not sufficient evidence to suggest that either x1 or x2 explain the variability in the response variable y.
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Ch. 14 Inference on the Least-Squares Regression Model and Multiple Regression Answer Key 14.1 Testing the Significance of the Least-Squares Regression Model 1 State the requirements of the least-squares regression model. 1) A 2) A 2 Compute the standard error of the estimate. 1) s = 15.35 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 3 Verify that residuals are normally distributed. 1) A 2) A 4 Conduct inference on the slope of the least-squares regression model. 1) A 2) Since t = -6.455 and 0.02 < P-value < 0.04, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 3) Since t = 19.510 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 4) Since t = -27.929 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 5) Since t = 13.031 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 6) Since t = 9.034 and P-value < 0.001, we reject the null hypothesis. There is sufficient evidence to support the claim of a linear relationship between the two variables. 5 Construct a confidence interval about the slope of the least-squares regression model. 1) A 2) A 3) A 4) A 5) A 6) b1 = 0.01087 sb1 = 0.0134 t0.05 = 2.015 90% confidence interval = (-0.0161, 0.0379) 7) A
14.2 Confidence and Prediction Intervals 1 Construct confidence intervals for mean responses. 1) A 2) A 3) A 4) A Page 31
5) A ^
6) For x = 5, y = -0.25 + 1.315(5) = 6.325 The confidence interval is of the form: ^ 1 (x - x)2 y ± tα/2s + n SSxx Confidence coefficient 0.90 = 1 - α ⇒ α = 1 - 0.90 = 0.10. α/2 = 0.10/2 = 0.05. From a Student's t table, t0.05 = 2.015 with n - 2 = 7 - 2 = 5 df. The confidence interval is: 1 (5 - 5.8571)2 ⇒ 6.325 ± 1.233 ⇒ (5.092, 7.558) 6.325 ± 2.015(1.577) + 7 94.8571 2 Construct prediction intervals for an individual response. 1) A 2) A 3) A 4) A 5) A 6) A 7) The prediction interval is of the form: ^ 1 (x - x)2 y ± tα/2s 1 + + n SSxx ^
y = 18.89 + 0.01087(300) = 22.151 Confidence coefficient 0.95 = 1 - α ⇒ α = 1 - 0.95 = 0.05. α/2 = 0.05/2 = 0.025. From a Student's t table, t0.025 = 2.571 with n - 2 = 7 - 2 = 5 df. The 95% prediction interval is: 1 (300 - 332)2 ⇒ 22.151 ± 5.528⇒ (16.623, 27.679) 22.151 ± 2.571(1.971) 1 + + 7 21,752 8) A 9) A
14.3 Introduction to Multiple Regression 1 Obtain the correlation matrix. 1) A 2) A 2 Use technology to find a multiple regression equation. 1) A 2) A 3) A 3 Interpret the coefficients of a multiple regression equation. 1) A 2) A 4 Determine R^2 and adjusted R^2. 1) A 2) A 3) A 4) A 5) A 6) A 7) A Page 32
5 Perform an F -test for lack of fit. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 6 Test individual regression coefficients for significance. 1) A 2) A 3) A 7 Construct confidence and prediction intervals. 1) A 2) A 3) A 4) A 5) A 6) A 7) (a) Happiness Job Sat Income Job Sat 0.699 Income 0.002 -0.183 Health 0.735 0.389 -0.175 There is no concern with multicollinearity. ^
(b) y = - 0.73 + 0.494 x1 + 0.0168 x2 + 0.301 x3
(c) Since the P-value is 0.003, we reject the null hypothesis and conclude that at least one of the explanatory variables is linearly associated with happiness. (d) For the explanatory variable "Job Satisfaction", t0 = 2.99 and the P-value is 0.015. For the explanatory variable "Income", t0 = 1.22 and the P-value is 0.254. For the explanatory variable "Health", t0 = 3.32 and the P-value is 0.009. We reject the null hypothesis for job satisfaction and health and conclude that job satisfaction and health are linearly related to happiness. Remove x2 = income as it does not appear to be linearly related to happiness. ^
(e) y = 0.25 + 0.468 x1 + 0.289 x3 (f) Both slope coefficients are significantly different from zero. (g) The P-value for the final equation (0.001) is lower than the P-value for the original equation (0.003) which indicates that the model has improved.
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8) (a)
Pulse Exercise Age Exercise -0.900 Age 0.495 -0.406 Body Fat 0.965 0.447 -0.886 Yes, there is a concern with multicollinearity. There is a strong negative correlation between exercise and percent body fat. This is not surprising - the more a person exercises, the lower their percent body fat is likely to be. Including both of these variables in the model might lead to strange results. ^
(b) y = 36.3 - 0.909 x1 + 0.0442 x2 + 1.76 x3 (c) Since the P-value is < 0.001, we reject the null hypothesis and conclude that at least one of the explanatory variables is linearly associated with resting pulse rate. (d) For the explanatory variable "Exercise", t0 = -1.17 and the P-value is 0.277. For the explanatory variable "Age", t0 = 0.84 and the P-value is 0.426. For the explanatory variable "Percent Body Fat", t0 = 4.15 and the P-value is 0.003. We reject the null hypothesis for percent body fat and conclude that percent body fat is linearly related to resting pulse rate. Remove x2 = "age" as the P-value for the slope coefficient of this variable is the largest and age doesn't appear to be linearly related to resting pulse rate. ^
(e) y = 36.7 - 0.926 x1 + 1.83 x3 (f) The slope coefficient for percent body fat is significantly different from zero, however the slope coefficient for exercise is not significantly different from zero so this variable should be removed. Exercise and resting pulse rate have a strong negative correlation, so a slope coefficient significantly different from zero might have been expected for exercise. However, we might be obtaining strange results because of multicollinearity. ^
Removing x2 : y = 21.4 + 2.27 x3 (g) The P-value for the final equation is < 0.001 which indicates that the model is good.
14.4 Interaction and Dummy Variables 1 Work with multiple regression models with interaction. 1) A 2) A 2 Build multiple regression models with indicator variables. 1) A 2) A 3) A 4) A 5) A 6) Answers will vary. We do not want the effect of x1 on the value of the response variable to depend on the value of x2.
14.5 Polynomial Regression 1 Find a quadratic regression equation. 1) A 2) A
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3) a) Miles per Gallon - 4-door sedan y 32 30 28 MPG
26 24 22 20 18 16
Speed 30
40
50
60
x
The relation appears quadratic that opens downward. ^ b) y = -0.0256x2 + 2.7533x - 43.5856 c) 91.2% of the variability in miles per gallon is explained by the least-squares regression model.
14.6 Building a Regression Model 1 Perform a partial F-test. 1) H0 : β 4 = β 5 = 0 vs. H 1 : at least one of β 4 and β 5 is different from zero. F0 = 0.42 < F0.05,2,4 = 6.94. Do not reject H 0. There is not sufficient evidence to suggest that either x4 or x5 explain the variability in the response variable y. 2) H0 : β 1 = β 2 = 0 vs. H 1 : at least one of β 1 and β 2 is different from zero. F0 = 1.76 < F0.05,2,5 = 5.79. Do not reject H 0. There is not sufficient evidence to suggest that either x1 or x2 explain the variability in the response variable y.
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Ch. 15 Nonparametric Statistics 15.1 An Overview of Nonparametric Statistics 1 Distinguish between parametric and nonparametric statistical procedures. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) ___________________ statistical processes are often called distribution free procedures. A) Nonparametric B) Parametric C) Free form D) Approximate Answer: A 2) Which of the below is an advantage of nonparametric statistical procedures? A) The computations are easy. B) The results are less powerful. C) They require a large sample size. D) Fewer requirements need to be met. Answer: A 3) Which of the below is a disadvantage of nonparametric statistical procedures? A) They require a larger sample size to have the same probability of a type I error. B) Fewer requirements need to be met. C) Computations are easy. D) Procedures are used only for continuous and normal data. Answer: A 4) The Kruskal-Wallis test is the nonparametric version of the parametric _________. A) One-way ANOVA test B) linear correlation of variable C) small sample z or t test D) inference test about the difference of the means for independent samples Answer: A
15.2 Runs Test for Randomness 1 Perform a runs test for randomness. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Test the hypothesis at the α = 0.05 level of significance that the sequence is not random. F F M M M F M F F F F M A) r = 6; lower critical value = 3; upper critical value = 11 Since 3 < r < 11, we do not reject H 0 . There is not sufficient evidence to support the hypothesis that the sequence is not random. B) r = 6; lower critical value = 4; upper critical value = 13 Since 4 < r < 13, we do not reject H 0 . There is not sufficient evidence to support the hypothesis that the sequence is not random. C) r = 3; lower critical value = 3; upper critical value = 11 Since r ≤ 3, we reject H0 . There is sufficient evidence to support the hypothesis that the sequence is not random. D) r = 3; lower critical value = 4; upper critical value = 13 Since r ≤ 4, we reject H0 . There is sufficient evidence to support the hypothesis that the sequence is not random. Answer: A Page 1
2) The two flavors of ice cream in a taste test are supposed to be administered to respondents in a random fashion. Suspecting that the administrators of the test did not follow the randomness rule, the project statistician examines the flavor order for the day's taste testing, shown below. peppermint neopolitan peppermint neopolitan neopolitan neopolitan peppermint neopolitan peppermint peppermint neopolitan peppermint neopolitan peppermint neopolitan peppermint neopolitan peppermint neopolitan peppermint Is there evidence at the α = 0.05 level of significance to support the hypothesis that the flavor order is not random? A) r = 17; lower critical value = 6; upper critical value = 16 Since r ≥ 16, we reject H 0 . There is sufficient evidence to support the hypothesis that the sequence is not random. B) r = 16; lower critical value = 7; upper critical value = 18 Since 7 < r < 18, we do not reject H 0 . There is not sufficient evidence to support the hypothesis that the sequence is not random. C) r = 18; lower critical value = 7; upper critical value = 18 Since r ≥ 18, we reject H 0 . There is sufficient evidence to support the hypothesis that the sequence is not random. D) r = 15; lower critical value = 6; upper critical value = 16 Since 7 < r < 18, we do not reject H 0 . There is not sufficient evidence to support the hypothesis that the sequence is not random. Answer: A 3) Wanting to enjoy the beautiful weather, Cassie took a walk with no destination in mind. At each four-way intersection, she randomly decided to head either east or south. A summary of her walk is shown below. E E E E E S S E S E E E E E E E S S S E E S S E S E S E S E S S S E E S S S E S Is there evidence at the α = 0.05 level of significance to support the hypothesis that Cassie's walk was not random? A) z = -0.26 Since |z| < z 0.025 = 1.96, we do not reject H 0. There is not sufficient evidence to support the hypothesis that her walk was not random. B) z = -0.91 Since |z| < z 0.025 = 1.96, we do not reject H 0. There is not sufficient evidence to support the hypothesis that her walk was not random. C) z = -5.43 Since |z| > z 0.025 = 1.96, we reject H 0. There is sufficient evidence to support the hypothesis that her walk was not random. D) z = -3.50 Since |z| > z 0.025 = 1.96, we reject H 0. There is sufficient evidence to support the hypothesis that her walk was not random. Answer: A
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4) A soda machine is set to fill bottles with 24 ounces of cola. As part of a quality control program, an inspector records the amount of fill over or under 24 ounces for one minute. His observations are shown below. -0.40 -0.45 -0.21 -0.15 -0.71 -0.05 -0.84 -0.12 -0.05 -0.58 0.50 0.08 -0.39 -0.77 -0.75 -0.85 -0.04 -1.01 -0.54 -1.00 0.70 0.25 0.12 0.99 0.59 0.05 0.57 0.31 0.14 0.67 0.45 0.60 0.01 0.98 0.22 0.72 0.77 0.48 0.13 0.39 0.80 0.12 0.16 0.37 0.27 -0.73 -0.78 -0.90 -0.14 -0.68 Is there evidence at the α = 0.01 level of significance to support the hypothesis that the filling process is not random with respect to amounts over and under 24 ounces? A) z = -5.71 Since |z| > z 0.005 = 2.58, we reject H 0. There is sufficient evidence to support the hypothesis that the filling process is not random. B) z = -6.28 Since |z| > z 0.005 = 2.58, we reject H 0. There is sufficient evidence to support the hypothesis that the filling process is not random. C) z = -1.25 Since |z| < z 0.005 = 2.58, we do not reject H 0. There is not sufficient evidence to support the hypothesis that the filling process is not random. D) z = 0.67 Since |z| < z 0.005 = 2.58, we do not reject H 0. There is not sufficient evidence to support the hypothesis that the filling process is not random. Answer: A 5) A runs test for randomness is a test to determine if the data ______________ . A) occurs randomly B) is independent C) is mutually exclusive D) has a correlation Answer: A
15.3 Inferences about Measures of Central Tendency 1 Conduct a one-sample sign test. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the sign test to test the given alternative hypothesis at the α = 0.05 level of significance. 1) Alternative Hypothesis: the median is less than 12. An analysis of the data reveals that there are 11 minus signs and 8 plus signs. Answer: H 0 : M = 12 versus H1 : M < 12; k = 8; critical value: 5. Do not reject H 0. 2) Alternative Hypothesis: the median is more than 60. An analysis of the data reveals that there are 3 minus signs and 16 plus signs. Answer: H 0 : M = 60 versus H1 : M > 60; k = 3; critical value: 5. Reject H 0 . 3) Alternative Hypothesis: the median is different from 84. An analysis of the data reveals that there are 22 minus signs and 44 plus signs. Answer: H 0 : M = 84 versus H1 : M ≠ 84; k = 22; z 0 = -2.58; -z 0.025 = -1.96. Reject H0 .
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 4) A convenience store owner believes that the median number of newspapers sold per day is 54. A random sample of 20 days yields the data below. Find the critical value to test the owner's hypothesis. Use α = 0.05. 37 53 64 69 36 60 75 32 38 43 52 59 59 49 49 54 54 64 59 43 A) 4 B) 2 C) 3 D) 5 Answer: A 5) A convenience store owner believes that the median number of newspapers sold per day is 66. A random sample of 20 days yields the data below. Find the test statistic x to test the owner's hypothesis. Use α = 0.05. 49 65 76 81 48 72 87 44 50 55 64 71 71 61 61 66 66 76 71 55 A) 8 B) 10 C) 2 D) 18 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A convenience store owner believes that the median number of newspapers sold per day is 67. A random sample of 20 days yields the data below. Test the owner's hypothesis. Use α = 0.05. 50 66 77 82 49 73 88 45 51 56 65 72 72 62 62 67 67 77 72 56 Answer: critical value 4; test statistic x = 8; fail to reject H 0; There is not sufficient evidence to reject the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) A real estate agent surmises that the median rent for a studio apartment in a mountain community in Colorado is less than $1,500 per month. The rents for a random sample of 15 studio apartments are listed below. Find the critical value to test the agent's hypothesis. Use α = 0.01. $1,800 $1,750 $1,200 $1,375 $1,235 $2,250 $1,675 $1,170 $1,890 $2,500 $1,495 $1,500 $1,575 $1,500 $1,280 A) 1 B) 2 C) 3 D) 4 Answer: A 8) A real estate agent surmises that the median rent for a studio apartment in a mountain community in Colorado is less than $1,400 per month. The rents for a random sample of 15 studio apartments are listed below. Find the test statistic x to test the agent's hypothesis. Use α = 0.01. $1,700 $1,650 $1,100 $1,275 $1,135 $2,150 $1,575 $1,070 $1,790 $2,400 $1,395 $1,400 $1,475 $1,400 $1,180 A) 6 B) 7 C) 13 D) 1 Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 9) A real estate agent surmises that the median rent for a studio apartment in a mountain community in Colorado is less than $1,600 per month. The rents for a random sample of 15 studio apartments are listed below. Test the agent's hypothesis. Use α = 0.01. $1,900 $1,850 $1,300 $1,475 $1,335 $2,350 $1,775 $1,270 $1,990 $2,600 $1,595 $1,600 $1,675 $1,600 $1,380 Answer: critical value 1; test statistic x = 6; fail to reject H 0; There is not sufficient evidence to support the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 10) The owner of a major golf course believes that the course is so tough that most golfers rarely break par of 73. The scores from a random sample of 20 golfers are listed below. Find the critical value to test the club owner's hypothesis. Use α = 0.05. 72 70 73 73 76 75 67 79 73 78 70 72 74 74 81 79 73 75 76 66 A) 4 B) 3 C) 2 D) 1 Answer: A 11) The owner of a major golf course believes that the course is so tough that most golfers rarely break par of 73. The scores from a random sample of 20 golfers are listed below. Find the test statistic x to test the club owner's hypothesis. Use α = 0.05. 72 70 73 73 76 75 67 79 73 78 70 72 74 74 81 79 73 75 76 66 A) 6 B) 4 C) 14 D) 10 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 12) The owner of a major golf course believes that the course is so tough that most golfers rarely break par of 74. The scores from a random sample of 20 golfers are listed below. Test the club owner's hypothesis. Use α = 0.05. 73 71 74 74 77 76 68 80 74 79 71 73 75 75 82 80 74 76 77 67 Answer: critical value 4; test statistic x = 6; fail to reject H 0; There is not sufficient evidence to support the hypothesis. 13) A consumer advocacy group believes a car manufacturer is falsely advertising that its new model car still gets at least 26 miles per gallon of gas. The advocacy group believes the median miles per gallon is less than 26. Ten cars are tested. The results are given below. Test the group's hypothesis. Use α = 0.05. 20.8 18.6 24.8 19.9 23 25.2 28.3 22.9 17.7 24 Answer: critical value 1; test statistic x = 1; reject H0 ; There is sufficient evidence to support the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 14) A government agency believes that the median hourly wages for retail store workers in the central 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. Find the critical values to test the government's hypothesis. Use α = 0.05. A) ±1.96 B) ±1.645 C) ±2.575 D) ±2.33 Answer: A
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15) A government agency believes that the median hourly wages for retail store workers in the central U.S. is $6.25. In a random sample of 100 workers, 68 were paid less than $6.25, 10 were paid $6.25, and the rest more than $ 6.25. Find the test statistic z to test the government's hypothesis. Use α = 0.05. A) -4.743 B) -3.912 C) -3.187 D) -2.386 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 16) A government agency believes that the median hourly wages for retail store workers in the central U.S. is $ 6.10. In a random sample of 100 workers, 68 were paid less than $6.10, 10 were paid $6.10, and the rest more than $6.10. Test the government's hypothesis. Use α = 0.05. Answer: critical values ±1.96; test statistic z ≈ -4.743; reject H0 ; There is sufficient evidence to reject the hypothesis. 17) Test the hypothesis that the median age of music teachers is 41 years. A random sample of 60 music teachers found 25 above 41 years and 35 below 41 years. Use α = 0.01. Answer: critical values ±2.575; test statistic z ≈ -1.162; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 18) A college dean believes that the median hours spent studying by full time students is at least 14 hours per week. In a random sample of 100 students, 64 studied more than 14 hours, 10 studied exactly 14 hours and the rest studied less than 14 hours. Find the critical value to test the dean's hypothesis. Use α = 0.05. A) -1.645 B) -1.96 C) -2.33 D) -2.575 Answer: A 19) A college dean believes that the median hours spent studying by full time students is at least 11 hours per week. In a random sample of 100 students, 60 studied more than 11 hours, 10 studied exactly 11 hours and the rest studied less than 11 hours. Find the test statistic to test the dean's hypothesis. Use α = 0.05. A) -3.057 B) -3.162 C) -3.9 D) -4 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 20) A college dean believes that the median hours spent studying by full time students is at least 15 hours per week. In a random sample of 100 students, 63 studied more than 15 hours, 10 studied exactly 15 hours and the rest studied less than 15 hours. Test the researcher's hypothesis. Use α = 0.05. Answer: critical value -1.645; test statistic z ≈ -3.689; reject H0 ; There is sufficient evidence to reject the hypothesis. 21) A labor organization believes that the monthly earnings of landscaping workers are less than $917. To test this, 100 workers are randomly selected and asked to provide their monthly earnings. The data is shown below. Test the labor organization's hypothesis. Use α = 0.05. Weekly Earnings Number of Workers More than $917 41 $917 5 Less than $917 54 Answer: critical value -1.645; test statistic z ≈ -1.231; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis.
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22) Test the hypothesis that the median age of art teachers is less than 45 years. A random sample of 60 art teachers found 25 above 45 years, 33 below 45 years, and the rest exactly 45 years. Use α = 0.01. Answer: critical value -2.33; test statistic z ≈ -0.919; Fail to reject H0 ; There is not sufficient evidence to support the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 23) A one-sample sign test is a test of the
.
A) median C) mode
B) mean D) standard deviation
Answer: A 24) What is the test statistic, k, if we have a small sample size and the alternative hypothesis of a one-sample sign test is H1 : M < M0 ? A) k is the number of + signs B) k is the number of - signs C) k is the smaller number of either the + or - signs D) k is the larger number of either the + or - signs Answer: A
15.4 Inferences about the Difference between Two Medians: Dependent Samples 1 Test a hypothesis about the difference between the medians of two dependent samples. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) Use the Wilcoxon matched-pairs signed-ranks test to test the given hypothesis at the α = 0.05 level of significance. The dependent samples were obtained randomly. Hypotheses: H0 : MD = 0 versus H1 : MD ≠ 0 with n = 45, T+ = 195, and T- = 840 Answer: z = -3.64; |z| > z 0.025 = 1.96; reject H 0; There is sufficient evidence that the medians are different. 2) Use the Wilcoxon matched-pairs signed-ranks test to test the given hypothesis at the α = 0.05 level of significance. The dependent samples were obtained randomly. Hypotheses: H0 : MD = 0 versus H1 : MD < 0 with n = 30, and T+ = 210 Answer: z = -0.46; |z| ≤ z0.05 = 1.645; do not reject H0 ; There is not sufficient evidence that the median of the first population is less than the median of the second population. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 3) Nine students took a standardized test. Their scores are listed below. Later on, they took a test preparation course and retook the test. Their new scores are listed below. Use the Wilcoxon signed-ranks test to find the test statistic T to test the hypothesis that the test preparation had no effect on their scores. Use α = 0.05. Student 1 2 3 4 5 6 7 8 9 Before Score 940 1,040 1,160 840 1,090 1,200 1,000 1,170 1,050 After Score 960 1,040 1,150 880 1,120 1,210 990 1,210 1,070 A) 4 B) 32 C) 41 D) 2 Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 4) Nine students took a standardized test. Their scores are listed below. Later on, they took a test preparation course and retook the test. Their new scores are listed below. Use the Wilcoxon signed-ranks test to test the hypothesis that the test preparation had no effect on their scores. Use α = 0.05. Student 1 2 3 4 5 6 7 8 9 Before Score 950 810 1,190 970 1,080 1,010 1,050 1,110 1,140 After Score 970 810 1,180 1,010 1,110 1,020 1,040 1,150 1,160 Answer: critical value 4; test statistic T = 4; reject H0 ; There is sufficient evidence to reject the hypothesis. The course has effect. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 5) A football coach believes that a player 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. The results are listed below. Thirty days later, after regular training supplemented by vitamin E, they are tested again. The new results are listed below. Use the Wilcoxon signed-ranks test to find the critical value to test the hypothesis that the vitamin E supplement is effective in increasing the athletes' strength. Use α = 0.05. Athlete 1 2 3 4 5 6 7 8 9 Before 223 252 230 250 251 181 198 227 196 After 233 257 230 248 258 196 203 222 201 A) 6 B) 4 C) 2 D) 3 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 6) A football coach believes that a player 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. The results are listed below. Thirty days later, after regular training supplemented by vitamin E, they are tested again. The new results are listed below. Use the Wilcoxon signed-ranks test to test the hypothesis that the vitamin E supplement is effective in increasing the athletes' strength. Use α = 0.05. Athlete 1 2 3 4 5 6 7 8 9 Before 192 220 251 233 230 262 242 209 232 After 202 225 251 231 237 277 247 204 237 Answer: critical value 5; test statistic T = 4.5; reject H0 ; There is sufficient evidence to support the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 7) A pharmaceutical company wishes to test a new drug with the expectation of weight reduction. Ten subjects are randomly selected and their weights are recorded. The subjects were placed on the drug for a period of 6 months, after which their weights were recorded again. The results are listed below. (All units are pounds.) Use the Wilcoxon signed-ranks test to find the test statistic T to test the company's hypothesis that the drug reduces weight. Use α = 0.05. Subject 1 2 3 4 5 6 7 8 9 10 Before 201 235 177 232 212 258 179 174 194 255 After 186 230 185 222 207 258 149 156 192 240 A) 4 B) 7.5 C) 41 D) 5.5 Answer: A
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8) A pharmaceutical company wishes to test a new drug with the expectation of weight reduction. Ten subjects are randomly selected and their weights are recorded. The subjects were placed on the drug for a period of 6 months, after which their weights were recorded again. The results are listed below. (All units are pounds.) Use the Wilcoxon signed-ranks test to test the company's hypothesis that the drug reduces weight. Use α = 0.05. Subject 1 2 3 4 5 6 7 8 9 10 Before 226 184 251 232 202 245 173 239 228 175 After 211 179 259 222 197 245 143 221 226 160 Answer: critical value 8; test statistic T = 4; reject H0 ; There is sufficient evidence to support the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 9) A physician believes that a person's diastolic blood pressure can be lowered, if, instead of taking a drug, the person performs yoga exercises each evening. Ten subjects are randomly selected. Their blood pressures, measured in millimeters of mercury, are listed below. The 10 patients are instructed in several yoga exercises and told to perform 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-ranks test to find the critical value to test the physician's hypothesis. Use α = 0.05. Patient 1 2 3 4 5 6 7 8 9 10 Before 83 96 98 81 97 84 93 80 94 85 After 80 90 98 73 91 73 96 70 90 69 A) 8 B) 6 C) 2 D) 4 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) A physician believes that a person's diastolic blood pressure can be lowered, if, instead of taking a drug, the person performs yoga exercises each evening. Ten subjects are randomly selected. Their blood pressures, measured in millimeters of mercury, are listed below. The 10 patients are instructed in several yoga exercises and told to perform 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-ranks test to test the physician's hypothesis. Use α = 0.05. Patient 1 2 3 4 5 6 7 8 9 10 Before 89 97 98 96 90 85 91 82 81 87 After 86 91 98 88 84 74 94 72 77 71 Answer: critical value 8; test statistic T = 1.5; reject H0 ; There is sufficient evidence to support the hypothesis. 11) A local company is concerned about the number of days missed by its employees due to illness. A random sample of 10 employees is selected. The numbers of days absent in one year is listed below. An incentive program is offered in an attempt to decrease the number of days absent. The number of days absent in one year after the incentive program is listed below. Use the Wilcoxon signed-ranks test to test the hypothesis that the incentive program cuts down on the number of days missed by employees. Use α = 0.05. Teacher 1 2 3 4 5 6 7 8 9 10 Days Absent Before Incentive 8 6 7 9 7 3 2 9 10 5 Days Absent After Incentive 6 5 7 7 6 1 0 10 8 5 Answer: critical value 5; test statistic T = 2; reject H0 ; There is sufficient evidence to support the hypothesis.
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12) In a study of the effectiveness of dietary restrictions on weight loss, 20 people were randomly selected to participate in a program for 30 days. Use the Wilcoxon signed-ranks test to test the hypothesis that dietary restrictions have no bearing on weight loss. Use α = 0.02. 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 Answer: critical value 40; test statistic T = 42.5; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 13) The Wilcoxon Matched-Pairs Signed-Ranks Test is a test of A) the median of the differences of the matched pairs. B) the mean of the differences of the matched pairs. C) the mode of the differences of the matched pairs. D) ranks of the matched differences. Answer: A
15.5 Inferences about the Difference between Two Medians: Independent Samples 1 Test a hypothesis about the difference between the medians of two independent samples. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the Mann–Whitney test to test the given hypotheses at the α = 0.05 level of significance. The independent samples were obtained randomly. 1) Hypotheses: H0 : Mx = My versus H 1 : Mx ≠ My with n 1 = 10, n 2 = 14, and S = 150. Answer: T = 95; critical values: 37, 103. Do not reject H 0 . 2) Hypotheses: H0 : Mx = My versus H1 : Mx < My with n 1= 13, n 2 = 16, and S = 125. Answer: T = 34; critical value: 66. Reject H0 . 3) Hypotheses: H0 : Mx = My versus H1 : Mx > My with n 1= 14, n 2 = 8, and S = 180. Answer: T = 75; critical value: 80. Do not reject H 0 . 4) Hypotheses: H0 : Mx = My versus H1 : Mx ≠ My with n 1 = 24, n 2 = 21, and S = 750. Answer: T = 450; z 0 = 4.5, -z0.025 = -1.96, z0.025 = 1.96; Reject H 0 .
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Provide an appropriate response. 5) Mathematics SAT scores for students randomly selected from two different schools are listed below. Test the hypothesis that the median scores in the schools are different. Use α = 0.05. School 1 School 2 540 510 760 480 430 670 470 740 520 420 700 580 570 770 600 680 540 520 580 720 740 620 630 530 Answer: critical values: w0.025 = 36 and w0.975 = 108 T=S-
n 1 (n 1 + 1) 2
= 171.5 -
12(13) = 93.5 2
Since w0.025 = 36 < T < w0.975 = 108, we do not reject H0 . There is not sufficient evidence that the median scores differ at the two schools. 6) A researcher wants to know if the time spent in prison for grand larceny was the same for men and women. A random sample of men and women convicted of grand larceny were each asked to give the length of sentence received. The data, in years, are listed below. Test the hypothesis that the median sentence for men is greater than the median sentence for women. Use α = 0.10. Men 29 41 35 37 38 45 Women 28 31 28 33 45 31 Men 33 41 31 38 42 43 Women 53 27 29 32 36 46 Answer: critical value: w0.90 = 94 T=S-
n 1 (n 1 + 1) 2
= 174.5 -
12(13) = 96.5 2
Since T > w0.90 = 94, we reject H0 . There is sufficient evidence that the median grand larceny sentence for men is greater than the median grand larceny sentence for women.
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7) A math teacher wanted to see whether there was a significant difference in commute time between day students and night students. A random sample of 35 students from each group was selected. The data (in minutes) are given below. Test the hypothesis that the median commute time of day students is different than the median commute time of evening students. Use α = 0.05. Day Students 25 27 27 26 22 22 26 25 21 24 24 21 21 28 32 27 26 25 25 24 23 23 23 30 20 22 21 24 23 26 29 33 27 24 28 Evening Students 21 26 28 26 24 24 26 27 23 23 26 22 28 27 30 26 28 27 26 31 23 22 26 27 32 33
30 34 37 23 23 24 23 30 24
Answer: critical values: z = ±1.96 n1n2 T2 (1093.5 - 630) - 612.5 ≈ -1.75 z= = n 1 n 2 (n 1 + n 2 + 1) 86975 12 12 Since |z| < z 0.05 = 1.96, we do not reject H0 . There is not sufficient evidence that the median commute times of day and evening students differ. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 8) The Mann-Whitney Test tests the equality of two population medians from A) independent samples C) mutually exclusive samples
.
B) dependent samples D) corresponding samples
Answer: A 9) True or False: In order to be able to use the Mann-Whitney Test, the samples must be the same size and independent. A) True B) False Answer: A 10) True or False: In order to be able to use the Mann-Whitney Test, the distributions must have the same shape. A) True B) False Answer: A
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15.6 Spearman's Rank-Correlation Test 1 Perform Spearman's rank-correlation test. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Draw a scatter diagram for the given data, compute Spearman's rank correlation coefficient rs, and determine if X and Y are associated at the α = 0.05 level of significance. 1) X 1 2 3 6 6 8 Y 1.2 3.7 5.1 5.8 6.2 6.5 Answer: y 7 6 5 4 3 2 1
1
2
3
4
5
6
7
8
x
rs = 0.986; H 0 : X and Y are not associated versus H0 : X and Y are associated. Critical values: -0.886 and 0.886; reject H 0 . 2) X 0 Y 6.5
2 5.1
3 5.3
3 4.2
6 2.4
2
3
8 2.1
Answer: y 7 6 5 4 3 2 1
1
4
5
6
7
8
x
rs = -0.871; H 0 : X and Y are not associated versus H0 : X and Y are associated. Critical values: -0.886 and 0.886; do not reject H 0 .
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 3) The table below lists the amounts (in dollars) spent on clothing and entertainment over the last year by 10 adults selected at random. Find the critical values to test the hypothesis of no correlation between spending in these two areas. Use α = 0.05. Clothing 395 480 485 390 470 Entertainment 480 550 575 510 560 Clothing 500 Entertainment 525 A) ± 0.648
400 610
450 520 530 400 B) ± 0.564
410 410 C) ± 0.794
D) ± 0.745
Answer: A 4) The table below lists the amounts (in dollars) spent on clothing and entertainment over the last year by 10 adults selected at random. Find the test statistic rs, to test the hypothesis of no correlation between spending in these two areas. Use α = 0.05. Clothing 495 580 Entertainment 580 650 Clothing 600 Entertainment 625 A) -0.006
500 710
585 675
490 610
570 660
550 620 630 500 B) -0.0192
510 510 C) -0.218
D) -0.326
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 5) The table below lists the amounts (in dollars) spent on clothing and entertainment over the last year by 10 adults selected at random. Test the claim of no correlation between spending in these two areas. Use α = 0.05. Clothing 445 530 535 440 520 Entertainment 530 600 625 560 610 Clothing 550 Entertainment 575
450 660
500 580
570 450
460 460
Answer: 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 spending on clothing and spending on entertainment.. 6) The maximum summer temperatures for nine randomly selected cities (in °F) and the number of summer power outages in those cities are listed below. Can you conclude that there is a correlation between the maximum temperatures and the number of power outages? Use α = 0.01. Temp 75 88 94 93 91 101 78 103 83 Outages 6 10 13 13 11 18 7 18 8 Answer: critical values ±0.833; test statistic rs ≈ 0.992; reject H 0 ; There is enough evidence to conclude that there is a significant correlation between the maximum temperature and the number of power outages. 7) A manager wishes to determine the relationship between the average number of new accounts opened by 9 sales representatives (per month) and their sales (in thousands of dollars) per month. Can you conclude that there is a correlation between the new accounts opened and sales generated? Use α = 0.05. Number of New Accounts 7 8 15 12 13 20 8 6 16 Sales 36 38 83 67 70 66 53 60 125 Answer: critical values ±0.700; test statistic rs ≈ 0.728; reject H 0 ; There is enough evidence to conclude that there is significant correlation between new accounts opened and sales generated. Page 14
8) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of soybeans (bushels per acre). Can you conclude that there is a correlation between rainfall and yield per acre? Use α = 0.01. Rainfall 13.5 11.8 16.4 15.5 21.8 13.3 10.0 18.6 19.0 Yield 53.5 49.2 61.8 62.0 85.4 52.2 34.9 79.0 81.8 Answer: critical values ±0.833; test statistic rs ≈ 0.983; reject H 0 ; There is enough evidence to conclude that there is a significant correlation between rainfall and yield per acre. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 9) The history department at a college asked its professors and students to rank 8 guest speakers according to their communication skills. The data are listed below. A 10 is the highest ranking and a 1 the lowest ranking. Find the critical values to test the hypothesis of no correlation between the rankings. Use α = 0.05. Speaker 1 2 3 4 5 6 7 8 Professors 2 3 6 10 8 1 5 4 Students 4 3 1 4 5 7 9 6 A) ±0.738 B) ±0.643 C) ±0.881 D) ±0.833 Answer: A 10) The history department at a college asked its professors and students to rank 8 guest speakers according to their communication skills. The data are listed below. A 10 is the highest ranking and a 1 the lowest ranking. Find the test statistic to test the hypothesis of no correlation between the rankings. Use α = 0.05. Speaker 1 2 3 4 5 6 7 8 Professors 2 3 6 10 8 1 5 4 Students 4 3 1 4 5 7 9 6 A) -0.208 B) -0.198 C) -0.354 D) -0.278 Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 11) The history department at a college asked its professors and students to rank 8 guest speakers according to their communication skills. The data are listed below. A 10 is the highest ranking and a 1 the lowest ranking. Test the hypothesis of no correlation between the rankings. Use α = 0.05. Speaker 1 2 3 4 5 6 7 8 Professors 2 3 6 10 8 1 5 4 Students 4 3 1 4 5 7 9 6 Answer: critical values ±0.738; test statistic rs ≈ -0.208; fail to reject H0 ; There is not enough evidence to conclude that there is a significant correlation between the rankings. 12) The ages and average monthly prescription drug bills (in dollars) of 9 randomly selected adults are given below. Can you conclude that there is a correlation between age and amount spent on prescription drugs? Use α = 0.05. Age 38 41 45 48 51 53 57 61 65 Drug Bill 116 120 123 131 142 145 148 150 152 Answer: critical values ±0.700; test statistic rs ≈ 1.0; reject H0 ; There is enough evidence to conclude that there is a significant correlation between age and amount spent on prescription drugs.
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13) The final exam scores of 10 randomly selected algebra students and the number of hours they studied for the exam are given below. Can you conclude that there is a positive correlation between the scores on the test and the times spend studying? Use α = 0.01. Hours 4 6 3 9 3 5 5 6 7 4 Scores 66 81 61 89 67 79 86 91 91 72 Answer: critical value 0.745; test statistic rs ≈ 0.889; reject H0 ; There is enough evidence to conclude that there is a significant positive correlation between the scores and the time spent studying. 14) The number of absences and the final grades of 9 randomly selected students from a French class are given below. Can you conclude that there is a negative correlation between the final grade and the number of absences? Use α = 0.01. Number of Absences 0 3 6 4 9 2 15 8 5 Final Grade 98 86 80 82 71 92 55 76 82 Answer: critical value -0.783; test statistic rs ≈ -0.996; reject H 0 ; There is enough evidence to conclude that there is a significant negative correlation between the final grade and the number of absences. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 15) Spearman's rank-correlation test is used to test hypotheses regarding _______________. A) the associations between two variables B) the median of a data set C) whether or not the coefficient of the mean is zero in a regression equation D) binomial relationships Answer: A
15.7 Kruskal-Wallis Test 1 Test a hypothesis using the Kruskal-Wallis test. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) The nonparametric test used in place of the one-way ANOVA is the A) Kruskal-Wallis test C) Spearman's rank-correlation test
.
B) Mann-Whitney test D) Runs Test
Answer: A 2) The decision to reject H0 using the Kruskal-Wallis test occurs when A) H is too large Answer: A
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B) H is too small
C) H ≠ K
. D) H ≠ 0
3) A medical researcher wishes to try three different techniques to lower the weight of obese patients. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, the reduction in each subject's weight (in pounds) is recorded. Use the Kruskal-Wallis test to find the critical value to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Group 1 Group 2 Group 3 11 8 6 12 5 12 9 2 4 15 3 8 13 4 9 8 0 4 A) 5.991 B) 4.153 C) 3.195 D) 1.960 Answer: A 4) A medical researcher wishes to try three different techniques to lower the weight of obese patients. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, the reduction in each subject's weight (in pounds) is recorded. Use the Kruskal-Wallis test to find the test statistic H to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Group 1 Group 2 Group 3 16 13 11 17 10 17 14 7 9 20 8 13 18 9 14 13 5 9 A) 10.187 B) 8.312 C) 6.813 D) 5.321 Answer: A 5) Four different watering schedules are used on bean plants. The number of beans on each randomly selected plant is given below. Use the Kruskal-Wallis test to find the test statistic H to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Schedule 1 Schedule 2 Schedule 3 Schedule 4 11 10 11 8 10 13 8 10 11 10 9 8 12 10 8 9 12 10 7 10 11 11 8 9 A) 12.833 B) 10.922 C) 15.364 D) 14.818 Answer: A
Page 17
6) Four different watering schedules are used on bean plants. The number of beans on each randomly selected plant is given below. Use the Kruskal-Wallis test to find the critical value χ 2 to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Schedule 1 Schedule2 Schedule 3 Schedule 4 9 8 9 6 8 11 6 8 9 8 7 6 10 8 6 7 10 8 5 8 9 9 6 7 A) 7.815 B) 7.352 C) 6.531
D) 5.198
Answer: A SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 7) Determine the Kruskall-Wallis test statistic H for the given data and the critical value at the α = 0.05 level of significance. Test whether the distributions of the populations are different. X 6 17 16 12
Y 3 8 10 5
Z 9 5 13 10
Answer: H = 3.846; critical value: 5.6923. Do not reject H0 . 8) A medical researcher wishes to try three different techniques to lower the weight of obese patients. The subjects are randomly selected and assigned to one of three groups. Group 1 is given medication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of six weeks, the reduction in each subject's weight (in pounds) is recorded. Use the Kruskal-Wallis test to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Group 1 Group 2 Group 3 12 9 7 13 6 13 10 3 5 16 4 9 14 5 10 9 1 5 Answer: critical value 5.991; test statistic H ≈ 10.187; reject H0 ; The data provide ample evidence that there is a difference in the distribution of the populations. 9) Four different watering schedules are used on bean plants. The number of beans on each randomly selected plant is given below. Use the Kruskal-Wallis test to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Schedule 1 Schedule 2 Schedule 3 Schedule 4 10 9 10 7 9 12 7 9 10 9 8 7 11 9 7 8 11 9 6 9 10 10 7 8 Answer: critical value 7.815; test statistic H ≈ 12.833; reject H0 ; There is enough evidence to conclude that there is a difference in the distribution of the populations.
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10) A researcher wishes to determine whether there is a difference in the average age of science, language, and art teachers. Teachers are randomly selected. Their ages are recorded below. Use the Kruskal-Wallis test to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Science Language Art Teachers Teachers Teachers 25 38 41 30 43 47 29 40 38 54 49 63 39 44 47 27 33 37 Answer: critical value 5.991; test statistic H ≈ 4.056; fail to reject H 0 ; There is not enough evidence to conclude that there is difference in the distribution of the populations. 11) The grade point averages of students participating in extracurricular activities at a college are to be compared. The data are listed below. Use the Kruskal-Wallis test to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Sports Community Service Religion Club 3.5 2.1 3 2.9 2.4 3.3 2.8 3.6 3.1 3.8 2.2 2.8 3.4 2.6 2.8 2.4 2.3 2.7 Answer: critical value 5.991; test statistic H ≈ 5.108; fail to reject; There is enough evidence to conclude that there is a difference in the distribution of the populations. 12) The time (in minutes) it takes to assemble a component for 3 different television models is listed below. Workers are randomly selected. Use the Kruskal-Wallis test to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. Model 1 Model 2 Model 3 37 45 33 36 34 30 37 43 34 35 38 36 38 40 35 36 37 32 37 41 42 Answer: critical value 5.991; test statistic H ≈ 7.482; reject H0 ; There is enough evidence to conclude that there is a difference in the distribution of the populations.
Page 19
13) A realtor wishes to compare the annual property taxes (in dollars) of houses in 4 different cities, all of which are priced approximately the same. The data are listed below. Use the Kruskal-Wallis test to test the hypothesis that there is no difference in the distribution of the populations. Use α = 0.05. City 1 City 2 City 3 City 4 2,160 1,790 1,540 2,410 1,990 1,550 1,680 2,360 2,010 1,700 1,590 2,610 2,220 1,660 1,610 2,160 1,910 1,710 1,510 2,010 2,060 1,760 2,210 1,660 2,360 2,260
Answer: critical value 7.815; test statistic H ≈ 20.657; reject H0 ; There is enough evidence to conclude that there is a difference in the distribution of the populations.
Page 20
Ch. 15 Nonparametric Statistics Answer Key 15.1 An Overview of Nonparametric Statistics 1 Distinguish between parametric and nonparametric statistical procedures. 1) A 2) A 3) A 4) A
15.2 Runs Test for Randomness 1 Perform a runs test for randomness. 1) A 2) A 3) A 4) A 5) A
15.3 Inferences about Measures of Central Tendency 1 Conduct a one-sample sign test. 1) H0 : M = 12 versus H1 : M < 12; k = 8; critical value: 5. Do not reject H 0 . 2) H0 : M = 60 versus H1 : M > 60; k = 3; critical value: 5. Reject H0 . 3) H0 : M = 84 versus H1 : M ≠ 84; k = 22; z 0 = -2.58; -z0.025 = -1.96. Reject H 0 . 4) A 5) A 6) critical value 4; test statistic x = 8; fail to reject H 0 ; There is not sufficient evidence to reject the hypothesis. 7) A 8) A 9) critical value 1; test statistic x = 6; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 10) A 11) A 12) critical value 4; test statistic x = 6; fail to reject H 0 ; There is not sufficient evidence to support the hypothesis. 13) critical value 1; test statistic x = 1; reject H0 ; There is sufficient evidence to support the hypothesis. 14) A 15) A 16) critical values ±1.96; test statistic z ≈ -4.743; reject H 0; There is sufficient evidence to reject the hypothesis. 17) critical values ±2.575; test statistic z ≈ -1.162; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 18) A 19) A 20) critical value -1.645; test statistic z ≈ -3.689; reject H 0 ; There is sufficient evidence to reject the hypothesis. 21) critical value -1.645; test statistic z ≈ -1.231; fail to reject H0 ; There is not sufficient evidence to support the hypothesis. 22) critical value -2.33; test statistic z ≈ -0.919; Fail to reject H 0; There is not sufficient evidence to support the hypothesis. 23) A 24) A
15.4 Inferences about the Difference between Two Medians: Dependent Samples 1 Test a hypothesis about the difference between the medians of two dependent samples. 1) z = -3.64; |z| > z 0.025 = 1.96; reject H 0 ; There is sufficient evidence that the medians are different. 2) z = -0.46; |z| ≤ z 0.05 = 1.645; do not reject H 0; There is not sufficient evidence that the median of the first population is less than the median of the second population. 3) A 4) critical value 4; test statistic T = 4; reject H 0; There is sufficient evidence to reject the hypothesis. The course has effect. 5) A Page 21
6) critical value 5; test statistic T = 4.5; reject H 0 ; There is sufficient evidence to support the hypothesis. 7) A 8) critical value 8; test statistic T = 4; reject H 0; There is sufficient evidence to support the hypothesis. 9) A 10) critical value 8; test statistic T = 1.5; reject H 0 ; There is sufficient evidence to support the hypothesis. 11) critical value 5; test statistic T = 2; reject H 0; There is sufficient evidence to support the hypothesis. 12) critical value 40; test statistic T = 42.5; fail to reject H0 ; There is not sufficient evidence to reject the hypothesis. 13) A
15.5 Inferences about the Difference between Two Medians: Independent Samples 1 Test a hypothesis about the difference between the medians of two independent samples. 1) T = 95; critical values: 37, 103. Do not reject H0 . 2) T = 34; critical value: 66. Reject H0 . 3) T = 75; critical value: 80. Do not reject H0 . 4) T = 450; z0 = 4.5, -z 0.025 = -1.96, z 0.025 = 1.96; Reject H0 . 5) critical values: w0.025 = 36 and w0.975 = 108 T=S-
n 1 (n 1 + 1) 2
= 171.5 -
12(13) = 93.5 2
Since w0.025 = 36 < T < w0.975 = 108, we do not reject H0 . There is not sufficient evidence that the median scores differ at the two schools. 6) critical value: w0.90 = 94 T=S-
n 1 (n 1 + 1) 2
= 174.5 -
12(13) = 96.5 2
Since T > w0.90 = 94, we reject H0 . There is sufficient evidence that the median grand larceny sentence for men is greater than the median grand larceny sentence for women. 7) critical values: z = ±1.96 n1n2 T2 (1093.5 - 630) - 612.5 ≈ -1.75 z= = n 1 n 2 (n 1 + n 2 + 1) 86975 12 12 Since |z| < z 0.05 = 1.96, we do not reject H 0. There is not sufficient evidence that the median commute times of day and evening students differ. 8) A 9) A 10) A
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15.6 Spearman's Rank-Correlation Test 1 Perform Spearman's rank-correlation test. 1) y 7 6 5 4 3 2 1
1
2
3
4
5
6
7
8
x
rs = 0.986; H0 : X and Y are not associated versus H 0 : X and Y are associated. Critical values: -0.886 and 0.886; reject H0 . 2) y 7 6 5 4 3 2 1
1
2
3
4
5
6
7
8
x
rs = -0.871; H0 : X and Y are not associated versus H 0 : X and Y are associated. Critical values: -0.886 and 0.886; do not reject H0 . 3) A 4) A 5) 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 spending on clothing and spending on entertainment.. 6) 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 maximum temperature and the number of power outages. 7) critical values ±0.700; test statistic rs ≈ 0.728; reject H0 ; There is enough evidence to conclude that there is significant correlation between new accounts opened and sales generated. 8) critical values ±0.833; test statistic rs ≈ 0.983; reject H0 ; There is enough evidence to conclude that there is a significant correlation between rainfall and yield per acre. 9) A 10) A Page 23
11) critical values ±0.738; test statistic rs ≈ -0.208; fail to reject H0 ; There is not enough evidence to conclude that there is a significant correlation between the rankings. 12) critical values ±0.700; test statistic rs ≈ 1.0; reject H0 ; There is enough evidence to conclude that there is a significant correlation between age and amount spent on prescription drugs. 13) critical value 0.745; test statistic rs ≈ 0.889; reject H0 ; There is enough evidence to conclude that there is a significant positive correlation between the scores and the time spent studying. 14) critical value -0.783; test statistic rs ≈ -0.996; reject H0 ; There is enough evidence to conclude that there is a significant negative correlation between the final grade and the number of absences. 15) A
15.7 Kruskal-Wallis Test 1 Test a hypothesis using the Kruskal-Wallis test. 1) A 2) A 3) A 4) A 5) A 6) A 7) H = 3.846; critical value: 5.6923. Do not reject H 0. 8) critical value 5.991; test statistic H ≈ 10.187; reject H0 ; The data provide ample evidence that there is a difference in the distribution of the populations. 9) critical value 7.815; test statistic H ≈ 12.833; reject H0 ; There is enough evidence to conclude that there is a difference in the distribution of the populations. 10) critical value 5.991; test statistic H ≈ 4.056; fail to reject H0 ; There is not enough evidence to conclude that there is difference in the distribution of the populations. 11) critical value 5.991; test statistic H ≈ 5.108; fail to reject; There is enough evidence to conclude that there is a difference in the distribution of the populations. 12) critical value 5.991; test statistic H ≈ 7.482; reject H0 ; There is enough evidence to conclude that there is a difference in the distribution of the populations. 13) critical value 7.815; test statistic H ≈ 20.657; reject H0 ; There is enough evidence to conclude that there is a difference in the distribution of the populations.
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Ch. 16 Appendix B: Lines (Online Only) 16.1 Lines 1 Calculate and interpret the slope of a line. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope of the line through the points and interpret the slope. 1) y 10
5 (2, 1) (0, 0) -10
-5
5
10
x
-5
-10
A)
1 ; for every 2-unit increase in x, y will increase by 1 unit 2
B) 2; for every 1-unit increase in x, y will increase by 2 units 1 C) - ; for every 2-unit increase in x, y will decrease by 1 unit 2 D) -2; for every 1-unit increase in x, y will decrease by 2 units Answer: A 2) y 10
5
(-7, 0) -10
-5
5
10
x
-5 (0, -7) -10
A) -1; for every 1-unit increase in x, y will decrease by 1 unit B) 1; for every 1-unit increase in x, y will increase by 1 unit C) -7; for every 1-unit increase in x, y will increase by -7 units D) 7; for every 1-unit increase in x, y will decrease by -7 units Answer: A
Page 1
3) y 10
5
(4, 0) -10
-5
5
10
x
(0, -4) -5
-10
A) 1; for every 1-unit increase in x, y will increase by 1 unit B) -1; for every 1-unit increase in x, y will decrease by 1 unit C) -4; for every 1-unit increase in x, y will increase by -4 units D) 4; for every 1-unit increase in x, y will decrease by -4 units Answer: A 4) 10
y
5
-10
-5
x
5
(-4, -4) -5 (-2, -5)
-10
A) -
1 ; for every 2-unit increase in x, y will decrease by 1 2
B) -2; for every 1-unit increase in x, y will decrease by 2 units 1 C) ; for every 2-unit increase in x, y will increase by 1 unit 2 D) 2; for every 1-unit increase in x, y will increase by 2 units Answer: A Find the slope of the line containing the two points. 5) (6, -6); (-3, 8) 14 14 A) B) 9 9
C)
9 14
D) -
9 14
C)
6 5
D) -
6 5
Answer: A 6) (6, 0); (0, 5) 5 A) 6 Answer: A Page 2
B)
5 6
7) (1, -5); (-8, -2) 1 A) 3
B)
1 3
C) - 3
D) 3
1 6
D) 0
Answer: A 8) (-4, -3); (-4, 3) A) undefined
C) -
B) 6
Answer: A 9) (-4, 5); (-2, 5) A) 0 Answer: A
Page 3
B) -
1 2
C) 2
D) undefined
2 Graph lines, given a point and the slope. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the line containing the point P and having slope m. 11 1) P = (6, -6); m = 9 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 4
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
2) P = (-2, 0); m = 1 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 5
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
3) P = (-3, 0); m = -2 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 6
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
4) P = (0, 5); m = 2 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 7
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
5) P = (0, 2); m = -
1 3 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 8
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
6) P = (-2, 0); m =
3 4 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 9
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
7) P = (4, 0); m = -
1 2 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 10
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
8) P = (5, 8); m = 0 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 11
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
9) P = (-7, 3); slope undefined y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
5
10
x
5
10
x
D) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
Answer: A The slope and a point on the line are given. Use this information to choose which three points are on the line. 10) Slope 4; point (5, 4) A) (6, 8), (7, 12), (3, -4) B) (9, 5), (13, 6), (-3, 2) C) (4, 8), (3, 12), (7, -4) D) (9, 3), (13, 2), (-3, 6) Answer: A 11) Slope
4 ; point (-6, -5) 3
A) (-3, -1), (-9, -9), (0, 3) C) (-2, -2), (-10, -8), (2, 1) Answer: A Page 12
B) (-3, -9), (-9, -1), (0, -13) D) (-2, -8), (-10, -2), (2, -11)
2 12) Slope - ; point (-4, 6) 5 A) (1, 4), (-9, 8), (6, 2) C) (-2, 1), (-6, 11), (0,-4)
B) (1, 8), (-9, 4), (6, 10) D) (-2, 11), (-6, 1), (0, 16)
Answer: A 13) Slope -2; point (-2, -2) A) (-3, 0), (-4, 2), (0, -6) C) (-1, 0), (0, 2), (0, 2)
B) (-4, -1), (-6, 0), (2, -4) D) (0, -1), (2, 0), (2, 0)
Answer: A 3 Find the equation of a line, given two points. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find an equation of the line shown. 1) 8
y
6 4
(3, 4)
2 (0, 0) -8
-6
-4
-2
2
4
6
8 x
-2 -4 -6 -8
A) y =
4 x 3
B) y = -
4 x 3
C) y =
3 x 4
D) y = -
3 x 4
Answer: A 2) 8
y
6 4 2 (0, 0) -8
-6
-4
-2
2
4
6
8 x
-2 -4
(5, -4)
-6 -8
A) y = Answer: A
Page 13
4 x 5
B) y =
4 x 5
C) y = -
5 x 4
D) y =
5 x 4
3) 8
y (2, 7)
6 4 2 -8
-6
-4
-2
(-4, -2)
2
4
6
8 x
-2 -4 -6 -8
A) y =
3 x+4 2
B) y =
3 10 x2 3
3 4 x+ 2 3
D) y =
6 23 x+ 7 3
D) y = -
C) y = -
3 x+2 2
Answer: A 4) 8
y
6 (-4, 3) 4 2 -8
-6
-4
-2
2
4
6
8 x
-2 -4
(2, -4)
-6 -8
A) y = -
7 5 x6 3
B) y = -
7 5 x+ 6 3
C) y =
6 5 x7 3
Answer: A 4 Identify the slope and y-intercept of a line given its equation. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the slope and the y-intercept of the line. 1) y = -7x + 8 A) slope = -7; y-intercept = 8 C) slope = 7; y-intercept = 8 Answer: A
Page 14
B) slope = 8; y-intercept = -7 D) slope = 8; y-intercept = 7
2)
1 x+y=6 2 B) slope = 6; y-intercept = -
1 2
1 ; y-intercept = 6 2
D) slope = -6; y-intercept =
1 2
A) slope = 4; y-intercept = 1
B) slope = 1; y-intercept = 4
A) slope = C) slope =
1 ; y-intercept = 6 2
Answer: A 3) y = 4x + 1
C) slope =
1 ; y-intercept = - 1 4
D) slope = - 4; y-intercept = - 1
Answer: A 4) x + 12y = 1 A) slope = C) slope =
1 1 ; y-intercept = 12 12
B) slope = 1; y-intercept = 1
1 1 ; y-intercept = 12 12
D) slope = -12; y-intercept = 12
1 ; y-intercept = 3 12
B) slope = -
Answer: A 5) -x + 12y = 36 A) slope =
C) slope = -1; y-intercept = 36
1 ; y-intercept = 3 12
D) slope = 12; y-intercept = -36
Answer: A 6) 19x + 3y = 16 A) slope = -
19 16 ; y-intercept = 3 3
C) slope = 19; y-intercept = 16
B) slope =
19 16 ; y-intercept = 3 3
D) slope =
19 16 ; y-intercept = 3 3
9 6 ; y-intercept = 7 7
Answer: A 7) 9x - 7y = 6 A) slope =
9 6 ; y-intercept = 7 7
B) slope =
C) slope =
7 6 ; y-intercept = 9 9
D) slope = 9; y-intercept = 6
Answer: A 8) x + y = 10 A) slope = -1; y-intercept = 10 C) slope = 0; y-intercept = 10 Answer: A
Page 15
B) slope = 1; y-intercept = 10 D) slope = -1; y-intercept = -10
9) y = 9 A) slope = 0; y-intercept = 9 C) slope = 1; y-intercept = 9
B) slope = 9; y-intercept = 0 D) slope = 0; no y-intercept
Answer: A 10) x = 2 A) slope undefined; no y-intercept C) slope = 2; y-intercept = 0
B) slope = 0; y-intercept = 2 D) slope undefined; y-intercept = 2
Answer: A 11) 2x - 4y = 0 A) slope =
1 ; y-intercept = 0 2
B) slope = -
1 ; y-intercept = 0 2
C) slope =
1 ; y-intercept = 4 2
D) slope = -
1 x; y-intercept = -4 2
Answer: A
Page 16
Graph the equation using the slope and the y-intercept. 12) y = 3x + 4 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 17
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
13) y = -3x + 4 10
y
5
-10
-5
x
5
-5
-10
A)
B) 10
y
10
5
-10
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C)
5
x
5
10 x
D) 10
y
10
-10
Answer: A
-5
y
5
5
Page 18
y
5
x
-10
-5
-5
-5
-10
-10
14)
1 y=x+1 3 10
y
5
-10
-5
x
5
-5
-10
A)
B) 10
y
10
5
-10
5
-5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D) 10
y
10
5
-10
Answer: A
Page 19
y
-5
y
5
5
x
-10
-5
-5
-5
-10
-10
15) y =
2 x-3 3 y 10
5
-10
-5
5
10
x
-5
-10
A)
B) y
-10
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
C)
10
x
5
10
x
D) y
-10
Answer: A
Page 20
5
y
10
10
5
5
-5
5
10
x
-10
-5
-5
-5
-10
-10
16) x - 2y = 8 10
y
5
-10
-5
x
5
-5
-10
A)
B) 10
y
10
5
-10
5
-5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D) 10
y
10
5
-10
Answer: A
Page 21
y
-5
y
5
5
x
-10
-5
-5
-5
-10
-10
17) 5x - 10y = 20 10
y
5
-10
-5
10 x
5
-5
-10
A)
B) 10
y
10
5
5
-10
-10
y
-5
5
-5
5
x
5
10 x
10 x -5
-5
-10
-10
C)
D) y
y
5
-10
-5
5
-5
Answer: A
Page 22
5
10 x
-10
-5
-5
18) x = 4 10
y
5
-10
-5
x
5
-5
-10
A)
B) 10
y
10
5
-10
5
-5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D) 10
y
10
5
-10
Answer: A
Page 23
y
-5
y
5
5
x
-10
-5
-5
-5
-10
-10
19) y = 4 10
y
5
-10
-5
x
5
-5
-10
A)
B) 10
y
10
5
-10
5
-5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D) 10
y
10
5
-10
Answer: A
Page 24
y
-5
y
5
5
x
-10
-5
-5
-5
-10
-10
20) x - y = 1 10
y
5
-10
-5
x
5
-5
-10
A)
B) 10
y
10
5
-10
5
-5
5
x
-10
-5
-5
-5
-10
-10
C)
5
x
5
x
D) 10
y
10
5
-10
Answer: A
Page 25
y
-5
y
5
5
x
-10
-5
-5
-5
-10
-10
21) 3x - 9y = 0 10
y
5
-10
-5
10 x
5
-5
-10
A)
B) 10
y
5
-10
5
-5
5
x
-10
-5
-5
5
10 x
-5
-10
-10
C)
D) 10
10
y
-10
Answer: A
-5
y
5
5
Page 26
y
10
5
10 x
-10
-5
5
-5
-5
-10
-10
x
Ch. 16 Appendix B: Lines (Online Only) Answer Key 16.1 Lines 1 Calculate and interpret the slope of a line. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 2 Graph lines, given a point and the slope. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 3 Find the equation of a line, given two points. 1) A 2) A 3) A 4) A 4 Identify the slope and y-intercept of a line given its equation. 1) A 2) A 3) A 4) A 5) A 6) A 7) A 8) A 9) A 10) A 11) A 12) A 13) A 14) A 15) A 16) A 17) A 18) A 19) A 20) A Page 27
21) A
Page 28