Essentials of Modern Business Statistics with Microsoft Excel 8th Edition Test Bank by Ace Test Banks

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Chapter 09: Hypothesis Tests Multiple Choice 1. More evidence against H0 is indicated by _____. a. lower levels of significance b. smaller p-values c. smaller critical values d. lower probabilities of a Type II error ANSWER: b 2. Two approaches to drawing a conclusion in a hypothesis test are _____. a. p-value and critical value b. one-tailed and two-tailed c. Type I and Type II d. null and alternative ANSWER: a 3. As a general guideline, the research hypothesis should be stated as the _____. a. null hypothesis b. alternative hypothesis c. tentative assumption d. hypothesis the researcher wants to disprove ANSWER: b 4. A Type I error is committed when _____. a. a true alternative hypothesis is not accepted b. a true null hypothesis is rejected c. the critical value is greater than the value of the test statistic d. sample data contradict the null hypothesis ANSWER: b 5. Which of the following hypotheses is not a valid null hypothesis? a. H0: µ ≤ 0 b. H0: µ ≥ 0 c. H0: µ = 0 d. H0: µ < 0 ANSWER: d 6. The practice of concluding “do not reject H0” is preferred over “accept H0” when we _____. a. are conducting a one-tailed test b. are testing the validity of a claim c. have an insufficient sample size d. have not controlled for the Type II error Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests ANSWER: d 7. If the cost of a Type I error is high, a smaller value should be chosen for the _____. a. critical value b. confidence coefficient c. level of significance d. test statistic ANSWER: c 8. When the rejection region is in the lower tail of the sampling distribution, the p-value is the area under the curve _____. a. less than or equal to the critical value b. less than or equal to the test statistic c. greater than or equal to the critical value d. greater than or equal to the test statistic ANSWER: b 9. In tests about a population proportion, p0 represents the _____. a. hypothesized population proportion b. observed sample proportion c. observed p-value d. probability of ANSWER: a 10. Which of the following is an improper form of the null and alternative hypotheses? a.

and

ANSWER: c 11. For a two-tailed hypothesis test about μ, we can use any of the following approaches EXCEPT compare the _____ to the _____. a. confidence interval estimate of μ; hypothesized value of μ b. p-value; value of α c. value of the test statistic; critical value d. level of significance; confidence coefficient ANSWER: d 12. An example of statistical inference is _____. a. a population mean b. descriptive statistics Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests c. calculating the size of a sample d. hypothesis testing ANSWER: d 13. In hypothesis testing, the hypothesis tentatively assumed to be true is _____. a. the alternative hypothesis b. the null hypothesis c. either the null or the alternative hypothesis, depending on the sample size d. always proven to be true ANSWER: b 14. In hypothesis testing, the alternative hypothesis is _____. a. the hypothesis tentatively assumed true in the hypothesis-testing procedure b. the hypothesis concluded to be true if the null hypothesis is rejected c. the maximum probability of a Type I error d. the maximum probability of a Type II error ANSWER: b 15. Your investment executive claims that the average yearly rate of return on the stocks she recommends is at least 10.0%. You plan on taking a sample to test her claim. The correct set of hypotheses is _____. a. H0: μ < 10.0% Ha: μ ≥ 10.0% b. H0: μ ≤ 10.0% Ha: μ > 10.0% c. H0: μ > 10.0% Ha: μ ≤ 10.0% d. H0: μ ≥ 10.0% Ha: μ < 10.0% ANSWER: d 16. A meteorologist stated that the average temperature during July in Chattanooga was 80 degrees. A sample of July temperatures over a 32-year period was taken. The correct set of hypotheses is _____. a. H0: μ < 80 Ha: μ ≤ 80 b. H0: μ ≤ 80 Ha: μ > 80 c. H0: μ ≠ 80 Ha: μ = 80 d. H0: μ = 80 Ha: μ ≠ 80 ANSWER: d 17. A student believes that the average grade on the final examination in statistics is at least 85. She plans on taking a sample to test her belief. The correct set of hypotheses is _____. a. H0: μ < 85 Ha: μ ≥ 85 b. H0: μ ≤ 85 Ha: μ > 85 c. H0: μ ≥ 85 Ha: μ < 85 d. H0: μ > 85 Ha: μ ≤ 85 ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests 18. The average life expectancy of tires produced by Whitney Tire Company has been 40,000 miles. Management believes that due to a new production process, the life expectancy of its tires has increased. In order to test the validity of this belief, the correct set of hypotheses is _____. a. H0: μ < 40,000 Ha: μ ≥ 40,000 b. H0: μ ≤ 40,000 Ha: μ > 40,000 c. H0: μ > 40,000 Ha: μ ≤ 40,000 d. H0: μ ≥ 40,000 Ha: μ < 40,000 ANSWER: b 19. A soft drink filling machine, when in perfect adjustment, fills the bottles with 12 ounces of soft drink. Any overfilling or underfilling results in the shutdown and readjustment of the machine. To determine whether or not the machine is properly adjusted, the correct set of hypotheses is _____. a. H0: μ < 12 Ha: μ ≤ 12 b. H0: μ ≤ 12 Ha: μ > 12 c. H0: μ ≠ 12 Ha: μ = 12 d. H0: μ = 12 Ha: μ ≠ 12 ANSWER: d 20. The manager of an automobile dealership is considering a new bonus plan in order to increase sales. Currently, the mean sales rate per salesperson is five automobiles per month. The correct set of hypotheses for testing the effect of the bonus plan is _____. a. H0: μ < 5 Ha: μ ≤ 5 b. H0: μ ≤ 5 Ha: μ > 5 c. H0: μ > 5 Ha: μ ≤ 5 d. H0: μ ≥ 5 Ha: μ < 5 ANSWER: b 21. In hypothesis testing if the null hypothesis is rejected, _____. a. no conclusions can be drawn from the test b. the alternative hypothesis must also be rejected c. the data must have been collected incorrectly d. the evidence supports the alternative hypothesis ANSWER: d 22. If a hypothesis test leads to the rejection of the null hypothesis, a _____. a. Type II error must have been committed b. Type II error may have been committed c. Type I error must have been committed d. Type I error may have been committed ANSWER: d 23. The error of rejecting a true null hypothesis is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests a. a Type I error b. a Type II error c. committed when the sample size is too small d. committed when not enough information is available ANSWER: a 24. In hypothesis testing, if the null hypothesis has been rejected when the alternative hypothesis has been true, _____. a. a Type I error has been committed b. a Type II error has been committed c. the sample size is too small d. the correct decision has been made ANSWER: d 25. A Type II error is committed when _____. a. a true alternative hypothesis is mistakenly rejected b. a true null hypothesis is mistakenly rejected c. the sample size has been too small d. not enough information has been available ANSWER: a 26. The probability of making a Type I error is denoted by _____. a. α b. β c. 1 − α d. 1 − β ANSWER: a 27. The level of significance is the _____. a. maximum allowable probability of a Type II error b. maximum allowable probability of a Type I error c. same as the confidence coefficient d. same as the p-value ANSWER: b 28. The level of significance in hypothesis testing is the probability of _____. a. accepting a true null hypothesis b. accepting a false null hypothesis c. rejecting a true null hypothesis d. rejecting a false null hypothesis ANSWER: c 29. In the hypothesis testing procedure, α is _____. a. the level of significance b. the critical value Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests c. the confidence level d. 1 − level of significance ANSWER: a 30. The level of significance is symbolized by _____. a.  b. ß c. σ d. p ANSWER: a 31. In hypothesis testing, the critical value is _____. a. a number that establishes the boundary of the rejection region b. the probability of a Type I error c. the probability of a Type II error d. the same as the p-value ANSWER: a 32. A one-tailed test is a hypothesis test in which rejection region is _____. a. in both tails of the sampling distribution b. in one tail of the sampling distribution c. only in the lower tail of the sampling distribution d. only in the upper tail of the sampling distribution ANSWER: b 33. A two-tailed test is a hypothesis test in which the rejection region is _____. a. in both tails of the sampling distribution b. in one tail of the sampling distribution c. only in the lower tail of the sampling distribution d. only in the upper tail of the sampling distribution ANSWER: a 34. A two-tailed test at a .0694 level of significance has z values of _____. a. –1.96 and 1.96 b. –1.48 and 1.48 c. –1.09 and 1.09 d. –.86 and .86 ANSWER: b 35. A one-tailed test (lower tail) at a .063 level of significance has a z value of _____. a. –1.86 b. –1.53 c. –1.96 d. –1.645 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests ANSWER: b 36. A one-tailed test (upper tail) at a .123 level of significance has a z value of _____. a. 1.54 b. 1.96 c. 1.645 d. 1.16 ANSWER: d 37. When the hypotheses H0: µ ≥ 100 and Ha: µ < 100 are being tested at a level of significance of α, the null hypothesis will be rejected if the test statistic z is _____. a. ≥ zα b. > –zα c. ≤ –zα d. < 100 ANSWER: c 38. In order to test the hypotheses H0: μ ≤ 100 and Ha: μ > 100 at an α level of significance, the null hypothesis will be rejected if the test statistic z is _____. a. ≥ zα b. < zα c. ≤ –zα d. < 100 ANSWER: a 39. For a one-tailed test (upper tail) with a sample size of 900, the null hypothesis will be rejected at the .05 level of significance if the test statistic is _____. a. less than or equal to –1.645 b. greater than or equal to 1.645 c. less than 1.645 d. less than –1.96 ANSWER: b 40. For a two-tailed test with a sample size of 40, the null hypothesis will NOT be rejected at a 5% level of significance if the test statistic is _____. a. between –1.96 and 1.96, exclusively b. greater than 1.96 c. less than 1.645 d. greater than –1.645 ANSWER: a 41. If a hypothesis is rejected at a 5% level of significance, it _____. a. will always be rejected at the 1% level b. will always be accepted at the 1% level Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests c. will never be tested at the 1% level d. may be rejected or not rejected at the 1% level ANSWER: d 42. If a hypothesis is not rejected at a 5% level of significance, it _____. a. will also not be rejected at the 1% level b. will always be rejected at the 1% level c. will sometimes be rejected at the 1% level d. may be rejected or not rejected at the 1% level ANSWER: a 43. A p-value is the _____. a. probability, when the null hypothesis is true, of obtaining a sample result that is at least as unlikely as what is observed b. value of the test statistic c. probability of a Type II error d. probability corresponding to the critical value(s) in a hypothesis test ANSWER: a 44. Which of the following does NOT need to be known in order to compute the p-value? a. knowledge of whether the test is one-tailed or two-tailed b. the value of the test statistic c. the level of significance d. All three pieces of information above are needed to compute a p-value. ANSWER: c 45. When the p-value is used for hypothesis testing, the null hypothesis is rejected if _____. a. p-value ≤ α b. α < p-value c. p-value > α d. p-value = z ANSWER: a 46. A two-tailed test is performed at a 5% level of significance. The p-value is determined to be .09. The null hypothesis _____. a. must be rejected b. should not be rejected c. may or may not be rejected, depending on the sample size d. has been designed incorrectly ANSWER: b 47. Excel's _____ function can be used to calculate a p-value for a hypothesis test. a. RAND b. NORM.S.DIST Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests c. NORM.S.INV d. COUNTIF ANSWER: b 48. When using Excel to calculate a p-value for an upper-tail hypothesis test, which of the following must be used? a. RAND b. 1 − NORM.S.DIST c. NORM.S.DIST d. COUNTIF ANSWER: b 49. When using Excel to calculate a p-value for a lower-tail hypothesis test, which of the following must be used? a. RAND b. 1 − NORM.S.DIST c. NORM.S.DIST d. COUNTIF ANSWER: c 50. For a sample size of 30, changing from using the standard normal distribution to using the t distribution in a hypothesis test _____. a. will result in the rejection region being smaller b. will result in the rejection region being larger c. would have no effect on the rejection region d. will result in the accepting region being smaller ANSWER: a 51. For a two-tailed test with a sample size of 20 and a .20 level of significance, the t value is _____. a. 1.328 b. 2.539 c. 1.325 d. 2.528 ANSWER: a 52. For a one-tailed test (upper tail) with a sample size of 18 and a .05 level of significance, the t value is _____. a. 2.12 b. 1.734 c. –1.740 d. 1.740 ANSWER: d 53. For a one-tailed test (lower tail) with a sample size of 10 and a .10 level of significance, the t value is _____. a. 1.383 b. –1.372 c. –1.383 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests d. –2.821 ANSWER: c 54. Excel's _____ function can be used to calculate a p-value for a hypothesis test when σ is unknown. a. RAND b. T.DIST c. NORM.S.DIST d. COUNTIF ANSWER: b 55. The school's newspaper reported that the proportion of students majoring in business is at least 30%. You plan on taking a sample to test the newspaper's claim. The correct set of hypotheses is _____. a. H0: p < .30 Ha: p ≥ .30 b. H0: p ≤ .30 Ha: p > .30 c. H0: p ≥ .30 Ha: p < .30 d. H0: p > .30 Ha: p ≤ .30 ANSWER: c 56. In the past, 75% of the tourists who visited Chattanooga went to see Rock City. The management of Rock City recently undertook an extensive promotional campaign. They are interested in determining whether the promotional campaign actually INCREASED the proportion of tourists visiting Rock City. The correct set of hypotheses is _____. a. H0: p > .75 Ha: p ≤ .75 b. H0: p < .75 Ha: p ≥ .75 c. H0: p ≥ .75 Ha: p < .75 d. H0: p ≤ .75 Ha: p > .75 ANSWER: d 57. The academic planner of a university thinks that at least 35% of the entire student body attends summer school. The correct set of hypotheses to test his belief is _____. a. H0: p > .35 Ha: p ≥ .35 b. H0: p ≤ .35 Ha: p > .35 c. H0: p ≥ .35 Ha: p < .35 d. H0: p > .35 Ha: p ≤ .35 ANSWER: c 58. Which Excel function would NOT be appropriate to use when conducting a hypothesis test for a population proportion? a. NORM.S.DIST b. COUNTIF c. STDEV d. Excel cannot be used to conduct a hypothesis test of a proportion. Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests ANSWER: c Exhibit 9-1

59. Refer to Exhibit 9-1. The test statistic equals _____. a. 2.3 b. .38 c. –2.3 d. –.38 ANSWER: a 60. Refer to Exhibit 9-1. The p-value is _____. a. .5107 b. .0214 c. .0107 d. 2.1 ANSWER: c 61. Refer to Exhibit 9-1. If the test is done at a .05 level of significance, the null hypothesis should _____. a. not be rejected b. be rejected c. be tested again d. Not enough information is given to answer this question. ANSWER: b Exhibit 9-2 The manager of a grocery store has taken a random sample of 100 customers. The average length of time it took the customers in the sample to check out was 3.1 minutes. The population standard deviation is known to be 0.5 minute. We want to test to determine whether or not the mean waiting time of all customers is significantly more than 3 minutes. 62. Refer to Exhibit 9-2. The test statistic is _____. a. 1.96 b. 1.64 c. 2.00 d. .056 ANSWER: c 63. Refer to Exhibit 9-2. The p-value is _____. a. .025 b. .0456 c. .05 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests d. .0228 ANSWER: d 64. Refer to Exhibit 9-2. At a .05 level of significance, it can be concluded that the mean of the population is _____. a. significantly greater than 3 b. not significantly greater than 3 c. significantly less than 3 d. significantly greater than 3.18 ANSWER: a Exhibit 9-3 n = 49 = 54.8

H0: μ = 50 Ha: μ ≠ 50

σ = 28 65. Refer to Exhibit 9-3. The test statistic equals _____. a. .1714 b. .3849 c. –1.2 d. 1.2 ANSWER: d 66. Refer to Exhibit 9-3. The p-value is equal to _____. a. .1151 b. .3849 c. .2698 d. .2302 ANSWER: d 67. Refer to Exhibit 9-3. If the test is done at a 5% level of significance, the null hypothesis should _____. a. not be rejected b. be rejected c. be tested again d. Not enough information is given to answer this question. ANSWER: a Exhibit 9-4 A random sample of 16 students selected from the student body of a large university had an average age of 25 years. We want to determine if the average age of all the students at the university is significantly different from 24. Assume the distribution of the population of ages is normal with a standard deviation of 2 years. 68. Refer to Exhibit 9-4. The test statistic is _____. a. 1.96 b. 2.00 c. 1.645 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests d. .05 ANSWER: b 69. Refer to Exhibit 9-4. At a .05 level of significance, it can be concluded that the mean age is _____. a. not significantly different from 24 b. significantly different from 24 c. significantly less than 24 d. significantly less than 25 ANSWER: a Exhibit 9-5 n = 16 = 75.607

H0: μ ≥ 80 Ha: μ < 80

σ = 8.246 Assume the population is normally distributed. 70. Refer to Exhibit 9-5. The test statistic equals _____. a. –2.131 b. –.53 c. .53 d. 2.131 ANSWER: a 71. Refer to Exhibit 9-5. The p-value is equal to _____. a. –.0166 b. .0166 c. .0332 d. .9834 ANSWER: b 72. Refer to Exhibit 9-5. If the test is done at a 2% level of significance, the null hypothesis should _____. a. not be rejected b. be rejected c. be tested again d. Not enough information is given to answer this question. ANSWER: a Exhibit 9-6 A random sample of 100 people was taken. Eighty of the people in the sample favored Candidate A. We are interested in determining whether or not the proportion of the population in favor of Candidate A is significantly more than 75%. 73. Refer to Exhibit 9-6. The test statistic is _____. a. .80 b. .05 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests c. 1.25 d. 2.00 ANSWER: c 74. Refer to Exhibit 9-6. The p-value is _____. a. .2112 b. .05 c. .025 d. .1251 ANSWER: d 75. Refer to Exhibit 9-6. At a .05 level of significance, it can be concluded that the proportion of the population in favor of candidate A is _____. a. significantly greater than 75% b. not significantly greater than 75% c. significantly greater than 80% d. not significantly greater than 80% ANSWER: b 76. For a two-tailed hypothesis test about a population mean, the null hypothesis can be rejected if the confidence interval _____. a. is symmetric b. is non-symmetric c. includes µ0 d. does not include µ0 ANSWER: d 77. In a two-tailed hypothesis test, the null hypothesis should be rejected if the p-value is _____. a. less than or equal to  b. less than or equal to 2 c. greater than or equal to  d. greater than or equal to 2 ANSWER: a 78. If a hypothesis test has a Type I error probability of .05, that means if the null hypothesis is _____. a. false, it will not be rejected 5% of the time b. false, it will be rejected 5% of the time c. true, it will not be rejected 5% of the time d. true, it will be rejected 5% of the time ANSWER: d 79. For a two-tailed hypothesis test with a test statistic value of z = 2.05, the p-value is _____. a. .0101 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests b. .0202 c. .0404 d. .4899 ANSWER: c 80. The rejection region for a one-tailed hypothesis test _____. a. has an area of 1 – ß b. has an area equal to the confidence coefficient c. is in the tail that supports the null hypothesis d. is in the tail that supports the alternative hypothesis ANSWER: d 81. The smaller the p-value, the _____. a. greater the evidence against H0 b. greater the chance of committing a Type II error c. greater the chance of committing a Type I error d. less likely one will reject H0 ANSWER: a Subjective Short Answer 82. A researcher is testing a new painkiller that claims to relieve pain in less than 15 minutes, on average. a. State the hypotheses associated with the researcher's test. b. Describe a Type I error for this situation. c. Describe a Type II error for this situation. ANSWER: a. H0: μ ≥ 15 b. c.

Ha: μ < 15 A Type I error for this situation would be to incorrectly conclude that the average pain relief time is less than 15 minutes. A Type II error for this situation would be to fail to conclude that the average relief time is less than 15 minutes when the average relief time actually is less than 15 minutes.

83. At a certain manufacturing plant, a machine produced ball bearings that should have a diameter of 0.50 mm. If the machine produces ball bearings that are either too small or too large, the ball bearings must be scrapped. Every hour, a quality control manager takes a random sample of 30 ball bearings to test to see if the process is "out of control" (i.e., to test to see if the average diameter differs from 0.50 mm). a. State the hypotheses associated with the manager's test. b. Describe a Type I error for this situation. c. Describe a Type II error for this situation. ANSWER: a. H0: μ = .50 b. c.

Ha: μ ≠ .50 A Type I error for this situation would be to incorrectly conclude that the process is out of control. A Type II error for this situation would be to fail to conclude that an out of control process is

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Chapter 09: Hypothesis Tests out of control.

84. A fast-food restaurant is considering a promotion that will offer customers to purchase a toy featuring a cartoon movie character. If more than 20% of the customers purchase the toy, the promotion will be profitable. A sample of 50 restaurants is used to test the promotion. a. State the hypotheses associated with the restaurant's test. b. Describe a Type I error for this situation. c. Describe a Type II error for this situation. ANSWER: a. H0: p ≤ .20 b. c.

Ha: p > .20 A Type I error for this situation would be to incorrectly conclude that the promotion would be profitable. A Type II error for this situation would be to conclude that the promotion would not be profitable when more than 20% of customers would purchase the toy.

85. The average gasoline price of one of the major oil companies has been $1.00 per gallon. Because of shortages in production of crude oil, it is believed that there has been a significant INCREASE in the average price. In order to test this belief, we randomly selected a sample of 36 of the company's gas stations and determined that the average price for the stations in the sample was $1.10. Assume that the standard deviation of the population (σ) is $0.12. a. State the null and alternative hypotheses. b. Test the claim at  = .05. c. What is the p-value associated with the above sample results? ANSWER: a. b. c.

H0: μ ≤ 1 Ha: μ > 1 z = 5; therefore, reject H0. There is sufficient evidence at  = .05 to conclude that there has been an increase in the average price. almost zero

86. "D" size batteries produced by MNM Corporation have had a life expectancy of 87 hours. Because of an improved production process, the company believes that there has been an INCREASE in the life expectancy of its D size batteries. A sample of 36 batteries showed an average life of 88.5 hours. Assume from past information that it is known that the standard deviation of the population is 9 hours. a. Use a .01 level of significance, and test to determine if there has been an increase in the life expectancy of the D size batteries. b. What is the p-value associated with the sample results? What is your conclusion, based on the p-value? ANSWER: a.

H0: μ ≤ 87 Ha: μ > 87 z = 1; therefore do not reject H0. There is not sufficient evidence at  = .01 to conclude that there has been an increase in the life expectancy in the D size batteries. p-value = 0.1587; therefore, do not reject H0 (same conclusion as part a)

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Chapter 09: Hypothesis Tests

87. At a local university, a sample of 49 evening students was selected in order to determine whether the average age of the evening students is significantly different from 21. The average age of the students in the sample was 23 years. The population standard deviation is known to be 3.5 years. Determine whether or not the average age of the evening students is significantly different from 21. Use a .1 level of significance. ANSWER: H0: μ = 21 Ha: μ ≠ 21 z = 4; therefore, reject H0. There is sufficient evidence at a = .1 to conclude that the average age of the evening students is significantly different from 21. 88. In order to determine the average price of hotel rooms in Atlanta, a sample of 64 hotels was selected. It was determined that the average price of the rooms in the sample was $112. The population standard deviation is known to be $16. Use a .05 level of significance and determine whether or not the average room price is significantly different from $108.50. ANSWER: H0: μ = 108.50 Ha: μ ≠ 108.50 z = 1.75; therefore, do not reject H0. There is not sufficient evidence at  = .05 to conclude that the average room price is significantly different from $108.50. 89. A sample of 81 account balances of a credit company showed an average balance of $1,200. The population standard deviation is $126. You want to determine if the mean of all account balances is significantly different from $1,150. Use a .05 level of significance. ANSWER: H0: μ = 1150 Ha: μ ≠ 1150 z = 3.57; therefore, reject H0. There is sufficient evidence at  = .05 to conclude that the mean of all account balances is significantly different from $1,150. 90. A lathe is set to cut bars of steel into lengths of 6 cm. The lathe is considered to be in perfect adjustment if the average length of the bars it cuts is 6 cm. A sample of 121 bars is selected randomly and measured. It is determined that the average length of the bars in the sample is 6.08 cm. The population standard deviation is 0.44 cm. Determine whether or not the lathe is in perfect adjustment. Use a .05 level of significance. ANSWER: H0: μ = 6 Ha: μ ≠ 6 z = 2; therefore, reject H0. There is sufficient evidence at  = .05 to conclude that the lathe is NOT in perfect adjustment. 91. Bastien, Inc. has been manufacturing small automobiles that have averaged 50 miles per gallon of gasoline in highway driving. The company has developed a more efficient engine for its small cars and now advertises that its new small cars average more than 50 miles per gallon in highway driving. An independent testing service road-tested 36 of the automobiles. The sample showed an average of 51.5 miles per gallon. The population standard deviation is 6 miles per gallon. a. With a .05 level of significance, test to determine whether or not the manufacturer's advertising campaign is legitimate. b. What is the p-value associated with the sample results? ANSWER: a.

H0: μ ≤ 50 Ha: μ > 50 z = 1.5, therefore, do not reject H0. There is not sufficient evidence at  = .05 to conclude that the new cars average more than 50 miles per gallon. .0668

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Chapter 09: Hypothesis Tests

92. A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. Assume the population standard deviation is known to be 5.6 days. a. State the null and alternative hypotheses. b. Using a critical value, test the null hypothesis at the 5% level of significance. c. Using a p-value, test the hypothesis at the 5% level of significance. d. What type of error may have been committed for this hypothesis test? ANSWER: a. H0: μ ≤ 15 b. c. d.

Ha: μ > 15 Do not reject H0, 1.5 < 1.645. There is not sufficient evidence at  = .05 to conclude that the average delivery time is more than what is advertised Do not reject H0, .0668 > 0.05 (same conclusion as part b) A Type II error may have been committed since we did not reject H0.

93. A student believes that the average grade on the statistics final examination is 87. A sample of 36 final examinations is taken. The average grade in the sample is 83.96. The population variance is 144. a. State the null and alternative hypotheses. b. Using a critical value, test the hypothesis at the 5% level of significance. c. Using a p-value, test the hypothesis at the 5% level of significance. d. Using a confidence interval, test the hypothesis at the 5% level of significance. e. Compute the probability of a Type II error if the average grade on the final is 85. ANSWER: a. H0: μ = 87 b. c. d. e.

Ha: μ ≠ 87 Do not reject H0, –1.96 < –1.52 < 1.96. There is not sufficient evidence at α = .05 to conclude that the average statistics final exam grade differs from 87. Do not reject H0, 0.1286 > 0.05 (same conclusion as part b) 80.04 to 87.88; do not reject H0 (same conclusion as part b) .8315

94. A carpet company advertises that it will deliver your carpet within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days. The population standard deviation is 5.6 days. a. State the null and alternative hypotheses. b. Using a critical value, test the null hypothesis at the 5% level of significance. c. Using a p-value, test the hypothesis at the 5% level of significance. Compute the probability of a Type II error if the true average delivery time is 17 days after d. purchase. ANSWER: a. H0: μ ≤ 15 b. c. d.

Ha: μ > 15 Do not reject H0, 1.5 < 1.645 Do not reject H0, .0668 > .05 .1949

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Chapter 09: Hypothesis Tests

95. The sponsors of televisions shows targeted at the market of 5- to 8-year olds want to test the hypothesis that children watch television AT MOST 20 hours per week. The population of viewing hours per week is known to be normally distributed with a standard deviation of 6 hours. A market research firm conducted a random sample of 30 children in this age group. The resulting data follow: 19.5 29.7 17.5 10.4 19.4 18.4 14.6 10.1 12.5 18.2 19.1 30.9 22.2 19.8 11.8 19.0 27.7 25.3 27.4 26.5 16.1 21.7 20.6 32.9 27.0 15.6 17.1 19.2 20.1 17.7 At a .10 level of significance, use Excel to test the sponsors' hypothesis. ANSWER: A B C D Values for D 1 Hours Sample Size =COUNT(A2:A31) 30 2 19.5 Sample Mean =AVERAGE(A2:A31) 20.266667 3 14.6 4 22.2 Popul. Std. Dev. 6 6 5 27.4 Hypothesized Value 20 20 6 27.0 7 29.7 Standard Error =D4/SQRT(D1) 1.095445 8 10.1 Test Statistic z =(D2-D5)/D7 0.243432 9 19.8 10 26.5 p-value (Lower Tail) =NORM.S.DIST(D8,TRUE) 0.596165 11 15.6 p-value (Upper Tail) =1-D10 0.403835 12 17.5 p-value (Two Tail) =2*(MIN(D10,D11)) 0.807670 13 12.5 Do not reject H0, .403835 > .10, cannot conclude that children watch TV more than 20hrs/week at .10 level of significance 96. At a certain manufacturing plant, a machine produces ball bearings that should have a diameter of 0.500 mm. If the machine produces ball bearings that are either too small or too large, the ball bearings must be scrapped. Every hour, a quality control manager takes a random sample of 36 ball bearings to test to see if the process is "out of control" (i.e., to test to see if the average diameter differs from 0.500 mm). Assume that the process is maintaining the desired standard deviation of 0.06 mm. The results from the latest sample follow: 0.468 0.521 0.421 0.476 0.448 0.346 0.452 0.513 0.465 0.395 0.558 0.526 0.354 0.474 0.447 0.405 0.411 0.453 0.456 0.477 0.529 0.440 0.570 0.319 0.471 0.480 0.499 0.446 0.405 0.557 0.468 0.521 0.421 0.476 0.448 0.346 At a .01 level of significance, use Excel to test whether the process is out of control. ANSWER: A B C D 1 Diameter Sample Size =COUNT(A2:A37) 2 0.468 Sample Mean =AVERAGE(A2:A37) 3 0.452 4 0.354 Popul. Std. Dev. 0.06 Copyright Cengage Learning. Powered by Cognero.

Values for D 36 0.457278 0.06 Page 19

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Chapter 09: Hypothesis Tests 5 0.456 Hypothesized Value 0.500 0.500 6 0.471 7 0.468 Standard Error =D4/SQRT(D1) 0.01 8 0.521 Test Statistic z =(D2-D5)/D7 -4.272223 9 0.513 10 0.474 p-value (Lower Tail) =NORM.S.DIST(D10,TRUE) 0.00001 11 0.477 p-value (Upper Tail) =1-NORM.S.DIST(D10,TRUE) 0.99999 12 0.480 p-value (TwoTail) =2*MIN(D12,D13) 0.00002 13 0.521 Reject H0, .00002 < .01. There is sufficient evidence at α = .01 to conclude that the process is out of control. 97. From a population of cans of coffee marked "12 ounces," a sample of 25 cans is selected and the contents of each can are weighed. The sample revealed a mean of 11.8 ounces and a standard deviation of 0.5 ounces. Test to see if the mean of the population is at least 12 ounces. (Assume the population is normally distributed.) Use a .05 level of significance. ANSWER: H0: μ ≥ 12 Ha: μ < 12 t = –2; therefore, reject H0. There is sufficient evidence at α = .05 to conclude that the population mean amount of coffee is less than 12 ounces. 98. In the past, the average age of employees of a large corporation has been 40 years. Recently, the company has been hiring older individuals. In order to determine whether there has been an INCREASE in the average age of all the employees, a sample of 25 employees was selected. The average age in the sample was 45 years with a standard deviation of 5 years. Assume the distribution of the population is normal. Let α = .05. a. State the null and the alternative hypotheses. Test to determine whether or not the mean age of all employees is significantly more than 40 b. years. ANSWER: a. H0: μ ≤ 40 b.

Ha: μ > 40 t = 5; therefore, reject H0. There is sufficient evidence at α = .05 to conclude that average age of all employees of the large corporation has increased.

99. A soft drink filling machine, when in perfect adjustment, fills bottles with 12 ounces of soft drink. A random sample of 25 bottles is selected, and the contents are measured. The sample yielded a mean content of 11.88 ounces, with a standard deviation of 0.24 ounces. With a .05 level of significance, test to see if the machine is in perfect adjustment. Assume the distribution of the population is normal. ANSWER: H0: μ = 12 Ha: μ ≠ 12 t = –2.5; therefore, reject H0. There is sufficient evidence at α = .05 to conclude that the machine is NOT in perfect adjustment. 100. A sample of 16 cookies is taken to test the claim that each cookie contains at least 9 chocolate chips. The average number of chocolate chips per cookie in the sample was 7.875 with a standard deviation of 1. Assume the distribution of the population is normal. a. State the null and alternative hypotheses. b. Using a critical value, test the hypothesis at the 1% level of significance. c. Using a p-value, test the hypothesis at the 1% level of significance. d. Compute the probability of a Type II error if the true number of chocolate chips per cookie is 8. ANSWER: a. H0: μ ≥ 9 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests b. c. d.

Ha: μ < 9 Reject H0, –4.5 < –2.602 Reject H0; the p-value is less than .005 A Type II error has not been committed since H0 was rejected.

101. Nancy believes that the average running time of movies is equal to 140 minutes. A sample of four movies was taken, and the following running times were obtained. Assume the distribution of the population is normally distributed. 150 a. b. c. d. e.

150

180

170

State the null and alternative hypotheses. Using a critical value, test the hypothesis at the 10% level of significance. Using a p-value, test the hypothesis at the 10% level of significance. Using a confidence interval, test the hypothesis at the 10% level of significance. Could a Type II error have been committed in this hypothesis test?

ANSWER: a. b. c. d. e.

H0: μ = 140 Ha: μ ≠ 140 Reject H0, 3 > 2.353. There is sufficient evidence at α = .10 to conclude that the average running time of movies differs from 140 minutes. The p-value is approximately equal to .06. Reject H0; .06 < 0.1 (same conclusion as part b) 144.85 to 180.15; reject H0 (same conclusion as part b) A Type II error could not have been committed since H0 was rejected.

102. You are given the following information obtained from a random sample of five observations. 20 18 17 22 18 At a 10% level of significance, use Excel to determine whether or not the mean of the population from which this sample was taken is significantly less than 21. (Assume the population is normally distributed.) ANSWER: A B C D Value of D 1 x Sample Size =COUNT(A2:A6) 5 2 20 Sample Mean =AVERAGE(A2:A6) 19 3 18 Sample Std. Dev. =STDEV(A2:A6) 2 4 17 5 22 Hypoth. Value 21 21 6 18 7 Standard Error =D3/SQRT(D1) 0.8944272 8 Test Statistic t =(D2-D5)/D7 -2.236068 9 Degr. of Freedom =D1-1 4 10 11 p-value (Low. Tail) =T.DIST(D8,D9,TRUE) 0.04450 12 p-value (Up. Tail) =1-D11 0.95550 13 p-value (TwoTail) =2*MIN(D11,D12) 0.08901 14 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests Reject H0, .04450 <.10, there is sufficient evidence at α = .10. To conclude the population mean is significantly less than 21. 103. You are given the following information obtained from a random sample of four observations. 25 47 32 56 At a .05 level of significance, use Excel to determine whether or not the mean of the population from which this sample was taken is significantly different from 48. (Assume the population is normally distributed.) ANSWER: A B C D Values for D 1 x Sample Size =COUNT(A2:A5) 4 2 25 Sample Mean =AVERAGE(A2:A5) 40 3 47 Sample Std. Dev. =STDEV(A2:A5) 14.07125 4 32 5 56 Hypothesized Value 48 48 6 7 Standard Error =D3/SQRT(D1) 7.03562 8 Test Statistic t =(D2-D5)/D7 -1.13707 9 Degrees of Freedom =D1-1 3 10 11 p-value (Lower Tail) =T.DIST(D8,D9,TRUE) 0.16906 12 p-value (Upper Tail) =1-D11 0.83094 13 p-value (Two Tail) =2*MIN(D11,D12) 0.33811 14 Do not reject H0, .33811 > .05. There is not sufficient evidence at α = .05 to conclude that the mean of the population is significantly different from 48. 104. A group of young businesswomen wish to open a high fashion boutique in a vacant store, but only if the average income of households in the area is more than $45,000. A random sample of nine households showed the following results. $48,000 $44,000 $46,000 $43,000 $47,000 $46,000 $44,000 $42,000 $45,000 Use the statistical techniques in Excel to advise the group on whether or not they should locate the boutique in this store. Use a .05 level of significance. (Assume the population is normally distributed.) ANSWER: A B C D Values for D 1 Income Sample Size =COUNT(A2:A10) 9 2 48000 Sample Mean =AVERAGE(A2:A10) 45000 3 44000 Sample Std. Dev. =STDEV(A2:A10) 1936.49167 4 46000 5 45000 Hypothesized Value 45000 45000 6 43000 7 47000 Standard Error =D3/SQRT(D1) 645.49722 8 46000 Test Statistic t =(D2-D5)/D7 0 9 42000 Degrees of Freedom =D1-1 8 10 44000 11 p-value (Lower Tail) =T.DIST(D8,D9,TRUE) 0.50000 Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests 12 p-value (Upper Tail) =1-D11 13 p-value (TwoTail) =2*MIN(D11,D12) 14 Do not reject H0, .5 > .05, and advise the group not to locate the boutique at this store.

0.50000 1.00000

105. In a television commercial, the manufacturer of a toothpaste claims that at least four out of five dentists recommend its product. A consumer-protection group wants to test that claim. Identify the hypotheses. ANSWER: H0: p ≥ .8 Ha: p < .8 106. A manufacturer is considering a new production method. The current method produces 94% non-defective (good) parts. The new method will be implemented if it produces more non-defectives than the current method. Identify the hypotheses. ANSWER: H0: p ≤ .94 Ha: p > .94 107. A new soft drink is being market tested. A sample of 400 individuals participated in the taste test and 80 indicated they like the taste. a. At a 5% significance level, test to determine if at least 22% of the population will like the new soft drink. b. Determine the p-value. ANSWER: a.

H0: p ≥ 0.22 Ha: p < 0.22 z = –.97; therefore, do not reject H0. There is not sufficient evidence at α = 5% to conclude that fewer than 22% of the population like the new soft drink. .1587

108. A student believes that no more than 20% (i.e., ≤ 20%) of the students who finish a statistics course get an A. A random sample of 100 students was taken. Twenty-four percent of the students in the sample received A's. a. State the null and alternative hypotheses. b. Using a critical value, test the hypothesis at the 1% level of significance. c. Using a p-value, test the hypothesis at the 1% level of significance. ANSWER: a. H0: p ≤ 0.2 b. c.

Ha: p > 0.2 Do not reject H0, 1 < 2.33. There is not sufficient evidence at α = 1% to conclude that more than 20% of the students get an A. Do not reject H0; .1587 > .01 (same conclusion as part b)

109. For each shipment of parts a manufacturer wants to accept only those shipments with at most 10% defective parts. A large shipment has just arrived. A quality control manager randomly selects 50 of the parts from the shipment and finds that 6 parts are defective. Is this sufficient evidence to reject the entire shipment? Use a .05 level of significance to conduct the appropriate hypothesis test. Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests ANSWER: H0: p ≤ 0.10 Ha: p > 0.10 Do not reject H0; .4714 < 1.645. There is not sufficient evidence at α = .05 to reject the entire shipment. 110. A national poll reported that 58% of those with Internet access have made purchases online. To investigate whether this percentage applies to its own state, a legislator commissions a study. A random sample of 400 state residents who have Internet access is taken. Of those 400 respondents, 215 said that they have made purchases online. Does this sample provide sufficient evidence to conclude that the state differs from the nation with respect to making purchases online? Use the p-value to conduct the hypothesis test and use a .05 level of significance. ANSWER: H0: p = .58 Ha: p ≠ .58 Reject H0; p-value = .0446 < .05. There is sufficient evidence at α = .05 to conclude the state differs from the nation with respect to making purchases online. 111. An official of a large national union claims that the fraction of women in the union is not significantly different from one-half. Using the sample information reported below, carry out a test of this statement. Use a .05 level of significance. sample size women men ANSWER: H0: p = .5 Ha: p ≠ .5

400 168 232 Reject H0; –3.2 < –1.96. There is sufficient evidence at α = .05 to refute the union official's claim.

112. A manufacturer claims that at least 40% of its customers use coupons. A study of 25 customers is undertaken to test that claim. The results of the study follow: yes no no yes yes no yes no no yes no no no no yes no no no no yes no no yes no yes At a .05 level of significance, use Excel to test the manufacturer's claim. ANSWER: A B C D Values for D 1 Use Coupons? Sample Size =COUNTA(A2:A26) 25 2 yes Response of Interest Yes yes 3 no Count of Response =COUNTIF(A2:A26,D2) 9 4 no Sample Proportion =D3/D1 0.36 5 no 6 no Hypothesized Value 0.4 0.4 7 no 8 yes Standard Error =SQRT(D4*(1-D4)/D1) 0.096 9 no Test Statistic =(D4-D6)/D8 -0.41666667 10 no 11 no p-value (Lower Tail) =NORM.S.DIST(D9,TRUE) 0.338461157 12 no p-value (Upper Tail) =1-D11 0.661538843 13 no p-value (TwoTail) =2*MIN(D11,D12) 0.67692 14 no Do not reject H0, .33846 > .05. There is not sufficient evidence at α = .05 to refute the manufacturer's claim. Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests 113. Several years ago the proportion of Americans aged 18–24 who invested in the stock market was 0.20. A random sample of 25 Americans in this age group was recently taken. They were asked whether or not they invested in the stock market. The results follow: yes no no yes no no yes no no yes no no no no no no no no yes no no no yes no no At a .05 level of significance, use Excel to determine whether or not the proportion of Americans 18–24 years old that invest in the stock market has changed. ANSWER: A B C D Values for D 1 Invest? Sample Size =COUNTA(A2:A26) 25 2 Yes Response of Interest yes yes 3 No Count of Response =COUNTIF(A2:A26,D2) 6 4 no Sample Proportion =D3/D1 0.24 5 no 6 no Hypothesized Value 0.2 0.2 7 no 8 yes Standard Error =SQRT(D6*(1-D6)/D1) 0.08 9 no Test Statistic =(D4-D6)/D8 0.50 10 no 11 no p-value (Lower Tail) =NORM.S.DIST(D9,TRUE) 0.6914625 12 no p-value (Upper Tail) =1-D11 0.3085375 13 no p-value (TwoTail) =2*MIN(D11,D12) 0.6170750 14 no 15 no Do not reject H0, .617 > .05. There is not sufficient evidence at α = .05 to conclude that the proportion of 18to 24-year-old Americans who invest in stocks has changed. 114. Identify the null and alternative hypotheses for the following problems. The manager of a restaurant believes that it takes a customer no more than 25 minutes to eat a. lunch. Economists have stated that the marginal propensity to consume is at least 90¢ out of every b. dollar. It has been stated that 75 out of every 100 people who go to the movies on Saturday night buy c. popcorn. ANSWER: a. H0: μ ≤ 25 b. c.

Ha: μ > 25 H0: p ≥ .9 Ha: p < .9 H0: p = .75 Ha: p ≠ .75

115. Fast ’n Clean operates 12 laundromats on the east side of the city. All of Fast ’n Clean’s clothes dryers have a label stating “20 minutes for $1.00.” You question the accuracy of the dryers’ clocks and decide to conduct an observational Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests study. You randomly select 36 dryers in several different Fast ‘n Clean locations, put $1.00 in each and time the drying cycle. The sample mean drying time is 20 minutes and 25 seconds. The manufacturer of the dryer states that the standard deviation for 20-minute drying cycles is 1 minute. a. Using the sample data and α = .05, test the validity of the label on the dryers. Apply the p-value and critical value approaches to conducting the two-tail hypothesis test. b. Conduct the same two-tail hypothesis test, but use the confidence interval approach to hypothesis testing. ANSWER: a. p-value = .0124 < .05, so reject H0: μ = 20 z-score = 2.50 > 1.96, so reject H0: μ = 20 b. confidence interval 20.09 to 20.74 does not include 20, so reject H0: μ = 20 116. The board of directors of a corporation has agreed to allow the human resources manager to move to the next step in planning day care service for employees’ children if the manager can prove that at least 25% of the employees have interest in using the service. The HR manager polls 300 employees and 90 say they would seriously consider utilizing the service. At the α = .10 level of significance, is there enough interest in the service to move to the next planning step? ANSWER: p-value = .0228 < .10, so reject H0: p = .25 z-score = 2.00 > 1.28, so reject H0: p = .25 yes, there is enough interest to move to the next step 117. Laura Naples, manager of Heritage Inn, periodically collects and tabulates information about a sample of the hotel’s overnight guests. This information aids her in pricing and scheduling decisions she must make. The table below lists data on 10 randomly selected hotel registrants, collected as the registrants checked out. The data listed are: • • • •

Number of people in the group Hotel’s shuttle service used: yes or no Total telephone charges incurred Reason for stay: business or personal Name of Registrant

Madam Sandler Michelle Pepper Claudia Shepler Annette Rodriquez Tony DiMarco Amy Franklin Julio Roberts Edward Blackstone Sara Goldman Todd Atherton

Number in Group

Shuttle Used

Telephone Charges ($)

Reason for Stay

1 2 1 2 1 3 2 4 1 1

yes no no no yes yes no yes no no

0.00 8.46 3.20 2.90 3.12 4.65 6.35 2.10 1.85 5.80

personal business business business personal business personal personal business business

a. Before cell telephones became so common, the average telephone charge per registered group was at least $5.00. Laura suspects that the average has dropped. Test H0:  > 5 and Ha:  < 5 using a .05 level of significance. Use both the critical value and p-value approaches to hypothesis testing. b. In the past, Laura has made some important managerial decisions based on the assumption that the average number of people in a registered group is 2.5. Now she wonders if the assumption is still valid. Test the assumption with  = .05 and Copyright Cengage Learning. Powered by Cognero.

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Chapter 09: Hypothesis Tests use both the critical value and p-value approaches. ANSWER: a. p-value = .0877 > .05, t = –1.471 > –1.833; do not reject H0: μ > 5 b. p-value = .0607 > .05, t = –2.1433 > –2.2622; do not reject H0: μ = 2.5 118. The average gasoline price of one of the major oil companies has been $3.00 per gallon. Because of shortages in production of crude oil, it is believed that there has been a significant INCREASE in the average price. In order to test this belief, we randomly selected a sample of 36 of the company's gas stations and determined that the average price for the stations in the sample was $3.06. Assume that the standard deviation of the population () is $0.09. a. State the null and alternative hypotheses. b. Test the claim at  = .05. c. What is the p-value associated with the above sample results ANSWER: a. H0: µ ≤ 3 b. c.

Ha: µ > 3 z = 4; therefore, reject H0. There is sufficient evidence at  = .05 to conclude that there has been an increase in the average price. less than .001

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations Multiple Choice 1. If the alternative hypothesis is that proportion of items in population 1 is larger than the proportion of items in population 2, then the null hypothesis should be _____. a. p1 – p2 < 0 b. p1 – p2 = 0 c. p1 – p2 > 0 d. p1 – p2 ≤ 0 ANSWER: d 2. To compute an interval estimate for the difference between the means of two populations, the t distribution _____. a. is restricted to small sample situations b. is not restricted to small sample situations c. can be applied when the populations have equal means d. cannot be applied ANSWER: b 3. When developing an interval estimate for the difference between two sample means, with sample sizes of n1 and n2, _____. a. n1 must be equal to n2 b. n1 must be smaller than n2 c. n1 must be larger than n2 d. n1 and n2 can be of different sizes ANSWER: d 4. To construct an interval estimate for the difference between the means of two populations when the standard deviations of the two populations are unknown, we must use a t distribution with _____ degrees of freedom. Let n1 be the size of sample 1 and n2 the size of sample 2. a. n1 + n2 b. n1 + n2 – 1 c. n1 + n2 – 2 d. n1 – n2 + 2 ANSWER: c 5. When each data value in one sample is matched with a corresponding data value in another sample, the samples are known as _____. a. corresponding samples b. matched samples c. independent samples d. dependent samples ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 6. Independent simple random samples are selected to test the difference between the means of two populations whose variances are not known. The sample sizes are n1 = 32 and n2 = 40. The correct distribution to use is the _____ distribution. a. binomial b. t c. normal d. uniform ANSWER: b 7. Independent simple random samples are selected to test the difference between the means of two populations whose standard deviations are not known. We are unwilling to assume that the population variances are equal. The sample sizes are n1 = 25 and n2 = 35. The correct distribution to use is the t distribution with ______ degrees of freedom. a. 25 b. 35 c. 58 d. The correct degrees of freedom cannot be calculated without being given the sample standard deviations. ANSWER: d 8. If two independent large samples are selected from two populations, the sampling distribution of the difference between the two sample means _____. a. can be approximated by a Poisson distribution b. will have a variance of 1 c. can be approximated by a normal distribution d. will have a mean of 1 ANSWER: c 9. The standard error of

is the _____.

a. variance of b. variance of the sampling distribution of c. standard deviation of the sampling distribution of d. difference between the two means ANSWER: c 10. The sampling distribution of

is approximated by a _____.

a. normal distribution b. t distribution with n1 + n2 degrees of freedom c. t distribution with n1 + n2 – 1 degrees of freedom d. t distribution with n1 + n2 + 2 degrees of freedom ANSWER: a Exhibit 10-1 Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations Salary information regarding two independent random samples of male and female employees of a large company is shown below. Sample size Sample mean salary (in $1000s) Population variance

Male 64 44 128

Female 36 41 72

11. Refer to Exhibit 10-1. The point estimate of the difference between the means of the two populations (Male – Female) is _____. a. –28 b. 3 c. 4 d. –4 ANSWER: b 12. Refer to Exhibit 10-1. The standard error for the difference between the two means is _____. a. 4 b. 7.46 c. 4.24 d. 2.0 ANSWER: d 13. Refer to Exhibit 10-1. At 95% confidence, the margin of error is _____. a. 1.96 b. 1.645 c. 3.920 d. 2.000 ANSWER: c 14. Refer to Exhibit 10-1. The 95% confidence interval for the difference between the means of the two populations is _____. a. 0 to 6.92 b. –2 to 2 c. –1.96 to 1.96 d. –.92 to 6.92 ANSWER: d 15. Refer to Exhibit 10-1. If you are interested in testing whether the average salary of males is significantly greater than that of females, the value of the test statistic is _____. a. 2.0 b. 1.5 c. 1.96 d. 1.645 ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 16. Refer to Exhibit 10-1. The p-value is _____. a. .0668 b. .0334 c. 1.336 d. 1.96 ANSWER: a 17. Refer to Exhibit 10-1. At 95% confidence, we have enough evidence to conclude that the _____. a. average salary of males is significantly greater than females b. average salary of males is significantly lower than females c. salaries of males and females are equal d. null hypothesis fails to be rejected ANSWER: d Exhibit 10-2 The following information was obtained from matched samples. The daily production rates for a random sample of workers before and after a training program are shown below. Worker 1 2 3 4 5 6 7

Before 20 25 27 23 22 20 17

After 22 23 27 20 25 19 18

18. Refer to Exhibit 10-2. The point estimate for the mean of the population of difference is _____. a. –1 b. –2 c. 0 d. 1 ANSWER: c 19. Refer to Exhibit 10-2. The null hypothesis to be tested is H0: μd = 0. The value of the test statistic is _____. a. –1.96 b. 1.96 c. 0 d. 1.645 ANSWER: c 20. Refer to Exhibit 10-2. Based on the results of the previous question, the _____. a. null hypothesis should be rejected b. null hypothesis should not be rejected c. alternative hypothesis should be accepted d. alternative hypothesis should be rejected Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations ANSWER: b Exhibit 10-3 A statistics teacher wants to see if there is any difference in the abilities of students enrolled in statistics today and those enrolled five years ago. Final examination scores from a random sample of students enrolled today and from a random sample of students enrolled five years ago were selected. You are given the following information.

σ2

Today 82 112.5 45

Five Years Ago 88 54 36

21. Refer to Exhibit 10-3. The point estimate for the difference between the means of the two populations is _____. a. 58.5 b. 9 c. –9 d. –6 ANSWER: d 22. Refer to Exhibit 10-3. The standard error of

is _____.

a. 12.9 b. 9.3 c. 4 d. 2 ANSWER: d 23. Refer to Exhibit 10-3. The 95% confidence interval for the difference between the two population means is _____. a. –9.92 to –2.08 b. –3.92 to 3.92 c. –13.84 to 1.84 d. –24.228 to 12.23 ANSWER: a 24. Refer to Exhibit 10-3. The test statistic for the difference between the two population means is _____. a. –.47 b. –.65 c. –1.5 d. –3 ANSWER: d 25. Refer to Exhibit 10-3. The p-value for the difference between the two population means is _____. a. .0013 b. .0036 Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations c. .4986 d. .9972 ANSWER: b 26. Refer to Exhibit 10-3. What conclusion can be reached about the difference in the average final examination scores between the two classes? (Use a .05 level of significance.) a. There is a statistically significant difference in the average final examination scores between the two classes. b. There is no statistically significant difference in the average final examination scores between the two classes. c. It is impossible to make a decision on the basis of the information given. d. There is a difference, but it is not significant. ANSWER: a Exhibit 10-4 The following information was obtained from independent random samples. Assume normally distributed populations with equal variances. Sample mean Sample variance Sample size

Sample 1 45 85 10

Sample 2 42 90 12

27. Refer to Exhibit 10-4. The point estimate for the difference between the means of the two populations is _____. a. 0 b. 2 c. 3 d. 15 ANSWER: c 28. Refer to Exhibit 10-4. The standard error of

is _____.

a. 3.0 b. 4.0 c. 8.372 d. 19.48 ANSWER: b 29. Refer to Exhibit 10-4. The degrees of freedom for the t distribution are _____. a. 22 b. 21 c. 20 d. 19 ANSWER: c 30. Refer to Exhibit 10-4. The 95% confidence interval for the difference between the two population means is _____. a. –5.367 to 11.367 b. –5 to 3 Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations c. –4.86 to 10.86 d. –2.65 to 8.65 ANSWER: a Exhibit 10-5 The following information was obtained from matched samples. Individual 1 2 3 4 5

Method 1 7 5 6 7 5

Method 2 5 9 8 7 6

31. Refer to Exhibit 10-5. The point estimate for the mean of the population of differences is _____. a. –1 b. 0 c. 1 d. 2 ANSWER: a 32. Refer to Exhibit 10-5. The 95% confidence interval for the mean of the population of differences is _____. a. –3.776 to 1.776 b. –2.776 to 2.776 c. –1.776 to 2.776 d. 0 to 3.776 ANSWER: a 33. Refer to Exhibit 10-5. The null hypothesis tested is H0: μd = 0. The test statistic for the mean of the population of differences is _____. a. 2 b. 0 c. –1 d. –2 ANSWER: c 34. Refer to Exhibit 10-5. If the null hypothesis is tested at the 5% level, the null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 10-6 The management of a department store is interested in estimating the difference between the mean credit purchases of customers using the store's credit card versus those customers using a national major credit card. You are given the Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations following information. Assume the samples were selected randomly. Sample size Sample mean Population standard deviation

Store's Card 64 $140 $10

Major Credit Card 49 $125 $8

35. Refer to Exhibit 10-6. A point estimate for the difference between the mean purchases of the users of the two credit cards (Store's Card – Major Credit Card) is _____. a. 2 b. 18 c. 265 d. 15 ANSWER: d 36. Refer to Exhibit 10-6. At 95% confidence, the margin of error is _____. a. 1.694 b. 3.32 c. 1.96 d. 15 ANSWER: b 37. Refer to Exhibit 10-6. A 95% confidence interval estimate for the difference (Store's Card – Major Credit Card) between the average purchases of the customers using the two different credit cards is _____. a. 49 to 64 b. 11.68 to 18.32 c. 125 to 140 d. 8 to 10 ANSWER: b Exhibit 10-7 In order to estimate the difference between the average hourly wages of employees of two branches of a department store, two independent random samples were selected and the following statistics were calculated. Sample size Sample mean Sample standard deviation

Downtown Store 25 $9 $2

North Mall Store 20 $8 $1

38. Refer to Exhibit 10-7. A point estimate for the difference between the two sample means (Downtown Store – North Mall Store) is _____. a. 1 b. 2 c. 3 d. 4 ANSWER: a 39. Refer to Exhibit 10-7. A 95% interval estimate for the difference between the two population means is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations a. .071 to 1.928 b. 1.922 to 2.078 c. 1.09 to 4.078 d. 1.078 to 2.922 ANSWER: a Exhibit 10-8 In order to determine whether or not there is a significant difference between the hourly wages of two companies, two independent random samples were selected and the following statistics were calculated. Sample size Sample mean Population standard deviation

Company A 80 $6.75 $1.00

Company B 60 $6.25 $0.95

40. Refer to Exhibit 10-8. A point estimate for the difference between the two sample means (Company A – Company B) is _____. a. 20 b. .50 c. .25 d. 1.00 ANSWER: b 41. Refer to Exhibit 10-8. The value of the test statistic is _____. a. .098 b. 1.645 c. 2.75 d. 3.01 ANSWER: d 42. Refer to Exhibit 10-8. The p-value is _____. a. .0013 b. .0026 c. .0042 d. .0084 ANSWER: b 43. Refer to Exhibit 10-8. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: a Exhibit 10-9 Two major automobile manufacturers have produced compact cars with the same size engines. We are interested in Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations determining whether or not there is a significant difference in the MPG (miles per gallon) of the two brands of automobiles. A random sample of eight cars from each manufacturer is selected, and eight drivers are selected to drive each automobile for a specified distance. The MPG for each manufacturer and driver is shown below. Driver 1 2 3 4 5 6 7 8

Manufacturer A 32 27 26 26 25 29 31 25

Manufacturer B 28 22 27 24 24 25 28 27

44. Refer to Exhibit 10-9. The mean of the differences (Manufacturer A – Manufacturer B) is _____. a. .50 b. 1.5 c. 2.0 d. 2.5 ANSWER: c 45. Refer to Exhibit 10-9. The value of the test statistic is _____. a. 1.645 b. 1.96 c. 2.096 d. 2.256 ANSWER: d 46. Refer to Exhibit 10-9. At 90% confidence, the null hypothesis _____. a. should not be rejected b. should be rejected c. should be revised d. should be retested ANSWER: b Exhibit 10-10 The results of a recent poll on the preference of shoppers regarding two products are shown below. Product A B

Shoppers Surveyed 800 900

Shoppers Favoring This Product 560 612

47. Refer to Exhibit 10-10. The point estimate for the difference between the two population proportions in favor of this product (Product A – Product B) is _____. a. 52 b. 100 Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations c. .44 d. .02 ANSWER: d 48. Refer to Exhibit 10-10. The standard error of

is _____.

a. 52 b. .044 c. .0225 d. 100 ANSWER: c 49. Refer to Exhibit 10-10. At 95% confidence, the margin of error is _____. a. .064 b. .044 c. .0225 d. 52 ANSWER: b 50. Refer to Exhibit 10-10. The 95% confidence interval estimate for the difference between the populations favoring the products is _____. a. –.024 to .064 b. .6 to .7 c. .024 to .7 d. .02 to .3 ANSWER: a Exhibit 10-11 An insurance company selected samples of clients under 18 years of age and over 18 and recorded the number of accidents they had in the previous year. The results are shown below. Under Age 18 Over Age 18 n1 = 500 n2 = 600 Number of accidents = 180 Number of accidents = 150 We are interested in determining if the accident proportions differ between the two age groups. 51. Refer to Exhibit 10-11 and let pu represent the proportion under and po the proportion over the age of 18. The null hypothesis is _____. a. pu – po ≤ 0 b. pu – po ≥ 0 c. pu – po ≠ 0 d. pu – po = 0 ANSWER: d 52. Refer to Exhibit 10-11. The pooled proportion is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations a. .305 b. .300 c. .027 d. .450 ANSWER: b 53. Refer to Exhibit 10-11. The value of the test statistic is _____. a. .96 b. 1.96 c. 2.96 d. 3.96 ANSWER: d 54. Refer to Exhibit 10.11. The p-value is _____. a. less than .001 b. more than .10 c. .0228 d. .3 ANSWER: a Exhibit 10-12 The results of a recent poll on the preference of teenagers regarding the types of music they listen to are shown below. Music Type Pop Rap

Teenagers Surveyed 800 900

Teenagers Favoring This Type 384 450

55. Refer to Exhibit 10-12. The point estimate for the difference between the proportions is _____. a. –.02 b. .048 c. 100 d. 66 ANSWER: a 56. Refer to Exhibit 10-12. The standard error of

is _____.

a. .48 b. .50 c. .03 d. .0243 ANSWER: d 57. Refer to Exhibit 10-12. The 95% confidence interval for the difference between the two proportions is _____. a. 384 to 450 Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations b. .48 to .5 c. .028 to .068 d. –.068 to .028 ANSWER: d Exhibit 10-13 In order to determine whether or not there is a significant difference between the hourly wages of two companies, the following data have been accumulated. Company 1 Company 2 n1 = 80 n2 = 60 1 = $10.80

σ1= $2.00

2 = $10.00

σ2= $1.50

58. Refer to Exhibit 10-13. The null hypothesis for this test is _____. a. μ1 – μ2 ≠ 0 b. μ1 – μ2 ≥ 0 c. μ1 – μ2 ≤ 0 d. μ1 – μ2 = 0 ANSWER: d 59. Refer to Exhibit 10-13. The point estimate of the difference between the means (Company 1 – Company 2) is _____. a. 20 b. .8 c. .50 d. –20 ANSWER: b 60. Refer to Exhibit 10-13. The test statistic has a value of _____. a. 1.96 b. 1.645 c. .80 d. 2.7 ANSWER: d 61. Refer to Exhibit 10-13. The p-value is _____. a. .0035 b. .007 c. .4965 d. 1.96 ANSWER: b

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 62. In testing the null hypothesis H0: μ1 – μ2 = 0, the computed test statistic is z = –1.66. The corresponding p-value is _____. a. .0485 b. .0970 c. .9515 d. .9030 ANSWER: b 63. A company wants to identify which of two production methods has the smaller completion time. One sample of workers is randomly selected and each worker first uses one method and then uses the other method. The sampling procedure being used to collect completion time data is based on _____ samples. a. cross b. pooled c. independent d. matched ANSWER: d Subjective Short Answer 64. In order to estimate the difference between the average miles per gallon of two different models of automobiles, two independent random samples are selected and the following statistics are calculated. Sample size Sample mean Sample variance a. b.

Model A 60 28 16

Model B 55 25 9

At 95% confidence, develop an interval estimate for the difference between the average miles per gallon for the two models. Is there conclusive evidence to indicate that one model gets a higher MPG than the other? Why or why not? Explain.

ANSWER:

a. b.

1.70 to 4.30 (Model A – Model B) Since the range of the interval is from a positive number to a positive number, it indicates that there is conclusive evidence that Model A has a larger mean.

65. The following statistics are given concerning the ACT scores of high school seniors from two local schools. School A School B n1 = 14 n2 = 15 2 = 23 1 = 25 = 16 = 10 Develop a 95% confidence interval estimate for the difference between the two populations. ANSWER: –.772 to 4.772 66. Independent random samples selected on two university campuses revealed the following information concerning the average amount of money spent on textbooks during the fall semester. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations University A University B Sample size 50 40 Average purchase $260 $250 Standard deviation (σ) $20 $23 We want to determine if, on average, students at University A spent more on textbooks than the students at University B. a. Compute the test statistic. b. Compute the p-value. c. What is your conclusion? Let α = .05. ANSWER:

a. b. c.

z = 2.17 p-value = .017 p-value = .017 < .05; reject H0 and conclude that students at University A spent more than students at University B

67. Maxforce, Inc. manufactures racquetball racquets by two different manufacturing processes (A and B). Because the management of this company is interested in estimating the difference between the average times needed to produce a racquet, they select independent samples from each process. The results of the samples are shown below. Sample size Sample mean (in minutes) Population variance (σ2) a. b.

Process A 32 43 64

Process B 35 47 70

Develop a 95% confidence interval estimate for the difference between the average times of the two processes. Is there conclusive evidence to conclude that one process takes longer than the other? If yes, which process appears to take longer? Explain.

ANSWER:

a. b.

–7.92 to –.08 (Process A – Process B) Yes, since the range of the interval is from a negative value to a negative value, we have reason to believe that process B takes longer.

68. The management of Recover Fast Hospital (RFH) claims that the average length of stay in their hospital after a major surgery is less than the average length of stay at General Hospital (GH). Independent random samples were selected, and the following statistics were calculated. Sample size Mean (in days) Standard deviation (σ) a. b. c.

RFH 45 0.6 0.5

GH 58 4.9 0.6

Formulate the hypotheses. Compute the test statistic. Using the p-value approach, determine if the average length of stay in RFH is significantly less than the average length of stay in GH. Let α = .05.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations ANSWER: a. b. c.

H0: μRFH – μGH ≥ 0 Ha: μRFH – μGH < 0 –39.65 p-value = .000. Reject H0, and conclude that the length of stay at RFH is significantly less than at GH.

69. In order to determine whether or not a driver's education course improves the scores on a driving exam, a random sample of six students was given the exam before and after taking the course. The results are shown below. Let d = Score After – Score Before. Score Before the Course 83 89 93 77 86 79

Student 1 2 3 4 5 6 a. b.

Score After the Course 87 88 91 77 93 83

Compute the test statistic. At 95% confidence using the p-value approach, test to see if taking the course actually increased scores on the driving exam.

ANSWER:

a. b.

Test statistic t = 1.391 p-value is between .1 and .2; do not reject H0 and conclude that there is not sufficient evidence to show that the course increased the scores

70. A random sample of 200 UTC seniors was surveyed and 60 indicated that they were planning on attending Graduate School. At UTK, a random sample of 400 seniors was surveyed and 100 indicated that they were planning to attend Graduate School. a. Determine a 95% confidence interval estimate for the difference between the proportions of seniors at the two universities that were planning to attend Graduate School. b. Is there conclusive evidence to suggest that the proportion of students from UTC who plan to go to Graduate School is significantly more than those from UTK? Explain. ANSWER:

a. b.

–.026 to .126 (UTC – UTK) No, the range of the interval is from a negative value to a positive value.

71. A random sample of 300 female registered voters was surveyed and 120 indicated they were planning to vote for the incumbent president. An independent random sample of 400 male registered voters indicated that 140 were planning to vote for the incumbent president. a. Compute the test statistic. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations b.

At α = .05, test to see if there is a significant difference between the proportions of females and males who plan to vote for the incumbent president. (Use the p-value approach.)

ANSWER:

a. b.

z = 1.35 p-value = .177; do not reject H0. There is not sufficient evidence to conclude that there is a significant difference between the proportions of females and males planning to vote for the incumbent president.

72. A random sample of 150 Chattanooga residents was surveyed and 60 indicated that they participated in a recycling program. In Knoxville, a random sample of 120 residents was surveyed and 36 claimed to recycle. a. Determine a 95% confidence interval estimate for the difference between the proportions of residents recycling in the two cities. b. From your answer in part (a), is there sufficient evidence to conclude that there is a significant difference in the proportion of residents participating in a recycling program? ANSWER:

a. b.

–.0134 to .2134 (Chattanooga – Knoxville) No, because the interval for the proportions ranges from a negative value to a positive value.

73. Consider the following results for two samples randomly selected from two normal populations with equal variances. Sample size Sample mean Population standard deviation a. b.

Sample I 28 48 9

Sample II 35 44 10

Develop a 95% confidence interval for the difference between the two population means. Is there conclusive evidence that one population has a larger mean? Explain.

ANSWER:

a. b.

–.70 to 8.70 (Sample I – Sample II) No, because the range of the interval is from a negative value to a positive value.

74. The business manager of a local health clinic is interested in estimating the difference between the fees for extended office visits in her center and the fees of a newly opened group practice. She selected an independent random sample from each location and calculated the following statistics: Health Clinic Group Practice Sample size 50 visits 45 visits Sample mean $21 $19 Standard deviation (σ) $2.75 $3.00 Develop a 95% confidence interval estimate for the difference between the average fees of the two locations. ANSWER: .8384 to 3.1617 (Health Clinic – Group Practice) 75. A random sample of 89 tourists in the Grand Bahamas showed that they spent an average of $2,860 (in a week) with a standard deviation of $126, and a sample of 64 tourists in New Province showed that they spent an average of $2,935 (in a week) with a standard deviation of $138. We are interested in determining if there is any significant difference between the average expenditures of those who visited the two islands. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations a. b. c. d.

Determine the degrees of freedom for this test. Compute the test statistic. Compute the p-value. What is your conclusion? Let α = .05.

ANSWER:

a. b. c. d.

128 Test statistic t = –3.438 p-value (two-tailed) < 0.005 (almost 0) p-value < .05; reject H0, there is a significant difference between the average expenditures of those who visited the Grand Bahamas and those who visited New Province.

76. Among a random sample of 50 MD's (medical doctors) in the city of Memphis, Tennessee, 10 indicated they make house calls. Among a random sample of 100 MD's in Atlanta, Georgia, 18 said they make house calls. Determine a 95% interval estimate for the difference between the proportions of doctors who make house calls in these two cities. ANSWER: –.114 to .154 (Memphis – Atlanta) 77. Consider the following results for two samples randomly selected from two populations. Sample size Sample mean Sample standard deviation a. b.

Sample A 31 106 8

Sample B 35 102 7

Determine the degrees of freedom for the t distribution. Develop a 95% confidence interval for the difference between the two population means.

ANSWER:

a. b.

60 .277 to 7.723 (Sample A – Sample B)

78. Consider the following results for two samples randomly selected from two populations. Sample size Sample mean Sample standard deviation a. b. c.

Sample A 25 66 5

Sample B 38 60 7

What are the degrees of freedom for the t distribution? At 95% confidence, compute the margin of error. Develop a 95% confidence interval for the difference between the two population means.

ANSWER:

a. b. c.

60 3.026 2.974 to 9.026 (Sample A – Sample B)

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 79. Consider the following results for two samples randomly selected from two populations. Sample A 20 28 5

Sample size Sample mean Sample standard deviation a. b. c.

Sample B 25 22 6

Determine the degrees of freedom for the t distribution. At 95% confidence, what is the margin of error? Develop a 95% confidence interval for the difference between the two population means.

ANSWER:

a. b. c.

42 3.31 2.69 to 9.31(Sample A – Sample B)

80. During the primary elections of 2004, candidate A showed the following pre-election voter support in Tennessee and Mississippi. Voters Surveyed Tennessee Mississippi a. b.

500 700

Voters Favoring Candidate A 295 357

Develop a 95% confidence interval estimate for the difference between the proportions of voters favoring candidate A in the two states. Is there conclusive evidence that one of the two states had a larger proportion of voters' support? If yes, which state? Explain.

ANSWER:

a. b.

.023 to .137 (Tennessee – Mississippi) Yes, the range of interval is from a positive value to a positive value, indicating Tennessee had the larger support.

81. Consider the following results for two samples randomly selected from two populations. Sample size Sample mean Sample standard deviation a. b. c.

Sample A 28 24 8

Sample B 30 22 6

Determine the degrees of freedom for the t distribution. Develop a 95% confidence interval for the difference between the two population means. Is there conclusive evidence that one population has a larger mean? Explain.

ANSWER:

a. b. c.

49 –1.753 to 5.753 (Sample A – Sample B) No, the interval ranges from negative to positive.

82. The results of a recent poll on the preference of voters regarding the presidential candidates are shown below. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations

Candidate A Candidate B a. b.

Voters Surveyed 200 300

Voters Favoring This Candidate 150 195

Develop a 90% confidence interval estimate for the difference between the proportions of voters favoring each candidate. (Candidate A – Candidate B) Does your confidence interval provide conclusive evidence that one of the candidates is favored more? Explain.

ANSWER:

a. b.

.032 to .168 Yes, the range of interval is from a positive value to a positive value, indicating candidate A had the larger support.

83. Consider the following hypothesis test: μ1 – μ2 ≤ 0 μ1 – μ2 > 0 The following results are for two independent samples selected from two populations. Sample 1 Sample 2 Sample size 35 37 Sample mean 43 37 Sample variance 140 170 a. b. c.

Determine the degrees of freedom for the t distribution. Compute the test statistic. Determine the p-value and test the above hypotheses.

ANSWER:

a. b. c.

69.88 Test statistic t = 2.05 p-value is between .01 and .025, reject H0

84. The following are the test scores of two independent random samples of students from University A and University B on a national statistics examination. Develop a 95% confidence interval estimate for the difference between the mean scores of the two populations. (University A – University B)

n σ

University A Scores 86 64 68

University B Scores 83 87 70

ANSWER: –19.22 to 25.224 (A – B) 85. In order to estimate the difference between the average mortgages in the South and the North of the United States, the Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations following information was gathered. South 40 $70 $5

Sample size Sample mean (in $1000s) Sample standard deviation (in $1000s) a. b.

North 45 $75 $7

Compute the degrees of freedom for the t distribution. Develop an interval estimate for the difference between the average of the mortgages in the South and North (South – North). Let α = .03.

ANSWER:

a. b.

79 –7.893 to –2.107 (in thousands)

86. The following information regarding the ages of two independent random samples, one from the population of all fulltime students and one from the population of all part-time students, is given. Using the following statistics, develop an interval estimate for the difference between the mean ages of the two populations (Full Time – Part Time). Use a 5% level of significance. Full Time 27 1.2 50

s n

Part Time 24 2 60

ANSWER: –.446 to 6.446 (Full-Time – Part Time) 87. Independent random samples of managers' yearly salaries (in $1000s) randomly selected from governmental and private organizations provided the following information. At 95% confidence, test to determine if there is a significant difference between the average salaries of the managers in the two sectors.

s n

Government 80 9 28

ANSWER:

Private 75 10 31

H0: μ1 – μ2 = 0 Ha: μ1 – μ2 ≠ 0 df = 56; test statistic t = 2.021; p-value (two-tailed) is between .02 and .05; reject H0 and conclude that there is a significant difference between the average salaries of the managers in the governmental and private sectors.

88. Independent random samples selected at two local malls provided the following information regarding purchases by patrons of the two malls. Sample size Average purchase

Hamilton Place 85 $143

Eastgate 93 $150 Page 21

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations Standard deviation $22 $18 We want to determine whether or not there is a significant difference between the average purchases by the patrons of the two malls. a. Give the hypotheses for the above. b. Compute the test statistic. c. At 95% confidence, test the hypotheses. ANSWER:

a. b. c.

H0: μ1 – μ2 = 0 Ha: μ1 – μ2 ≠ 0 t = –2.31 p-value is between .02 and .05 (two-tailed), reject H0 (using the normal distribution, p-value = .0208) and conclude that there is a significant difference between the average purchases by patrons of Hamilton Place and Eastgate.

89. Recently, a local newspaper reported that part-time students are older than full-time students. In order to test the validity of its statement, two independent samples of students were selected.

s n a. b. c. d.

Full Time 26 2 42

Part Time 24 3 31

Give the hypotheses for the above. Determine the degrees of freedom. Compute the test statistic. At 95% confidence, test to determine whether or not the average age of part-time students is significantly more than full-time students.

ANSWER: a. b. c. d.

H0: μf – μp ≤ 0 Ha: μf – μp > 0 49 test statistic t = 3.221 p-value is less than .005, reject H0 and conclude that the average age of part-time students is significantly more than full-time students.

90. The daily production rates for a sample of factory workers before and after a training program are shown below. Let d = After – Before. Worker 1 2 3 4

Before 6 10 9 8

After 9 12 10 11 Page 22

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 5 7 9 We want to determine if the training program was effective. a. Give the hypotheses for this problem. b. Compute the test statistic. c. At 95% confidence, test the hypotheses. That is, did the training program actually increase the production rates? ANSWER:

a. b. c.

H0: μd ≤ 0 Ha: μd > 0 test statistic t = 5.8 p-value is less than .005, reject H0 and conclude that the training program increased production rates.

91. In a sample of 40 Democrats, 6 opposed the president's foreign policy, while of 50 Republicans, 8 were opposed to his policy. Determine a 90% confidence interval estimate for the difference between the proportions of the opinions of the individuals in the two parties. ANSWER: –.136 to .116 92. In a sample of 100 Republicans, 60 favored the president's anti-drug program. While in a sample of 150 Democrats, 84 favored his program. At 95% confidence, test to see if there is a significant difference in the proportions of the Democrats and the Republicans who favored the president's anti-drug program. ANSWER: H0: p1 – p2 = 0 Ha: p1 – p2 ≠ 0 z = .63; p-value = .5286; do not reject H0. There is not sufficient evidence to conclude that there is a significant difference in the proportions of the Democrats and Republicans who favored the president’s anti-drug program.

93. In a random sample of 200 Republicans, 160 opposed the new tax laws. While in a random sample of 120 Democrats, 84 opposed the new tax laws. Determine a 95% confidence interval estimate for the difference between the proportions of Republicans and Democrats opposed to this new law. ANSWER: .001 to .199 94. Two independent random samples of annual starting salaries for individuals with master's and bachelor's degrees in business were selected and the results are shown below.

Sample size Sample mean (in $1000s) Sample standard deviation (in $1000s) a. b.

Master's Degree 33 38 2.4

Bachelor's Degree 30 34 2

What are the degrees of freedom for the t distribution? Provide a 95% confidence interval estimate for the difference between the salaries of the two

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations groups. (Master's Degree – Bachelor's Degree) ANSWER:

a. b.

60 2.89 to 5.11 (thousands)

95. During the recent primary elections, the Democrat presidential candidate showed the following pre-election voter support in Alabama and Mississippi. State Alabama Mississippi a. b. c.

Voters Surveyed 800 600

Voters Favoring the Democrat Candidate 440 360

We want to determine whether or not the proportions of voters favoring the Democrat candidate were the same in both states. Provide the hypotheses. Compute the test statistic. Determine the p-value; and at 95% confidence, test the above hypotheses.

ANSWER:

a. b. c.

H0: p1 – p2 = 0 Ha: p1 – p2 ≠ 0 z = –1.87 p-value = .0614 (using the normal distribution); do not reject H0. There is not sufficient evidence to conclude that the proportion of voters favoring the Democrat candidate were the same in both states.

96. A test on world history was given to a group of individuals before and after a film on the history of the world was presented. The results are given below. Determine if the film significantly increased the test scores. (For the following matched samples, let d = After – Before.) Individual 1 2 3 4 5 6 7

After 92 86 89 90 93 88 97

Before 86 88 84 90 85 90 91

a. Give the hypotheses for this problem. b. Compute the test statistic. c. At 95% confidence, test the hypotheses. ANSWER: a. H0: μd ≤ 0 Ha: μd > 0 b. t = 1.89 c. p-value is between .05 and .10; do not reject H0. There is not sufficient evidence to conclude Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations that the film significantly increased the test scores.

97. The Dean of Students at UTC has said that the average grade of UTC students is higher than that of the students at GSU. Random samples of grades from the two schools are selected, and the results are shown below. Sample size Sample mean Sample standard deviation a. b. c.

UTC 14 2.85 0.40

GSU 12 2.61 0.35

Give the hypotheses. Compute the test statistic. At a .1 level of significance, test the Dean of Students' statement.

ANSWER:

a. b. c.

H0: μ1 – μ2 ≤ 0 Ha: μ1 – μ2 > 0 t = 1.632 p-value is between .05 and .1, reject H0 and conclude that the average grade of UTC students is higher than that of the students at GSU.

98. Two independent random samples of employees of Company A and Company B were selected. The following statistics are provided regarding the ages of employees. Company A Company B Sample size 32 36 Average age 42 47 Variance 16 36 Develop a 97% confidence interval for the difference between the average ages of the employees of the two companies. (Company A – Company B) ANSWER: –7.721 to –2.279 99. A random sample of test scores was taken from University A and University B. The following statistics were calculated: UA UB Sample size 28 41 Average test score 84 82 Variance 64 100 Provide a 98% confidence interval estimate for the difference between the test scores of the two universities. (University A – University B) ANSWER: –2.521 to 6.521 100. The following shows the monthly sales in units of six salespersons before and after a bonus plan was introduced. At Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 95% confidence, determine whether the bonus plan has increased sales significantly. (For the following matched samples, let d = After – Before.) Salesperson 1 2 3 4 5 6 ANSWER:

After 94 82 90 76 79 85

H0: μd ≤ 0 Ha: μd > 0

Before 90 84 84 70 80 80

t = 2.05; p-value is between .025 and .05; reject H0 and conclude that the bonus plan has increased sales significantly.

101. The office of records at a university has stated that the proportion of incoming female students who major in business has increased. A random sample of female students selected several years ago is compared with a random sample of female students selected this year. Results are summarized below. Has the proportion increased significantly? Test at α = .10. Previous sample Present sample ANSWER:

Sample Size 250 300

H0: p1 – p2 ≤ 0 Ha: p1 – p2 > 0

No. Majoring in Business 50 69

z = 1.86; p-value = .0628; reject H0 and conclude that the proportion of incoming female students who major in business has increased significantly.

102. The following information regarding the number of semester hours selected from random samples of day and evening students is provided. Day Evening 16 12 s 4 3 n 40 37 Develop a 95% confidence interval estimate for the difference between the mean semester hours selected by the two groups of students. (Day – Evening) ANSWER: 3.269 to 4.731 103. The following data give the number of computer units sold per day by a random sample of six salespersons before Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations and after a bonus plan was implemented. Salesperson Before After 1 3 6 2 7 5 3 6 6 4 8 7 5 7 8 6 9 8 At 95% confidence, test to see if the bonus plan was effective. That is, did the bonus plan actually increase sales? ANSWER: H0: μ ≤ 0 t = 0; p-value = 1.0, do not reject H0. We do not Ha: μ > 0 have enough evidence to conclude that the bonus plan actually increased sales. 104. Zip, Inc. manufactures Zip drives on two different manufacturing processes. Because the management of this company is interested in determining if process 1 takes less manufacturing time, they selected independent random samples from each process. The results of the samples are shown below. Sample size Sample mean (in minutes) Sample variance a. b. c. d.

Process 1 27 10 16

Process 2 22 14 25

State the null and alternative hypotheses. Determine the degrees of freedom for the t test. Compute the test statistic. At 95% confidence, test to determine if there is sufficient evidence to indicate that process 1 takes a significantly shorter time to manufacture the Zip drives.

ANSWER:

a. b. c. d.

H0: μ1 – μ2 ≥ 0 Ha: μ1 – μ2 < 0 39 –3.042 p-value is less than .005; thus, reject H0 and conclude that process 1 takes less manufacturing time than process 2.

105. A credit company has gathered information regarding the average amount owed by people under 30 years old and by people over 30 years. Independent random samples were selected from both age groups. You are given the following information. Under 30 Over 30 600 550 n 200 300 2 361 400 σ Construct a 95% confidence interval for the difference between the average amounts owed by the two age groups. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations ANSWER: 46.528 to 53.472 106. In order to estimate the difference between the average age of male and female employees at Young Corporation, the following information was gathered: Male Female Sample size 32 36 Sample mean 25 23 Sample standard deviation 4 6 Develop a 95% confidence interval estimate for the difference between the average age of male and female employees at Young Corporation. ANSWER: –.449 to 4.449 107. A recent Time magazine reported the following information about a sample of workers in Germany and the United States. United States Germany Average length of workweek (hours) 42 38 Sample standard deviation 5 6 Sample size 600 700 Determine whether or not there is a significant difference between the average workweek in the United States and the average workweek in Germany. a. State the null and alternative hypotheses. b. Compute the test statistic. c. Compute the p-value. What is your conclusion? ANSWER:

a. b. c.

H0: μU – μG = 0 Ha: μU – μG ≠ 0 Test statistic t = 13.1 p-value is less than .005 (almost 0); reject H0 and conclude that there is a significant difference between the average workweek in the United States and the average work week in Germany.

108. Allied Corporation is trying to determine whether to purchase Machine A or Machine B. It has leased the two machines for a month. A random sample of five employees has been selected. These employees have gone through a training session on both machines. Below is information on their productivity rate on both machines. (Let d = A – B.) Person 1 2 3 4 5 a. b. c.

Machine A 47 53 50 55 45

Machine B 52 58 47 60 53

State the null and alternative hypotheses for a two-tailed test. Find the mean and standard deviation for the difference. Compute the test statistic.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations d.

Test the null hypothesis stated in part (a) at the 10% level.

ANSWER:

a. b. c. d.

H0: μd = 0 Ha: μd ≠ 0 –4 and 4.123 t = –2.169 p-value is between .05 and .10, reject H0 and conclude that there is a significant difference between the productivity rate of Machine A and B.

109. The reliability of two types of machines used in the same manufacturing process is to be tested. The first machine failed to operate correctly in 90 out of 300 trials, while the second type failed to operate correctly in 50 out of 250 trials. a. Give a point estimate for the difference between the population proportions of these machines. b. Calculate the pooled estimate of the population proportion. Carry out a hypothesis test to check whether there is a statistically significant difference in the c. reliability for the two types of machines using a .10 level of significance. ANSWER:

a. b. c.

.10 .2545 test statistic z = 2.68, p-value = .0074, reject H0 and conclude that there is a significant difference in the reliability of the two types of machines.

110. A company attempts to evaluate the potential for a new bonus plan by selecting a sample of four salespersons to use the bonus plan for a trial period. The weekly sales volume before and after implementing the bonus plan is shown below. (For the following matched samples, let d = After – Before.) Salesperson 1 2 3 4 a. b. c.

Before 48 48 38 44

After 44 40 36 50

State the hypotheses. Compute the test statistic. Use α = 0.05 and test to see if the bonus plan will result in an increase in the mean weekly sales.

ANSWER:

a. b. c.

H0: μd ≤ 0 Ha: μd > 0 t = .68 p-value > .2, do not reject H0. There is not sufficient evidence to conclude that the bonus plan will result in an increase in the mean weekly sales.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations

111. The following information was obtained from matched samples regarding the productivity of four individuals using two different methods of production. Individual Method 1 Method 2 1 6 8 2 9 5 3 7 6 4 7 5 5 8 6 6 9 5 7 6 3 Let d = Method 1 – Method 2. Is there a significant difference between the productivity of the two methods? Let α = .05. ANSWER: H0: μd = 0 Ha: μd ≠ 0 Test statistic t = 2.54, p-value is between .02 and .05; do not reject H0. There is not sufficient evidence to conclude that there is a significant difference between the productivity of the two methods. 112. The results of a recent poll on the preference of a random sample of Republican and Democrat voters regarding whether they preferred the Republican presidential candidates are shown below. Party Democrat Republican

Voters Surveyed 400 450

Voters Favoring the Republican Candidate 192 225

At 95% confidence, test to determine whether or not there is a significant difference between the preferences for the two candidates. ANSWER: H0: p1 – p2 = 0 Ha: p1 – p2 ≠ 0 z = –.58; p-value = .562; do not reject H0. There is not sufficient evidence to conclude that there is a significant difference between the preferences for the two candidates.

113. A potential investor collected attendance data over a period of 49 days at the North Mall and South Mall theaters in order to determine the difference between the average daily attendances. The North Mall Theater averaged 720 patrons per day with a variance of 100, while the South Mall Theater averaged 700 patrons per day with a variance of 96. Develop an interval estimate for the difference between the average daily attendances at the two theaters. Use a confidence coefficient of .95. ANSWER: 16.03 to 23.97 114. From production line A, a sample of 500 items is selected at random, and it is determined that 30 items are defective. In a sample of 300 items from production process B (which produces identical items to line A), there are 12 defective items. Determine a 95% confidence interval estimate for the difference between the proportions of defectives in the two lines. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations ANSWER: –.01 to .05 115. Two independent samples are drawn from two populations, and the following information is provided. Population 1 34 55 s 14 We want to test the following hypotheses:

Population 2 52 65 18

H0: μ1 – μ2 ≥ 0 Ha: μ1 – μ2 < 0 a. b. c.

Determine the degrees of freedom. Compute the test statistic. At 95% confidence, test the hypotheses. Assume the two populations are normally distributed and have equal variances.

ANSWER:

a. b. c.

81 t = –2.887 p-value < .005, reject H0

116. In order to estimate the difference between the average yearly salaries of top managers in private and governmental organizations, the following information was gathered: Sample size Sample mean (in $1000s) Sample standard deviation (in $1000s)

Private 50 90 6

Governmental 60 80 8

Develop an interval estimate for the difference between the average salaries of the two sectors. Let α = .05. ANSWER: $7,350 to $12,650 117. Independent random samples selected at two companies provided the following information regarding annual salaries of the employees. Sample size Sample mean (in $1000s) Sample standard deviation (in $1000s) a. b.

Whitney Co. 72 48 12

Max Co. 50 43 10

Determine whether there is a significant difference between the average salaries of the employees at the two companies. Compute the test statistic. Compute the p-value; at 95% confidence, test the hypotheses.

ANSWER:

test statistics t = 2.5

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations b.

p-value = .0124 (using the normal distribution); thus, reject H0 and conclude that there is a significant difference in the average salaries of the employees at the two companies.

118. In order to estimate the difference between the yearly incomes of marketing managers in the East and West of the United States, the following information was gathered: East n1 = 40 1 = 72 (in $1000s)

s1 = 6 (in $1000s) a. b.

West n2 = 45 2 = 78 (in $1000s)

s2 = 8 (in $1000s)

Develop an interval estimate for the difference between the average yearly incomes of the marketing managers in the East and West. Use α = .05. At 95% confidence, use the p-value approach and test to determine if the average yearly income of marketing managers in the East is significantly different from the West.

ANSWER: a. b.

–9.084 to –2.916 test statistic t = –3.937 (df = 80); p-value < .005; reject H0 and conclude that there is a significant difference in the average yearly income of marketing managers in the East and in the West.

119. In order to estimate the difference between the average daily sales of two branches of a department store, the following data have been gathered: Sample size Sample mean (in $1000s) Sample standard deviation (in $1000s) a. b. c. d.

Downtown Store n1 = 23 days 1= 37 s1 = 4

North Mall Store n2 = 26 days 2= 34 s2 = 5

Determine the point estimate of the difference between the means. Determine the degrees of freedom for this interval estimation. Compute the margin of error. Develop a 95% confidence interval for the difference between the two population means.

ANSWER: a. b. c. d.

3 46 (rounded down from 46.56) 2.591 3 ± 2.591 (in dollars it will be from $409 to $5,591)

120. Babies weighing less than 5.5 pounds at birth are considered “low-birth-weight babies.” In the United States, 7.6% of Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations newborns are low-birth-weight babies. The following information was accumulated from samples of new births selected from two counties: Sample size Number of low-birth-weight babies a. b.

Hamilton 150 18

Shelby 200 22

Develop a 95% confidence interval estimate for the difference between the proportions of low-birth-weight babies in the two counties. Is there conclusive evidence that one of the proportions is significantly more than the other? If yes, which county? Explain, using the results of part (a). Do not perform any test.

ANSWER: a. b.

–.0577 to .0777 Since the range of the interval is from negative to positive, there is no indication that one proportion is significantly different (at 95% confidence) from the other.

121. A poll was administered this year asking college students if they considered themselves overweight. A similar poll was administered five years ago. Results are summarized below. Has the proportion increased significantly? Let α = 0.05.

Present sample Previous sample

Sample Size 300 275

Number Considered Themselves Overweight 150 121

ANSWER: H0: p1 – p2 ≤ 0 Ha: p1 – p2 > 0 z = 1.44; p-value = 0.0749; do not reject H0. There is not sufficient evidence to conclude that the proportion of college students who consider themselves overweight has increased significantly. 122. A potential investor conducted a 144-day survey in each theater in order to determine the difference between the average daily attendance at the North Mall and South Mall theaters. The North Mall Theater averaged 630 patrons per day, while the South Mall Theater averaged 598 patrons per day. From past information, it is known that the variance for North Mall is 1,000, while the variance for the South Mall is 1,304. Develop a 95% confidence interval for the difference between the average daily attendances at the two theaters. ANSWER: 24.16 to 39.84 123. A comparative study of organic and conventionally grown produce was checked for the presence of E. coli. Results are summarized below. Is there a significant difference in the proportion of E. coli in organic versus conventionally grown produce? Test at α = .10. Organic Conventional

Sample Size 200 500

E. Coli Prevalence 3 20

ANSWER: H0: p1 – p2 = 0 Ha: p1 – p2 ≠ 0 z = –1.68, p-value = .093; reject H0 and conclude that there is a significant difference in the proportion of E.coli in organic versus conventionally grown produce. Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations 124. Starting annual salaries for business school graduates majoring in finance and management information systems (MIS) were collected in two independent random samples summarized below. Based on previous studies, the population standard deviations for finance and MIS salaries are estimated to be $2,100 and $2,600, respectively. Finance

MIS

n1 = 60

n2 = 50

1 = $43,200

2 = $46,500

a. Develop a 95% confidence interval estimate of the difference between the starting salaries for the two majors. b. Using α = .10, test to determine if the average starting salary for an MIS graduate is $4,000 more than the starting salary for a finance graduate. Use both the critical value and p-value approaches to hypothesis testing. (Hint: The null hypothesis is H0: μ1 – μ2 = $4,000, where μ1 is the average starting salary of MIS graduates.) ANSWER: a. MIS is higher by $2,404.62 to $4,195.38. b. z = –1.53 > –1.645; do not reject H0. There is not sufficient evidence to conclude that the average starting salary for an MIS graduate is $4,000 more than the starting salary for a finance graduate. 125. A manager is thinking of providing, on a regular basis, in-house training for employees preparing for an inventory management certification exam. In the past, some employees received the in-house training before taking the exam, while others did not. Independent random samples selected from the company’s records provided the following exam scores for 10 workers who did not receive in-house training and eight workers who did receive training. (The manager is confident that the distributions of both populations’ exam scores are approximately normal.) No Training 76 80 60 91 73 77 82 68 75 86

Training 80 66 71 79 94 74 83 78

a. Develop a 95% confidence interval estimate for the difference between the average test scores for the two populations of employees. b. Using α = .05, test for any difference between the average test scores for the two populations of employees. ANSWER: a. –10.02 to 7.37 (thus, μ1 – μ2 could be 0) b. t = –.325 > –2.131, p-value = .75 > .05; do not reject H0: μ1 – μ2 = 0. There is not sufficient evidence to conclude that there is a difference between the average test scores. 126. A survey was recently conducted to determine if consumers spend more on computer-related purchases via the Internet or store visits. Assume a random sample of eight respondents provided the following data on their computerrelated purchases during a 30-day period. Using a .05 level of significance, can we conclude that consumers spend more on computer-related purchases by way of the Internet than by visiting stores? Copyright Cengage Learning. Powered by Cognero.

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Chapter 10: Statistical Inferences About Means and Proportions with Two Populations Respondent 1 2 3 4 5 6 7 8

Expenditures (dollars) In-Store 132 90 119 16 85 248 64 49

Internet 225 24 95 55 13 105 57 0

ANSWER: t = 1.12 < 1.89, p-value = .15 > .05; do not reject H0: μd < 0. There is not sufficient evidence to conclude that consumers spend more on computer-related purchases by way of the Internet than by visiting stores. 127. A movie based on a best-selling novel was recently released. Six hundred viewers of the movie, 235 of whom had previously read the novel, were asked to rate the quality of the movie. The survey showed that 141 of the novel readers gave the movie a rating of excellent, while 248 of the non-readers gave the movie an excellent rating. a. Develop an interval estimate of the difference between the proportions of the two populations, using a .05 level of significance, as the basis for your decision. b. Can we conclude, on the basis of a hypothesis test about p1 – p2, that the proportion of the non-readers of the novel who thought the movie was excellent is greater than the proportion of readers of the novel who thought the movie was excellent? Use a 0.05 level of significance. (Hint: This is a one-tailed test.) ANSWER: a. .0118 to .1690 (note that 0 is not in the interval) b. H0: pNR – pR < 0; z = 2.274 > 1.645; p-value = 0.0115 < .05; reject H0 and conclude that the proportion of the non-readers of the novel who thought the movie was excellent is greater than the proportion of readers of the novel who thought the movie was excellent.

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Chapter 11: Inferences About Population Variances Multiple Choice 1. A sample of 28 elements is selected to estimate a 95% confidence interval for the variance of the population. The χ2 values to be used for this interval estimation are _____. a. –1.96 and 1.96 b. 14.573 and 43.195 c. 16.151 and 40.113 d. 15.308 and 44.461 ANSWER: b 2. We are interested in testing whether the variance of a population is significantly less than 1.44. The null hypothesis for this test is _____. a. H0: σ2 < 1.44 b. H0: s2 ≥ 1.44 c. H0: σ ≤ 1.20 d. H0: σ2 ≥ 1.44 ANSWER: d 3. A sample of 41 observations yielded a sample standard deviation of 5. If we want to test H0: σ2 = 20, the test statistic is _____. a. 100 b. 10 c. 51.25 d. 50 ANSWER: d 4. The value of F.05 with 8 numerator and 19 denominator degrees of freedom is _____. a. 2.48 b. 2.58 c. 3.63 d. 2.96 ANSWER: a 5. To avoid the problem of not having access to tables of F distribution with values given for the lower tail, the numerator of the test statistic should be the one with the _____. a. larger sample size b. smaller sample size c. larger sample variance d. smaller sample variance ANSWER: c 6. The symbol used for the variance of the population is _____. a. σ Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances b. σ2 c. s d. s2 ANSWER: b 7. The symbol used for the variance of the sample is _____. a. σ b. σ2 c. s d. s2 ANSWER: d 8. The random variable for a chi-square distribution may assume any _____. a. value between –1 and 1 b. value between –infinity and +infinity c. negative value d. value greater than 0 ANSWER: d 9. A sample of n observations is taken from a population. When performing statistical inference about a population variance, the appropriate χ2 distribution has _____. a. n degrees of freedom b. n – 1 degrees of freedom c. n – 2 degrees of freedom d. n – 3 degrees of freedom ANSWER: b 10. For an F distribution, the number of degrees of freedom for the numerator _____. a. must be larger than the number of degrees for the denominator b. must be smaller than the number of degrees of freedom for the denominator c. must be equal to the number of degrees of freedom for the denominator d. can be larger, smaller, or equal to the number of degrees of freedom for the denominator ANSWER: d 11. The bottler of a certain soft drink claims its equipment to be accurate and that the variance of all filled bottles is .05 or less. The null hypothesis in a test to confirm the claim would be written as _____. a. H0: σ2 ≥ .05 b. H0: σ2 > .05 c. H0: σ2 < .05 d. H0: σ2 ≤ .05 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances 12. A sample of 20 cans of tomato juice showed a standard deviation of 0.4 ounce. A 95% confidence interval estimate of the variance for the population is _____. a. .2313 to .8533 b. .2224 to .7924 c. .0889 to .3169 d. .0925 to .3413 ANSWER: d 13. The manager of the service department of a local car dealership has noted that the service times of a sample of 15 new automobiles has a standard deviation of 4 minutes. A 95% confidence interval estimate for the variance of service times for all its new automobiles is _____. a. 8.58 to 39.79 b. 4 to 16 c. 4 to 15 d. 1.64 to 1.96 ANSWER: a 14. The manager of the service department of a local car dealership has noted that the service times of a sample of 30 new automobiles has a standard deviation of 6 minutes. A 95% confidence interval estimate for the standard deviation of the service times for all its new automobiles is _____. a. 16.05 to 45.72 b. 4.78 to 8.07 c. 2.93 to 6.31 d. 22.83 to 65.06 ANSWER: b 15. The producer of a certain medicine claims that its bottling equipment is very accurate and that the standard deviation of all its filled bottles is 0.1 ounce or less. A sample of 20 bottles showed a standard deviation of .11. The test statistic to test the claim is _____. a. 400 b. 22.99 c. 4.85 d. 20 ANSWER: b 16. The producer of a certain bottling equipment claims that the variance of all its filled bottles is .027 or less. A sample of 30 bottles showed a standard deviation of .2. The p-value for the test is _____. a. between .025 and .05 b. between .05 and .01 c. .05 d. .025 ANSWER: a 17. We are interested in testing to see if the variance of a population is less than 7. The correct null hypothesis is _____. a. σ < 7 Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances b. σ2 ≥ 49 c. s < 49 d. s > 49 ANSWER: b 18. A random sample of 31 charge sales showed a sample standard deviation of $50. A 90% confidence interval estimate of the population standard deviation is _____. a. 1,715.10 to 4,055.68 b. 1,596.45 to 4,466.73 c. 39.96 to 66.83 d. 41.39 to 63.68 ANSWER: d 19. The χ2 values (for interval estimation) for a sample size of 10 at 95% confidence are _____. a. 3.32511 and 16.9190 b. 2.70039 and 19.0228 c. 4.16816 and 14.6837 d. 3.24697 and 20.4831 ANSWER: b 20. The χ2 value for a one-tailed (upper tail) hypothesis test at 95% confidence and a sample size of 25 is _____. a. 33.1963 b. 36.4151 c. 39.3641 d. 37.6525 ANSWER: b 21. The χ2 value for a one-tailed test (lower tail) when the level of significance is .1 and the sample size is 15 is _____. a. 21.0642 b. 23.6848 c. 7.78453 d. 6.57063 ANSWER: c 22. The critical value of F at 95% confidence when there is a sample size of 21 for the sample with the smaller variance and there is a sample size of 9 for the sample with the larger sample variance is _____. a. 2.45 b. 2.94 c. 2.37 d. 2.10 ANSWER: a

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Chapter 11: Inferences About Population Variances 23. The sampling distribution of the

quantity is the _____.

a. χ2 distribution b. normal distribution c. F distribution d. t distribution ANSWER: a 24. The sampling distribution of the ratio of independent sample variances extracted from two normal populations with equal variances is the _____. a. χ2 distribution b. normal distribution c. Z distribution d. t distribution ANSWER: c 25. The 95% confidence interval estimate for a population variance when a sample variance of 30 is obtained from a sample of 12 items is _____. a. 14.14 to 74.94 b. 15.05 to 86.48 c. 16.42 to 94.35 d. 16.77 to 72.13 ANSWER: b 26. The 99% confidence interval estimate for a population variance when a sample standard deviation of 12 is obtained from a sample of 10 items is _____. a. 4.58 to 62.25 b. 46.53 to 422.17 c. 54.94 to 747.01 d. 62.04 to 562.89 ANSWER: c 27. The 90% confidence interval estimate for a population standard deviation when a sample variance of 50 is obtained from a sample of 15 items is _____. a. 4.18 to 15.07 b. 5.18 to 11.15 c. 5.44 to 10.32 d. 29.55 to 106.53 ANSWER: c 28. To avoid the problem of not having access to tables of the F distribution with values given for the lower tail when a two-tailed test is required, let the smaller sample variance be _____. a. the denominator of the test statistic Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances b. the numerator of the test statistic c. at least 1 d. less than 1 ANSWER: a 29. To avoid the problem of having access to tables of the F distribution with values for the lower tail when a one-tailed test is required, let the _____ variance be the numerator of the test statistic. a. smaller sample b. larger sample c. sample variance from the population with the smaller hypothesized d. sample variance from the population with the larger hypothesized ANSWER: c 30. Which of the following has a χ2 distribution? a. (n – 1)σ2/s2 b. (n – 2)σ2/s2 c. (n – 1)s/σ d. (n – 1)s2/σ2 ANSWER: a 31. Which of the following has an F distribution? a. b. c.

ANSWER: a 32. The sampling distribution used when making inferences about a single population's variance is a(n) _____ distribution. a. F b. t c. χ2 d. normal ANSWER: c 33. The sampling distribution of the ratio of two independent sample variances taken from normal populations with equal variances is a(n) _____ distribution. Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances a. F b. χ2 c. t d. normal ANSWER: a 34. In Excel, which of the following functions is used to construct a confidence interval for a population variance? a. CHISQ.DIST b. F-Test c. CHI.INV d. NORM.S.INV ANSWER: c 35. In Excel, which of the following functions is used to conduct a hypothesis test (using the p-value) for a population variance? a. CHISQ.DIST b. F-Test c. CHI.INV d. NORM.S.INV ANSWER: a 36. In Excel, which of the following functions is used to conduct a hypothesis test for comparing two population variances? a. CHISQ.DIST b. F-Test c. CHI.INV d. NORM.S.INV ANSWER: b 37. The value of F0.01 with 9 numerator and 20 denominator degrees of freedom is _____. a. 2.39 b. 2.94 c. 2.91 d. 3.46 ANSWER: d 38. A sample of 60 items from population 1 has a sample variance of 8, while a sample of 40 items from population 2 has a sample variance of 10. If we test whether the variances of the two populations are equal, the test statistic will have a value of _____. a. .8 b. 1.56 c. 1.5 d. 1.25 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances 39. A sample of 21 elements is selected to estimate a 90% confidence interval for the variance of the population. The χ2 value(s) to be used for this interval estimation is(are) _____. a. –1.96 and 1.96 b. 12.443 c. 10.851 and 31.410 d. 12.443 and 28.412 ANSWER: d Exhibit 11-1 Last year, the standard deviation of the ages of the students at UA was 1.81 years. Recently, a sample of 10 students had a standard deviation of 2.1 years. We are interested in testing to see if there has been a significant change in the standard deviation of the ages of the students at UA. 40. Refer to Exhibit 11-1. The test statistic is _____. a. 14.2 b. 12.1 c. 3.28 d. 2.1 ANSWER: b 41. Refer to Exhibit 11-1. At 95% confidence, the null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 11-2 We are interested in determining whether the variances of the sales at two music stores (A and B) are equal. A sample of 25 days of sales at store A has a sample standard deviation of 30, while a sample of 16 days of sales from store B has a sample standard deviation of 20. 42. Refer to Exhibit 11-2. The test statistic is _____. a. 1.50 b. .67 c. 1.56 d. 2.25 ANSWER: c 43. Refer to Exhibit 11-2. At 95% confidence, the null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances Exhibit 11-3 The contents of a sample of 26 cans of apple juice showed a standard deviation of 0.06 ounce. We are interested in testing to determine whether the variance of the population is significantly more than .003. 44. Refer to Exhibit 11-3. The null hypothesis is _____. a. s2 > .003 b. s2 ≤ .003 c. σ2 > .003 d. σ2 ≤ .003 ANSWER: d 45. Refer to Exhibit 11-3. The test statistic is _____. a. 1.2 b. 31.2 c. 30 d. 500 ANSWER: c 46. Refer to Exhibit 11-3. At 95% confidence, the critical value(s) from the table is(are) _____. a. 13.1197 and 40.6465 b. 37.6525 c. 14.6114 and 37.6525 d. 14.6114 ANSWER: b 47. Refer to Exhibit 11-3. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b 48. Refer to Exhibit 11-3. The p-value for this test is _____. a. .05 b. greater than .10 c. less than .10 d. 1.96 ANSWER: b Exhibit 11-4 n = 30 s2 = 625

H0: σ2 = 500 Ha: σ2 ≠ 500

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Chapter 11: Inferences About Population Variances 49. Refer to Exhibit 11-4. The test statistic for this problem equals _____. a. 23.2 b. 24 c. 36.25 d. 37.5 ANSWER: c 50. Refer to Exhibit 11-4. The null hypothesis is to be tested at the 5% level of significance. The critical value(s) from the table is(are) _____. a. 42.5569 b. 43.7729 c. 16.0471 and 45.7222 d. 16.7908 and 46.9792 ANSWER: c 51. Refer to Exhibit 11-4. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 11-5 n = 14 s = 20

H0: σ2 ≤ 410 Ha: σ2 > 410

52. Refer to Exhibit 11-5. The test statistic for this problem equals _____. a. .63 b. 12.68 c. 13.33 d. 13.66 ANSWER: b 53. Refer to Exhibit 11-5. The null hypothesis is to be tested at the 5% level of significance. The critical value(s) from the table is(are) _____. a. 22.3621 b. 23.6848 c. 5.00874 and 24.7356 d. 5.62872 and 26.119 ANSWER: a 54. Refer to Exhibit 11-5. The null hypothesis _____. a. should be rejected b. should not be rejected Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances c. should be revised d. should be retested ANSWER: b Exhibit 11-6 2

s n

Sample A 32 24

Sample B 38 16

We want to test the hypothesis that the population variances are equal. 55. Refer to Exhibit 11-6. The test statistic for this problem equals _____. a. .67 b. .84 c. 1.19 d. 1.50 ANSWER: c 56. Refer to Exhibit 11-6. The null hypothesis is to be tested at the 10% level of significance. The critical value from the table is _____. a. 2.11 b. 2.13 c. 2.24 d. 2.29 ANSWER: b 57. Refer to Exhibit 11-6. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 11-7 s2 n

Sample A 22 10

Sample B 25 8

We want to test the hypothesis that population B has a smaller variance than population A. 58. Refer to Exhibit 11-7. The test statistic for this problem equals _____. a. .77 b. .88 c. 1.14 d. 1.29 Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances ANSWER: c 59. Refer to Exhibit 11-7. The null hypothesis is to be tested at the 5% level of significance. The critical value from the table is _____. a. 3.07 b. 3.29 c. 3.35 d. 3.68 ANSWER: b 60. Refer to Exhibit 11-7. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 11-8 n = 23 s2 = 60

H0: σ2 ≥ 66 Ha: σ2 < 66

61. Refer to Exhibit 11-8. The test statistic has a value of _____. a. 20.91 b. 24.20 c. 24.00 d. 20.00 ANSWER: d 62. Refer to Exhibit 11-8. If the test is to be performed at 95% confidence, the critical value(s) from the table is(are) _____. a. 10.9823 and 36.7897 b. 33.9244 c. 12.3380 d. 43.7729 ANSWER: c 63. Refer to Exhibit 11-8. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b 64. Refer to Exhibit 11-8. The p-value is _____. a. less than .025 Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances b. less than .05 c. less than .10 d. greater than .10 ANSWER: d Exhibit 11-9 n = 14

s = 20

H0: σ2 ≤ 500 Ha: σ2 ≥ 500

65. Refer to Exhibit 11-9. The test statistic for this problem equals _____. a. .63 b. 12.68 c. 13.33 d. 13.66 ANSWER: b 66. Refer to Exhibit 11-9. The null hypothesis is to be tested at the 5% level of significance. The critical value(s) from the table is(are) _____. a. 22.362 b. 23.685 c. 5.009 and 24.736 d. 5.629 and 26.119 ANSWER: a 67. Refer to Exhibit 11-9. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 11-10 n = 81

s2 = 625

H0: σ2 = 500 Ha: σ2 ≠ 500

68. Refer to Exhibit 11-10. The test statistic for this problem equals _____. a. 100 b. 101.88 c. 101.25 d. 64 ANSWER: a 69. Refer to Exhibit 11-10. The p-value is between _____. a. .025 and .05 b. .05 and .1 Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances c. .1 and .2 d. .2 and .3 ANSWER: c 70. Refer to Exhibit 11-10. At 95% confidence, the null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b 71. In a hypothesis test about two population variances, the test statistic F is computed as _____. a.

ANSWER: b 72. Which of the following rejection rules is proper? a. Reject H0 if p-value ≥ Fα b. Reject H0 if p-value ≤ c. Reject H if p-value ≥ 0

d. Reject H0 if p-value ≤  ANSWER: d 73. There is a .90 probability of obtaining a χ2 value such that _____. a. b. c. Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances d. ANSWER: d 74.

= 8.9066 indicates that _____. a. 97.5% of the chi-square values are greater than 8.9066 b. 97.5% of the chi-square values are less than 8.9066 c. 2.5% of the chi-square values are greater than 8.9066 d. 5% of the chi-square values are more than 8.9066 from the mean

ANSWER: a 75. In practice, the most frequently encountered hypothesis test about a population variance is a _____. a. one-tailed test, with rejection region in lower tail b. one-tailed test, with rejection region in upper tail c. two-tailed test, with equal-size rejection regions d. two-tailed test, with unequal-size rejection regions ANSWER: b Subjective Short Answer 76. A sample of 15 items provides a sample mean of 18 and a sample variance of 16. Compute a 95% confidence interval estimate for the standard deviation of the population. ANSWER: 2.93 to 6.31 (rounded) 77. A sample of 30 items provided a sample mean of 28 and a sample standard deviation of 6. Test the following hypotheses using α = .05. What is your conclusion? H0: σ2 ≤ 25 Ha: σ2 > 25 ANSWER: χ2 = 41.76 < 45.7222; do not reject H0, there is not sufficient evidence to conclude that the population variance exceeds 25 78. A sample of 61 items provided a sample mean of 932, a sample mode of 900, and a sample standard deviation of 11. Test the following hypotheses using α = .05. What is your conclusion? H0: α2 ≤ 80 Ha: α2 > 80 ANSWER: χ2 = 90.75 > 83.2976; reject H0, there is sufficient evidence to conclude that the population variance exceeds 80 79. We are interested in determining whether or not the variances of the sales at two small grocery stores are equal. A sample of 16 days of sales at each store indicated the following: Store A Store B n1 = 16 n2 = 16 s1 = $125 s2 = $105 Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances Are the variances of the populations (from which these samples came) equal? Use α = .05. ANSWER: F = 1.42; do not reject H0, there is not sufficient evidence to conclude that the variance of the sales at two small grocery stores are different 80. A random sample of 25 employees of a local utility firm showed that their monthly incomes had a sample standard deviation of $112. Provide a 90% confidence interval estimate for the standard deviation of the incomes for all the firm's employees. ANSWER: 90.93 to 147.43 (rounded) 81. A random sample of 41 scores of students taking the ACT test showed a standard deviation of 8 points. Provide a 98% confidence interval estimate for the standard deviation of all the ACT test scores. ANSWER: 6.34 to 10.74 (rounded) 82. A lumber company has claimed that the standard deviation for the lengths of its six-foot boards is 0.5 inch or less. To test its claim, a random sample of 17 six-foot boards is selected, and it is determined that the standard deviation of the sample is .43. Do the results of the sample support the company's claim? Use α = .1. ANSWER: Since χ2 = 11.83 < 23.54, do not reject H0, there is not sufficient evidence to refute the company's claim. 83. An egg-packing company has stated that the standard deviation of the weights of its grade A large eggs is 0.07 ounce or less. The sample variance for 51 eggs was 0.0065 ounce. Can this sample result confirm the company's claim? Use α = .1. ANSWER: Since χ2 = 66.33 > 63.17, reject H0, there is sufficient evidence to refute the company's claim. 84. The standard deviation of the daily temperatures in Honolulu last year was 3.2 degrees Fahrenheit. A random sample of 19 days resulted in a standard deviation of 4 degrees Fahrenheit. Has there been a significant change in the variance of the temperatures? Use α = .02. ANSWER: Since χ2 = 28.125, do not reject H0, there is not sufficient evidence to conclude that there has been a significant change in the variance of the temperatures. 85. Do the following data indicate that the variance of the population from which this sample has been drawn is 17? Use α = .05. 12 5 9 14 10 2 ANSWER: Since χ = 2.70, do not reject H0, there is not sufficient evidence to conclude that the variance of the population from which this sample has been drawn differs from 17. 86. At α = .1, test to see if the population variances from which the following samples were drawn are equal. Group 1 n1 = 21 s1 = 18

Group 2 n2 = 19 s2 = 16

ANSWER: Since F = 1.26 < 2.19, do not reject H0, there is not sufficient evidence to conclude that the population variances from which the following samples were drawn are different. 87. The standard deviation of the ages of a sample of 16 executives from the northern states was 8.2 years, while the standard deviation of the ages of a sample of 25 executives from the southern states was 12.8 years. At α = .1, test to see if Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances there is any difference in the standard deviations of the ages of all the northern and southern executives. ANSWER: Since F = 2.44 > 2.29, reject H0, there is sufficient evidence to conclude that there is a difference in the standard deviations of the ages of all the northern and southern executives. 88. Student advisors are interested in determining if the variances of the grades of day students and night students are the same. The following samples are drawn: Day Night n1 = 25 n2 = 31 s1 = 9.8 s2 = 14.7 Test the equality of the variances of the populations at 95% confidence. ANSWER: Since F = 2.25 > 2.21, reject H0, there is sufficient evidence to conclude that there is a difference in the variances of the grades of day students and night students. 89. The grades of a sample of five students, selected from a large population, are given below. Grade 70 80 60 90 75 a. b. c.

Determine a point estimate for the variance of the population. Determine a 95% confidence interval for the variance of the population. At 90% confidence, test to determine if the variance of the population is significantly less than 130.

ANSWER:

a. b. c.

125 44.87 to 1032.17 (rounded) H0: σ2 ≥ 130 χ2 = 3.85 > 1.06; reject H0, there is sufficient evidence to conclude that the Ha: σ2 < 130 variance of the population is significantly less than 130

90. A machine produces pipes used in airplanes. The average length of the pipe is 16 inches. The acceptable variance for the length is 0.3 inch. A sample of 25 pipes was taken. The average length in the sample was 15.95 inches with a variance of 0.4 inch. a. Construct a 95% confidence interval for the population variance. b. State the null and alternative hypotheses to be tested. c. Compute the test statistic. d. The null hypothesis is to be tested at the 5% level of significance. State the decision rule for the test. e. What do you conclude about the population variance? ANSWER:

.2439 to .7741

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Chapter 11: Inferences About Population Variances b. c. d. e.

H0: σ2 = .3 Ha: σ2 ≠ .3 32 Do not reject H0 if 12.4011 < χ2 < 39.3641 Reject H0 if χ2 > 39.3641 or χ2 < 12.4011 Do not reject H0, there is not sufficient evidence to conclude that the population variance is unacceptable.

91. You are given the following results from a sample of five observations. 4 a. b. c. d.

Construct a 99% confidence interval for the population variance. The null and alternative hypotheses are H0: σ2 ≥ 2 and Ha: σ2 < 2. Compute the test statistic. Perform the test of the hypothesis at the 1% level. What do you conclude about the population variance?

ANSWER:

a. b. c. d.

.4038 to 28.9869 3 Reject H0 if χ2 < 0.29711 Do not reject H0, there is not sufficient evidence to conclude that the population variance is less than 2.

92. It is crucial that the variance of a production process be less than or equal to 25. A sample of 22 is taken. The sample variance equaled 26. a. Construct a 90% confidence interval for the population variance. b. Construct a 90% confidence interval for the population standard deviation. c. State the null and alternative hypotheses to be tested. d. Compute the test statistic. e. The null hypothesis is to be tested at the 10% level of significance. State the decision rule for the test. f. What do you conclude about the population variance? ANSWER:

a. b. c. d. e. f.

16.7123 to 47.1043 4.0881 to 6.8633 H0: σ2 ≤ 25 Ha: σ2 > 25 21.84 Reject H0 if χ2 > 29.6151. Do not reject H0, there is not sufficient evidence to conclude that the variance exceeds 25.

93. It has been suggested that night shift workers show more variability in their output levels than do day workers. Below are the results of two independent random samples. Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances Night Shift 9 520 25

Sample size Sample mean Sample variance a. b. c. d.

Day Shift 8 540 23

State the null and alternative hypotheses to be tested. Compute the test statistic. The null hypothesis is to be tested at the 5% level of significance. State the decision rule for the test. What do you conclude?

ANSWER: a. b. c. d.

H0:

≤

Ha: > where population 1 is the day shift and population 2 is the night shift .92 Reject H0 if F > 3.5. Do not reject the null hypothesis since .92 < 3.5, there is not sufficient evidence to conclude that there is more variability in night shift workers' output levels than that of day workers.

94. A coach is deciding on whether to buy stopwatches from company A or company B. A test was set up to see how many seconds each stopwatch was off in a precise 10-minute test period. For samples of watches from companies A and B, the following information on stopwatch errors was found. Company A Company B Sample size 5 8 Sample mean (seconds) .18 .15 Sample variance .3 1.1 Test the hypothesis of equal variances using a .05 level of significance. Be sure to state the null and alternative hypotheses being tested and the final conclusions of the test. ANSWER: H0:

Ha:

≠

Do not reject H0; 3.667 < 9.07, there is not sufficient evidence to conclude that the variances differ between the two companies.

95. The president of a bank believes that the variance of the deposits of suburban customers is less than the variance of city customers. Below are the results of samples taken from suburban and city customers. Customers s n a. b. c. d.

Suburban Customers $780 $100 5

City Customers $600 $90 7

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Chapter 11: Inferences About Population Variances ANSWER: a. b. c. d.

H0: ≤ Ha: > 1.235 Do not reject H0 if F < 4.53. Do not reject H0; 1.235 < 4.53, there is not sufficient evidence to conclude that the variance of the deposits of suburban customers is less than the variance of city customers.

96. Two classes in business statistics showed the following results on a recent test. Class 1 Class 2 Sample size 25 21 Sample mean 82 84 Sample standard deviation 6.2 7.6 Carry out a test to determine whether the difference in the variance of the scores on this test is due to chance variation or is statistically significant with a .02 level of significance. ANSWER: H0: Do not reject H0; 1.226 < 2.74, there is not sufficient evidence to conclude that the difference in the variance of the scores on this test is statistically significant.

Ha:

97. Test scores of two independent samples of students from UA and UB on a national examination are given below. At a .05 level of significance, test to determine if there is a significant difference between the variances of the two populations. UA 82 90 65 83 80 ANSWER:

UB 70 80 60 90 75 H0: Ha:

F = 125/84.5 = 1.479

Do not reject H0, there is not sufficient evidence to conclude that there is a significant difference between the variances of the two populations. 98. A random sample of 21 checking accounts at a bank showed an average daily balance of $430 with a standard deviation of $50. Provide a 95% confidence interval estimate of the variance of the population of the checking a. accounts. Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances Provide a 95% confidence interval estimate of the standard deviation of the population of the checking accounts. ANSWER: a. 1463.27 to 5213.76 b. 38.25 to 72.21 b.

99. A company has claimed that the standard deviation of the monthly incomes of its employees is less than or equal to $120. To test its claim, a random sample of 15 employees of the company was taken, and it was determined that the standard deviation of their incomes was $135. At a 5% level of significance, test the company's claim. ANSWER: H0: σ2 ≤ 14400 Ha: σ2 > 14400 χ2 = 17.72 < 23.68 Do not reject H0, there is not sufficient evidence to refute the company's claim. 100. A sample of 16 students showed that the variance in the number of hours they spend studying is 25. At a 5% level of significance, test to see if the variance of the population is significantly different from 30. ANSWER: H0: σ2 = 30 Ha: σ2 ≠ 30 χ2 = 12.5; do not reject H0 critical values: 6.26 and 27.49, there is not sufficient evidence to conclude that the variance is significantly different from 30

101. We are interested in determining whether or not the variances of the starting salaries of accounting majors are significantly different from the starting salaries of management majors. The following information was gathered from two samples: Accounting Management Sample size 16 16 Average monthly income $2,400 $2,500 Standard deviation $105 $125 At a 5% level of significance, test to determine whether or not the variances are equal. ANSWER: H0: Variances are equal Ha: Variances are not equal F = 1.42 < 2.86; do not reject H0, there is not sufficient evidence to conclude that the variance of the starting salaries of accounting majors is significantly different from that of management majors 102. The average grade of a sample of 25 students on their second statistics examination was 85 with a standard deviation of 9. Is the variance of the population significantly more than 80? Use a .05 level of significance. ANSWER: H0: σ2 ≤ 80 χ2 = 24.3 < 36.41; do not reject H0, there is not sufficient evidence to conclude Ha: σ2 > 80 that the variance of the population is significantly more than 80

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Chapter 11: Inferences About Population Variances 103. A company claims that the standard deviation in its delivery time is less than 5 days. A sample of 27 past customers is taken. The average delivery time in the sample was 14 days with a standard deviation of 4.5 days. At a 5% level of significance, test the company's claim. ANSWER: H0: σ2 ≥ 25 χ2 = 21.06 > 15.37; do not reject H0, there is not sufficient evidence to confirm Ha: σ < 25 the company's claim 104. A sample of 22 bottles of soft drink showed a variance of .64 in their contents. At a 5% level of significance, determine whether or not the standard deviation of the population is significantly different from .7. ANSWER: H0: σ2 = .49 Ha: σ2 ≠ .49 χ2 = 27.43 which is between 10.28 and 35.48 Do not reject H0, there is not sufficient evidence to conclude that the standard deviation of the population is significantly different from .7. 105. A random sample of 20 observations showed a standard deviation of 8. At a 5% level of significance, test to see if the variance of the population is significantly less than 65. ANSWER: H0: σ2 ≥ 65 χ2 = 18.71 > 10.11; do not reject H0, there is not sufficient evidence to Ha: σ2 < 65 conclude that the variance of the population is significantly less than 65

106. Do the following data indicate that the variance of the population from which this sample has been drawn is significantly more than 12? Use α = .05. 16 12 21 10 13 18 ANSWER: Since the test statistic χ2 = 7 < 11.075, do not reject H0 and conclude there is not sufficient evidence to show the variance is significantly more than 12. 107. A sample of 10 earnings per share estimates is shown below. 2.92 4.60 4.20 3.10 3.66 7.22 2.54 1.45 2.88 3.64 Use Excel to estimate the variance with a 95% level of confidence. ANSWER: 1 2 3 4 5 6 7 8 9 10

A Earnings 2.92 4.60 4.20 3.10 3.66 7.22 2.54 1.45 2.88

C Sample Size Variance

D =COUNT(A2:A11) =VAR(A2:A11)

Value for D 10 2.38245

Confidence Coefficient Level of Significance Chi-Square Value (lower tail) Chi-Square Value (upper tail)

0.95 =1-D4 =CHISQ.INV(D5/2,D1-1) =CHISQ.INV.RT(D5/2,D1-1)

0.95 0.05 2.70039 19.02278

Point Estimate Lower Limit

=D2 =((D1-1)*D2)/D7

2.38245 1.12718

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Chapter 11: Inferences About Population Variances 11

3.64

Upper Limit

=((D1-1)*D2)/D6

7.94037

108. The time it takes to complete a test was recorded for a sample of 15 students. The results follow. 25 41 39

45 41 49

50 48 46

32 30 44

38 40 36

Use Excel to estimate the variance with a 90% level of confidence. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12

A Times 25 45 50 32 38 41 41 48 30 40 32

C Sample Size Variance

D =COUNT(A2:A16) =VAR(A2:A16)

Value for D 15 50.40952

Confidence Coefficient Level of Significance (alpha) Chi-Square Value (lower tail) Chi-Square Value (upper tail)

0.9 =1-D4 =CHISQ.INV(D5/2,D1-1) =CHISQ.INV.RT(D5/2,D1-1)

0.9 0.1 6.57063 23.68478

Point Estimate Lower Limit Upper Limit

=D2 =((D1-1)*D2)/D7 =((D1-1)*D2)/D6

50.40952 29.79691 107.40723

109. A professor believes the variability in time to complete tests has increased. In the past, the variance in minutes to complete a particular test was 25. The time it takes to complete a test was recorded for a sample of 15 students. The results follow. 25 45 50 32 38 41 41 48 30 40 39 49 46 44 36 Use Excel to determine if the variance in times has increased. Use a .02 level of significance. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12

A Times 25 45 50 32 38 41 41 48 30 40 32

C Sample Size Sample Mean Variance

D =COUNT(A2:A16) =AVERAGE(A2:A16) =VAR(A2:A16)

Value for D 15 40.26667 52.35238

Hypothesized Value

Test Statistic Degrees of Freedom

=((D1-1)*D3)/D5 =D1-1

29.31733 14

p-value (Lower Tail) p-value (Upper Tail) p-value (Two Tail)

=CHISQ.DIST(D7,D8,TRUE) =CHISQ.DIST.RT(D7,D8) =2*MIN(D10,D9)

0.990536 0.009464 0.018928

Reject H0, there is sufficient evidence to conclude that the variance in times has increased. 110. A manufacturing company is considering changing suppliers of a particular raw material. The standard deviation of days until delivery for the current supplier is 3. A sample of five delivery times is taken from the new supplier. The data Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances follow: 2 2 1 5 3 The manufacturing company will use the new supplier if the variance in delivery time is less than that of the current supplier. Use Excel to determine whether the company should use the new supplier. Use a 10% level of significance. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12

A Days 2 2 1 5 3

C Sample Size Sample Mean Variance

D =COUNT(A2:A6) =AVERAGE(A2:A6) =VAR(A2:A6)

Value for D 5 2.6 2.3

Hypothesized Value

Test Statistic Degrees of Freedom

=((D1-1)*D3)/D5 =D1-1

1.02222 4

p-value (Lower Tail) p-value (Upper Tail) p-value (Two Tail)

=CHISQ.DIST(D7,D8,TRUE) =CHISQ.DIST.RT(D7,D8) =2*MIN(D10,D11)

0.093592 0.906408 0.187184

Reject H0, there is sufficient evidence to conclude that the company should use the new supplier. 111. The specifications for the filling of soft drink bottles is a variance of .05 (ounces)2. A sample of 20 bottles of soft drink showed the following results: 12.03 12.01 11.89 12.15 12.23 11.97 12.10 11.85 12.33 12.02 11.96 12.00 11.82 12.02 11.99 11.96 11.83 11.84 12.21 12.38 At a 5% level of significance, use Excel to determine whether or not the variance of the population is significantly different from .05. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12

A Ounces 12.03 11.97 11.96 11.96 12.01 12.10 12.00 11.83 11.89 11.85 11.82

C Sample Size Sample Mean Sample Variance

D =COUNT(A2:A21) =AVERAGE(A2:A21) =VAR(A2:A21)

Value for D 20 12.0295 0.025942

Hypothesized Value

0.05

Test Statistic Degrees of Freedom

=(D1-1)*D3/D5 =D1-1

9.8579 19

p-value (Lower Tail) p-value (Upper Tail) p-value (Two Tail)

=CHISQ.DIST(D7,D8,TRUE) =CHISQ.DIST.RT(D7,D8) =2*MIN(D10,D11)

0.04364 0.95636 0.08727

Do not reject H0, there is not sufficient evidence to conclude that the variance of the population is significantly different from .05. 112. The data below shows samples of annual salaries (in $1000s) for accountants in two different firms. Company A

Company B

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Chapter 11: Inferences About Population Variances 74.8 78.2 69.9 84 65.7 66.8 72.1 77.4 79.3 69.1

83.3 65.7 82.1 78.5 69.7 77.7 66.7 78.4 79.2 80.1 74.9 69.8 Use Excel to determine whether the variance in salaries in Company A differs from that of Company B. Use a .10 level of significance. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13

A Salaries A 74.8 78.2 69.9 84.0 65.7 66.8 72.1 77.4 79.3 69.1

B Salaries B 83.3 65.7 82.1 78.5 69.7 77.7 66.7 78.4 79.2 80.1 74.9 69.8

D F-Test 2-Sample for Variances

Mean Variance Observations Df F P(F<=f) one-tail F Critical one-tail

Salaries A 73.73 35.81789 10 9 0.98139 0.49707 0.32232

Salaries B 75.50833 36.49720 12 11

Do not reject H0, there is not sufficient evidence to conclude that the variance in salaries in Company A differs from that of Company B. 113. In a manufacturing production process, two machines are being compared. The data below show the diameters, measured in millimeters, of a sample of ball bearings produced by each machine. Machine A 1.95 2.45 2.50 2.75 2.38 2.26 2.33 2.20 2.16 2.20

Machine B 2.22 2.30 2.34 2.28 2.29 2.25 2.30 2.27 2.38 2.34

Use Excel to determine whether the variance in diameters differs between the two machines. Use a .05 level of significance. ANSWER: 1

A Mach. A

B Mach. B

D F-Test 2-Sample for Variances

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Chapter 11: Inferences About Population Variances 2 3 4 5 6 7 8 9 10 11 12 13

1.95 2.45 2.5 2.75 2.38 2.26 2.33 2.2 2.16 2.2

2.22 2.3 2.34 2.28 2.29 2.25 2.3 2.27 2.38 2.34

Machine A 2.31800 0.04808 10.00000 9.00000 21.84553 0.00004 4.02599

Mean Variance Observations Df F P(F<=f) one-tail F Critical one-tail

Machine B 2.29700 0.00220 10.00000 9.00000

Reject H0, there is sufficient evidence to conclude that the variance in diameters differs between the two machines. 114. The State Highway Patrol (SHP) periodically samples vehicle speeds at Milepost 92 on Interstate 17. The SHP is concerned about the dispersion of speeds of vehicles sharing the same highway because significant difference in speed is a known cause of accidents. The speeds of a random sample of 16 vehicles are shown below. Vehicle Speed Vehicle Speed

1 69.6 9 71.1

2 73.5 10 70.8

3 74.1 11 64.6

4 64.4 12 67.4

5 66.3 13 69.9

6 68.7 14 66.3

7 69.0 15 68.3

8 65.2 16 70.6

a. Use Excel to develop a 90% confidence interval estimate of the population variance (i.e., the speed variance of vehicles at Milepost 92 on Interstate 17). b. Develop a 90% confidence interval estimate of the population standard deviation. ANSWER: a. 6.4358 to 22.1553 b. 2.537 to 4.707

115. The State Highway Patrol (SHP) periodically samples vehicle speeds at Milepost 92 on Interstate 17. The SHP is concerned about the dispersion of speeds of vehicles sharing the same highway because significant difference in speed is a known cause of accidents. The speeds of a random sample of 16 vehicles are shown below. Copyright Cengage Learning. Powered by Cognero.

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Chapter 11: Inferences About Population Variances Vehicle Speed Vehicle Speed

1 69.6 9 71.1

2 73.5 10 70.8

3 74.1 11 64.6

4 64.4 12 67.4

5 66.3 13 69.9

6 68.7 14 66.3

7 69.0 15 68.3

8 65.2 16 70.6

The SHP’s policy is to position a patrol car at Milepost 92 on Interstate 17 if the vehicle speed variance at that location is believed to be greater than 6 mph. Use Excel to conduct a hypothesis test (with a = .10) to determine whether the speed variance for all vehicles passing Milepost 92 exceeds 6 mph. ANSWER: H0: σ2 ≤ 6.0 and Ha: σ2 > 6.0 Do not reject H0 because p-value (.12) is not ≤ α (.10).

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions Multiple Choice 1. A population where each element of the population is assigned to one and only one of several classes or categories is a(n) _____. a. multinomial population b. Poisson population c. normal population d. independent population ANSWER: a 2. The sampling distribution for a goodness of fit test is the _____. a. Poisson distribution b. t distribution c. normal distribution d. chi-square distribution ANSWER: d 3. A goodness of fit test is always conducted as a(n) _____. a. lower-tail test b. upper-tail test c. middle test d. two-tailed test ANSWER: b 4. An important application of the chi-square distribution is _____. a. testing for equality of three or less population proportions b. testing for goodness of fit c. testing for the dependence of two variables d. testing for the acceptance of the null hypothesis ANSWER: b 5. The number of degrees of freedom for the appropriate chi-square distribution in a test of independence is _____. a. n – 1 b. k – 1 c. number of rows minus 1 times number of columns minus 1 d. dependent upon the statement of the null hypothesis ANSWER: c 6. In order NOT to violate the requirements necessary to use the chi-square distribution, each expected frequency in a goodness of fit test must be _____. a. at least 5 b. at least 10 c. no more than 5 d. less than 2 Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions ANSWER: a 7. A statistical test conducted to determine whether to reject or not reject a hypothesized probability distribution for a population is known as a _____. a. contingency test b. probability test c. goodness of fit test d. dependence test ANSWER: c 8. The degrees of freedom for a contingency table with 12 rows and 12 columns is _____. a. 144 b. 121 c. 12 d. 120 ANSWER: b 9. The degrees of freedom for a contingency table with 6 rows and 3 columns is _____. a. 18 b. 15 c. 6 d. 10 ANSWER: d 10. The degrees of freedom for a contingency table with 10 rows and 11 columns is _____. a. 100 b. 110 c. 21 d. 90 ANSWER: d 11. Excel's _____ function is used to perform a goodness of fit test. a. ZTEST b. TTEST c. CHISQ.TEST d. NORM.S.DIST ANSWER: c 12. In a goodness of fit test, Excel's CHISQ.TEST function returns a _____. a. chi-square critical value b. chi-square test statistic c. p-value d. confidence interval estimate ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions 13. Excel's _____ function is used to perform a test of independence. a. ZTEST b. TTEST c. CHISQ.TEST d. NORM.S.DIST ANSWER: c 14. Excel's CHISQ.TEST function can be used to perform _____. a. a test for equality of population means b. a test for equality of population proportions c. a goodness of fit test d. a test for the difference in population means ANSWER: c Exhibit 12-1 Individuals in a random sample of 150 were asked whether they supported capital punishment. The following information was obtained. Do You Support Number of Capital Punishment? Individuals Yes 40 No 60 No Opinion 50 We are interested in determining whether the opinions of the individuals (as to Yes, No, and No Opinion) are uniformly distributed. 15. Refer to Exhibit 12-1. If the opinions are uniformly distributed, the expected frequency for each group would be _____. a. .333 b. .50 c. 1/3 d. 50 ANSWER: d 16. Refer to Exhibit 12-1. The calculated value for the test statistic equals _____. a. 2 b. –2 c. 20 d. 4 ANSWER: d 17. Refer to Exhibit 12-1. The number of degrees of freedom associated with this problem is _____. a. 150 b. 149 c. 2 Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions d. 3 ANSWER: c 18. Refer to Exhibit 12-1. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals _____. a. 7.37776 b. 7.81473 c. 5.99147 d. 9.34840 ANSWER: c 19. Refer to Exhibit 12-1. What conclusion should be made? a. There is enough evidence to conclude that the distribution is uniform. b. There is enough evidence to conclude that the distribution is not uniform. c. The test is inconclusive. d. The test should be done again to be certain of the results. ANSWER: a Exhibit 12-2 Last school year, the student body of a local university consisted of 30% freshmen, 24% sophomores, 26% juniors, and 20% seniors. A random sample of 300 students taken from this year's student body showed the following number of students in each class. Freshmen 83 Sophomores 68 Juniors 85 Seniors 64 We are interested in determining whether there has been a significant change in the distribution of class between the last school year and this school year. 20. Refer to Exhibit 12-2. If the distribution is the same as the previous year, the expected number of freshmen is _____. a. 83 b. 90 c. 30 d. 10 ANSWER: b 21. Refer to Exhibit 12-2. If the distribution is the same as the previous year, the expected frequency of seniors is _____. a. 60 b. 20 c. 68 d. 64 ANSWER: a 22. Refer to Exhibit 12-2. The calculated value for the test statistic equals _____. a. .5444 Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions b. 300 c. 1.6615 d. 6.6615 ANSWER: c 23. Refer to Exhibit 12-2. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals _____. a. 1.645 b. 1.96 c. 2.75 d. 7.815 ANSWER: d 24. Refer to Exhibit 12-2. The null hypothesis _____. a. should not be rejected b. should be rejected c. was designed wrong d. should be retested ANSWER: a Exhibit 12-3 In order to determine whether a particular medication was effective in curing the common cold, one group of patients was given the medication, while another group received sugar pills. The results of the study are shown below. Patients Patients Cured Not Cured Received medication 70 10 Received sugar pills 20 50 We are interested in determining whether the medication was effective in curing the common cold. 25. Refer to Exhibit 12-3. If the proportion of patients that are cured is independent of whether the patient received medication then the expected frequency of those who received medication and were cured is _____. a. 70 b. 150 c. 28 d. 48 ANSWER: d 26. Refer to Exhibit 12-3. The test statistic is _____. a. 10.08 b. 54.02 c. 1.96 d. 1.645 ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions 27. Refer to Exhibit 12-3. The number of degrees of freedom associated with this problem is _____. a. 4 b. 149 c. 1 d. 3 ANSWER: c 28. Refer to Exhibit 12-3. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals _____. a. 3.84 b. 7.81 c. 5.99 d. 9.34 ANSWER: a 29. Refer to Exhibit 12-3. The null hypothesis _____. a. should not be rejected b. should be rejected c. should be revised d. should be retested ANSWER: b Exhibit 12-4 In the past, 35% of the students at ABC University were in the Business College, 35% of the students were in the Liberal Arts College, and 30% of the students were in the Education College. To see whether the proportions have changed, a random sample of 300 students from ABC University was selected. Ninety of the sample students are in the Business College, 120 are in the Liberal Arts College, and 90 are in the Education College. 30. Refer to Exhibit 12-4. This problem is an example of a _____. a. normally distributed variable b. test for independence c. uniformly distributed variable d. multinomial population ANSWER: d 31. Refer to Exhibit 12-4. If the proportions are the same as they were in the past, the expected frequency for the Business College is _____. a. .3 b. .35 c. 90 d. 105 ANSWER: d 32. Refer to Exhibit 12-4. The calculated value for the test statistic equals _____. a. .01 Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions b. .75 c. 4.29 d. 4.38 ANSWER: c 33. Refer to Exhibit 12-4. The hypothesis is to be tested at the 5% level of significance. The critical value from the table equals _____. a. 1.645 b. 19.6 c. 5.99 d. 7.80 ANSWER: c 34. Refer to Exhibit 12-4. Based upon this test, what can be concluded? a. There is enough evidence to conclude that the proportions have changed significantly. b. There is enough evidence to conclude that the proportions have not changed significantly. c. The test is inconclusive. d. The test should be done again to be certain of the results. ANSWER: b Exhibit 12-5 The table below gives beverage preferences for random samples of teens and adults. Beverage Coffee Tea Soft drink Other

Teens Adults Total 50 200 250 100 150 250 200 200 400 50 50 100 400 600 1,000 We are asked to test for independence between age (i.e., adult and teen) and drink preferences. 35. Refer to Exhibit 12-5. With a .05 level of significance, the critical value for the test is _____. a. 1.645 b. 7.815 c. 14.067 d. 15.507 ANSWER: b 36. Refer to Exhibit 12-5. If age and drink preference is independent then the expected number of adults who prefer coffee would be _____. a. .25 b. .33 c. 150 d. 200 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions 37. Refer to Exhibit 12-5. The value of the test statistic for this test for independence is _____. a. 0 b. 8.4 c. 62.5 d. 82.5 ANSWER: c 38. Refer to Exhibit 12-5. What can be concluded from this test? a. There is enough evidence to conclude that age and drink preference is dependent. b. There is not enough evidence to conclude that age and drink preference is dependent. c. The test is inconclusive. d. The test should be done again to be certain of the results. ANSWER: a Exhibit 12-6 The following shows the number of individuals in a random sample of 300 adults who indicated they support the new tax proposal. Political Party Support Democrats 100 Republicans 120 Independents 80 We are interested in determining whether the opinions of the individuals of the three groups are uniformly distributed. 39. Refer to Exhibit 12-6. If the opinions of the individuals of the three groups are uniformly distributed, the expected frequency for each group is _____. a. .333 b. .50 c. 50 d. 100 ANSWER: d 40. Refer to Exhibit 12-6. The calculated value for the test statistic equals _____. a. 300 b. 4 c. 0 d. 8 ANSWER: d 41. Refer to Exhibit 12-6. The number of degrees of freedom associated with this problem is _____. a. 2 b. 3 c. 300 d. 299 ANSWER: a Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions 42. Refer to Exhibit 12-6. The test statistic for goodness of fit has a chi-square distribution with k – 1 degrees of freedom provided that the expected frequencies for all categories are _____. a. 5 or more b. 10 or more c. k or more d. 2k ANSWER: a 43. Refer to Exhibit 12-6. This test for goodness of fit _____. a. is a lower-tail test b. is an upper-tail test c. is a two-tailed test d. can be a lower-tail or upper-tail test ANSWER: b 44. The number of categories of outcomes per trial for a multinomial probability distribution is _____. a. two or more b. three or more c. four or more d. five or more ANSWER: b 45. The test for goodness of fit, test of independence, and test of multiple proportions are designed for use with _____. a. categorical data b. bivariate data c. quantitative data d. ordinal data ANSWER: a 46. The properties of a multinomial experiment include all of the following EXCEPT _____. a. the experiment consists of a sequence of n identical trials b. three or more outcomes are possible on each trial c. the probability of each outcome can change from trial to trial d. the trials are independent ANSWER: c Subjective Short Answer 47. Before the presidential debates, it was expected that the percentages of registered voters in favor of various candidates would be as follows: Democrats Republicans Independent Undecided

Percentages 48% 38% 4% 10%

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions After the presidential debates, a random sample of 1200 voters showed that 540 favored the Democrat candidate; 480 were in favor of the Republican candidate; 40 were in favor of the Independent candidate, and 140 were undecided. At a 5% level of significance, carry out a test to determine if the proportion of voters has changed. ANSWER: Since 8.18 > 7.81, reject H0. Therefore, there has been a change. 48. Last school year, in the school of Business Administration, 30% were Accounting majors, 24% Management majors, 26% Marketing majors, and 20% Economics majors. A random sample of 300 students taken this year showed the following number of students in each major: Accounting 83 Management 68 Marketing 85 Economics 64 Total 300 Has there been a significant change in the number of students in each major between the last school year and this school year? Use α = .05. ANSWER: Chi-square = 1.66 < 7.815; no significant change. 49. A medical journal reported the following frequencies of deaths due to cardiac arrest for each day of the week: Cardiac Death by Day of the Week Day Frequency Monday 40 Tuesday 17 Wednesday 16 Thursday 29 Friday 15 Saturday 20 Sunday 17 At a 5% level of significance, determine whether the number of deaths is uniform over the week. ANSWER: Chi-square = 23.25 > 12.59; distribution is not uniform. 50. The personnel department of a large corporation reported 60 resignations during the last year. The following table groups these resignations according to the season in which they occurred: Season Resignations Winter 10 Spring 22 Summer 19 Fall 9 Carry out a test to determine if the number of resignations is uniform over the four seasons. Let α = .05. ANSWER: Chi-square = 8.41 > 7.815; distribution is not uniform. 51. In 1996, forty percent of the students at a major university were Business majors, 35% were Engineering majors and the rest of the students were majoring in other fields. In a random sample of 600 students from the same university taken in 1997, 200 were Business majors, 220 were Engineering majors, and the remaining students in the sample were majoring in other fields. At a 5% significance level, test to see if there has been a significant change in the proportions Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions between 1996 and 1997. ANSWER: Chi-square = 13.15 > 5.99; thus, reject H0 and conclude there has been a significant change. 52. In the last presidential election before the candidates began their major campaigns, the percentages of registered voters who favored the various candidates were as follows: Registered Voters Percentages Republicans 34 Democrats 43 Independents 23 After the major campaigns began, a random sample of 400 voters showed that 172 favored the Republican candidate, 164 were in favor of the Democrat candidate, and 64 favored the Independent candidate. Use α = .01 to determine if the proportion of voters who favored the various candidates changed. ANSWER: Chi-square = 18.42 > 9.21; proportion has changed. 53. Before the rush began for Christmas shopping, a department store noted that the percentage of its customers who use the store's credit card, the percentage of those who use a major credit card, and the percentage of those who pay cash are the same. During the Christmas rush, a random sample of 150 shoppers was selected. Of those, 46 used the store's credit card, 43 used a major credit card, and 61 paid cash. Use α = .05 to determine if the methods of payment have changed during the Christmas rush. ANSWER: Chi-square = 3.72 < 5.99; has not changed 54. A major automobile manufacturer claimed that the frequencies of repairs on all five models of its cars are the same. A random sample of 200 repair services showed the following frequencies on the various makes of cars: Model of Car Frequency A 32 B 45 C 43 D 34 E 46 At α = .05, test the manufacturer's claim. ANSWER: Chi-square = 4.25 < 9.487; no difference 55. A lottery is conducted that involves the random selection of numbers from 0 to 4. To make sure that the lottery is fair, a random sample of 250 observations was taken. The following results were obtained: Value 0 1 2 3 4 a. b. c. d.

Frequency 40 45 55 60 50

State the null and alternative hypotheses to be tested. Compute the test statistic. The null hypothesis is to be tested at the 5% level of significance. Determine the critical value from the table. What do you conclude about the fairness of this lottery?

ANSWER: Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions a. b. c. d.

H0: p0 = p1 = p2 = p3 = p4 = .2 Ha: The population proportions are not p0 = p1 = p2 = p3 = p4 = .2 5 9.48773 Do not reject the null hypothesis. There is not enough evidence to conclude that the lottery is unfair.

56. The makers of Compute-All know that in the past, 40% of their sales were from people under 30 years old, 45% of their sales were from people who are between 30 and 50 years old, and 15% of their sales were from people who are over 50 years old. A random sample of 300 customers was taken to see if the market shares had changed. In the sample, 100 of the people were under 30 years old, 150 people were between 30 and 50 years old, and 50 people were over 50 years old. a. State the null and alternative hypotheses to be tested. b. Compute the test statistic. The null hypothesis is to be tested at the 1% level of significance. Determine the critical value c. from the table. d. What do you conclude? ANSWER: a. H0: p1 = .4, p2 = .45, and p3 = .15 Ha: The population proportions are not p1 = .4, p2 = .45, and p3 = .15 b. 5.55 c. 9.21034 Do not reject the null hypothesis. There is not enough evidence to conclude that the market d. share has changed.

57. Shown below is a 3 × 2 contingency table with observed values from a sample of 1,500. At 95% confidence, test for independence of the row and column factors. Column Factor Row Factor x y Total A 450 300 750 B 300 300 600 C 150 0 150 Total 900 600 1,500 ANSWER: Chi-square = 24.37 > 3.84; we do not have enough evidence to conclude that the row and column factors are independent. 58. Shown below is a 2 × 3 contingency table with observed values from a random sample of size 500. At 95% confidence, test for independence of the row and column factors. Column Factor Row Factor x y z A 40 50 110 B 60 100 140 ANSWER: Chi-square = 4.44 < 5.99; we have enough evidence to conclude that the row and column variables are independent. 59. A random sample of 150 individuals (males and females) was surveyed, and the individuals were asked to indicate their yearly incomes. The results of the survey are shown below. Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions Income Category Male Female Category 1: $20,000 up to $40,000 10 30 Category 2: $40,000 up to $60,000 35 15 Category 3: $60,000 up to $80,000 15 45 Test at α = .05 to determine if the yearly income is independent of the gender. ANSWER: Chi-square = 28.125 > 5.99; thus, reject H0 and conclude income is not independent of gender. 60. A random sample of 2,000 individuals from three different cities was asked whether they owned a foreign or a domestic car. The following contingency table shows the results of the survey. City Type of Car Detroit Atlanta Denver Total Domestic 80 200 520 800 Foreign 120 600 480 1,200 Total 200 800 1,000 2,000 At α = .05, test to determine if the type of car purchased is independent of the city in which the purchasers live. ANSWER: Chi-square = 135 > 5.99; thus, reject H0 and conclude that the type of car purchased is not independent of the city. 61. Dr. Ross's diet pills are supposed to cause significant weight loss. The following table shows the results of a recent study where a random sample of individuals took part in a placebo controlled study. No Weight Loss Weight Loss Total

Diet Pills 80 100 180

No Diet Pills 20 100 120

Total 100 200 300

With 95% confidence, determine if weight loss is dependent upon taking the diet pills. ANSWER: Chi-square = 25 > 5.99; weight loss appears to be dependent upon taking the diet pills. 62. Five hundred randomly selected automobile owners were questioned about the main reason they had purchased their current automobile. The results are given below. You would like to know if the reason for purchase is independent of gender. Main Reason for Purchase Male Female Total a. b. c. d.

Styling 70 30 100

Engineering 130 20 150

Fuel Economy 150 100 250

Total 350 150 500

State the null and alternative hypotheses for a test of independence. State the decision rule, using a .10 level of significance. Calculate the chi-square test statistic. Give your conclusion for this test.

ANSWER: a.

H0: Reason for purchase is independent of gender Ha: Reason for purchase is not independent of gender

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions b. c. d.

Do not reject H0 if chi-square ≤ 4.60517 Reject H0 if chi-square > 4.60517 31.746 Reject the null hypothesis and conclude that the reason for purchase is not independent of gender.

63. A random sample of 500 individuals was asked to cast their votes regarding a particular issue of the Equal Rights Amendment. The following contingency table shows the results of the votes: Vote Cast Gender Favor Undecided Oppose Total Female 180 80 40 300 Male 150 20 30 200 Total 330 100 70 500 Test at α = .05 to determine if voting preference is independent of the gender of the individuals. ANSWER: Chi-square = 20.99 > 5.99; votes are not independent of gender. 64. A random sample of 1,000 managers with degrees in Business Administration classified each person according to their field of concentration and management level as shown below. Major Top Management Middle Management Total Management 300 200 500 Marketing 200 0 200 Accounting 100 200 300 Total 600 400 1,000 Test at α = .01 to determine if the position in management is independent of the major of concentration. ANSWER: Chi-square = 222.2 > 9.21; the position is not independent of major 65. From a poll of a random sample of 800 television viewers, the following data have been collected. The table below classifies each individual by their level of education and preference of television station. Level of Education High School Bachelor Graduate Total Public broadcasting 150 150 100 400 Commercial stations 50 250 100 400 Total 200 400 200 800 Test at α = .05 to determine if the selection of a TV station is dependent upon the level of education. ANSWER: Chi-square = 75 > 5.99; selection of station is not independent of the level of education. 66. The data below represent the fields of specialization for a randomly selected sample of undergraduate students. Test to determine whether there is a significant difference in the fields of specialization between regions of the country. Use a .05 level of significance. Specialization Business Engineering Liberal Arts

Northeast 54 15 65

Region of United States Midwest South 65 28 24 8 84 33

West 93 33 98

Total 240 80 280 Page 14

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions Fine Arts Health Sciences Total

13 3 150

15 12 200

7 4 80

25 21 270

60 40 700

State the critical value of the chi-square random variable for this test of independence of categories. b. Calculate the value of the test statistic. c. What is the conclusion for this test? ANSWER: a. 21.0261 b. 8.674 c. Do not reject the null hypothesis that fields of specialization and region are independent. a.

67. During "sweeps week" last year, the viewing audience was distributed as follows: 36% NBC, 22% ABC, 24% CBS, and 18% FOX. This year during sweeps week, a random sample of 50 homes yielded the following data. Use Excel to test at α = .05 to determine if the audience proportions have changed. ABC FOX ABC FOX NBC ABC CBS ABC ABC FOX NBC CBS NBC NBC NBC NBC CBS CBS FOX FOX ANSWER: Value Sheet:

ABC NBC CBS FOX NBC

ABC NBC NBC ABC CBS

CBS NBC NBC FOX FOX

NBC CBS ABC NBC CBS

FOX FOX FOX FOX FOX

FOX ABC FOX CBS NBC

Home

Network

Hypoth. Obs. Category Propor. Freq.

2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8

ABC NBC ABC NBC CBS FOX ABC FOX

ABC NBC CBS FOX

NBC

11 12 13 14 15

10 11 12 13 14

CBS ABC CBS NBC NBC

Number of Categories

0.22 0.36 0.24 0.18 Total

10 15 10 15 50

Exp. Freq.

J Sqd. Diff. Sqd. Divided by Diff. Differ. Exp. Freq.

11 18 12 9

-1 -3 -2 6

1 9 4 36

0.09091 0.50000 0.33333 4.00000 4.92424

Test Statistic 4.92424 Degrees of Freedom 3 p-value

0.17743

Formula Sheet: D

1 2 3

Category

Hyp. Propor.

Obs. Frequency

Exp. Freq.

Diff.

Sqrd. Diff.

ABC

0.22

=COUNTIF(B2:B51,"ABC") =E4*F8

NBC

0.36

=F4G4 =COUNTIF(B2:B51,"NBC") =E5*F8 =F5-

J Sqrd. Diff. Divided by Exp. Freq.

=H4^2 =I4/G4 =H5^2 =I5/G5 Page 15

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions 6

CBS

0.24

FOX

0.18

8 9 10

G5 =F6=COUNTIF(B2:B51,"CBS") =E6*F8 G6 =F7=COUNTIF(B2:B51,"FOX") =E7*F8 G7 =SUM(F4:F7)

Total No. of Categories

=H6^2 =I6/G6 =H7^2 =I7/G7 =SUM(J4:J7)

11 12 Test Statistic Degr. of 13 Freedom 14 15 p-value

=J8 =F10-1 =CHISQ.TEST(F12,F13)

Do not reject the null hypothesis; there is not sufficient evidence to conclude the audience proportions have changed 68. Members of a focus group stated their preferences between three possible slogans. The results follow. Use Excel to test at α = .05 to determine any difference in preference among the three slogans. Slogan Preferences A A C C C B B A B C A B C C C B C B C C A A C A ANSWER: Value Sheet:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Person

Pref.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A C B C C A A B C C B A C B A

B A C B A B

Categ.

Hypoth. Propor.

Obs. Freq.

Exp. Freq.

Diff.

Sqrd. Diff.

0.33333 0.33333 0.33333 Total

9 9 12 30

9.99999 9.99999 9.99999

-1 -1 2

0.99998 0.99998 4.00004

A B C

J Sqrd. Diff. Divided by Exp. Freq.

0.099998 0.099998 0.400004 0.600001

No. of Categories 3 Test Statistic 0.600001 Degr. of Freedom 2 p-value

0.74082

Formula Sheet: Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions D

Categ.

Hypoth. Proportion

Observed Frequency

Exp. Freq.

Diff.

Sqrd. Diff.

0.33333 0.33333 0.33333 Total

=COUNTIF(B2:B31,"A") =COUNTIF(B2:B31,"B") =COUNTIF(B2:B31,"C") =SUM(F4:F6)

=E4*F7 =F4-G4 =H4^2 =I4/G4 =E5*F7 =F5-G5 =H5^2 =I5/G5 =E6*F7 =F6-G6 =H6^2 =I6/G6 =SUM(J4:J6)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A B C

J Sqrd. Diff. Divided by Exp. Freq.

No. of Categories 3 Test Statistic =J7 Degr. of Freedom =F10-1 p-value

=CHISQ.TEST(F12,F13)

Do not reject the null hypothesis; there is not sufficient evidence to conclude that there is a difference in preference among the three slogans.

69. A study of wage discrimination at a local store compared employees' race and status. Partial results of the study follow. Use Excel and test at α = .05 to determine if race is independent of status. Employee 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Race white non-white white white white non-white non-white white non-white white non-white non-white white non-white white white non-white non-white white non-white white non-white non-white

Status manager associate district mgr. manager manager associate associate associate associate manager manager associate associate associate district mgr. district mgr. associate associate associate manager district mgr. district mgr. manager

Employee 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Race non-white white non-white white non-white non-white white white non-white white non-white non-white non-white white non-white non-white non-white white white non-white non-white non-white white

Status associate district mgr. manager associate district mgr. district mgr. district mgr. district mgr. associate district mgr. associate manager associate district mgr. associate manager district mgr. manager district mgr. associate associate district mgr. manager Page 17

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions 24 25

non-white non-white

associate associate

49 50

non-white non-white

manager associate

ANSWER: Value Sheet: E Count of Employee Race white non-white Grand Total

Status associate 4 18 22

G Grand Total manager 6 7 13

district mgr. 10 5 15

20 30 50

associate 8.8 13.2

manager 5.2 7.8

district mgr. 6 9

p-value

0.01104

Expected Frequencies white non-white

Formula Sheet:

5 6 7 8 9 10 11 12 13 14

E Count of Employee Race white non-white Grand Total

Status associate 4 18 22

G Grand Total manager 6 7 13

district mgr. 10 5 15

20 30 50

associate =F7*I5/I7 =F7*I6/I7

manager =G7*I5/I7 =G7*I6/I7

district mgr. =H7*I5/I7 =H7*I6/I7

p-value

=CHISQ.TEST(F5:H6,F11:H12)

Expected Frequencies white non-white

Reject the null hypothesis; there is sufficient evidence to conclude that race is dependent of status. 70. City planners are evaluating three proposed alternatives for relieving the growing traffic congestion on a north-south highway in a booming city. The proposed alternatives are: (1) designate high-occupancy vehicle (HOV) lanes on the existing highway, (2) construct a new, parallel highway, and (3) construct a light (passenger) rail system. In an analysis of the three proposals, a citizen group has raised the question of whether preferences for the three alternatives differ among residents near the highway and nonresidents. A test of independence will address this question, with the hypotheses being: H0: Proposal preference is independent of the residency status of the individual Ha: Proposal preference is not independent of the residency status of the individual A simple random sample of 500 individuals has been selected. A crosstabulation of the residency statuses and proposal Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions preferences of the individuals sampled is shown below. RESIDENCY STATUS Nearby Resident Distant Resident

HOV Lanes 110 140

PROPOSAL New Highway 45 75

Light Rail 70 60

Conduct a test of independence using α = .05 to address the question of whether residency status is independent of the proposal preference. ANSWER: p-value = .0031 < .05; reject H0 (proposal preference is not independent of residency status). 71. Employee panel preferences for three proposed company logo designs follow. Design A 78

Design B 59

Design C 66

Use  = .05 and test to determine any difference in preference among the three logo designs. ANSWER: p-value = 0.2667 > 0.05; do not reject H0 (no apparent preferences) 72. A random sample of shoppers were asked where they do their regular grocery shopping. The table below shows the responses of the sampled shoppers. We are interested in determining if the proportions of females in the three categories are different from each other. Grocery Chain 230 80 310

Gender Female Male Total a. b. c. d. e.

Discount Store 80 50 130

Membership Warehouse 100 60 160

Total 410 190 600

Provide the null and alternative hypotheses. Determine the expected frequencies. Compute the sample proportions. Compute the critical values (CVij). Give your conclusions by providing numerical reasoning.

ANSWER: a. H : p = p = p 0 1 2 3 Ha: Not all population proportions are equal b.

Female Male Total

Grocery Chain 211.83 98.167 310

Discount Store 88.83 41.17 130

Membership Warehouse 109.3 50.67 160

Total 410 190 600

c. p1= .7419 p2= .6154 p3= .6250 Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions d. CV12 = .12087 CV13 = .11170 CV23 = .14030 e. Pairwise comparison 1 vs. 2 1 vs. 3 2 vs. 3

|pi – pj| .1266 .1169 .0096

CVij .12087 .11170 .14030

Sig. if |pi – pj| > CVij Significant Significant Not significant

73. The following table shows the results of a study on smoking and three illnesses. We are interested in determining if the proportions of smokers in the three categories are different from each other. Emphysema 150 50 200

Smoker Nonsmoker Total a. b. c. d. e.

Heart Problem 70 130 200

Cancer 100 500 600

Total 320 680 1,000

Provide the null and the alternative hypotheses. Determine the expected frequencies. Compute the sample proportions. Compute the critical values (CVij). Give your conclusions by providing numerical reasoning.

ANSWER: a. H0: p1 = p2 = p3 Ha: Not all population proportions are equal b.

Smoker Non-smoker Total

Emphysema 64 136 200

Heart Problem 64 136 200

Cancer 192 408 600

Total 320 680 1,000

c. p1= .7500 p2= .3500 p3= 01667 d. CV12 = .11150 CV13 = .08369 Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions CV23 = .09057 e. Pairwise Comparison 1 vs. 2 1 vs. 3 2 vs. 3

|pi – pj| .4000 .5833 .1833

CVij .11150 .08369 .09057

Sig. if |pi – pj| > CVij Significant Significant Significant

74. Prior to the start of the season, it was expected that audience proportions for the four major news networks would be CBS 18.6%, NBC 12.5%, ABC 28.9%, and BBC 40%. A random sample of households yielded the following viewing audience data: Observed Frequencies (fi) CBS NBC ABC BBC Total

400 230 560 810 2000

We want to determine whether the recent sample supports the expectations of the number of households of the viewing audience of the four networks. a. b. c. d.

State the null and alternative hypotheses to be tested. Compute the test statistic. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. What do you conclude?

ANSWER: a. b. c. d.

H0: p1 = .186, p2 = .125, p3 = .289, p4 = .4 Ha: Proportions are not as stated in H0 Test statistic chi-square = 4.393 Critical value = 7.8147 Do not reject H0; there is no evidence that the proportions are different from those stated in H0.

75. Prior to the start of the season, it was expected that audience proportions for the four major news networks would be CBS 28%, NBC 35%, ABC 22%, and BBC 15%. A random sample of homes yielded the following viewing audience data: Network CBS NBC ABC BBC

Number of Homes 850 980 670 500

We want to determine whether the recent sample supports the expectations of the number of homes of the viewing audience of the four networks. Copyright Cengage Learning. Powered by Cognero.

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Chapter 12: Tests of Goodness of Fit, Independence, and Multiple Proportions a. State the null and alternative hypotheses to be tested. b. Compute the test statistic. c. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. d. What do you conclude? ANSWER: a. H0: p1 = .28, p2 = .35, p3 = .22, p4 = .5 b. c. d.

Ha: Proportions are not as stated in H0 Test statistic chi-square = 10.943 Critical chi-square = 7.8147 Reject H0 and conclude that the proportions are different from those stated in H0.

76. Before the start of the Winter Olympics, it was expected that the percentages of medals awarded to the top contenders would be as follows: United States Germany Norway Austria Russia France

Percentages 25 22 18 14 11 10

Midway through the Olympics, of the 120 medals awarded, the following distribution was observed. United States Germany Norway Austria Russia France

Number of Medals 33 36 18 15 12 6

We want to test to see if there is a significant difference between the expected and actual awards given. a. Compute the test statistic. b. Using the p-value approach, test to see if there is a significant difference between the expected and actual values. Let α = .05. c. At 95% confidence, test for a significant difference using the critical value approach. ANSWER: a. b. c.

x2 = 7.69 p-value is larger than .10; do not reject H0; no significant difference. critical 2 = 7.69 = 11.070; do not reject H0.

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Chapter 13: Experimental Design and Analysis of Variance Multiple Choice 1. In an analysis of variance problem, if SST = 120 and SSTR = 80, then SSE is _____. a. 200 b. 40 c. 80 d. 120 ANSWER: b 2. In the analysis of variance procedure (ANOVA), factor refers to _____. a. the dependent variable b. the independent variable c. different levels of a treatment d. the critical value of F ANSWER: b 3. In an analysis of variance problem involving three treatments and 10 observations per treatment, SSE = 399.6. The MSE for this situation is _____. a. 133.2 b. 13.32 c. 14.8 d. 30.0 ANSWER: c 4. When an analysis of variance is performed on samples drawn from k populations, the mean square between treatments (MSTR) is _____. a. SSTR/nT b. SSTR/(nT – 1) c. SSTR/k d. SSTR/(k – 1) ANSWER: d 5. In an analysis of variance where the total sample size for the experiment is nT and the number of populations is k, the mean square within treatments is _____. a. SSE/(nT – k) b. SSTR/(nT – k) c. SSE/(k – 1) d. SSE/k ANSWER: a 6. The F ratio in a completely randomized ANOVA is the ratio of _____. a. MSTR/MSE b. MST/MSE c. MSE/MSTR Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance d. MSE/MST ANSWER: a 7. The critical F value with 6 numerator and 60 denominator degrees of freedom at α = .05 is _____. a. 3.74 b. 2.25 c. 2.37 d. 1.96 ANSWER: b 8. The ANOVA procedure is a statistical approach for determining whether the means of _____. a. two samples are equal b. two or more samples are equal c. more than two samples are equal d. two or more populations are equal ANSWER: d 9. The independent variable of interest in an ANOVA procedure is called _____. a. a partition b. a treatment c. either a partition or a treatment d. a factor ANSWER: d 10. An ANOVA procedure is applied to data obtained from 6 samples where each sample contains 20 observations. The degrees of freedom for the critical value of F are _____. a. 6 numerator and 20 denominator degrees of freedom b. 5 numerator and 20 denominator degrees of freedom c. 5 numerator and 114 denominator degrees of freedom d. 6 numerator and 20 denominator degrees of freedom ANSWER: c 11. In the ANOVA, treatment refers to _____. a. experimental units b. different levels of a factor c. a factor d. applying antibiotic to a wound ANSWER: b 12. The mean square is the sum of squares divided by _____. a. the total number of observations b. its corresponding degrees of freedom c. its corresponding degrees of freedom minus 1 d. the sample size Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance ANSWER: b 13. In factorial designs, the response produced when the treatments of one factor interact with the treatments of another in influencing the response variable is known as _____. a. the main effect b. replication c. interaction d. a factor ANSWER: c 14. An experimental design where the experimental units are randomly assigned to the treatments is known as _____. a. factor block design b. random factor design c. completely randomized design d. systematic sampling ANSWER: c 15. The number of times each experimental condition is observed in a factorial design is known as a(n) _____. a. partition b. replication c. experimental condition d. factor ANSWER: b 16. The required condition for using an ANOVA procedure on data from several populations is that the _____. a. selected samples are dependent on each other b. sampled populations are all uniform c. sampled populations have equal variances d. sampled populations have equal means ANSWER: c 17. An ANOVA procedure is used for data that were obtained from four sample groups each comprised of five observations. The degrees of freedom for the critical value of F are _____. a. 3 and 20 b. 3 and 16 c. 4 and 17 d. 3 and 19 ANSWER: b 18. In ANOVA, which of the following is NOT affected by whether or not the population means are equal? a. b. between-samples estimate of 2 c. within-samples estimate of 2 Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance d. Population means must be equal in ANOVA. ANSWER: c 19. A term that means the same as the term "variable" in an ANOVA procedure is _____. a. factor b. treatment c. replication d. variance within ANSWER: a 20. To determine whether the means of two populations are equal, _____. a. a t test must be performed b. an analysis of variance must be performed c. either a t test or an analysis of variance can be performed d. a chi-square test must be performed ANSWER: c 21. The process of allocating the total sum of squares and degrees of freedom is called _____. a. factoring b. blocking c. replicating d. partitioning ANSWER: d 22. An experimental design that permits statistical conclusions about two or more factors is a _____. a. randomized block design b. factorial design c. completely randomized design d. randomized design ANSWER: b 23. In a completely randomized design involving three treatments, the following information is provided: Sample size Sample mean

Treatment 1 5 4

Treatment 2 10 8

Treatment 3 5 9

The overall mean for all the treatments is _____. a. 7.00 b. 6.67 c. 7.25 d. 4.89 ANSWER: c Exhibit 13-1 Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance SSTR = 6,750 SSE = 8,000 nT = 20

H0: μ1 = μ2 = μ3 = μ4 Ha: At least one mean is different

24. Refer to Exhibit 13-1. The mean square between treatments (MSTR) equals _____. a. 400 b. 500 c. 1,687.5 d. 2,250 ANSWER: d 25. Refer to Exhibit 13-1. The mean square within treatments (MSE) equals _____. a. 400 b. 500 c. 1,687.5 d. 2,250 ANSWER: b 26. Refer to Exhibit 13-1. The test statistic to test the null hypothesis equals _____. a. .22 b. .84 c. 4.22 d. 4.5 ANSWER: d 27. Refer to Exhibit 13-1. The null hypothesis is to be tested at the 5% level of significance. The critical value from the table is _____. a. 2.87 b. 3.24 c. 4.08 d. 8.7 ANSWER: b 28. Refer to Exhibit 13-1. The null hypothesis _____. a. should be rejected b. should not be rejected c. was designed incorrectly d. should be retested ANSWER: a Exhibit 13-2 Source of Variation

Sum of Squares

Degrees of Freedom

Mean Square

F Page 5

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Chapter 13: Experimental Design and Analysis of Variance Between treatments Between blocks Error Total

2,073.6 6,000.0

4 5 20 29

1,200 288

29. Refer to Exhibit 13-2. The null hypothesis for this ANOVA problem is _____. a. μ1 = μ2 = μ3 = μ4 b. μ1 = μ2 = μ3 = μ4 = μ5 c. μ1 = μ2 = μ3 = μ4 = μ5 = μ6 d. μ1 = μ2 = ... = μ20 ANSWER: b 30. Refer to Exhibit 13-2. The mean square between treatments equals _____. a. 288 b. 518.4 c. 1,200 d. 8,294.4 ANSWER: b 31. Refer to Exhibit 13-2. The sum of squares due to error equals _____. a. 14.4 b. 2,073.6 c. 5,760 d. 6,000 ANSWER: c 32. Refer to Exhibit 13-2. The test statistic to test the null hypothesis equals _____. a. .432 b. 1.8 c. 4.17 d. 28.8 ANSWER: b 33. Refer to Exhibit 13-2. The null hypothesis is to be tested at the 5% level of significance. The critical value from the table is _____. a. 2.71 b. 2.87 c. 5.19 d. 5.8 ANSWER: b 34. Refer to Exhibit 13-2. The null hypothesis _____. a. should be rejected Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 13-3 To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the three treatments. You are given the results below. Treatment A B C

20 22 40

Observation 30 25 26 20 30 28

33 28 22

35. Refer to Exhibit 13-3. The null hypothesis for this ANOVA problem is _____. a. μ1 = μ2 b. μ1 = μ2 = μ3 c. μ1 = μ2 = μ3 = μ4 d. μ1 = μ2 = ... = μ12 ANSWER: b 36. Refer to Exhibit 13-3. The mean square between treatments (MSTR) equals _____. a. 1.872 b. 5.86 c. 34 d. 36 ANSWER: d 37. Refer to Exhibit 13-3. The mean square within treatments (MSE) equals _____. a. 1.872 b. 5.86 c. 34 d. 36 ANSWER: c 38. Refer to Exhibit 13-3. The test statistic to test the null hypothesis equals _____. a. .944 b. 1.059 c. 3.13 d. 19.231 ANSWER: b 39. Refer to Exhibit 13-3. The null hypothesis is to be tested at the 1% level of significance. The critical value from the table is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance a. 4.26 b. 8.02 c. 16.69 d. 99.39 ANSWER: b 40. Refer to Exhibit 13-3. The null hypothesis _____. a. should be rejected b. should not be rejected c. should be revised d. should be retested ANSWER: b Exhibit 13-4 In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments. The following information is provided. SSTR = 200 (Sum Square Between Treatments) SST = 800 (Total Sum Square) 41. Refer to Exhibit 13-4. The sum of squares within treatments (SSE) is _____. a. 1,000 b. 600 c. 200 d. 1,600 ANSWER: b 42. Refer to Exhibit 13-4. The number of degrees of freedom corresponding to between treatments is _____. a. 60 b. 59 c. 5 d. 4 ANSWER: d 43. Refer to Exhibit 13-4. The number of degrees of freedom corresponding to within treatments is _____. a. 60 b. 59 c. 5 d. 4 ANSWER: a 44. Refer to Exhibit 13-4. The mean square between treatments (MSTR) is _____. a. 3.34 b. 10.00 c. 50.00 Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance d. 12.00 ANSWER: c 45. Refer to Exhibit 13-4. The mean square within treatments (MSE) is _____. a. 50 b. 10 c. 200 d. 600 ANSWER: b 46. Refer to Exhibit 13-4. If at a 5% level of significance we want to determine whether or not the means of the five populations are equal, the critical value of F is _____. a. 2.53 b. 19.48 c. 4.98 d. 39.48 ANSWER: a 47. Refer to Exhibit 13-4. The conclusion of the test is that the five means _____. a. are equal b. may be equal c. are not equal d. have some equal values ANSWER: c Exhibit 13-5 Part of an ANOVA table is shown below. Source of Variation Between treatments Within treatments (Error) Total

Sum of Squares 180

Degrees of Freedom 3

480

Mean Square

48. Refer to Exhibit 13-5. The mean square between treatments (MSTR) is _____. a. 20 b. 60 c. 300 d. 15 ANSWER: b 49. Refer to Exhibit 13-5. The mean square within treatments (MSE) is _____. a. 60 b. 15 Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance c. 300 d. 20 ANSWER: d 50. Refer to Exhibit 13-5. If at a 5% level of significance, we want to determine whether the means of the populations are equal, the critical value of F is _____. a. 2.53 b. 19.48 c. 3.29 d. 5.86 ANSWER: c 51. Refer to Exhibit 13-5. The conclusion of the test is that the means _____. a. are equal to 50 b. may be equal c. are not equal d. have some equal values ANSWER: b Exhibit 13-6 Part of an ANOVA table is shown below. Source of Variation Between treatments Within treatments (Error) Total

Sum of Squares 64

Degrees of Freedom

Mean Square

F 8

2 100

52. Refer to Exhibit 13-6. The number of degrees of freedom corresponding to between treatments is _____. a. 18 b. 2 c. 4 d. 3 ANSWER: c 53. Refer to Exhibit 13-6. The number of degrees of freedom corresponding to within treatments is _____. a. 22 b. 4 c. 5 d. 18 ANSWER: d 54. Refer to Exhibit 13-6. The mean square between treatments (MSTR) is _____. a. 36 Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance b. 16 c. 64 d. 15 ANSWER: b 55. Refer to Exhibit 13-6. If at a 5% significance level we want to determine whether or not the means of the populations are equal, the critical value of F is _____. a. 5.80 b. 2.93 c. 3.16 d. 2.90 ANSWER: b 56. Refer to Exhibit 13-6. The conclusion of the test is that the means _____. a. are equal b. may be equal c. are not equal d. have some equal values ANSWER: c Exhibit 13-7 The following is part of an ANOVA table, which was the result of three treatments and a total of 15 observations. Source of Variation Between treatments Within treatments (Error) Total

Sum of Squares 64 96

Degrees of Freedom

Mean Square

57. Refer to Exhibit 13-7. The number of degrees of freedom corresponding to between treatments is _____. a. 12 b. 2 c. 3 d. 4 ANSWER: b 58. Refer to Exhibit 13-7. The number of degrees of freedom corresponding to within treatments is _____. a. 12 b. 2 c. 3 d. 15 ANSWER: a 59. Refer to Exhibit 13-7. The mean square between treatments (MSTR) is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance a. 36 b. 16 c. 8 d. 32 ANSWER: d 60. Refer to Exhibit 13-7. If at a 5% level of significance, we want to determine whether or not the means of the populations are equal, the critical value of F is _____. a. 4.75 b. 19.41 c. 3.16 d. 1.96 ANSWER: a 61. Refer to Exhibit 13-7. The computed test statistic is _____. a. 32 b. 8 c. .667 d. 4 ANSWER: d 62. Refer to Exhibit 13-7. The conclusion of the test is that the means _____. a. are equal b. may be equal c. are not equal d. have some equal values ANSWER: b 63. In a completely randomized design involving four treatments, the following information is provided. Treatment 1 Treatment 2 Sample size 50 18 Sample mean 32 38 The overall mean (the grand mean) for all treatments is _____. a. 40.0 b. 37.3 c. 48.0 d. 37.0 ANSWER: b

Treatment 3 15 42

Treatment 4 17 48

64. An ANOVA procedure is used for data obtained from five populations. Five samples, each comprised of 20 observations, were taken from the five populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are _____. a. 5 and 20 b. 4 and 20 Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance c. 4 and 99 d. 4 and 95 ANSWER: d 65. The critical F value with 8 numerator and 29 denominator degrees of freedom at α = .01 is _____. a. 2.28 b. 3.20 c. 3.33 d. 3.64 ANSWER: b 66. An ANOVA procedure is used for data obtained from four populations. Four samples, each comprised of 30 observations, were taken from the four populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are _____. a. 3 and 30 b. 4 and 30 c. 3 and 119 d. 3 and 116 ANSWER: d 67. Which of the following is NOT a required assumption for the analysis of variance? a. The random variable of interest for each population has a normal probability distribution. b. The variance associated with the random variable must be the same for each population. c. At least two populations are under consideration. d. Populations have equal means. ANSWER: d 68. In an analysis of variance, one estimate of a. means of each sample b. overall sample mean c. sum of observations d. populations have equal means ANSWER: d

is based upon the differences between the treatment means and the _____.

69. In testing for the equality of k population means, the number of treatments is _____. a. k b. k-1 c. nT d. nT - k ANSWER: d 70. If we are testing for the equality of 3 population means, we should use the _____. a. test statistic F b. test statistic t Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance c. test statistic z d. test statistic χ2 ANSWER: d Subjective Short Answer 71. Information regarding the ACT scores of samples of students in three different majors is given below. Management 28 26 25 27 21 19 27 17 17 23

Sum Mean Variance a. b.

230 23 18

Student's Major Finance Accounting 22 29 23 27 24 26 22 28 24 25 26 26 27 28 29 20 28 24 28 28 29 225 338 25 26 6.75 9.33

Set up the ANOVA table for this problem. At a 5% level of significance, test to determine whether there is a significant difference in the means of the four populations.

ANSWER: a. ANOVA Source of Variation Between Treatments Error Total b.

SS 51.468 328.000 379.468

df 2 29 31

MS 25.73 11.31

F 2.27

F critical 3.3276

Since the test statistic F = 2.27< 3.3276 do not reject H0, cannot conclude that there is a difference in the means of the four populations

72. Information regarding the ACT scores of samples of students in four different majors is given below. Management 29 27 21 28 22 28

Student's Major Marketing Finance 22 29 22 27 25 27 26 28 27 24 20 20

Accounting 28 26 25 20 21 19 Page 14

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Chapter 13: Experimental Design and Analysis of Variance

Sum Mean Variance

28 23 28 24 29 31

23 25 27 28

20 30 29

318 26.50 10.99

245 24.50 6.94

234 26.00 14.50

27 24 21 23 27 27 24 312 24.00 9.00

Set up the ANOVA table for this problem. At a 5% level of significance, test to determine whether there is a significant difference in the b. means of the four populations. ANSWER: a. ANOVA Source of Variation Between Treatments Error Total b.

SS 49.659 397.500 447.159

df 3 40 43

MS 16.533 9.937

F 1.6657

F critical 2.8387

Since the test statistic F = 1.6657< 2.8387do not reject Ho, cannot conclude that there is a difference in the means of the four populations

73. Guitars R. US has three stores located in three different areas. Random samples of the sales of the three stores (in $1,000s) are shown below. Store 1 80 80

a. b.

Store 2 85

Store 3 79 86 85 76 81 88 89 80

Set up the ANOVA table for this problem. At a 5% level of significance, test to see if there is a significant difference in the average sales of the three stores. (Please note that the sample sizes are not equal.)

At a 5% level of significance, test to see if there is a significant difference in the average sales of the three stores. (Please note that the sample sizes are not equal.) ANSWER: a. ANOVA Source of Variation Between Treatments Within Treatments (Error) Total Copyright Cengage Learning. Powered by Cognero.

Sum of Squares 20.55 152 172.55

Degrees of Freedom 2 8 10

Mean Square 10.27 19

F 0.54

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Chapter 13: Experimental Design and Analysis of Variance b. Since the test statistic F = 0.54 < 4.46 do not reject H0, cannot conclude that there is a significant difference in the average sales of the three stores 74. In a completely randomized experimental design, 18 experimental units were used for the first treatment, 10 experimental units for the second treatment, and 15 experimental units for the third treatment. Part of the ANOVA table for this experiment is shown below. Source of Variation Between Treatments Within Treatments (Error) Total

Sum of Squares _____? _____? _____?

Degrees of Freedom _____? _____? _____?

Mean Square _____? 6

F 3.0

a. Fill in all the blanks in the above ANOVA table. b. At a 5% level of significance, test to see if there is a significant difference among the means. ANSWER: a. Sum of Degrees of Mean Source of Variation F Squares Freedom Square Between Treatments 36 2 18 3.0 Within Treatments (Error) 240 40 6 Total 276 42 b.

Since the test statistic F = 3 < 3.23 do not reject H0, and conclude there is not a significant difference among the means.

75. Random samples were selected from three populations. The data obtained are shown below. Treatment 1 37 33 36 38

Treatment 2 Treatment 3 43 28 39 32 35 33 38 40 At a 5% level of significance, test to see if there is a significant difference in the means of the three populations. (Please note that the sample sizes are not equal.) ANSWER: Since the test statistic F = 4.27> 4.26 reject H0 and conclude at least one mean is different from others. 76. In a completely randomized experimental design, 7 experimental units were used for the first treatment, 9 experimental units for the second treatment, and 14 experimental units for the third treatment. Part of the ANOVA table for this experiment is shown below. Source of Variation Between Treatments Within Treatments (Error) Total a.

Sum of Squares _____? _____? _____?

Degrees of Freedom _____? _____? _____?

Mean Square _____? 4

F 4.5

Fill in all the blanks in the above ANOVA table.

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Chapter 13: Experimental Design and Analysis of Variance b.

At a 5% level of significance, test to see if there is a significant difference among the means.

ANSWER: a. Source of Variation Between Treatments Within Treatments (Error) Total b.

Sum of Squares 36 108 144

Degrees of Freedom 2 27 29

Mean Square 18 4

F 4.5

Since the test statistic F = 3.26 < 3.89 do not reject H0. We cannot conclude there is a significant difference among the means.

77. Random samples were selected from three populations. The data obtained are shown below. Treatment 1 Treatment 2 Treatment 3 45 30 39 41 34 35 37 35 38 40 40 42 At a 5% level of significance, test to see if there is a significant difference in the means of the three populations. (Please note that the sample sizes are not equal.) ANSWER: Since the test statistic F = 4.27 > 4.26 reject H0, conclude at least one mean is different from others 78. The manager of Young Corporation wants to determine whether or not the type of work schedule for her employees has any effect on their productivity. She has selected 15 production employees at random and then randomly assigned five employees to each of the three proposed work schedules. The following table shows the units of production (per week) under each of the work schedules. Work Schedule (Treatments) Work Schedule 1 Work Schedule 2 Work Schedule 3 50 60 70 60 65 75 70 66 55 40 54 40 45 57 55 At a 5% level of significance, determine if there is a significant difference in the mean weekly units of production for the three types of work schedules. ANSWER: SSTR = 154.53 MSTR = 77.27 SSE = 1,455.20 MSE = 121.27 F = 0.64 < 3.89; do not reject H0, cannot conclude that there is a significant difference in the mean weekly units of production for the three types of work schedules 79. Six observations were selected from each of three populations. The data obtained are shown below. Sample 1 31

Sample 2 37

Sample 3 37

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Chapter 13: Experimental Design and Analysis of Variance 28 32 31 34 34 32 32 24 39 26 32 30 29 33 35 Test at α = .05 level to determine if there is a significant difference in the means of the three populations. ANSWER:

SSTR = 48 MSTR = 24 SSE = 200 MSE = 13.33 F = 1.80 < 3.89; do not reject H0, cannot conclude that there is a significant difference in the means of the three populations

80. The test scores for selected samples of sociology students who took the course from three different instructors are shown below. Instructor A Instructor B Instructor C 83 90 85 60 55 90 80 84 90 85 91 95 71 85 80 At α = .05, test to see if there is a significant difference among the averages of the three groups. ANSWER: SSTR = 374.8 MSTR = 187.4 SSE = 1,438.8 MSE = 119.9 F = 1.56 < 3.89; do not reject H0, cannot conclude that there is a significant difference among the averages of the three groups 81. Three universities administer the same comprehensive examination to the recipients of MS degrees in psychology. From each institution, a random sample of MS recipients was selected, and these recipients were then given the exam. The following table shows the scores of the students from each university. University A University B University C 89 60 81 95 95 70 75 89 90 92 80 78 99 66 77 At α = .01, test to see if there is any significant difference in the average scores of the students from the three universities. (Note that the sample sizes are not equal.) ANSWER: SSTR = 302.02 MSTR = 151.01 SSE = 1563.58 MSE = 130.30 F = 1.16 < 6.93; do not reject H0, cannot conclude that there is any significant difference in the average scores of the students from the three universities 82. In a completely randomized experimental design, 11 experimental units were used for each of the three treatments. Part of the ANOVA table is shown below. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance Source of Variation Between treatments Within treatments (Error) Total a. b.

Sum of Squares 1,500 _____? 6,000

Degrees of Freedom _____? _____? _____?

Mean Square _____? _____?

F _____?

Fill in the blanks in the above ANOVA table. At a 5% level of significance, test to determine whether or not the means of the three populations are equal.

ANSWER: a. Source of Variation Between treatments Within treatments (Error) Total b.

Sum of Squares 1,500 4,500 6,000

Degrees of Freedom 2 30 32

Mean Square 750 150

F 5.00

F = 5.00 > 3.32; reject H0, conclude that there is a significant difference in the means of the three populations

83. MNM, Inc. has three stores located in three different areas. Random samples of the sales of the three stores (in $1,000s) are shown below. Store 1 Store 2 Store 3 88 76 85 84 78 67 88 60 55 82 58 92 At a 5% level of significance, test to see if there is a significant difference in the average sales of the three stores. Show your complete work and the ANOVA table. (Please note that the sample sizes are not equal.) ANSWER: MSTR = 493.06 MSE = 93.87 F = 5.25 > 4.26; reject H0, conclude that there is a significant difference in the average sales of the three stores 84. Three different brands of tires were compared for wear characteristics. For each brand of tire, 10 tires were randomly selected and subjected to standard wear testing procedures. The average mileage obtained for each brand of tire and sample standard deviations (both in 1000 miles) are shown below. Brand A Brand B Brand C Average mileage 37 38 33 Sample variance 3 4 2 Use the above data and test to see if the mean mileage for all three brands of tires is the same. Let α = .05. ANSWER: SSTR = 140 MSTR = 70 SSE = 90 MSE = 3.33 F = 21.00 > 3.34; reject H0, conclude that there is a significant difference in the mean mileage among the three brands of tires Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance 85. Three different models of automobiles (A, B, and C) were compared for gasoline consumption. For each model of car, fifteen cars were randomly selected and subjected to standard driving procedures. The average miles/gallon obtained for each model of car and sample standard deviations are shown below. Car A Car B Car C Average miles per gallon 42 49 44 Sample standard deviation 4 5 3 Use the above data and test to see if the mean gasoline consumption for all three models of cars is the same. Let α = .05. ANSWER:

SSTR = 390 MSTR = 195 SSE = 700 MSE = 16.67 F = 11.7 > 3.21; reject H0, conclude that there is a significant difference in the mean gasoline consumption among the three models of cars

86. At α = .05, test to determine if the means of the three populations (from which the following samples are selected) are equal. Sample 1 Sample 2 Sample 3 60 84 60 78 78 57 72 93 69 66 81 66 ANSWER: SSTR = 936 MSTR = 468 SSE = 396 MSE = 44 F = 10.64 > 4.26; reject H0, conclude that there is a significant difference in the means of the three populations 87. To test to determine whether there is any significant difference in the mean number of units produced per week by each of three production methods, the following data were collected: Method I Method II Method III 182 170 162 170 192 166 180 190 At the α = .05 level of significance, is there any difference in the mean number of units produced per week by each method? (Please note that the sample sizes are not equal.) ANSWER: SSTR = 483.88 MSTR = 241.67 SSE = 386.67 MSE = 77.33 F = 3.12 < 5.79; do not reject H0, cannot conclude that there is a significant difference in the mean number of units produced per week by each method 88. A dietitian wants to see if there is any difference in the effectiveness of three diets. Eighteen people, comprising a sample, were randomly assigned to the three diets. Below is the total amount of weight lost in a month by each person. Diet A 14 18 20 12 20

Diet B 12 10 22 12 16

Diet C 25 32 18 14 17

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Chapter 13: Experimental Design and Analysis of Variance 18 a. b. c.

State the null and alternative hypotheses. Calculate the test statistic. What would you advise the dietician about the effectiveness of the three diets? Use a .05 level of significance.

ANSWER:

a. b. c.

H0: μ1 = μ2 = μ3 Ha: At least one mean is different 2.005 Do not reject the null hypothesis of no difference since 2.00 < 3.68. Thus, there is not sufficient evidence to conclude that there is any difference in the effectiveness of the three diets.

89. Allied Corporation wants to increase the productivity of its line workers. Four different programs have been suggested to help increase productivity. Twenty employees, making up a sample, have been randomly assigned to one of the four programs and their output for a day's work has been recorded. The results are shown below. Program A 150 130 120 180 145 a. b. c. d.

Program B 150 120 135 160 110

Program C 185 220 190 180 175

Program D 175 150 120 130 175

State the null and alternative hypotheses. Construct an ANOVA table. As the statistical consultant to Allied, what would you advise? Use a .05 level of significance. Use Fisher's LSD procedure and determine which population mean (if any) is different from the others. Let α = .05.

ANSWER: a.

H0: μ1 = μ2 = μ3 = μ4 Ha: At least one mean is different

b. Source of Variation Treatment Error Total c. d.

Sum of Squares 8,750 7,600 16,350

Degrees of Freedom 3 16 19

Mean Square 2,916.67 475.00

F 6.14

Reject H0; 6.14 > 3.24, conclude that there is a significant difference in the mean output among the four programs LSD = 29.22; the mean of population C is different from the others.

90. The marketing department of a company has designed three different boxes for its product. It wants to determine which box will produce the largest amount of sales. Each box will be test marketed in five different stores for a period of a month. Below is the information on sales. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance Store 1 210 195 295

Box 1 Box 2 Box 3 a. b. c. d.

Store 2 230 170 275

Store 3 190 200 290

Store 4 180 190 275

Store 5 190 193 265

State the null and alternative hypotheses. Construct an ANOVA table. What conclusion do you draw? Use Fisher's LSD procedure and determine which mean (if any) is different from the others. Let α = .01.

ANSWER: a.

H0: μ1 = μ2 = μ3 Ha: At least one mean is different

b. Source of Variation Treatment Block Error Total c. d.

Sum of Squares 24,667.20 711.07 2,022.14 27,400.41

Degrees of Freedom 2 4 8 14

Mean Square 12,333.60 177.77 252.77

F 48.4

Reject the null hypothesis; 48.4 > 8.65; at least one mean is different from the others. LSD = 33.73; the mean of box 3 is different from the others.

91. You are given an ANOVA table below with some missing entries. Source of Variation

Sum of Squares

Between treatments Between blocks Error Total

5,040 5,994

a. b. c. d. e. f.

Degrees of Freedom 3 6 18 27

Mean Square 1,198.8 840.0

State the null and alternative hypotheses. Compute the sum of squares between treatments. Compute the mean square due to error. Compute the total sum of squares. Compute the test statistic F. Test the null hypothesis stated in part (a) at the 1% level of significance. Be sure to state your conclusion.

ANSWER: a. b. c. d. e.

H0: μ1 = μ2 = μ3 = μ4 Ha: At least one mean is different 3596.4 333 14630.4 3.6

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Chapter 13: Experimental Design and Analysis of Variance f.

Do not reject the null hypothesis; at least one mean is different from the others.

92. For four populations, the population variances are assumed to be equal. Random samples from each population provide the following data. Population Sample Size Sample Mean Sample Variance 1 11 40 23.4 2 11 35 21.6 3 11 39 25.2 4 11 37 24.6 Using a .05 level of significance, test to see if the means for all four populations are the same. ANSWER: Do not reject the null hypothesis of equal means since 2.282 < 2.84. 93. A research organization wishes to determine whether four brands of batteries for transistor radios perform equally well. Three batteries of each type were randomly selected and installed in the three test radios. The number of hours of use for each battery is given below. Radio A B C

Brand 2 27 38 28

1 25 29 21

3 20 24 16

4 28 37 19

Use the analysis of variance procedure for completely randomized designs to determine whether there is a significant difference in the mean useful life of the four types of batteries. (Ignore the fact that there are different test radios.) Use the .05 level of significance and be sure to construct the ANOVA table. b. Now consider the three different test radios and carry out the analysis of variance procedure for a randomized block design. Include the ANOVA table. c. Compare the results in parts (a) and (b). ANSWER: a. Sum of Degrees of Mean Source of Variation F Squares Freedom Square Treatment 198 3 66.0 1.76 Error 300 8 37.5 Total 498 11 b.

Do not reject the null hypothesis of equal means since1.76 < 4.07, cannot conclude that there is a significant difference in the mean useful life of the four types of batteries.

Source of Variation Treatment Block Error Total c.

Sum of Squares 198 248 52 498

Degrees of Freedom 3 2 6 11

Mean Square 66.000 124.000 8.667

F 7.62

Reject the null hypothesis since 7.62 > 4.76, conclude that there is a significant difference in the mean useful life of the four types of batteries. Controlling for the differences among radios has made a difference.

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Chapter 13: Experimental Design and Analysis of Variance 94. Employees of MNM Corporation are about to undergo a retraining program. Management is trying to determine which of three programs is the best. They believe that the effectiveness of the programs may be influenced by gender. A factorial experiment was designed. You are given the following information. Factor A: Program Program 1 Program 2 Program 3

Factor B: Gender Male Female 320 380 240 300 160 240 180 210 240 360 290 380

a. Set up the ANOVA table. b. What advice would you give MNM? Use a .05 level of significance. ANSWER: a. Sum of Degrees of Mean Source of Variation F Squares Freedom Square Factor A 36,150.000 2 18,075.000 12.76 Factor B 16,133.333 1 16,133.333 11.39 Interaction 1,516.667 2 758.334 0.54 Error 8,500.000 6 1,416.667 Total 62,300.000 11 b.

There is a significant difference in the programs since 12.76 > 5.14. There is a significant difference in gender since 11.39 > 5.99. There is no significant interaction effect since .54 < 5.14.

95. The final examination grades of random samples of students from three different classes are shown below. Class A Class B Class C 92 91 85 85 85 93 96 90 82 95 86 84 At the α = .05 level of significance, is there any difference in the mean grades of the three classes? ANSWER: MSTR = 37.34 MSE = 18.89 F = 1.977 < 4.26; do not reject H0, cannot conclude that there is a significant difference in the mean grades of the three classes 96. Individuals were randomly assigned to three different production processes. The hourly units of production for the three processes are shown below. Process 1 33 30 28 29

Production Process Process 2 Process 3 33 28 35 36 30 30 38 34

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Chapter 13: Experimental Design and Analysis of Variance Use the analysis of variance procedure with α = .05 to determine if there is a significant difference in the mean hourly units of production for the three types of production processes. ANSWER: MSTR = 16.00 MSE = 9.78 F = 1.636 < 4.26; do not reject H0, cannot conclude that there is a significant difference in the mean hourly units of production for the three types of production processes 97. Random samples of employees from three different departments of MNM Corporation showed the following yearly incomes (in $1,000s). Department A Department B Department C 40 46 46 37 41 40 43 43 41 41 33 48 35 41 39 38 42 45 At α = .05, test to determine if there is a significant difference among the average incomes of the employees from the three departments. ANSWER: MSTR = 26.06 MSE = 13.52 F = 1.927 < 3.68; do not reject H0, cannot conclude that there is a significant difference in the average incomes of the employees from the three departments 98. The heating bills for a selected sample of houses using various forms of heating are given below (values are in dollars). Gas-Heated Homes Central Electric Heat Pump 83 90 81 80 88 83 82 87 80 83 82 82 82 83 79 At α = .05, test to see if there is a significant difference among the average bills of the homes. ANSWER: MSTR = 35 MSE = 5.17 F = 6.774 > 3.89; reject H0, conclude that there is a significant difference among the average bills of the homes 99. Three universities in your state decided to administer the same comprehensive examination to the recipients of MBA degrees from the three institutions. From each institution, MBA recipients were randomly selected and were given the test. The following table shows the scores of the students from each university. Northern University Central University Southern University 75 85 80 80 89 81 84 86 84 85 88 79 81 83 85 At α = .01, test to see if there is any significant difference in the average scores of the students from the three universities. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance (Note that the sample sizes are not equal.) ANSWER: MSTR = 38.93 MSE = 39.34 F = 4.17 < 6.39; do not reject H0, cannot conclude that there is a significant difference in the average scores of the students from the three universities 100. Three major automobile manufacturers have entered their cars in the Indianapolis 500 race. The speeds of the tested cars are given below. Manufacturer A Manufacturer B Manufacturer C 180 177 175 175 180 176 179 167 177 176 172 190 At α = .05, test to see if there is a significant difference in the average speeds of the cars of the auto manufacturers. ANSWER: MSTR = 42 MSE = 26.89 F = 1.562 < 4.26; do not reject H0, cannot conclude that there is a significant difference in the average speeds of the cars of the auto manufacturers 101. Part of an ANOVA table is shown below. Source of Variation Between treatments Within treatments (Error) Total a. b. c.

Sum of Squares 90 120 _____?

Degrees of Freedom 3 20 _____?

Mean Square _____? _____?

F _____?

Compute the missing values and fill in the blanks in the above table. Use α = .01 to determine if there is any significant difference among the means. How many groups have there been in this problem? What has been the total number of observations?

ANSWER: a. Source of Variation Between treatments Within treatments (Error) Total

b. c.

Sum of Squares 90 120 210

Degrees of Freedom 3 20 23

Mean Square 30 6

F 5.00

F = 5.00 > 4.94; reject H0, conclude that there is a significant difference among the means 4 24

102. Part of an ANOVA table involving eight groups for a study is shown below. Source of Variation Between treatments

Sum of Squares 126

Degrees of Freedom _____?

Mean Square _____?

F _____? Page 26

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Chapter 13: Experimental Design and Analysis of Variance Within treatments (Error) Total a. b.

240 _____?

_____? 67

_____?

Complete all the missing values in the above table and fill in the blanks. Use α = .01 to determine if there is any significant difference among the means of the eight groups.

ANSWER: a. Source of Variation Between treatments Within treatments (Error) Total b.

Sum of Squares 126 240 366

Degrees of Freedom 7 60 67

Mean Square 18 4

F 4.50

F = 4.5 > 2.95; reject H0, conclude that there is a significant difference in the means of the eight groups

103. MNM, Inc. has three stores located in three different areas. Random samples of the daily sales of the three stores (in $1,000s) are shown below. Store 1 Store 2 Store 3 9 10 6 8 11 7 7 10 8 8 13 11 At a 5% level of significance, test to see if there is a significant difference in the average sales of the three stores. ANSWER: MSTR = 12.00 MSE = 2.44 F = 4.909 > 4.26; reject H0, conclude that there is a significant difference in the average sales of the three stores 104. Five drivers were selected to test drive two makes of automobiles. The following table shows the number of miles per gallon for each driver driving each car. Driver Automobile 1 2 3 4 5 A 30 31 30 27 32 B 36 35 28 31 30 Consider the makes of automobiles as treatments and the drivers as blocks and test to see if there is any difference in the miles/gallon of the two makes of automobiles. Let α = .05. ANSWER: MSTR = 10 MSE = 7 F = 1.43 < 7.71; do not reject H0, cannot conclude that there is a significant difference in the mean miles/gallon of the two makes of automobiles 105. A factorial experiment involving two levels of factor A and two levels of factor B resulted in the following. Factor B Factor A Level 1

Level 1 14

Level 2 18

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Chapter 13: Experimental Design and Analysis of Variance 16 12 Level 2 18 16 20 14 Using α = .05, test to see if there is any significant main effect and any interaction effect. ANSWER: Factor A Treatment F = 1.33 < 7.71; do not reject H0 Factor B Treatment F = 1.33 < 7.71; do not reject H0 Interaction (AB) F = 1.33 < 7.71; do not reject H0 106. The following are the results from a completely randomized design consisting of three treatments. Source of Variation Between treatments Within treatments (Error) Total a. b.

Sum of Squares 390.58 158.40 548.98

Degrees of Freedom

Mean Square

Using α = .05, test to see if there is a significant difference among the means of the three populations. The sample sizes for the three treatments are equal. If in part (a) you concluded that at least one mean is different from the others, determine which mean(s) is(are) different. The three sample means are = 17.000, = 21.625, and 3 = 26.875. Use Fisher's LSD procedure and let α = .05.

ANSWER: a.

H0: μ1 = μ2 = μ3 Ha: At least one mean is different F = 25.89 > 3.47; reject H0, conclude that at least one mean is different from the others LSD = 2.856 = 4.625; = 9.875; = 5.25 All three means are different from one another.

107. Eight observations were selected from each of three populations, and an analysis of variance was performed on the data. The following are the results: Source of Variation Between treatments Within treatments (Error) Total

Sum of Squares 195.58 10.77

Degrees of Freedom

Mean Square

Using α = .05, test to see if there is a significant difference among the means of the three populations. The sample sizes for the three treatments are equal. ANSWER:

H0: μ1 = μ2 = μ3 Ha: At least one mean is different F = 190.68 > 3.47; reject H0, conclude that there is a significant difference among the means of the three populations

108. Random samples of individuals from three different cities were asked how much time they spend per day watching Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance television. The results (in minutes) for the three groups are shown below. City I City II City III 260 178 211 280 190 190 240 220 250 260 240 300 At α = .05, test to see if there is a significant difference in the averages of the three groups. ANSWER: SSTR = 9,552.92 MSTR = 4,776.46 SSE = 6,322 MSE = 702.44 F = 6.8 > 4.26; reject H0, conclude that there is a significant difference in the averages of the three groups 109. Three different brands of tires were compared for wear characteristics. From each brand of tire, 10 tires were randomly selected and subjected to standard wear-testing procedures. The average mileage obtained for each brand of tire and sample variances (both in 1,000 miles) are shown below. Average mileage Sample variance

Brand A 37 3

Brand B 38 4

Brand C 33 2

Use the above data to compute the following: SSRT, MSTR, SSE, MSE, and F. ANSWER: SSTR = 140 MSTR = 70 SSE = 81 MSE = 3 F = 23.3 110. Halls, Inc. has three stores located in three different areas. Random samples of the sales of the three stores (in $1,000s) are shown below. Store 1 Store 2 Store 3 46 34 33 47 36 31 45 35 35 42 39 45 At a 5% level of significance, test to see if there is a significant difference in the average sales of the three stores. ANSWER: SSTR = 324 MSTR= 162 SSE = 36 MSE = 4 F = 40.5 > 4.26; reject H0, conclude that there is a significant difference in the average sales of the three stores 111. In a completely randomized experimental design, 11 experimental units were used for each of the four treatments. Part of the ANOVA table is shown below. Sum of Degrees of Squares Freedom Between treatments 1,500 _____? Within treatments (Error) _____? _____? Total 5,500 Fill in the blanks in the above ANOVA table. Source of Variation

Mean Square _____? _____?

F _____?

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Chapter 13: Experimental Design and Analysis of Variance ANSWER: Source of Variation Between treatments Within treatments (Error) Total

Sum of Squares 1,500 4,000 5,500

Degrees of Freedom 3 40 43

Mean Square 500 100

F 5.00

112. Ten observations were selected from each of three populations, and an analysis of variance was performed on the data. The following are part of the results. Sum of Degrees of Mean Source of Variation F Squares Freedom Square Between treatments 82.4 Within treatments (Error) 158.4 Total a. Using α = .05, test to see if there is a significant difference among the means of the three populations. b. If in part (a) you concluded that at least one mean is different from the others, determine which mean is different. The three sample means are 1 = 24.8, 2 = 23.4, and 3 = 27.4. Use Fisher's LSD procedure and let α = .05. ANSWER: a.

H0: μ1 = μ2 = μ3 Ha: At least one mean is different

F = 7.02 > 3.35; reject H0, conclude that at least one mean is different from the others LSD = 2.22

The mean of the third population is different from both the first and second populations. 113. Samples were selected from three populations. The data obtained are shown below. Sample 1 10 13 12 13

Sample 2 16 14 13 14 16 17

Sample 3 15 15 16 14 10

Sample mean ( j)

Sample variance ( )

2.4

5.5

Set up the ANOVA table for this problem.

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Chapter 13: Experimental Design and Analysis of Variance b.

At a 5% level of significance, test to determine whether there is a significant difference in the means of the three populations.

ANSWER: a. Source of Variation Between treatments Within treatments (Error) Total b.

Sum of Squares 21.73 40.00

Degrees of Freedom 2 12

Mean Square 10.87 3.33

F 3.26

Since the test statistic F = 3.26 < 3.89 do not reject H0, cannot conclude that there is a significant difference in the means of the three populations

114. In a completely randomized experimental design, 14 experimental units were used for each of the five levels of the factor (i.e., five treatments). Fill in the blanks in the following ANOVA table. Source of Variation Between treatments Within treatments (Error) Total

Sum of Squares _____? _____? 10,600

ANSWER: Source of Variation Between treatments Within treatments (Error) Total

Degrees of Freedom _____? _____? _____?

Sum of Squares 3,200 7,400 10,600

Mean Square 800.00 _____?

Degrees of Freedom 4 65 69

F _____?

Mean Square 800.00 113.85

F 7.03

115. Samples were selected from three populations. The data obtained are shown below. Sample 1 10 13 12 13

Sample 2 Sample 3 16 15 14 15 13 16 14 14 16 10 17 At a 5% level of significance, use Excel to test to determine whether there is a significant difference in the means of the three populations. ANSWER: 1 2 3 4 5 6 7

A Observation 1 2 3 4

B Sample 1 10 13 12 13

C Sample 2 16 14 13 14 16 17

D Sample 3 15 15 16 14 10

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Chapter 13: Experimental Design and Analysis of Variance 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Anova: Single Factor SUMMARY Groups Sample 1 Sample 2 Sample 3

Count 4 6 5

Sum 48 90 70

Average 12 15 14

Variance 2.0 2.4 5.5

ANOVA Source of Variation Between Groups Within Groups

SS 21.73333 40.00000

df 2 12

MS 10.86667 3.33333

F 3.26

Total

61.73333

P-value 0.07400

F crit 3.88529

Do not reject H0, cannot conclude that there is a significant difference in the means of the three populations 116. Halls, Inc. has three stores located in three different areas. Random samples of the sales of the three stores (in $1,000s) are shown below. Store 1 Store 2 Store 3 46 34 33 47 36 31 45 35 35 42 39 45 At a 5% level of significance, use Excel to determine whether there is a significant difference in the average sales of the three stores. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

A Observation 1 2 3 4 5

B Store 1 46 47 45 42 45

C Store 2 34 36 35 39

D Store 3 33 31 35

SUMMARY Groups Store 1 Store 2 Store 3

Count 5 4 3

Sum 225 144 99

Average 45 36 33

Variance 3.50000 4.66667 4.00000

ANOVA Source of Variation Between Groups Within Groups

SS 324 36

df 2 9

MS 162 4

F 40.5

P-value 0.00003

F crit 4.25649

Anova: Single Factor

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Chapter 13: Experimental Design and Analysis of Variance 20 21

Total

360

Reject H0, conclude that there is a significant difference in the average sales of the three stores 117. Random samples of individuals from three different cities were asked how much time they spend per day watching television. The results (in minutes) for the three groups are shown below. City I City II City III 260 178 211 280 190 190 240 220 250 260 240 300 At α = .05, use Excel to test to see if there is a significant difference in the averages of the three groups. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

A Observation 1 2 3 4 5

B City I 260 280 240 260 300

C City II 178 190 220 240

D City III 211 190 250

SUMMARY Groups City I City II City III

Count 5 4 3

Sum 1340 828 651

Average 268 207 217

Variance 520 796 927

ANOVA Source of Variation Between Groups Within Groups

SS 9552.92 6322.00

df 2 9

MS 4776.458 702.444

F 6.79977

Total

15874.92

P-value 0.01587

F crit 4.25649

Anova: Single Factor

Reject H0, conclude that there is a significant difference in the averages of the three groups 118. Three major automobile manufacturers have entered their cars in the Indianapolis 500 race. The speeds of the tested cars are given below. Manufacturer A 180 175 179 176 190

Manufacturer B 177 180 167 172

Manufacturer C 175 176 177

At α = .05, use Excel to determine whether there is a significant difference in the average speeds of the cars of the auto manufacturers. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

A Observation 1 2 3 4 5

B Mfr. A 180 175 179 176 190

C Mfr. B 177 180 167 172

D Mfr. C 175 176 177

SUMMARY Groups Manufacturer A Manufacturer B Manufacturer C

Count 5 4 3

Sum 900 696 528

Average 180 174 176

Variance 35.50000 32.66667 1.00000

ANOVA Source of Variation Between Groups Within Groups

SS 84 242

df 2 9

MS 42.00000 26.88889

F 1.56198

Total

326

P-value 0.26163

F crit 4.25649

Anova: Single Factor

Do not reject H0, cannot conclude that there is a significant difference in the average speeds of the cars of the auto manufacturers 119. A dietitian wants to see if there is any difference in the effectiveness of three diets. Eighteen people, comprising a sample, were randomly assigned to the three diets. Below you are given the total amount of weight lost in a month by each person. Diet A Diet B Diet C 14 12 25 18 10 32 20 22 18 12 12 14 20 16 17 18 12 14 What would you advise the dietician about the effectiveness of the three diets? Use Excel and a .05 level of significance. ANSWER: 1 2 3 4 5 6 7 8 9

A Observation 1 2 3 4 5 6

B Diet A 14 18 20 12 20 18

C Diet B 12 10 22 12 16 12

D Diet C 25 32 18 14 17 14

Anova: Single Factor

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Chapter 13: Experimental Design and Analysis of Variance 10 11 12 13 14 15 16 17 18 19 20 21 22

SUMMARY Groups Diet A Diet B Diet C

Count 6 6 6

Sum 102 84 120

Average 17 14 20

Variance 10.8 19.2 50.8

ANOVA Source of Variation Between Groups Within Groups

SS 108 404

df 2 15

MS 54.00000 26.93333

F 2.00495

Total

512

P-value 0.16918

F crit 3.68232

Conclude that the diets are equally effective. 120. Individuals were randomly assigned to three different production processes. The hourly units of production for the three processes are shown below. Production Process Process 1 Process 2 Process 3 33 33 28 30 35 36 28 30 30 29 38 34 Use Excel with α = .05 to determine whether there is a significant difference in the mean hourly units of production for the three types of production processes. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A Observation 1 2 3 4

B Process 1 33 30 28 29

C Process 2 33 35 30 38

D Process 3 28 36 30 34

SUMMARY Groups Process 1 Process 2 Process 3

Count 4 4 4

Sum 120 136 128

Average 30 34 32

Variance 4.66667 11.33333 13.33333

ANOVA Source of Variation Between Groups Within Groups

SS 32 88

df 2 9

MS 16.00000 9.77778

F 1.63636

Total

120

P-value 0.24766

F crit 4.25649

Anova: Single Factor

We cannot conclude that there is a significant difference in the mean hourly units of production for the three types of production processes. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance 121. A research organization wishes to determine whether four brands of batteries for transistor radios perform equally well. Three batteries of each type were randomly selected and installed in the three test radios. The number of hours of use for each battery is given below. Brand Radio 1 2 3 4 A 25 27 20 28 B 29 38 24 37 C 21 28 16 19 Consider the three different test radios and use Excel to carry out the analysis of variance procedure for a randomized block design. Use a .05 level of significance. ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

A Radio A B C

B Brand 1 25 29 21

C Brand 2 27 38 28

D Brand 3 20 24 16

E Brand 4 28 37 19

SUMMARY A B C

Count 4 4 4

Sum 100 128 84

Average 25 32 21

Variance 12.66667 44.66667 26.00000

Brand 1 Brand 2 Brand 3 Brand 4

3 3 3 3

75 93 60 84

25 31 20 28

16 37 16 81

ANOVA Source of Variation Rows Columns Error

SS 248 198 52

df 2 3 6

MS 124.0000 66.0000 8.6667

F 14.30769 7.61539

Total

498

P-value 0.00521 0.01808

F crit 5.14325 4.75706

Anova: Two-Factor Without Replication

Conclude that there is a significant difference in the mean useful life of the four brands of batteries. 122. Five drivers were selected to test drive two makes of automobiles. The following table shows the number of miles per gallon for each driver driving each car. Driver Automobile 1 2 3 4 5 A 30 31 30 27 32 B 36 35 28 31 30 Consider the makes of automobiles as treatments and the drivers as blocks and use Excel to determine whether there is any difference in the miles/gallon of the two makes of automobiles. Let α = .05. Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

A Automobile A B

B Driver 1 30 36

C Driver 2 31 35

D Driver 3 30 28

E Driver 4 27 31

SUMMARY A B

Count 5 5

Sum 150 160

Average 30 32

Variance 3.5 11.5

Driver 1 Driver 2 Driver 3 Driver 4 Driver 5

2 2 2 2 2

66 66 58 58 62

33 33 29 29 31

18 8 2 8 2

ANOVA Source of Variation Rows Columns Error

SS 10 32 28

df 1 4 4

MS 10 8 7

F 1.42857 1.14286

Total

F Driver 5 32 30

P-value 0.29802 0.45007

F crit 7.70865 6.38823

Anova: Two-Factor Without Replication

We cannot conclude that there is a significant difference in the mean miles/gallon of the two makes of automobiles. 123. A factorial experiment involving two levels of factor A and two levels of factor B resulted in the following. Factor B Level 1 Level 2 14 18 16 12 Level 2 18 16 20 14 Use Excel and test for any significant main effect and any interaction effect. Use α = .05. ANSWER: Factor A Level 1

A 1 2 3 4 5 6 7 8 9 10

A Level 1 A Level 2

B B Level 1 14 16 18 20

C B Level 2 18 12 16 14

B Level 1

B Level 2

Total

Anova: Two-Factor With Replication SUMMARY A Level 1

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Chapter 13: Experimental Design and Analysis of Variance 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Count Sum Average Variance

2 30 15 2

2 30 15 18

4 60 15 6.66667

A Level 2 Count Sum Average Variance

2 38 19 2

2 30 15 2

4 68 17 6.66667

Total Count Sum Average Variance

4 68 17 6.66667

4 60 15 6.66667

ANOVA Source of Variation Sample Columns Interaction Within

SS 8 8 8 24

df 1 1 1 4

Total

MS 8 8 8 6

F 1.33333 1.33333 1.33333

P-value 0.3125 0.3125 0.3125

F crit 7.70865 7.70865 7.70865

Factor A Treatment: Do not reject H0 Factor B Treatment: Do not reject H0 Interaction (AB): Do not reject H0 124. Employees of MNM Corporation are about to undergo a retraining program. Management is trying to determine which of three programs is the best. They believe that the effectiveness of the programs may be influenced by gender. A factorial experiment was designed. You are given the following information. Factor B: Gender Male Female 320 380 240 300 Program B 160 240 180 210 Program C 240 360 290 380 What advice would you give MNM? Use Excel and a .05 level of significance. ANSWER: Factor A: Program Program A

A 1 2 3 4 5

Program A Program B

B Male 320 240 160 180

C Female 380 300 240 210

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Chapter 13: Experimental Design and Analysis of Variance 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

Program C

240 290

360 380

Male

Female

Total

2 560 280 3200

2 680 340 3200

4 1240 310 3333.333

Program B Count Sum Average Variance

2 340 170 200

2 450 225 450

4 790 197.5 1225

Program C Count Sum Average Variance

2 530 265 1250

2 740 370 200

4 1270 317.5 4158.333

Total Count Sum Average Variance

6 1430 238.333 3776.667

6 1870 311.667 5456.667

ANOVA Source of Variation Sample Columns Interaction Within

SS 36150.00 16133.33 1516.67 8500.00

df 2 1 2 6

Total

62300.00

Anova: Two-Factor With Replication SUMMARY Program A Count Sum Average Variance

MS 18075.00 16133.33 758.33 1416.67

F 12.75882 11.38824 0.53529

P-value 0.00690 0.01496 0.61107

F crit 5.14325 5.98737 5.14325

There is a significant difference in the programs, and there is a significant difference in gender. There is no significant interaction effect. 125. Regional Manager Sue Collins would like to know if the mean number of telephone calls made per eight-hour shift is the same for the telemarketers at her three call centers (Austin, Las Vegas, and Albuquerque). A simple random sample of six telemarketers from each of the three call centers was taken, and the number of telephone calls made in eight hours by each observed employee is shown below. Observation 1

Center 1 Austin 82

Center 2 Las Vegas 72

Center 3 Albuquerque 71 Page 39

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Chapter 13: Experimental Design and Analysis of Variance 2 3 4 5 6 Sample mean Sample variance a. b.

68 77 80 69 78 75.667 33.867

63 74 60 70 73 68.667 33.467

81 73 68 76 80 74.833 26.167

Using α = .10, test for any significant difference in number of telephone calls made at the three call centers. Apply Fisher’s least significant difference (LSD) procedure to determine where the differences occur. Use α = .05.

ANSWER: a. Reject H0, because F = 2.815 > 2.695 (or because p-value = .092 < α = .10). Conclusion: Not all call center means are equal. ANOVA Source of Variation Between groups Within groups Total b.

SS 175.444 467.500 642.944

df 2 15 17

MS 87.722 31.167

F 2.814637

P-value 0.091645

F crit 2.69517

Centers 1 & 2: Reject H0 Using α: 7.000 > 6.869 Using t: 2.172 > 2.1317 Using p-value: .0463 < .05 Centers 1 & 3: Do not reject H0 Using α: 0.834 < 6.869 Using t: 0.259 < 2.1317 Using p-value: .7992 > .05 Centers 2 & 3: Do not reject H0 Using α: 6.166 < 6.869 Using t: 1.913 < 2.1317 Using p-value: .0750 > .05

126. To test whether the time required to fully load a standard delivery truck is the same for three work shifts (day, evening, and night), NatEx obtained the following data on the time (in minutes) needed to pack a truck. Use these data to test whether the population mean times for loading a truck differ for the three work shifts. Use α = .05. Observation 1 2 3 4 5 Sample mean Sample variance

Day Shift 92 81 103 77 82 87.0 110.5

Evening Shift 83 93 79 102 84 88.2 85.7

Night Shift 89 97 95 88 106 95.0 52.5

ANSWER: We cannot reject H0: µ1 = µ2 = µ3 because F = 2.073 < F = 3.885 (or p-value = .17 > α = .05). Conclusion: Copyright Cengage Learning. Powered by Cognero.

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Chapter 13: Experimental Design and Analysis of Variance There is little evidence in the three shifts’ mean loading times. ANOVA Source of Variation Between groups Within groups Total

SS 260.8 754.8 1015.6

df 2 12 14

MS 130.4 62.9

F 2.073132

P-value 0.168521

F crit 3.88529

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Chapter 14: Simple Linear Regression Multiple Choice 1. The proportion of the variation in the dependent variable y that is explained by the estimated regression equation is measured by the _____. a. correlation coefficient b. standard error of the estimate c. coefficient of determination d. confidence interval estimate ANSWER: c 2. The least squares criterion is _____. a. min b. min c. min d. min ANSWER: d 3. In a residual plot against x that does NOT suggest we should challenge the assumptions of our regression model, we would expect to see a _____. a. horizontal band of points centered near 0 b. widening band of points c. band of points having a slope consistent with that of the regression equation d. parabolic band of points ANSWER: a 4. The difference between the observed value of the dependent variable and the value predicted by using the estimated regression equation is called _____. a. the standard error b. a residual c. a prediction interval d. the variance ANSWER: b 5. As the goodness of fit for the estimated regression equation increases, the _____. a. absolute value of the regression equation’s slope increases b. value of the regression equation’s y-intercept decreases c. value of the coefficient of determination increases d. value of the correlation coefficient increases ANSWER: c 6. A measure of the strength of the relationship between two variables is the _____. a. confidence interval estimate Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression b. slope b1 of the estimated regression line c. standard error of the estimate d. correlation coefficient ANSWER: d 7. Regression analysis is a statistical procedure for developing a mathematical equation that describes how _____. a. one independent and one or more dependent variables are related b. several independent and several dependent variables are related c. one dependent and one or more independent variables are related d. two independent variables are related ANSWER: c 8. The interval estimate of the mean value of y for a given value of x is the _____. a. confidence interval b. prediction interval c. residual interval d. correlation interval ANSWER: a 9. In regression analysis, the variable that is being predicted is the _____. a. dependent variable b. independent variable c. intervening variable d. controlled variable ANSWER: a 10. In a regression analysis, the variable that is used to predict the dependent variable _____. a. must have the same units as the variable doing the predicting b. is the independent variable c. is the dependent variable d. usually is denoted by y ANSWER: b 11. In regression analysis, the independent variable is typically plotted on the _____. a. y-axis of a scatter diagram b. x-axis of a scatter diagram c. y-axis of a histogram d. x-axis of a histogram ANSWER: b 12. The equation that describes how the dependent variable (y) is related to the independent variable (x) is called _____. a. the correlation model b. the regression model Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression c. correlation analysis d. the least squares criterion ANSWER: b 13. In a simple regression analysis (where y is a dependent and x an independent variable), if the slope is positive, then it must be true that _____. a. there is a positive correlation between x and y b. there is a negative correlation between x and y c. there is no correlation between x and y d. the y-intercept is 0 ANSWER: a 14. A procedure used for finding the equation of a straight line that provides the best approximation for the relationship between the independent and dependent variables is _____. a. correlation analysis b. the mean squares method c. the least squares method d. the most squares method ANSWER: c 15. Application of the least squares method results in values of the y-intercept and the slope that minimizes the sum of the squared deviations between the _____. a. observed values of the independent variable and the predicted values of the independent variable b. actual values of the independent variable and the predicted values of the dependent variable c. observed values of the dependent variable and the predicted values of the dependent variable d. predicted values of the independent variable and the actual values of the dependent variable ANSWER: c 16. A least squares regression line _____. a. may be used to predict a value of y if the corresponding x value is given b. implies a cause-effect relationship between x and y c. can only be determined if a good linear relationship exists between x and y d. is only used for positively correlated data ANSWER: a 17. In regression analysis, if the dependent variable is measured in dollars, the independent variable _____. a. must also be in dollars b. must be in some unit of currency c. can be any units d. cannot be in dollars ANSWER: c 18. Regression analysis was applied between sales (in $1000s) and advertising (in $100s), and the following regression function was obtained. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression = 80 + 6.2x Based on the above estimated regression line, if advertising is $10,000, then the point estimate for sales (in dollars) is _____. a. $62,080 b. $142,000 c. $700 d. $700,000 ANSWER: d 19. Regression analysis was applied between sales (in $1000s) and advertising (in $100s), and the following regression function was obtained. = 500 + 4x Based on the above estimated regression line, if advertising is $10,000, then the point estimate for sales (in dollars) is _____. a. $900 b. $900,000 c. $40,500 d. $505,000 ANSWER: b 20. A regression analysis between demand (y in 1000 units) and price (x in dollars) resulted in the following equation: = 9 − 3x The above equation implies that if the price is increased by $1, the demand is expected to _____. a. increase by 6 units b. decrease by 3 units c. decrease by 6,000 units d. decrease by 3,000 units ANSWER: d 21. A regression analysis between sales (in $1000s) and price (in dollars) resulted in the following equation: = 50,000 − 8x The above equation implies that an increase of _____. a. $1 in price is associated with a decrease of $8 in sales b. $8 in price is associated with an increase of $8,000 in sales c. $1 in price is associated with a decrease of $42,000 in sales d. $1 in price is associated with a decrease of $8,000 in sales ANSWER: d 22. A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation: = 50,000 + 6x Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression The above equation implies that an increase of _____. a. $6 in advertising is associated with an increase of $6,000 in sales b. $1 in advertising is associated with an increase of $6 in sales c. $1 in advertising is associated with an increase of $56,000 in sales d. $1 in advertising is associated with an increase of $6,000 in sales ANSWER: d 23. SSE can never be _____. a. larger than SST b. smaller than SST c. equal to 1 d. equal to 0 ANSWER: a 24. Which of the following is correct? a. SSE = SSR + SST b. SSR = SSE + SST c. SST = SSR + SSE d. SST = (SSR)2 ANSWER: c 25. In simple linear regression, r2 is the _____. a. estimated regression equation b. correlation coefficient c. sum of the squared residuals d. coefficient of determination ANSWER: d 26. If all the points of a scatter diagram lie on the least squares regression line, then the coefficient of determination for these variables based on these data _____. a. is 0 b. is 1 c. is either 1 or –1, depending upon whether the relationship is positive or negative d. could be any value between –1 and 1 ANSWER: b 27. Larger values of r2 imply that the observations are more closely grouped about the _____. a. average value of the independent variables b. average value of the dependent variable c. least squares line d. origin ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression 28. In a regression analysis, if r2 = 1, then _____. a. SSE must also be equal to 1 b. SSE must be equal to 0 c. SSE can be any positive value d. SSE must be negative ANSWER: b 29. In a regression analysis, if r2 = 1, then _____. a. SSE = SST b. SSE = 1 c. SSR = SSE d. SSR = SST ANSWER: d 30. In regression and correlation analysis, if SSE and SST are known, then with this information the _____. a. coefficient of determination can be computed b. slope of the line can be computed c. y-intercept can be computed d. regression equation can be computed ANSWER: a 31. In a regression analysis, if SSE = 200 and SSR = 300, then the coefficient of determination is _____. a. .667 b. .600 c. .400 d. 1.500 ANSWER: b 32. In a regression analysis, if SSE = 500 and SSR = 300, then the coefficient of determination is _____. a. .600 b. .166 c. 1.666 d. .375 ANSWER: d 33. If a data set has SST = 2,000 and SSE = 800, then the coefficient of determination is _____. a. .4 b. .6 c. .5 d. .8 ANSWER: b 34. In a regression analysis, if SST = 4500 and SSE = 1575, then the coefficient of determination is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression a. .35 b. .65 c. 2.85 d. .45 ANSWER: b 35. If the coefficient of determination is a positive value, then the regression equation _____. a. must have a positive slope b. must have a negative slope c. could have either a positive or a negative slope d. must have a positive y-intercept ANSWER: c 36. It is possible for the coefficient of determination to be _____. a. larger than 1 b. less than 1 c. less than 0 d. between –1 and 1 ANSWER: b 37. If the correlation coefficient is .8, then the percentage of variation in the dependent variable explained by the estimated regression equation is _____. a. 0.80% b. 80% c. 0.64% d. 64% ANSWER: d 38. If the correlation coefficient is .4, the percentage of variation in the dependent variable explained by the estimated regression equation is _____. a. 40% b. 16% c. 4% d. 2% ANSWER: b 39. If the coefficient of determination is equal to 1, then the correlation coefficient _____. a. must also be equal to 1 b. can be either –1 or 1 c. can be any value between –1 and 1 d. must be –1 ANSWER: b 40. If there is a very strong negative correlation between two variables, then the correlation coefficient must be _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression a. very close to 1 b. very close to –0.5 c. exactly 0 d. very close to –1 ANSWER: d 41. If the correlation coefficient is a positive value, then the slope of the regression line _____. a. must also be positive b. is dependent on the value of the y-intercept c. can be 0 d. must be negative ANSWER: a 42. If the correlation coefficient is a negative value, then the coefficient of determination _____. a. must also be negative b. must be 0 c. can be either negative or positive d. must be positive ANSWER: d 43. The numerical value of the coefficient of determination _____. a. is always larger than the correlation coefficient b. is always smaller than the correlation coefficient c. is negative if the correlation coefficient is negative d. is positive if the correlation coefficient is negative ANSWER: d 44. Find the correlation coefficient given the estimated regression equation a. .6561 b. –.9356 c. .9356 d. –.6561 ANSWER: c

and

45. If two variables, x and y, have a strong linear relationship, then _____. a. there may or may not be any causal relationship between x and y b. x causes y to happen c. y causes x to happen d. the slope of the regression equation must be negative ANSWER: a 46. In regression analysis, which of the following is NOT a required assumption about the error term ε? a. The expected value of the error term is 0. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression b. The variance of the error term is the same for all values of x. c. The values of the error term are independent. d. The values of the error term are dependent. ANSWER: d 47. In simple linear regression analysis, which of the following is NOT true? a. The F test and the t test yield the same results. b. The F test and the t test may or may not yield the same results. c. The relationship between x and y is represented by means of a straight line. d. The value of F = t2. ANSWER: b 48. Compared to the confidence interval estimate for a particular value of y (in a linear regression model), the interval estimate for an average value of y will be _____. a. narrower b. wider c. the same d. Not enough information is given. ANSWER: a 49. A data point (observation) that does not fit the trend shown by the remaining data is called a(n) _____. a. residual b. outlier c. point estimate d. y-intercept ANSWER: b 50. An observation that has a strong effect on the regression results is called a(n) _____. a. residual b. sum of squares error c. influential observation d. point estimate ANSWER: c 51. Data points having high leverage are often _____. a. residuals b. a sum of squares error c. influential d. point estimates ANSWER: c Exhibit 14-1 A regression analysis resulted in the following information regarding a dependent variable (y) and an independent variable (x). Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression n = 10 ∑x = 55 ∑y = 55 ∑x2 = 385 ∑y2 = 385 ∑xy = 220 52. Refer to Exhibit 14-1. The least squares estimate of b1 equals _____. a. 1 b. –1 c. 5.5 d. 11 ANSWER: b 53. Refer to Exhibit 14-1. The least squares estimate of b0 equals _____. a. 1 b. –1 c. 5.5 d. 11 ANSWER: d 54. Refer to Exhibit 14-1. The point estimate of y when x = 20 is _____. a. 0 b. 31 c. 9 d. –9 ANSWER: d 55. Refer to Exhibit 14-1. The sample correlation coefficient equals _____. a. 0 b. –1 c. 1 d. –0.5 ANSWER: b 56. Refer to Exhibit 14-1. The coefficient of determination equals _____. a. 0 b. –1 c. 1 d. –0.5 ANSWER: c Exhibit 14-2 You are given the following information about x and y. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression x Independent Variable 15 12 10 7

y Dependent Variable 5 7 9 11

57. Refer to Exhibit 14-2. The least squares estimate of b1 equals _____. a. –0.7647 b. –0.13 c. 21.4 d. 16.412 ANSWER: a 58. Refer to Exhibit 14-2. The least squares estimate of b0 equals _____. a. –7.647 b. –1.3 c. 21.4 d. 16.41176 ANSWER: d 59. Refer to Exhibit 14-2. The sample correlation coefficient equals _____. a. –86.667 b. –.99705 c. .9941 d. .99705 ANSWER: b 60. Refer to Exhibit 14-2. The coefficient of determination equals _____. a. –.99705 b. –.9941 c. .9941 d. .99705 ANSWER: c Exhibit 14-3 Regression analysis was applied between sales data (in $1000s) and advertising data (in $100s), and the following information was obtained. = 12 + 1.8x n = 17 SSR = 225 SSE = 75 Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression = 0.2683

61. Refer to Exhibit 14-3. Based on the above estimated regression equation, if advertising is $3,000, then the point estimate for sales (in dollars) is _____. a. $66,000 b. $5,412 c. $66 d. $17,400 ANSWER: a 62. Refer to Exhibit 14-3. The F statistic computed from the above data is _____. a. 3 b. 45 c. 48 d. 38 ANSWER: b 63. Refer to Exhibit 14-3. The critical F value at α = .05 is _____. a. 3.59 b. 3.68 c. 4.45 d. 4.54 ANSWER: d 64. Refer to Exhibit 14-3. The t statistic for testing the significance of the slope is _____. a. 1.80 b. 1.96 c. 6.709 d. .555 ANSWER: c 65. Refer to Exhibit 14-3. Using α = .05, the critical t value for testing the significance of the slope is _____. a. 1.753 b. 2.131 c. 1.746 d. 2.120 ANSWER: b Exhibit 14-4 The following information regarding a dependent variable (y) and an independent variable (x) is provided. x 2 1

y 4 3

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Chapter 14: Simple Linear Regression 4 4 3 6 5 8 SSE = 6 SST = 16 66. Refer to Exhibit 14-4. The least squares estimate of the y-intercept is _____. a. 1 b. 2 c. 3 d. 4 ANSWER: b 67. Refer to Exhibit 14-4. The least squares estimate of the slope is _____. a. 1 b. 2 c. 3 d. 4 ANSWER: a 68. Refer to Exhibit 14-4. The coefficient of determination is _____. a. .7096 b. –.7906 c. .625 d. .375 ANSWER: c 69. Refer to Exhibit 14-4. The sample correlation coefficient is _____. a. .7906 b. –.7906 c. .625 d. .375 ANSWER: a 70. Refer to Exhibit 14-4. The MSE is _____. a. 1 b. 2 c. 3 d. 4 ANSWER: b Exhibit 14-5 You are given the following information about x and y. x

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Chapter 14: Simple Linear Regression Independent Variable 1 2 3 4 5

Dependent Variable 5 4 3 2 1

71. Refer to Exhibit 14-5. The least squares estimate of b1 (slope) equals _____. a. 1 b. –1 c. 6 d. 5 ANSWER: b 72. Refer to Exhibit 14-5. The least squares estimate of b0 (intercept) equals _____. a. 1 b. –1 c. 6 d. 5 ANSWER: c 73. Refer to Exhibit 14-5. The point estimate of y when x = 10 is _____. a. –10 b. 10 c. –4 d. 4 ANSWER: c 74. Refer to Exhibit 14-5. The sample correlation coefficient equals _____. a. 0 b. 1 c. –1 d. –.5 ANSWER: c 75. Refer to Exhibit 14-5. The coefficient of determination equals _____. a. 0 b. –1 c. 1 d. –.5 ANSWER: c Exhibit 14-6 You are given the following information about x and y. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression x Independent Variable 4 6 2 4

y Dependent Variable 12 3 7 6

76. Refer to Exhibit 14-6. The least squares estimate of b1 equals _____. a. 1 b. –1 c. –11 d. 11 ANSWER: b 77. Refer to Exhibit 14-6. The least squares estimate of b0 equals _____. a. 1 b. –1 c. –11 d. 11 ANSWER: d 78. Refer to Exhibit 14-6. The sample correlation coefficient equals _____. a. –.4364 b. .4364 c. –.1905 d. .1905 ANSWER: a 79. Refer to Exhibit 14-6. The coefficient of determination equals _____. a. –.4364 b. .4364 c. –.1905 d. .1905 ANSWER: d 80. The primary tool or measure for determining whether the assumed regression model is appropriate is _____. a. the F test b. residual analysis c. the r2 value d. the correlation coefficient ANSWER: b 81. The standardized residual is provided by dividing each residual by its _____. a. mean residual Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression b. corresponding x value c. standard deviation d. z-score ANSWER: c Subjective Short Answer 82. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). x 2 3 6 7 8 7 9 a. b. c. d.

y 12 9 8 7 6 5 2

Develop the least squares estimated regression equation. At 95% confidence, perform a t test and determine whether the slope is significantly different from zero. Perform an F test to determine whether the model is significant. Let α = 0.05. Compute the coefficient of determination.

ANSWER: a. b. c. d.

= 13.75 − 1.125x t = -5.196 < –2.571; reject H0 F = 27 > 6.61; reject H0 .844

83. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel to develop a scatter diagram and to compute the least squares estimated regression equation. x 2 3 6 7 8 7 9

y 12 9 8 7 6 5 2

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Chapter 14: Simple Linear Regression ANSWER:

84. A company has recorded data on the daily demand for its product (y in thousands of units) and the unit price (x in hundreds of dollars). A sample of 11 days' demand and associated price resulted in the following data: ∑ x = 154 a. b. c. d. e.

∑ x 2 = 2,586

∑ y = 451

∑ y 2 = 18,901

∑ xy = 5,930

Using the above information, develop the least-squares estimated regression line. Compute the coefficient of determination. Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let α = 0.05. Perform a t test to determine whether the slope is significantly different from zero. Let α = 0.05. Would the demand ever reach zero? If yes, at what price would the demand be zero? Show your complete work.

ANSWER: a. b. c. d. e.

= 53.502 − 0.893x .836 Since the test statistic F = 46.011 > 5.12, reject H0. The test statistic t = –6.765 Critical t = –2.262 to + 2.262; therefore reject H0. Yes, at $5,991

85. A company has recorded data on the daily demand for its product (y in thousands of units) and the unit price (x in hundreds of dollars). A sample of 15 days' demand and associated prices resulted in the following data: ∑ x = 75 a. b. c. d.

∑ x 2 = 469

∑ y = 135

∑ y 2 = 1315

∑ xy = 616

Using the above information, develop the least squares estimated regression line and write the equation. Compute the coefficient of determination. Perform an F test and determine whether there is a significant relationship between demand and unit price. Let α = .05. Would the demand ever reach zero? If yes, at what price would the demand be zero?

ANSWER: a.

= 12.138 – 0.6276x

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Chapter 14: Simple Linear Regression b. c. d.

r2 = .3703 Since the test statistic F = 7.65 > 4.67, reject H0 and conclude that demand and unit price are related. Yes, at $1,934

86. A company has recorded data on the daily demand for its product (y in thousands of units) and the unit price (x in hundreds of dollars). A sample of 15 days' demand and associated prices resulted in the following data: ∑ x 2 = 437

∑ x = 75 a. b. c. d.

∑ y = 180

∑ y 2 = 2266

∑ xy = 844

Using the above information, develop the least squares estimated regression line and write the equation. Compute the coefficient of determination. Perform an F test and determine whether there is a significant relationship between demand and unit price. Let α = 0.05. Would the demand ever reach zero? If yes, at what price would the demand be zero?

ANSWER: a. b. c. d.

= 16.515 – 0.903x r = .477 Since the test statistic F = 11.87 > 4.67, reject H0. Yes, at $1,828.68 2

87. A company has recorded data on the weekly sales for its product (y) and the unit price of the competitor's product (x). The data resulting from a random sample of seven weeks follows. Use Excel to develop a scatter diagram and to compute the least squares estimated regression equation and the coefficient of determination. Week 1 2 3 4 5 6 7

Price 0.33 0.25 0.44 0.40 0.35 0.39 0.29

Sales 20 14 22 21 16 19 15

ANSWER:

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Chapter 14: Simple Linear Regression

88. We are interested in determining the relationship between daily supply (y) and the unit price (x) for a particular item. A sample of 10 days’ supply and associated price resulted in the following data: ∑x2 = 526

∑x = 66 a. b. c.

∑y = 71

∑y2= 605

∑xy = 557

Develop the least square estimated regression equation. Compute the coefficient of determination and fully explain its meaning. At α = 0.05, perform a t test and determine if the slope is significantly different from zero.

ANSWER: a. b. c.

= 0.646 + 0.978x .8567; 85.67% of variation in supply is explained by variation in price t = 6.917 > 2.306; reject H0

89. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel to develop a scatter diagram and to compute the least squares estimated regression equation and the coefficient of determination. x 2 3 6 7 8 7 9 ANSWER:

y 12 9 8 7 6 5 2

90. Shown below is a portion of a computer output for a regression analysis relating y (dependent variable) and x (independent variable). ANOVA Regression Residual Total

df 1 13 14

SS 50.58 55.42 106.00

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Chapter 14: Simple Linear Regression Coefficients 16.156 -0.903

Intercept Variable x a. b.

Standard Error 1.42 0.26

t Stat

Perform a t test and determine whether y and x are related. Use α = .05. Compute the coefficient of determination and fully interpret the meaning.

ANSWER: a. b.

Since the test statistic t = –3.47 < –2.11, reject H0. r2 = .477 47.7% of variability in y is explained by variability in x.

91. Shown below is a portion of a computer output for regression analysis relating y (dependent variable) and x (independent variable). ANOVA

a. b. c. d. e.

Regression Residual Total

df 1 20 21

SS 882 4000 4882

Intercept Variable x

Coefficients 5.00 6.30

Standard Error 3.56 3.00

t Stat

What is the sample size? Perform a t test and determine whether x and y are related. Use α = .05. Perform an F test and determine whether x and y are related. Use α = .05. Compute the coefficient of determination. Interpret the meaning of the value of the coefficient of determination that you found in part (d).

ANSWER: a. b. c. d. e.

22 Since the test statistic t = 2.1 > 2.086, reject H0. Therefore, x and y are related. Since the test statistic F = 4.41 > 4.35, reject H0. .1807 18.07% of the variation in y is due to the variation in x.

92. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel's Regression tool to answer the following questions: x 2 3 6 7 8

y 12 9 8 7 6

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Chapter 14: Simple Linear Regression 7 9 a. b. c. d.

5 2

What is the estimated regression equation? Perform a t test and determine whether x and y are related. Use α = .05. Perform an F test and determine whether x and y are related. Use α = .05. Find and interpret the coefficient of determination.

ANSWER:

x 2 3 6 7 8 7 9

y 12 9 8 7 6 5 2

SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Sq. Standard Error Observations

0.9185587 0.84375 0.8125 1.3693064 7

ANOVA Regression Residual Total

df 1 5 6 Coefficients

Intercept x

13.75 -1.125

a. b. c. d.

SS 50.625 9.375 60

MS 50.625 1.875

Standard t Stat Error 1.398341 9.833082 0.216506 -5.19615

F 27

Significance F 0.00348

P-value

Lower 95%

0.0001853 10.1555 0.0034782 -1.68155

= 13.75 − 1.125x Since the p-value .003478165 < .05, reject H0. Therefore, x and y are related. Since the p-value .00348 < .05, reject H0. r2 = .84375; 84.375% of the variation in y is explained by the variation in x.

93. A company has recorded data on the weekly sales for its product (y) and the unit price of the competitor's product (x). The data resulting from a random sample of seven weeks follows. Use Excel's Regression tool to answer the following Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression questions: Week 1 2 3 4 5 6 7 a. b. c. d.

Price 0.33 0.25 0.44 0.40 0.35 0.39 0.29

Sales 20 14 22 21 16 19 15

ANSWER:

Price 0.33 0.25 0.44 0.4 0.35 0.39 0.29

Sales 20 14 22 21 16 19 15

SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Sq. Standard Error Observations

0.877760967 0.770464315 0.724557178 1.643764862 7

ANOVA Regression Residual Total

df 1 5 6

Standard t Stat Error 3.581788441 3.608215 0.992676 41.60305344 10.15521 4.096719 Coefficients

Intercept Price

SS MS 45.34733 45.34733 13.50981 2.701963 58.85714

a. b. c. d. Copyright Cengage Learning. Powered by Cognero.

F 16.78311

Significance F 0.009385

P-value

Lower 95%

0.366447 0.009385

-5.69341 15.49829

= 3.581788441 + 41.60305344x Since the p-value .009385 < .05, reject H0. Therefore, the competitor's price and sales are related. Since the p-value .009385 < .05, reject H0. r2 = .770464315; 77.05% of the variation in sales is explained by Page 22

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Chapter 14: Simple Linear Regression the variation in the competitor's price.

94. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel to Compute a 95% confidence interval for E(y) when x = 5. Compute a 95% prediction interval for y when x = 5.

a. b. x 2 3 6 7 8 7 9

y 12 9 8 7 6 5 2

ANSWER: A

Value Sheet B C

2 3 4 5

2 3 6 7

12 9 8 7

7 8 9 10 11 12 13 14

7 9

5 2

Multiple R

0.91855865

16 17 18 19 20 21 22 23 24 25 26

R Square Adjusted R Square Standard Error Observations

0.84375 0.8125 1.36930639 7

SUMMARY OUTPUT

D E Confidence Interval Given value of x 5 xbar 6 x-xbar -1 (x-xbar)sq 1 Sum of (x40 xbar)sq Var of yhat 0.3147321 Stdev of yhat 0.5610099 t value 2.5705776 Margin of Error 1.4421196 Point Estimate 8.125 Lower Limit 6.6828804 Upper Limit 9.5671196

Regression Statistics

Prediction Interval Var of yind 2.1897321 Stdev of yind 1.4797744 Margin of Error 3.8038749 Lower Limit 4.3211251 Upper Limit 12.446125

ANOVA Regression Residual Total

df 1 5 6

SS 50.625 9.375 60

MS 50.625 1.875

F 27

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Chapter 14: Simple Linear Regression 27

Coefficients

28 29

Intercept x

13.75 -1.125

Standard Error 1.39834 0.21651

t Stat

P-value

9.833081802 -5.196152423

0.0001853 0.0034782

Formula Sheet Confidence Interval Given value of x 5 xbar =AVERAGE(A2:A8) x-xbar =E2-E3 (x-xbar)sq =E4^2 Sum of (x=DEVSQ(A2:A8) xbar)sq Var of yhat =D24*(1/B19+E5/E6) Stdev of yhat =SQRT(E7) t value =T.INV(0.05,5) Margin of Error =E9*E8 Point Estimate =B28+B29*E2 Lower Limit =E11-E10 Upper Limit =E11+E10 Prediction Interval Var of yind Stdev of yind Margin of Error Lower Limit Upper Limit

=D24+E7 =SQRT(E16) =E9*E17 =E11-E18 =E11+E19

a. b.

6.68 to 9.57 4.32 to 12.45

95. A company has recorded data on the weekly sales for its product (y) and the unit price of the competitor's product (x). The data resulting from a random sample of seven weeks follows. Use Excel to Compute a 95% confidence interval for expected sales for all weeks when the competitor’s price is 0.30. Compute a 95% prediction interval for sales for a week when the competitor's price is 0.30.

a. b. Week 1 2 3 4 5 6 7

Price 0.33 0.25 0.44 0.40 0.35 0.39 0.29

Sales 20 14 22 21 16 19 15

ANSWER: Copyright Cengage Learning. Powered by Cognero.

Value Sheet Page 24

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Chapter 14: Simple Linear Regression A

Price

Sales

2 3 4 5

0.33 0.25 0.44 0.4

20 14 22 21

0.35

7 8 9 10 11 12 13 14

0.39 0.29

19 15

Multiple R

0.877761

16 17 18 19 20

R Square Adjusted R Square Standard Error Observations

0.7704643 0.7245572 1.6437649 7

SUMMARY OUTPUT

Lower Limit

D Confidence Interval Given value of x xbar x-xbar (x-xbar)sq Sum of (xxbar)sq Var of yhat Stdev of yhat t value Margin of Error Point Estimate 14.00012 Upper Limit

0.3 0.35 -0.05 0.0025 0.0262 0.643816 0.802381 2.570578 2.062583 16.0627 18.12529

Regression Statistics

Prediction Interval Var of yind Stdev of yind Margin of Error Lower Limit Upper Limit

3.345779 1.829147 4.701964 11.36074 27.42344

Formula Sheet Confidence Interval Given value of x 0.3 Xbar =AVERAGE(A2:A8) x-xbar =E2-E3 (x-xbar)sq =E4^2 Sum of (x-xbar)sq =DEVSQ(A2:A8) Var of yhat =D24*(1/B19+E5/E6) Stdev of yhat =SQRT(E7) t value =T.INV(0.05,5) Margin of Error =E9*E8 Point Estimate =B28+B29*E2 Lower Limit =E11-E10 Upper Limit =E11+E10 Prediction Interval Var of yind Stdev of yind Margin of Error Lower Limit Upper Limit

=D24+E7 =SQRT(E16) =E9*E17 =E11-E18 =E11+E19

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Chapter 14: Simple Linear Regression a. b.

14.00 to 18.13 11.36 to 27.42

96. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel's Regression tool to construct a residual plot and use it to determine if any model assumptions may have been violated. x 2 3 6 7 8 7 9 ANSWER:

y 12 9 8 7 6 5 2

The constant variance assumption may be violated. 97. A company has recorded data on the weekly sales for its product (y) and the unit price of the competitor's product (x). The data resulting from a random sample of seven weeks follows. Use Excel's Regression tool to construct a residual plot and use it to determine if any model assumptions may have been violated. Week 1 2 3 4 5 6 7

Price 0.33 0.25 0.44 0.40 0.35 0.39 0.29

Sales 20 14 22 21 16 19 15

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Chapter 14: Simple Linear Regression ANSWER:

The constant variance assumption may be violated. 98. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel's Regression tool to construct a residual plot and use it to determine if any model assumptions may have been violated. x 2 3 6 7 8 7 9 ANSWER:

y 12 9 8 7 6 5 2

The constant variance assumption may be violated. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression

99. Connie Harris, in charge of office supplies at First Capital Mortgage Corp., would like to predict the quantity of paper used in the office photocopying machines per month. She believes that the number of loans originated in a month influences the volume of photocopying performed. She has compiled the following recent monthly data: Number of Loans Originated in Month 45 25 50 60 40 25 35 40

Sheets of Photocopy Paper Used (1000's) 22 13 24 25 21 16 18 25

a. Develop the least squares estimated regression equation that relates sheets of photocopy paper used to loans originated. b. Use the regression equation developed in part (a) to forecast the amount of paper used in a month when 42 loan originations are expected. c. Compute SSE, SST, and SSR. d. Compute the coefficient of determination r2. Comment on the goodness of fit. e. Compute the correlation coefficient. f. Compute the mean square error MSE. g. Compute the standard error of the estimate. h. Compute the estimated standard deviation of b1. i. Use the t test to test the following hypothesis β1 = 0 at α = .05. j. Develop a 95% confidence interval estimate for β1 to test the hypothesis β1 = 0. k. Use the F test to test the hypothesis β1 = 0 at a .05 level of significance. l. Develop a 95% confidence interval estimate of the mean number of sheets of paper used when 38 mortgages are originated. m. Develop a 95% prediction interval estimate for the number of sheets of paper used when 38 mortgages are originated. ANSWER: a. b. c. SSE = 32.38, SST = 138.00, SSR = 105.63 d. r2 = .7654; very good fit e. r = .8749 f. 5.3958 g. 2.3229 h. .0735 i. p-value = .0045; reject H0 j. .145 to .505; reject H0 k. F = 19.575 > 5.99; reject H0 l. 17.83 to 21.87 m. 13.82 to 25.88 100. Scott Bell Builders would like to predict the total number of labor hours spent framing a house based on the square footage of the house. The following data have been compiled on 10 houses recently built. Copyright Cengage Learning. Powered by Cognero.

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Chapter 14: Simple Linear Regression Square Footage (100s) 20 21 23 23 26

Framing Labor Hours 195 170 220 200 230

Square Footage (100s) 27 29 31 32 35

Framing Labor Hours 225 240 225 275 260

a. Develop the least squares estimated regression equation that relates framing labor hours to house square footage. b. Use the regression equation developed in part (a) to predict framing labor hours when the house size is 3350 square feet. ANSWER: a. b. 101. Assume you have noted the following prices for books and the number of pages that each book contains. Book A B C D E F G a. b. c.

Pages (x) 500 700 750 590 540 650 480

Price (y) $7.00 7.50 9.00 6.50 7.50 7.00 4.50

Develop a least squares estimated regression line. Compute the coefficient of determination and explain its meaning. Compute the correlation coefficient between the price and the number of pages. Test to see if x and y are related. Use α = .10.

ANSWER: a. b. c.

= 1.0416 + 0.0099x r = .5629; the regression equation has accounted for 56.29% of the total sum of squares rxy = .75 t = 2.54 > 2.015 (df = 5); p-value is between .05 and .1; (Excel’s results: p-value of .052); reject H0, and conclude x and y are related 2

102. Assume you have noted the following prices for books and the number of pages that each book contains. Book A B C D E F G a.

Pages (x) 500 700 750 590 540 650 480

Price (y) $7.00 7.50 9.00 6.50 7.50 7.00 4.50

Perform an F test and determine if the price and the number of pages of the books are related.

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Chapter 14: Simple Linear Regression b. c. d. ANSWER: a. b. c. d.

Let α = .01. Perform a t test and determine if the price and the number of pages of the books are related. Let α = .01. Develop a 90% confidence interval for estimating the average price of books that contain 800 pages. Develop a 90% confidence interval to estimate the price of a specific book that has 800 pages. F = 6.439 < 16.26; p-value is between .05 and .1 (Excel’s result: p-value = .052); do not reject H0; conclude x and y are not related t = 2.5376 < 4.032; p-value is between .05 and .1. (Excel’s result: p-value = .052); do not reject H0; conclude x and y are not related $7.29 to $10.63 (rounded) $5.62 to $12.31 (rounded)

103. The following data represent the number of flash drives sold per day at a local computer shop and their prices. Price (x) $34 36 32 35 30 38 40 a. b. c.

Units Sold (y) 3 4 6 5 9 2 1

Develop a least squares regression line and explain what the slope of the line indicates. Compute the coefficient of determination and comment on the strength of relationship between x and y. Compute the sample correlation coefficient between the price and the number of flash drives sold. Use α = .01 to test the relationship between x and y.

ANSWER: a.

b. c.

= 29.7857 – 0.7286x The slope indicates that as the price goes up by $1, the number of units sold goes down by .7286 units. r 2 = .8556; the regression equation has accounted for 85.56% of the total sum of squares rxy = –.92 t = –5.44 < –4.032 (df = 5); p-value < .01 (Excel’s result: p-value = .0028); reject H0, and conclude x and y are related

104. The following data represent the number of flash drives sold per day at a local computer shop and their prices. Price (x) $34 36 32 35 Copyright Cengage Learning. Powered by Cognero.

Units Sold (y) 3 4 6 5 Page 30

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Chapter 14: Simple Linear Regression 30 38 40 a. b.

9 2 1

Perform an F test and determine if the price and the number of flash drives sold are related. Let α = .01. Perform a t test and determine if the price and the number of flash drives sold are related. Let α = .01.

ANSWER:

a. b.

F = 29.624 > 16.26; p-value < .01 (Excel’s result: p-value = .0028); reject H0, x and y are related t = –5.4428 < –4.032; p-value < .01 (Excel’s result: p-value = .0028); reject H0, x and y are related

105. Shown below is a portion of an Excel output for regression analysis relating y (dependent variable) and x (independent variable). ANOVA Regression Residual Total

df 1 8 9

SS 110 74 184

Intercept x

Coefficients 39.222 -0.5556

Standard Error 5.943 0.1611

a. b. c. d. e.

What is the sample size? Perform a t test and determine whether x and y are related. Let α = .05. Perform an F test and determine whether x and y are related. Let α = .05. Compute the coefficient of determination. Interpret the meaning of the value of the coefficient of determination that you found in part (d).

ANSWER: a. through d. Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.7732 0.5978 0.5476 3.0414 10

ANOVA df Copyright Cengage Learning. Powered by Cognero.

Significance F Page 31

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Chapter 14: Simple Linear Regression Regression Residual Total

1 8 9

110 74 184

110 9.25

11.892

Intercept x

Coefficients 39.222 -0.556

Standard Error 5.942 0.161

t Stat 6.600 -3.448

P-value 0.000 0.009

0.009

59.783% of the variability in y is explained by the variability in x.

106. Shown below is a portion of a computer output for regression analysis relating y (dependent variable) and x (independent variable). ANOVA Regression Residual

df 1 8

SS 24.011 67.989

Intercept x

Coefficients 11.065 -0.511

Standard Error 2.043 0.304

a. b. c. d. e.

ANSWER: a. through d. Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.511 0.261 0.169 2.915 10

ANOVA Regression Residual Total

df 1 8 9

SS 24.011 67.989 92

MS 24.011 8.499

F 2.825

Intercept x

Coefficients 11.065 -0.511

Standard Error 2.043 0.304

t Stat 5.415 -1.681

P-value 0.001 0.131

Significance F 0.131

26.1% of the variability in y is explained by the variability in x.

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Chapter 14: Simple Linear Regression 107. Part of an Excel output relating x (independent variable) and y (dependent variable) is shown below. Fill in all the blanks marked with a question mark. Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.1347 ? ? 3.3838 ?

ANOVA Regression Residual Total

df ? ? 14

SS 2.750 ? ?

MS ? 11.45

F ?

Intercept x

Coefficients 8.6 0.25

Standard Error 2.2197 0.5101

t Stat ? ?

P-value 0.0019 0.6322

Significance F 0.6322

ANSWER: Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.1347 0.0181 –0.0574 3.3838 15

ANOVA Regression Residual Total

df 1 13 14

SS 2.750 148.850 151.600

MS 2.75 11.45

F 0.2402

Intercept x

Coefficients 8.6 0.25

Standard Error 2.2197 0.5101

t Stat 3.8744 0.4901

P-value 0.0019 0.6322

Significance F 0.6322

108. Shown below is a portion of a computer output for a regression analysis relating Y (dependent variable) and X (independent variable). ANOVA Regression Residual Total

df 1 13

SS 115.064 82.936

Coefficients

Standard Error

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Chapter 14: Simple Linear Regression Intercept x a. b. c.

15.532 –1.106

1.457 0.261

Perform a t test using the p-value approach and determine whether x and y are related. Let α = .05. Using the p-value approach, perform an F test and determine whether x and y are related. Compute the coefficient of determination and fully interpret its meaning.

ANSWER: a. and b. Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.7623 0.5811 0.5489 2.5258 15

ANOVA Regression Residual Total

df 1 13 14

SS 115.064 82.936 198

MS 115.064 6.380

F 18.036

Intercept x

Coefficients 15.532 –1.106

Standard Error 1.457 0.261

t Stat 10.662 –4.247

P-value 0.000 0.001

Significance F 0.001

.5811; 58.11% of the variability in y is explained by the variability in x.

109. Part of an Excel output relating x (independent variable) and y (dependent variable) is shown below. Fill in all the blanks marked with a question mark. Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

? 0.5149 ? 7.3413 11

ANOVA Regression Residual Total

df ? ? ?

SS ? ? 1000.0000

MS ? ?

F ?

Coefficients

Standard Error

t Stat

P-value

Significance F 0.0129

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Chapter 14: Simple Linear Regression Intercept x

? ?

29.4818 0.7000

3.7946 –3.0911

0.0043 0.0129

ANSWER: Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.7176 0.5149 0.4611 7.3413 11

ANOVA Regression Residual Total

df 1 9 10

SS 514.9455 485.0545 1000.0000

MS 514.9455 53.8949

F 9.5546

Intercept x

Coefficients 111.8727 –2.1636

Standard Error 29.4818 0.7000

t Stat 3.7946 –3.0911

P-value 0.0043 0.0129

Significance F 0.0129

110. Shown below is a portion of a computer output for a regression analysis relating y (demand) and x (unit price). ANOVA Regression Residual Total

df 1 46 47

SS 5048.818 3132.661 8181.479

Intercept x

Coefficients 80.390 –2.137

Standard Error 3.102 0.248

a. b. c. d.

Perform a t test and determine whether demand and unit price are related. Let α = .05. Perform an F test and determine whether demand and unit price are related. Let α = .05. Compute the coefficient of determination and fully interpret its meaning. Compute the correlation coefficient and explain the relationship between demand and unit price.

ANSWER: a. and b. Summary Output Regression Statistics Multiple R R Square Adjusted R Square

0.786 0.617 0.609

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Chapter 14: Simple Linear Regression Standard Error Observations

8.252 48

ANOVA Regression Residual Total

df 1 46 47

SS 5048.818 3132.661 8181.479

MS 5048.818 68.101

F 74.137

Intercept x

Coefficients 80.390 –2.137

Standard Error 3.102 0.248

t Stat 25.916 –8.610

P-value 0.000 0.000

c. d.

Significance F 0.000

r2 = .617; 61.7% of the variability in demand is explained by the variability in price. r = –.786; since the slope is negative, the coefficient of correlation is also negative, indicating that as unit price increases demand decreases.

111. Shown below is a portion of a computer output for a regression analysis relating supply (y in thousands of units) and unit price (x in thousands of dollars). ANOVA Regression Residual

df 1 39

SS 354.689 7035.262

Intercept x

Coefficients 54.076 0.029

Standard Error 2.358 0.021

a. b. c. d. e. f.

What is the sample size? Perform a t test and determine whether or not supply and unit price are related. Let α = .05. Perform and F test and determine whether or not supply and unit price are related. Let α = .05. Compute the coefficient of determination and fully interpret its meaning. Compute the correlation coefficient and explain the relationship between supply and unit price. Predict the supply (in units) when the unit price is $50,000.

ANSWER: a. through c. Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.219 0.048 0.024 13.431 41

ANOVA Regression

df 1

SS 354.689

MS 354.689

F 1.966

Significance F 0.169 Page 36

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Chapter 14: Simple Linear Regression Residual Total

39 40

7035.262 7389.951

180.391

Intercept x

Coefficients 54.076 0.029

Standard Error 2.358 0.021

t Stat 22.938 1.402

d. e. f.

P-value 0.000 0.169

r2 = .048; 4.8% of the variability in supply is explained by the variability in price. r = .219; since the slope is positive, as unit price increases so does supply. Supply = 54.076 + 0.029(50) = 55.526 (55,526 units)

112. Coyote Cable has been experiencing an increase in cable service subscribers in recent months due to increased advertising and an influx of new residents to the region. The number of subscribers (in 1000s) for the last 16 months is as follows: Month 1 2 3 4 5 6

Sales 12.8 14.6 15.2 16.1 15.8 17.2

Month 7 8 9 10 11

Sales 20.6 18.5 19.9 23.6 24.2

Month 12 13 14 15 16

Sales 23.8 25.1 24.7 26.5 28.9

Using simple linear regression, forecast the number of subscribers for months 17, 18, 19, and 20. ANSWER:

= 11.93 + 1.0046x. Month 17: 29.0; Month 18: 30.0; Month 19: 31.0; Month 20: 32.0

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Chapter 15: Multiple Regression Multiple Choice 1. If a qualitative variable has k levels, the number of dummy variables required is _____. a. k − 1 b. k c. k + 1 d. 2k ANSWER: a 2. As the goodness of fit for the estimated multiple regression equation increases, _____. a. the value of the adjusted multiple coefficient of determination decreases b. the value of the regression equation’s constant b0 decreases c. the value of the multiple coefficient of determination increases d. the value of the correlation coefficient increases ANSWER: c 3. For a multiple regression model, SSR = 600 and SSE = 200. The multiple coefficient of determination is _____. a. .333 b. .275 c. .300 d. .75 ANSWER: d 4. In a multiple regression analysis involving 15 independent variables and 200 observations, SST = 800 and SSE = 240. The coefficient of determination is _____. a. .300 b. .192 c. .500 d. .700 ANSWER: d 5. A regression model involved 5 independent variables and 126 observations. The critical value of t for testing the significance of each of the independent variable's coefficients will have _____. a. 131 degrees of freedom b. 125 degrees of freedom c. 130 degrees of freedom d. 4 degrees of freedom ANSWER: c 6. To test for the significance of a regression model involving 3 independent variables and 47 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are _____. a. 47 and 3 b. 3 and 47 c. 2 and 43 Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression d. 3 and 43 ANSWER: d 7. In regression analysis, an outlier is an observation whose _____. a. mean is larger than the standard deviation b. residual is 0 c. mean is 0 d. residual is much larger than the rest of the residual values ANSWER: d 8. A variable that cannot be measured in terms of how much or how many but instead is assigned values to represent categories is called a(n) _____. a. interaction b. constant variable c. category variable d. qualitative variable ANSWER: d 9. A variable that takes on the values of 0 or 1 and is used to incorporate the effect of qualitative variables in a regression model is called a(n) _____. a. interaction b. constant variable c. dummy variable d. outlier ANSWER: c 10. In a multiple regression model, the error term ε is assumed to be a random variable with a mean of _____. a. 0 b. –1 c. 1 d. any value ANSWER: a 11. In regression analysis, the response variable is the _____. a. independent variable b. dependent variable c. slope of the regression function d. intercept ANSWER: b 12. The multiple coefficient of determination is _____. a. MSR/MST b. MSR/MSE c. SSR/SST Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression d. SSE/SSR ANSWER: c 13. A multiple regression model has the form = 7 + 2 x1 + 9 x2 As x1 increases by 1 unit (holding x2 constant), is expected to _____. a. increase by 9 units b. decrease by 9 units c. increase by 2 units d. decrease by 2 units ANSWER: c 14. A multiple regression model has _____. a. only one independent variable b. more than one dependent variable c. more than one independent variable d. at least two dependent variables ANSWER: c 15. A measure of goodness of fit for the estimated regression equation is the _____. a. multiple coefficient of determination b. mean square due to error c. mean square due to regression d. sample size ANSWER: a 16. The numerical value of the coefficient of determination _____. a. is always larger than the correlation coefficient b. is always smaller than the correlation coefficient c. is negative if the coefficient of determination is negative d. can be larger or smaller than the correlation coefficient ANSWER: d 17. The correct relationship between SST, SSR, and SSE is given by _____. a. SSR = SST + SSE b. SSR = SST – SSE c. SSE = SSR – SST d. SST = SSE – SSR ANSWER: b 18. In a multiple regression analysis, SSR = 1,000 and SSE = 200. The F statistic for this model is _____. a. 5.0 Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression b. 1,200 c. 800 d. Not enough information is provided to answer this question. ANSWER: d 19. The ratio of MSR/MSE yields _____. a. SST b. the F statistic c. SSR d. the t statistic ANSWER: b 20. In a multiple regression model, the variance of the error term ε is assumed to be _____. a. the same for all values of the dependent variable b. 0 c. the same for all values of the independent variable d. –1 ANSWER: c 21. The adjusted multiple coefficient of determination is adjusted for _____. a. the number of dependent variables b. the number of independent variables c. the number of equations d. detrimental situations ANSWER: b 22. In multiple regression analysis, the correlation among the independent variables is termed _____. a. collinearity b. linearity c. multicollinearity d. adjusted coefficient of determination ANSWER: c 23. In a multiple regression model, the values of the error term, ε, are assumed to be _____. a. 0 b. dependent on each other c. independent of each other d. always negative ANSWER: c 24. In multiple regression analysis, _____. a. there can be any number of dependent variables but only one independent variable b. there must be only one independent variable c. the coefficient of determination must be larger than 1 Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression d. there can be several independent variables, but only one dependent variable ANSWER: d 25. In a multiple regression model, the error term ε is assumed to _____. a. have a mean of 1 b. have a variance of 0 c. have a standard deviation of 1 d. be normally distributed ANSWER: d 26. In a multiple regression analysis involving 12 independent variables and 166 observations, SSR = 878 and SSE = 122. The coefficient of determination is _____. a. .1389 b. .1220 c. .878 d. .7317 ANSWER: c 27. A regression analysis involved 17 independent variables and 697 observations. The critical value of t for testing the significance of each of the independent variable's coefficients will have _____. a. 696 degrees of freedom b. 16 degrees of freedom c. 713 degrees of freedom d. 714 degrees of freedom ANSWER: c 28. To test for the significance of a regression model involving 14 independent variables and 255 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are _____. a. 14 and 255 b. 255 and 14 c. 13 and 240 d. 14 and 240 ANSWER: d Exhibit 15-1 In a regression model involving 44 observations, the following estimated regression equation was obtained: = 29 + 18x1 +43x2 + 87x3 For this model, SSR = 600 and SSE = 400. 29. Refer to Exhibit 15-1. The coefficient of determination for the above model is _____. a. .667 b. .600 c. .336 d. .400 Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression ANSWER: b 30. Refer to Exhibit 15-1. MSR for this model is _____. a. 200 b. 10 c. 1,000 d. 43 ANSWER: a 31. Refer to Exhibit 15-1. The computed F statistic for testing the significance of the above model is _____. a. 1.500 b. 20.00 c. .600 d. .6667 ANSWER: b Exhibit 15-2 A regression model between sales (y in $1000s), unit price (x1 in dollars) and television advertisement (x2 in dollars) resulted in the following function: = 7 – 3x1 + 5x2 For this model, SSR = 3500, SSE = 1500, and the sample size is 18. 32. Refer to Exhibit 15-2. The coefficient of the unit price indicates that if the unit price is _____. a. increased by $1 (holding advertising constant), sales are expected to increase by $3 b. decreased by $1 (holding advertising constant), sales are expected to decrease by $3 c. increased by $1 (holding advertising constant), sales are expected to increase by $4000 d. increased by $1 (holding advertising constant), sales are expected to decrease by $3000 ANSWER: d 33. Refer to Exhibit 15-2. The coefficient of x2 indicates that if television advertising is increased by $1 (holding the unit price constant), sales are expected to _____. a. increase by $5 b. increase by $12,000 c. increase by $5000 d. decrease by $2000 ANSWER: c 34. Refer to Exhibit 15-2. If we want to test for the significance of the regression model, the critical value of F at 95% confidence is _____. a. 3.68 b. 3.29 c. 3.24 d. 4.54 Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression ANSWER: a 35. Refer to Exhibit 15-2. If SSR = 600 and SSE = 300, the test statistic F is _____. a. 2.33 b. .70 c. 17.5 d. 1.75 ANSWER: c 36. Refer to Exhibit 15-2. The multiple coefficient of determination for this problem is _____. a. .4368 b. .6960 c. .3040 d. .2289 ANSWER: b Exhibit 15-3 In a regression model involving 30 observations, the following estimated regression equation was obtained: = 17 + 4x1 – 3x2 + 8x3 + 8x4 For this model, SSR = 700 and SSE = 100. 37. Refer to Exhibit 15-3. The coefficient of determination for the above model is approximately _____. a. –.875 b. .875 c. .125 d. .144 ANSWER: b 38. Refer to Exhibit 15-3. The computed F statistic for testing the significance of the above model is _____. a. 43.75 b. .875 c. 50.19 d. 7.00 ANSWER: a 39. Refer to Exhibit 15-3. The critical F value at 95% confidence is _____. a. 2.53 b. 2.69 c. 2.76 d. 2.99 ANSWER: c 40. Refer to Exhibit 15-3. The conclusion is that the _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression a. model is not significant b. model is significant c. slope of x1 is significant d. slope of x2 is significant ANSWER: b Exhibit 15-4 a. y = β0 + β1x1 + β2x2 + ε b. E(y) = β0 + β1x1 c. = b0 + b1 x1 + b2 x2 d. E(y) = β0 + β1x1 + β2x2

41. Refer to Exhibit 15-4. Which equation describes the multiple regression model? a. equation a b. equation b c. equation c d. equation d ANSWER: a 42. Refer to Exhibit 15-4. Which equation gives the estimated regression line? a. equation a b. equation b c. equation c d. equation d ANSWER: c 43. Refer to Exhibit 15-4. Which equation describes the multiple regression equation? a. equation a b. equation b c. equation c d. equation d ANSWER: d Exhibit 15-5 Below is a partial Excel output based on a sample of 25 observations. Intercept x1 x2 x3

Coefficients 145.321 25.625 –5.720 0.823

Standard Error 48.682 9.150 3.575 0.183

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Chapter 15: Multiple Regression 44. Refer to Exhibit 15-5. The estimated regression equation is _____. a. y = β0 + β1x1 + β2x2 + β3x3 + ε b. E(y) = β0 + β1x1 + β2x2 + β3x3 c.

= 145.321 + 25.625x1 - 5.720x2 + 0.823x3

d. = 48.682 + 9.15x1 + 3.575x2 + 0.183x3 ANSWER: c 45. Refer to Exhibit 15-5. The interpretation of the coefficient on x1 is that _____. a. a one-unit change in x1 will lead to a 25.625-unit change in y b. a one-unit change in x1 will lead to a 25.625-unit increase in y when all other variables are held constant c. a one-unit change in x1 will lead to a 25.625-unit increase in x2 when all other variables are held constant d. a one-unit change in x1 will lead to a 25.625-unit decrease in x2 when all other variables are held constant ANSWER: b 46. Refer to Exhibit 15-5. We want to test whether the parameter β1 is significant. The test statistic equals _____. a. .357 b. 2.8 c. 14 d. 1.96 ANSWER: b 47. Refer to Exhibit 15-5. The t value obtained from the table to test an individual parameter at the 5% level is _____. a. 2.06 b. 2.069 c. 2.074 d. 2.080 ANSWER: d 48. Refer to Exhibit 15-5. Carry out the test of significance for the parameter β1 at the 5% level. The null hypothesis should _____. a. be rejected b. not be rejected c. be revised d. be retested ANSWER: a Exhibit 15-6 Below you are given a partial Excel output based on a sample of 16 observations. ANOVA df Regression Residual

SS 4,853

MS 2,426.5 485.3

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Chapter 15: Multiple Regression

Intercept x1 x2

Coefficients 12.924 -3.682 45.216

Standard Error 4.425 2.630 12.560

49. Refer to Exhibit 15-6. The estimated regression equation is _____. a. y = β0 + β1x1 + β2x2 + ε b. E(y) = β0 + β1x1 + β2x2 c.

= 12.924 - 3.682 x1 + 45.216 x2

d. = 4.425 + 2.63 x1 + 12.56 x2 ANSWER: c 50. Refer to Exhibit 15-6. The interpretation of the coefficient of x1 is that _____. a. a one-unit change in x1 will lead to a 3.682-unit decrease in y b. a one-unit increase in x1 will lead to a 3.682-unit decrease in y when all other variables are held constant c. a one-unit increase in x1 will lead to a 3.682-unit decrease in x2 when all other variables are held constant d. a one-unit increase in x1 will lead to a 3.682-unit increase in x2 when all other variables are held constant ANSWER: b 51. Refer to Exhibit 15-6. We want to test whether the parameter β1 is significant. The test statistic equals _____. a. –1.4 b. 1.4 c. 3.6 d. 5 ANSWER: a 52. Refer to Exhibit 15-6. The t value obtained from the table that is used to test an individual parameter at the 1% level is _____. a. 2.65 b. 2.921 c. 2.977 d. 3.012 ANSWER: d 53. Refer to Exhibit 15-6. Carry out the test of significance for the parameter β1 at the 1% level. The null hypothesis should _____. a. be rejected b. not be rejected c. be revised d. be retested ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression 54. Refer to Exhibit 15-6. The degrees of freedom for the sum of squares explained by the regression (SSR) are _____. a. 2 b. 3 c. 13 d. 15 ANSWER: a 55. Refer to Exhibit 15-6. The sum of squares due to error (SSE) equals _____. a. 37.33 b. 485.3 c. 4,853 d. 6,308.9 ANSWER: d 56. Refer to Exhibit 15-6. The test statistic used to determine if there is a relationship among the variables equals _____. a. –1.4 b. .2 c. .77 d. 5 ANSWER: d 57. Refer to Exhibit 15-6. The F value obtained from the table used to test if there is a relationship among the variables at the 5% level equals _____. a. 3.41 b. 3.63 c. 3.81 d. 19.41 ANSWER: c 58. Refer to Exhibit 15-6. Carry out the test to determine if there is a relationship among the variables at the 5% level. The null hypothesis should _____. a. be rejected b. not be rejected c. be revised d. be retested ANSWER: a Exhibit 15-7 A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of squares: SSR = 165 SSE = 60 59. Refer to Exhibit 15-7. The coefficient of determination is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression a. .3636 b. .7333 c. .275 d. .5 ANSWER: b 60. Refer to Exhibit 15-7. If we want to test for the significance of the model at 95% confidence, the critical F value (from the table) is _____. a. 3.06 b. 3.48 c. 3.34 d. 3.11 ANSWER: b 61. Refer to Exhibit 15-7. The test statistic from the information provided is _____. a. 2.110 b. 3.480 c. 4.710 d. 6.875 ANSWER: d Exhibit 15-8 The following estimated regression model was developed relating yearly income (y in $1000s) of 30 individuals with their age (x1) and their gender (x2) (0 if male and 1 if female). = 30 + 0.7x1 + 3x2 Also provided are SST = 1200 and SSE = 384. 62. Refer to Exhibit 15-8. From the above function, it can be said that the expected yearly income for _____. a. male is $3 more than females b. female is $3 more than males c. male is $3,000 more than females d. female is $3,000 more than males ANSWER: d 63. Refer to Exhibit 15-8. The yearly income of a 24-year-old female individual is _____. a. $19.80 b. $19,800 c. $49.80 d. $49,800 ANSWER: d 64. Refer to Exhibit 15-8. The yearly income of a 24-year-old male individual is _____. a. $13.80 Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression b. $13,800 c. $46,800 d. $49,800 ANSWER: c 65. Refer to Exhibit 15-8. The multiple coefficient of determination is _____. a. .32 b. .42 c. .68 d. .50 ANSWER: c 66. Refer to Exhibit 15-8. If we want to test for the significance of the model, the critical value of F at a 5% significance level is _____. a. 3.33 b. 3.35 c. 3.34 d. 2.96 ANSWER: b 67. Refer to Exhibit 15-8. The test statistic for testing the significance of the model is _____. a. .73 b. 1.47 c. 28.69 d. 5.22 ANSWER: c 68. Refer to Exhibit 15-8. The model _____. a. is significant b. is not significant c. would be significant if the sample size was larger than 30 d. is inconclusive ANSWER: a 69. Refer to Exhibit 15-8. The estimated income of a 30-year-old male is _____. a. $51,000 b. $5100 c. $510 d. $51 ANSWER: a 70. A multiple regression model has the form: = 5 + 6x + 7w Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression As x increases by 1 unit (holding w constant), y is expected to _____. a. increase by 11 units b. decrease by 11 units c. increase by 6 units d. decrease by 6 units ANSWER: c 71. A variable that cannot be measured in numerical terms is called a _____. a. non-measurable random variable b. constant variable c. dependent variable d. qualitative variable ANSWER: d 72. A term used to describe the case when the independent variables in a multiple regression model are correlated is _____. a. regression b. correlation c. multicollinearity d. linearity ANSWER: c 73. A regression model in which more than one independent variable is used to predict the dependent variable is called a(n) _____. a. simple linear regression model b. multiple regression model c. independent model d. dependent model ANSWER: b 74. For a multiple regression model, SST = 200 and SSE = 50. The multiple coefficient of determination is _____. a. .25 b. 4.00 c. 250 d. .75 ANSWER: d 75. In a multiple regression analysis involving 10 independent variables and 81 observations, SST = 120 and SSE = 42. The coefficient of determination is _____. a. .81 b. .11 c. .35 d. .65 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression 76. A regression model involved 18 independent variables and 200 observations. The critical value of t for testing the significance of each of the independent variable's coefficients will have _____. a. 18 degrees of freedom b. 200 degrees of freedom c. 199 degrees of freedom d. 181 degrees of freedom ANSWER: d 77. To test for the significance of a regression model involving 8 independent variables and 121 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are _____. a. 8 and 121 b. 7 and 120 c. 8 and 112 d. 7 and 112 ANSWER: c 78. In a multiple regression analysis involving 5 independent variables and 30 observations, SSR = 360 and SSE = 40. The coefficient of determination is _____. a. .80 b. .90 c. .25 d. .15 ANSWER: b 79. A regression analysis involved 6 independent variables and 27 observations. The critical value of t for testing the significance of each of the independent variable's coefficients will have _____. a. 27 degrees of freedom b. 26 degrees of freedom c. 21 degrees of freedom d. 20 degrees of freedom ANSWER: d 80. To test for the significance of a regression model involving 4 independent variables and 36 observations, the numerator and denominator degrees of freedom (respectively) for the critical value of F are _____. a. 4 and 36 b. 3 and 35 c. 4 and 31 d. 4 and 32 ANSWER: c 81. In a residual plot that does not suggest we should challenge the assumptions of our regression model, we would expect to see a _____. a. horizontal band of points centered near 0 b. widening band of points Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression c. band of points having a slope consistent with that of the regression equation d. parabolic band of points ANSWER: a 82. The difference between the observed value of the dependent variable and the value predicted by using the estimated regression equation is the _____. a. standard error b. residual c. predicted interval d. variance ANSWER: b 83. The mathematical equation that explains how the dependent variable y is related to several independent variables x1, x2, ..., xp and the error term ε is a(n) _____. a. simple nonlinear regression model b. multiple regression model c. estimated multiple regression equation d. multiple regression equation ANSWER: b 84. The least squares criterion is _____. a. min ∑(xi – yi)2 b. min ∑(yi – i)2 c. min ∑(yi – i)2 d. min ∑(yi – i) ANSWER: c Subjective Short Answer 85. Multiple regression analysis was used to study how an individual's income (y in thousands of dollars) is influenced by age (x1 in years), level of education (x2 ranging from 1 to 5), and the person's gender (x3 where 0 = female and 1= male). The following is a partial Excel output that was used on a sample of 20 individuals. ANOVA df

SS 84 112

Coefficients 0.6251 0.9210 -0.510

Standard Error 0.094 0.190 0.920

Regression Residual

x1 x2 x3

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Chapter 15: Multiple Regression a. b. c. d.

Compute the coefficient of determination. Perform a t test and determine whether the coefficient of the variable "level of education" (i.e., x2) is significantly different from zero. Let α = .05. At α = .05, perform an F test and determine whether or not the regression model is significant. The coefficient of x3 is –.510. Fully interpret the meaning of this coefficient.

ANSWER:

a. b. c. d.

.4286 t = 4.84 > 2.12; significantly different from zero. F = 4 > 3.24; yes, the model is significant. Male income is lower than female income by $510.

86. A multiple regression analysis between yearly income (y in $1000s), college grade point average (x1), age of the individuals (x2), and gender of the individual (x3; 0 representing female and 1 representing male) was performed on a sample of 10 people, and the following results were obtained using Excel: ANOVA df

SS 360.59 23.91

Coefficients 4.0928 10.0230 0.1020 -4.4811

Standard Error 1.4400 1.6512 0.1225 1.4400

Regression Residual

Intercept x1 x2 x3 a. b. c. d. e. f. g.

Write the regression equation for the above. Interpret the meaning of the coefficient of x3. Compute the coefficient of determination. Is the coefficient of x1 significant? Use α = .05. Is the coefficient of x2 significant? Use α = .05. Is the coefficient of x3 significant? Use α = .05. Perform an F test and determine whether or not the model is significant.

ANSWER: a. b. c. d. e. f. g.

= 4.0928 +10.0230x1 + 0.102x2 – 4.481x3 This coefficient indicates that male income is lower than female income by 4.4811 (in thousands). .9378 t = 6.07 > 2.447; yes, the coefficient is significant. t = 0.83 < 2.447; no, the coefficient is not significant. t = –3.11 < -2.447; yes, the coefficient is significant. F = 30.16 > 4.76; yes, the model is significant.

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Chapter 15: Multiple Regression

87. The following results were obtained from a multiple regression analysis: Source of Variation Regression Error Total a. b. c.

Degrees of Freedom 35 39

Sum of Squares 900

Mean Square

4,980

How many independent variables were involved in this model? How many observations were involved? Determine the F statistic.

ANSWER:

a. b. c.

4 40 1.93

88. Shown below is a partial Excel output from a regression analysis. ANOVA df Regression Residual Total

SS 60

140

Intercept x1 x2 x3

Coefficients 10.00 –2.00 6.00 –4.00

Standard Error 2.00 1.50 2.00 1.00

a. b. c. d. e.

Use the above results and write the regression equation. Compute the coefficient of determination and fully interpret its meaning. Is the regression model significant? Perform an F test and let α = .05. At α = .05, test to see if there is a relationship between x1 and y. At α = .05, test to see if there is a relationship between x3 and y.

ANSWER: a. b. c. d. e.

= 10 – 2x1 + 6x2 – 4x3 .4286; 42.86% of variation in dependent variable is explained by variation in the three independent variables. F = 4.0 > 3.24; yes, the regression model is significant. t = –1.33 > –2.12; no, there is not a relationship between x1 and y. t = –4.0 < –2.12; yes, there is a relationship x3 and y.

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Chapter 15: Multiple Regression 89. In order to determine whether or not the sales volume of a company (y in $ millions) is related to advertising expenditures (x1 in $ millions) and the number of salespeople (x2), data were gathered for 10 years. Part of the Excel output is shown below. ANOVA df

SS 321.11 63.39

Coefficients 7.0174 8.6233 0.0858

Standard Error 1.8972 2.3968 0.1845

Regression Residual

Intercept x1 x2 a. b. c. d. e. f.

Use the above results and write the regression equation that can be used to predict sales. Estimate the sales volume for an advertising expenditure of $3.5 million and 45 salespeople. Give your answer in dollars. At α = .01, test to determine if the fitted equation developed in part (a) represents a significant relationship between the independent variables and the dependent variable. At α = .05, test to see if β1 is significantly different from 0. Determine the multiple coefficient of determination. Compute the adjusted coefficient of determination.

ANSWER: a. b. c. d. e. f.

= 7.0174 + 8.6233x1 + .0858x2 $41,059,950 F = 17.73 > 9.55; yes, the equation from part (a) represents a significant relationship between the independent variables and dependent variable. t = 3.598 > 2.365; yes, β1 is significantly different from 0. .8351 .7879

90. To determine whether the number of automobiles sold per day (y) is related to price (x1 in $1000s) and the number of advertising spots (x2), data were gathered for seven days. Part of the Excel output is shown below. ANOVA df

SS 40.700 1.016

Coefficients 0.8051 0.4977 0.4733

Standard Error

Regression Residual

Intercept x1 x2

0.4617 0.0387

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Chapter 15: Multiple Regression a. b. c. d. e.

Determine the least squares regression function relating y to x1 and x2. If the company charges $20,000 for each car and uses 10 advertising spots, how many cars would you expect it to sell in a day? At α = .05, test to determine if the fitted equation developed in part (a) represents a significant relationship between the independent variables and the dependent variable. At α = .05, test to see if β1 is significantly different from 0. Determine the multiple coefficient of determination.

ANSWER: a. b. c. d. e.

= 0.8051 + 0.4977x1 + 0.4773x2 16 (rounded from 15.49) F = 80.12 > 6.94; yes, the equation in part (a) represents a significant relationship between the independent variables and dependent variable. t = 1.078 < 2.776; no, β1 is not significantly different from 0. 0.9756

91. The following is part of the results of a regression analysis involving sales (y in $ millions), advertising expenditures (x1 in $1000s), and number of salespeople (x2) for a corporation. The regression was performed on a sample of 10 observations. Coefficient –11.340 0.798 0.141

Constant x1 x2 a. b. c. d. e.

Standard Error 20.412 0.332 0.278

Write the regression equation. Interpret the coefficients of the estimated regression equation found in part (a). At α = .05, test for the significance of the coefficient of advertising. At α = .05, test for the significance of the coefficient of number of salespeople. If the company uses $50,000 in advertisement and has 800 salespersons, what are the expected sales? Give your answer in dollars.

ANSWER: a. b.

c. d. e.

= –11.34 + .798x1 + .141x2 As advertising increases by 1 unit ($1000) (holding the number of salespersons constant), sales are expected to increase by 0.798 units ($798,000). As the number of salespersons increases by 1 (holding advertising constant), sales are expected to increase by $141,000. t = 2.404 > 2.365; advertising is significant. t = 0.507 < 2.365; the number of salespersons is not significant. $141.36 million

92. The following is part of the results of a regression analysis involving sales (y in $ millions), advertising expenditures (x1 in $1000s), and number of salespeople (x2) for a corporation: Source of Variation Regression

Degrees of Freedom 2

Sum of Squares 822.088

Mean Square

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Chapter 15: Multiple Regression Error a.

b. c. d.

736.012

At α = .05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable. Determine the multiple coefficient of determination. Determine the adjusted multiple coefficient of determination. What has been the sample size for this regression analysis?

ANSWER:

a. b. c. d.

F = 3.91 < 4.74; no, the model is not significant. There is not a significant relationship between the independent variables and the dependent variable. .528 .393 10

93. Below is a partial Excel output based on a sample of 12 observations relating the number of personal computers sold by a computer shop per month (y), unit price (x1 in $1000s), and the number of advertising spots (x2) used on a local television station. Coefficient 17.145 –.104 1.376

Intercept x1 x2 a. b. c. d. e.

Standard Error 7.865 3.282 0.250

Use the output shown above and write an equation that can be used to predict the monthly sales of computers. Interpret the coefficients of the estimated regression equation found in part (a). If the company charges $2000 for each computer and uses 10 advertising spots, how many computers would you expect it to sell? At α = .05, test to determine if the price is a significant variable. At α = .05, test to determine if the number of advertising spots is a significant variable.

ANSWER:

a. b. c. d. e.

= 17.145 – .104x1 + 1.376x2 As unit price increases by $1000, the number of units sold decreases by 17.145 units. As advertising spots increase by 1, the number of units sold increases by 1.376. 30.697 t = –.032 < 2.262; price is not significant. t = 5.504 > 2.262; advertising is significant.

94. Below is a partial ANOVA table based on a sample of 12 observations relating the number of personal computers sold by a computer shop per month (y), unit price (x1 in $1000s), and number of advertising spots (x2) it used on a local Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression television station. Source of Variation Regression Error Total a.

b. c.

Degrees of Freedom 2 9

Sum of Squares 655.955

Mean Square

838.917

ANSWER:

a. b. c.

F = 16.133 > 4.26; yes, the model is significant. There is a significant relationship between the independent variables and the dependent variable. .782 .733

95. Below is a partial Excel output based on a sample of 30 days of the price of a company's stock (y in dollars), the Dow Jones Industrial Average (x1), and the stock price of the company's major competitor (x2 in dollars). Coefficient 20.000 0.030 –0.70

Intercept x1 x2 a. b. c. d.

Standard Error 5.455 0.010 0.200

Use the output shown above and write an equation that can be used to predict the price of the stock. If the Dow Jones Industrial Average is 2650 and the price of the competitor is $45, what would you expect the price of the stock to be? At α = .05, test to determine if the Dow Jones Industrial Average is a significant variable. At α = .05, test to determine if the stock price of the major competitor is a significant variable.

ANSWER: a. b. c. d.

= 20 + .03x1 015 – .7x2 $68 t = 3 > 2.052; Dow Jones is significant. t = –3.5 < –2.052; competitor's price is significant.

96. Below is a partial ANOVA table relating the price of a company's stock (y in dollars), the Dow Jones Industrial Average (x1), and the stock price of the company's major competitor (x2 in dollars). Source of Variation

Degrees of

Sum of

Mean

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Chapter 15: Multiple Regression Regression Error Total a. b.

Freedom

Squares

40 800

Square

What has been the sample size for this regression analysis? At α = .05 level of significance, test to determine if the model is significant. That is, determine if there exists a significant relationship between the independent variables and the dependent variable. Determine the multiple coefficient of determination.

ANSWER:

a. b. c.

23 F = 190 > 3.49; yes, the model is significant. There is a significant relationship between the independent variables and the dependent variable. .95

97. A regression was performed on a sample of 16 observations. The estimated equation is = 23.5 – 14.28x1 + 6.72x2 + 15.68x3. The standard errors for the coefficients are = 4.2, = 5.6, and = 2.8. For this model, SST = 3809.6 and SSR = 3285.4. a. Compute the appropriate t ratios. b. Test for the significance of β1, β2, and β3 at the 5% level of significance. c. Do you think that any of the variables should be dropped from the model? Explain. d. Compute R2 and Ra2. Interpret R2. e. Test the significance of the relationship among the variables at the 5% level of significance. ANSWER: a. b. c. d. e.

t1 = –3.4, t2 = 1.2, t3 = 5.6 t.025 = 2.179; x1 and x3 are statistically significant. Yes, x2 should be dropped since it is not statistically significant. .8624; .828; 86.24% of the variability in y is explained by the independent variables. Reject the hypothesis of no relationship since 25.07 > 3.49.

98. The following results were obtained from a multiple regression analysis of supermarket profitability. The dependent variable, y, is the profit (in $1000s), and the independent variables, x1 and x2, are the food sales and nonfood sales (also in $1000s). ANOVA Regression Error

df 2 9

SS 562.363 225.326

Coefficients –15.0620

Standard Error

Intercept

F 11.23

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Chapter 15: Multiple Regression 0.0972 0.054 x1 0.2484 0.092 x2 Coefficient of determination = .7139 a. b. c. d. e.

Write the estimated regression equation for the relationship between the variables. What can you say about the strength of this relationship? Carry out a test of whether y is significantly related to the independent variables. Use a .01 level of significance. Carry out a test of whether x1 and y are significantly related. Use a .05 level of significance. How many supermarkets are in the sample used here?

ANSWER: a. b. c. d. e.

= –15.0621 + .0972x1 + .248x2 71.39% of the variability in y is explained by the independent variables. Reject the hypothesis of no relationship since 11.23 > 8.02. Do not reject the hypothesis of no relationship since –2.262 < 1.8 < 2.262. 12

99. A regression was performed on a sample of 20 observations. Two independent variables were included in the analysis, x and z. The relationship between x and z is z = x2. The following estimated equation was obtained. = 23.72 + 12.61x + .798z The standard errors for the coefficients are Sb1 = 4.85 and Sb2 = .21. For this model, SSR = 520.2 and SSE = 340.6. a. Estimate the value of y when x = 5. b. Compute the appropriate t ratios. c. Test for the significance of the coefficients at the 5% level. Which variable(s) is (are) significant? d. Compute the coefficient of determination and the adjusted coefficient of determination. Interpret the meaning of the coefficient of determination. e. Test the significance of the relationship among the variables at the 5% level of significance.

ANSWER:

a. b. c. d. e.

106.72 2.6, 3.8 t.025 = 2.11; both x and z are significant. .6043; .5578; 60.43% of the variability in y is explained by the independent variables. Reject the hypothesis of no relationship; 12.98 > 3.59.

100. A student used multiple regression analysis to study how family spending (y) is influenced by income (x1), family size (x2), and additions to savings (x3). The variables y, x1, and x3 are measured in thousands of dollars. The following Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression results were obtained: ANOVA df 3 11

Regression Error

SS 45.9634 2.6218

F 64.28

Coefficients Standard Error Intercept 0.0136 0.7992 0.074 x1 0.2280 0.190 x2 –0.5796 0.920 x3 Coefficient of determination = .946 a. b. c. d. e.

Write out the estimated regression equation for the relationship between the variables. What can you say about the strength of this relationship? Carry out a test of whether y is significantly related to the independent variables. Use a .05 level of significance. Carry out a test to see if x3 and y are significantly related. Use a .05 level of significance. Why would a coefficient of determination very close to 1.0 be expected here?

ANSWER:

= .0136 + .7992x1 + .228x2 – .5796x3

a. b. c. d. e.

94.6% of the variability in y is explained by the independent variables. Reject the hypothesis of no relationship; 64.28 > 3.59. Do not reject the hypothesis of no relationship; –2.201 < –0.63 < 2.201. y is x1 – x3

101. A regression model involving three independent variables for a sample of 20 periods resulted in the following sum of squares:

Regression Residual (Error) a. b.

Sum of Squares 90 100

Compute the coefficient of determination and fully explain its meaning. At α = .05 level of significance, test to determine whether there is a significant relationship between the independent variables and the dependent variable.

ANSWER:

a. b.

.4737; 47.37% of variation in the dependent variable is explained by variations in the independent variables. F = 4.8 > 3.24; yes, there is a significant relationship between the independent variables and the dependent variable.

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Chapter 15: Multiple Regression

102. A regression model involving eight independent variables for a sample of 69 periods resulted in the following sum of squares: SSE = 306 SST = 1800 a. b.

Compute the coefficient of determination. At α = .05, test to determine whether or not the model is significant.

ANSWER:

a. b.

.83 F = 36.62 > 2.10; the model is significant.

103. In a regression model involving 46 observations, the following estimated regression equation was obtained: = 17 + 4x1 – 3x2 + 8x3 + 5x4 + 8x5 For this model, SST = 3410 and SSE = 510. a. Compute the coefficient of determination. b. Perform an F test and determine whether or not the regression model is significant. ANSWER:

a. b.

.85 F = 45.49 > 2.45; the model is significant.

104. A microcomputer manufacturer has developed a regression model relating his sales (y in $10,000s) with three independent variables. The three independent variables are price per unit (Price in $100s), advertising (ADV in $1000s), and the number of product lines (Lines). Part of the regression results is shown below. ANOVA df Regression Error

SS 2708.61 2840.51

Intercept Price ADV Lines

Coefficients 1.0211 -0.1524 0.8849 -0.1463

Standard Error 22.8752 0.1411 0.2886 1.5340

a. b. c. d.

Use the above results and write the regression equation that can be used to predict sales. If the manufacturer has 10 product lines, advertising of $40,000, and price per unit of $3000, what is an estimate of its sales? Give the answer in dollars. Compute the coefficient of determination and fully interpret its meaning. At α = .05, test to see if there is a significant relationship between sales and unit price.

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Chapter 15: Multiple Regression e. f. g. h.

At α = .05, test to see if there is a significant relationship between sales and the number of product lines. Is the regression model significant? (Perform an F test.) Fully interpret the meaning of the regression price per unit, that is, the slope for the price per unit. What has been the sample size for this analysis?

ANSWER: a. b. c. d. e. f. g. h.

= 1.0211 – 0.1524Price + 0.8849ADV – 0.1463 Lines $303,821.9 .488; 48.8% of the variation in sales is explained by variations in the three independent variables. t = –1.08; there is not a significant relationship between sales and unit price. t = –.095; there is not a significant relationship between sales and the number of product lines. F = 4.45 > 3.34; yes, the model is significant. As the price is increased by $100, sales are expected to decrease by $1,524.50. 18

105. The following is part of the results of a regression analysis involving sales (y in $ millions), advertising expenditures (x1 in $1000s), and number of salespeople (x2) for a corporation. The regression was performed on a sample of 10 observations. Coefficient 40.00 8.00 6.00

Intercept x1 x2 a. b. c.

Standard Error 7.00 2.50 3.00

If the company uses $40,000 in advertisement and has 30 salespeople, what are the expected sales? Give your answer in dollars. At α = .05, test for the significance of the coefficient of advertising. At α = .05, test for the significance of the coefficient of the number of salespeople.

ANSWER:

a. b. c.

$540,000,000 t = 3.2 > 2.365; advertising is significant. t = 2.0 < 2.365; number of salespeople is not significant.

106. The Natural Drink Company has developed a regression model relating its sales (y in $10,000s) with four independent variables. The four independent variables are price per unit (PRICE, in dollars), competitor's price (COMPRICE, in dollars), advertising (ADV, in $1000s), and type of container used (CONTAIN; 1 = Cans and 0 = Bottles). Part of the regression results is shown below. (Assume n = 25.) Intercept PRICE

Coefficient 443.143 –57.170

Standard Error 20.426

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Chapter 15: Multiple Regression COMPRICE ADV CONTAIN a. b. c. d. e.

27.681 0.025 –95.353

19.991 0.023 91.027

If the manufacturer uses can containers, his price is $1.25, advertising is $200,000, and his competitor's price is $1.50, what is your estimate of his sales? Give your answer in dollars. Test to see if there is a significant relationship between sales and unit price. Let α = .05. Test to see if there is a significant relationship between sales and advertising. Let α = .05. Is the type of container a significant variable? Let α = .05. Test to see if there is a significant relationship between sales and competitor's price. Let α = .05.

ANSWER:

a. b. c. d. e.

$3,228,490 t = –2.8 < –2.086; yes, there is a significant relationship between sales and unit price. t = 1.087 < 2.086; no, there is not a significant relationship between sales and advertising. t = –1.047 > –2.086; no, the type of container is not significant. t = 1.38 < 2.086; there is not a significant relationship between sales and competitor’s price.

107. The Very Fresh Juice Company has developed a regression model relating sales (y in $10,000s) with four independent variables. The four independent variables are price per unit (x1, in dollars), competitor's price (x2, in dollars), advertising (x3, in $1000s), and type of container used (x4) (1 = Cans and 0 = Bottles). Part of the regression results is shown below. Source of Variation Regression Error Total a. b. c.

Degrees of Freedom 4 18

Sum of Mean Squares Square 283,940.60 621,735.14

Compute the coefficient of determination and fully interpret its meaning. Is the regression model significant? Let α = .05. What has been the sample size for this analysis?

ANSWER:

a. b. c.

.3135; 31.35% of variation in sales is explained by variation of the independent variables. F = 2.055 < 2.93; no, the model is not significant. 23

108. The following regression model has been proposed to predict sales at a furniture store: = 10 – 4x1 + 7x2 + 18x3 where x1 = competitor's previous day's sales (in $1000s) Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression x2 = population within 1 mile (in 1000s) x3 = 1 if any form of advertising was used, 0 if otherwise = sales (in $1000s) a. b.

Fully interpret the meaning of the coefficient of x3. Predict sales (in dollars) for a store with competitor's previous day's sale of $3000, a population of 10,000 within 1 mile, and six radio advertisements.

ANSWER:

a. b.

When advertising was used, sales were higher by $18,000. $86,000

109. A sample of 30 houses that were sold in the last year was taken. The value of the house (y) was estimated. The independent variables included in the analysis were the number of rooms (x1), the size of the lot (x2), the number of bathrooms (x3), and a dummy variable (x4), which equals 1 if the house has a garage and 0 if the house does not have a garage. The following results were obtained: ANOVA df

SS 204,242.88 205,890.00

Coefficients 15,232.5 2,178.4 7.8 2,675.2 1,157.8

Standard Error 8,462.5 778.0 2.2 2,229.3 463.1

Regression Error

Intercept x1 x2 x3 x4 a. b. c. d. e.

f. g. h.

MS 51,060.72 8,235.60

Write out the estimated equation. Interpret the coefficient on the number of rooms (x1). Interpret the coefficient on the dummy variable (x4). What are the degrees of freedom for the sum of squares explained by the regression (SSR) and the sum of squares due to error (SSE)? Test whether or not there is a significant relationship between the value of a house and the independent variables. Use a .05 level of significance. Be sure to state the null and alternative hypotheses. Test the significance of β1 at the 5% level. Be sure to state the null and alternative hypotheses. Compute the coefficient of determination and interpret its meaning. Estimate the value of a house that has 9 rooms, a lot with an area of 7,500, 2 bathrooms, and a garage.

ANSWER: Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression a. b. c. d. e.

g. h.

= 15232.5 + 2178.4x1 + 7.8x2 + 2675.2x3 + 1157.8x4 Each additional room increases the value of the house by $2178.4, holding all other variables constant. The value of a house increases by $1157.8 if the house has a garage when compared to a house that does not have a garage, holding all other variables constant. 4, 25 H0: β1 = β2 = β3 = β4 = 0 Ha: At least one of the β's does not equal 0 Reject H0; 6.2 > 2.76; there is a significant relationship between the value of a house and the independent variables. H0: β1 = 0 Ha: β1 ≠ 0 Reject H0; 2.8 > 2.06 .498; 49.8% of the variability in y is explained by the independent variables. 99,846.3

110. A sample of 25 families was taken. The objective of the study was to estimate the factors that determine the monthly expenditure on food for families. The independent variables included in the analysis were the number of members in the family (x1), the number of meals eaten outside the home (x2), and a dummy variable (x3) that equals 1 if a family member is on a diet and 0 if no family member is on a diet. The following results were obtained: ANOVA df

SS 3,078.39 2,013.90

Coefficients 150.08 49.92 10.12 –.60

Standard Error 53.6 9.6 2.2 12.0

Regression Error

Intercept x1 x2 x3 a. b. c. d. e. f.

g. h. i.

MS 1,026.13 95.90

Write out the estimated regression equation. Interpret all coefficients. Compute the appropriate t ratios. Test for the significance of β1, β2, and β3 at the 1% level of significance. What are the degrees of freedom for the sum of squares explained by the regression (SSR) and the sum of squares due to error (SSE)? Test whether or not there is a significant relationship between the monthly expenditure on food and the independent variables. Use a .01 level of significance. Be sure to state the null and alternative hypotheses. Compute the coefficient of determination and explain its meaning. Estimate the monthly expenditure on food for a family that has 4 members, eats out 3 times, and does not have any member of the family on a diet. At 95% confidence, determine which parameter is not statistically significant.

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Chapter 15: Multiple Regression

ANSWER: a.

= 150.08 + 49.92x1 + 10.12x2 – 3.6x3

An increase in the number of family members by 1 increases monthly expenditure on food by $49.92, holding all other variables constant. An increase in the number of meals eaten outside the home by 1 increases the monthly expenditure on food by $10.12, holding all other variables constant. Monthly food expenditures decrease by $0.60 if a family member is on a diet, holding all other variables constant. c. and d. t.005 = 2.831 t1 = 5.2 reject H0 t2 = 4.6 reject H0; t3 = –.3 do not reject H0 e. 3, 21 f. H0: β1 = β2 = β3 = 0 Ha: At least one of the β's does not equal 0 Reject H0; 10.7 > 4.87; there is a significant relationship between the monthly expenditure on food and the independent variables. g. .6045; 60.45% of the variability in y is explained by the independent variables h. 380.12 i. β3 is not statistically significant.

111. The following regression model has been proposed to predict sales at a fast-food outlet: = 18 – 2x1 + 7x2 + 15x3 where x1 = the number of competitors within 1 mile x2 = the population within 1 mile (in 1000s) x3 = 1 if drive-up windows are present, 0 otherwise = sales (in $1000s) a. b. c.

What is the interpretation of 15 (the coefficient of x3) in the regression equation? Predict sales for a store with 2 competitors, a population of 10,000 within one mile, and one drive-up window (give the answer in dollars). Predict sales for the store with 2 competitors, a population of 10,000 within one mile, and no drive-up window (give the answer in dollars).

ANSWER: Copyright Cengage Learning. Powered by Cognero.

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Chapter 15: Multiple Regression a. b. c.

Sales of stores with drive-up windows are $15,000 higher than those without drive-up windows. $99,000 $84,000

112. The following regression model has been proposed to predict sales at a computer store: = 50 – 3x1 + 20x2 + 10x3 where x1 = competitor's previous day's sales (in $1000s) x2 = population within 1 mile (in 1000s)

= sales (in $1000s) Predict sales (in dollars) for a store with the competitor's previous day's sale of $5000, a population of 20,000 within 1 mile, and nine radio advertisements. ANSWER: $445,000 113. The following regression model has been proposed to predict monthly sales at a shoe store. = 40 – 3x1 + 12x2 + 10x3 where x1 = competitor's previous month's sales (in $1000s) x2 = stores previous month's sales (in $1000s)

= sales (in $1000s) a. b.

Predict sales (in dollars) for the shoe store if the competitor's previous month's sales were $9000, the store's previous month's sales were $30,000, and no radio advertisements were run. Predict sales (in dollars) for the shoe store if the competitor's previous month's sales were $9000, the store's previous month's sales were $30,000, and 10 radio advertisements were run.

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Chapter 15: Multiple Regression ANSWER:

a. b.

$373,000 $383,000

114. A company has recorded data on the weekly sales for its product (y), the unit price of its competitor's product (x1), and advertising expenditures (x2). The data resulting from a random sample of seven weeks follows. Use Excel's Regression tool to answer the following questions: Week 1 2 3 4 5 6 7 a. b. c. d. e.

Price 0.33 0.25 0.44 0.40 0.35 0.39 0.29

Advertising 5 2 7 9 4 8 9

Sales 20 14 22 21 16 19 15

What is the estimated regression equation? Determine whether the model is significant overall. Use α = .10. Determine if price is significantly related to sales. Use α = .10. Determine if advertising is significantly related to sales. Use α = .10. Find and interpret the multiple coefficient of determination.

ANSWER:

Price 0.33 0.25 0.44 0.40 0.35 0.39 0.29

Advertising 5 2 7 9 4 8 9

Sales 20 14 22 21 16 19 15

SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.877814352 0.770558037 0.655837056 1.837409752 7

ANOVA Regression Residual Total

df 2 4 6

SS 45.35284 13.5043 58.85714

MS 22.67642 3.376075

F 6.716801

Significance F 0.052644

Intercept Price

Coefficients 3.597615086 41.32002219

Standard Error t Stat 4.052244 0.887808 13.33736 3.098065

P-value 0.424805 0.036289

Lower 95% -7.65324 4.289491

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Chapter 15: Multiple Regression Advertising a. b. c. d. e.

0.013241819

0.327592

0.040422

0.969694

-0.8963

y = 3.597615086 + 41.32002219x1 +.013241819x2 Since the p-value .052644 < .10, reject H0. Therefore, a significant relationship exists between sales and the two independent variables, competitor's price and advertising. Since the p-value .036289 < .10, reject H0. Therefore, price is significantly related to sales. Since the p-value .969694 < .10, do not reject H0. Therefore, advertising is not significantly related to sales. R2 = .877814352. Therefore, 87.78% of the variability in sales is explained by the estimated regression equation.

115. The prices of Rawlston, Inc. stock (y) over a period of 12 days, the number of shares (in 100s) of the company's stocks sold (x1), and the volume of exchange (in millions) on the New York Stock Exchange (x2) are shown below. Day (y) (x1) (x2) 1 87.50 950 11.00 2 86.00 945 11.25 3 84.00 940 11.75 4 83.00 930 11.75 5 84.50 935 12.00 6 84.00 935 13.00 7 82.00 932 13.25 8 80.00 938 14.50 9 78.50 925 15.00 10 79.00 900 16.50 11 77.00 875 17.00 12 77.50 870 17.50 Excel was used to determine the least squares regression equation. Part of the computer output is shown below. ANOVA df Regression 2 Residual 9 Total 11

Intercept (x1) (x2) a. b. c. d.

Coefficients 118.5059 –0.0163 –1.5726

SS 118.8474 13.0692 131.9167

MS F 59.4237 40.9216 1.4521

Standard Error 33.5753 0.0315 0.3590

t Stat 3.5296 –0.5171 –4.3807

Significance F 0.0000

P-value 0.0064 0.6176 0.0018

Use the output shown above and write an equation that can be used to predict the price of the stock. Interpret the coefficients of the estimated regression equation that you found in part (a). At 95% confidence, determine which variables are significant and which are not. If on a given day, the number of shares of the company that were sold was 94,500 and the volume of exchange on the New York Stock Exchange was 16 million, what would you expect the price of the stock to be?

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Chapter 15: Multiple Regression

ANSWER: a. b.

c. d.

= 118.5055 – 0.0163x1 – 1.5726x2 As the number of shares of the stock sold goes up by 1 unit, the stock price goes down by $0.0163 (holding the volume of exchange on the NYSE constant). As the volume of exchange on the NYSE goes up by 1 unit, the stock price goes down by $1.5726 (holding the number of shares of the stock sold constant). x1 is not significant; the p-value = .6176 > α = .05 x2 is significant; the p-value = .0018 < α = .05 $77.94

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Essentials of Modern Business Statistics with Microsoft Excel 8th Edition Test Bank

richard@qwconsultancy.com

1|Pa ge

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Chapter 01: Data and Statistics Multiple Choice 1. Which of the following is an example of quantitative data? a. the player’s number on a baseball uniform b. the serial number on a one-dollar bill c. the part number of an inventory item d. the number of people in a waiting line ANSWER: d 2. Which of the following is NOT an example of descriptive statistics? a. a histogram depicting the age distribution for 30 randomly selected students b. an estimate of the number of Alaska residents who have visited Canada c. a table summarizing the data collected in a sample of new-car buyers d. the proportion of mailed-out questionnaires that were returned ANSWER: b 3. Which of the following is an example of categorical data? a. social security number b. score on a multiple-choice exam c. height, in meters, of a diving board d. number of square feet of carpet ANSWER: a 4. The number of observations in a complete data set having 10 elements and 5 variables is _____. a. 5 b. 10 c. 25 d. 50 ANSWER: b 5. Facts and figures collected, analyzed, and summarized for presentation and interpretation are called _____. a. data b. variables c. elements d. variables and elements ANSWER: a 6. The entities on which data are collected are _____. a. elements b. populations c. sets d. samples ANSWER: a Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics 7. The set of measurements collected for an element is called a(n) _____. a. census b. variable c. observation d. sample ANSWER: c 8. A characteristic of interest for the elements is called a(n) _____. a. sample b. data set c. variable d. observation ANSWER: c 9. All data collected in a study are referred to as the _____. a. census b. inference c. variable d. data set ANSWER: d 10. In a data set, the number of observations will always be the same as the number of _____. a. variables b. elements c. data sets d. data ANSWER: b 11. Which of the following is NOT a scale of measurement? a. nominal b. ordinal c. interval d. categorical ANSWER: d 12. When the data are labels or names used to identify an attribute of the elements, the variable has which scale of measurement? a. nominal b. ordinal c. interval d. ratio ANSWER: a 13. When the data are labels or names used to identify an attribute of the elements and the rank of the data is meaningful, Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics the variable has which scale of measurement? a. nominal b. ordinal c. interval d. ratio ANSWER: b 14. When the data have the properties of ordinal data and the interval between observations is expressed in terms of a fixed unit of measure, the variable has which scale of measurement? a. nominal b. ordinal c. interval d. ratio ANSWER: c 15. When the data have the properties of interval data and the multiplication or division of two values is meaningful, the variable has which scale of measurement? a. nominal b. ordinal c. interval d. ratio ANSWER: d 16. Which two scales of measurement can be either numeric or nonnumeric? a. nominal and ratio b. ordinal and interval c. interval and ordinal d. nominal and ordinal ANSWER: d 17. Which of the following variables uses the interval scale of measurement? a. name of stock exchange b. time c. SAT scores d. social security number ANSWER: c 18. Which of the following variables uses the ratio scale of measurement? a. name of stock exchange b. time c. SAT scores d. social security number ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics 19. Quantitative data _____. a. are always nonnumeric b. may be either numeric or nonnumeric c. are always numeric d. are always labels ANSWER: c 20. Categorical data _____. a. are always nonnumeric b. may be either numeric or nonnumeric c. are always numeric d. indicate either how much or how many ANSWER: b 21. _____ analytics is the set of analytical techniques that yield a course of action. a. Descriptive b. Predictive c. Prescriptive d. Data ANSWER: c 22. What organization developed the report, “Ethical Guidelines for Statistical Practice”? a. Ethics Committee for Statistical Practices b. American Statistical Association c. International Statistical Organization d. Federal Bureau for Ethical Practices in Statistics ANSWER: b 23. Arithmetic operations are inappropriate for _____. a. categorical data b. quantitative data c. both categorical and quantitative data d. large data sets ANSWER: a 24. In a questionnaire, respondents are asked to mark their gender as Male, Female, Transgender MtoF, Transgender FtoM, Non-binary, or Intersex. Gender is an example of a(n) _____ variable. a. categorical b. quantitative c. interval d. ratio ANSWER: a 25. In a questionnaire, respondents are asked to record their age in years. Age is an example of a _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics a. categorical variable b. quantitative variable c. categorical or quantitative variable, depending on how the respondents answered the question d. ratio variable ANSWER: b 26. In an application for a credit card, potential customers are asked for their social security numbers. A social security number is an example of a _____. a. categorical variable b. quantitative variable c. categorical or quantitative variable, depending on how the respondents answered the question d. ratio variable ANSWER: a 27. Temperature is an example of which scale of measurement? a. nominal b. ordinal c. interval d. ratio ANSWER: c 28. For ease of data entry into a university database, 1 denotes the student is enrolled in an undergraduate degree program, 2 indicates the student is enrolled in a master’s degree program, and 3 indicates the student is enrolled in a doctoral degree program. In this case, the data are which scale of measurement? a. nominal b. ordinal c. interval d. ratio ANSWER: b 29. Income is an example of _____. a. categorical data b. either categorical or quantitative data c. currency data d. quantitative data ANSWER: d 30. The birth weight of newborns, measured in grams, is an example of _____. a. categorical data b. either categorical or quantitative data c. neither categorical nor quantitative data d. quantitative data ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics 31. The graph below best exemplifies a _____.

(Graph courtesy of Robert Allison.)

a. bar graph b. time series graph c. cross-sectional graph d. line graph ANSWER: b 32. The graph below best exemplifies a _____.

a. bar graph b. time series graph c. cross-sectional graph d. line graph ANSWER: c 33. Data collected at the same, or approximately the same, point in time are _____ data. a. time series b. static c. cross-sectional d. one-dimensional ANSWER: c

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Chapter 01: Data and Statistics 34. Data collected over several time periods are _____ data. a. time series b. time-controlled c. cross-sectional d. time dependent ANSWER: a 35. Statistical studies in which researchers do not control variables of interest are _____. a. experimental studies b. uncontrolled experimental studies c. not of any value d. observational studies ANSWER: d 36. Statistical studies in which researchers control variables of interest are _____ studies. a. experimental b. control observational c. non-experimental d. observational ANSWER: a 37. _____ analytics encompasses the set of analytical techniques that describe what has happened in the past. a. Descriptive b. Predictive c. Prescriptive d. Data ANSWER: a 38. Which of the following is NOT an example of an existing source of data? a. the Internet b. internal company records c. U.S. Census Bureau d. data from an experiment ANSWER: d 39. Which of the following is NOT an example of a firm that sells or leases business database services to clients? a. Dun & Bradstreet b. Bloomberg c. U.S. Census Bureau d. Dow Jones and Company ANSWER: c 40. The most common type of observational study is a(n) _____. a. experiment Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics b. survey c. debate d. statistical inference ANSWER: b 41. The summaries of data, which may be tabular, graphical, or numerical, are referred to as _____. a. inferential statistics b. descriptive statistics c. statistical inference d. report generation ANSWER: b 42. A Scanner Data User Survey of 50 companies found that the average amount spent on scanner data per category of consumer goods was $387,325 (Mercer Management Consulting, Inc., April 24, 1997). The $387,325 is an example of _____. a. categorical data b. a categorical variable c. a descriptive statistic d. time series data ANSWER: c 43. Statistical inference _____. a. refers to the process of drawing inferences about the sample based on the characteristics of the population b. is the same as descriptive statistics c. is the process of drawing inferences about the population based on the information taken from the sample d. is the same as a census ANSWER: c 44. The collection of all elements of interest in a study is _____. a. the population b. the sampling c. statistical inference d. descriptive statistics ANSWER: a 45. A portion of the population selected to represent the population is called _____. a. statistical inference b. descriptive statistics c. a census d. a sample ANSWER: d 46. Of 800 students in a university, 360, or 45%, live in the dormitories. The 800 is an example of _____. a. a sample Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics b. a population c. statistical inference d. descriptive statistics ANSWER: b 47. In a sample of 800 students in a university, 160, or 20%, are Business majors. Based on the above information, the school's paper reported, "20% of all students at the university are Business majors." This report is an example of _____. a. a sample b. a population c. statistical inference d. descriptive statistics ANSWER: c 48. Six hundred residents of a city are polled to obtain information on voting intentions in an upcoming city election. The 600 residents in this study is an example of a(n) _____. a. census b. sample c. observation d. population ANSWER: b 49. A statistics professor asked students in a class their ages. Based on this information, the professor states that the average age of students in the university is 21 years. This is an example of _____. a. a census b. descriptive statistics c. an experiment d. statistical inference ANSWER: d 50. The owner of a factory regularly requests a graphical summary of all employees' salaries. The graphical summary of salaries is an example of _____. a. a sample b. descriptive statistics c. statistical inference d. an experiment ANSWER: b 51. The Department of Transportation of a city has noted that on the average there are 14 accidents per day. The average number of accidents is an example of _____. a. descriptive statistics b. statistical inference c. a sample d. a population ANSWER: a Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics Exhibit 1-1 In a recent study based upon an inspection of 200 homes in Daisy City, 120 were found to violate one or more city codes. 52. Refer to Exhibit 1-1. The city manager released a statement that 60% of Daisy City's 3,000 homes are in violation of city codes. The manager's statement is an example of _____. a. a census b. an experiment c. descriptive statistics d. statistical inference ANSWER: d 53. Refer to Exhibit 1-1. The Daisy City study is an example of the use of a _____. a. census b. sample c. probability d. population ANSWER: b 54. Refer to Exhibit 1-1. The manager's statement that 60% of Daisy City's 3,000 homes are in violation of city codes is _____. a. an accurate statement b. only an approximation, since it is based upon sample information c. obviously wrong, since it is based upon a study of only 200 homes d. wrong. All 3,000 homes need to be surveyed to make that statement ANSWER: b Exhibit 1-2 In a sample of 3,200 registered voters, 1,440, or 45%, approve of the way the president is doing his job. 55. Refer to Exhibit 1-2. The 45% approval is an example of _____. a. a sample b. descriptive statistics c. statistical inference d. a population ANSWER: b 56. Refer to Exhibit 1-2. A political pollster states, "Forty five percent of all voters approve of the president." This statement is an example of _____. a. a sample b. descriptive statistics c. statistical inference d. a population ANSWER: c 57. The process of analyzing sample data to draw conclusions about the characteristics of a population is called _____. a. descriptive statistics Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics b. statistical inference c. data analysis d. data summarization ANSWER: b 58. In a post office, the mailboxes are numbered from 1 to 5,000. These numbers represent _____. a. categorical data b. time series data c. cross-sectional data d. quantitative data ANSWER: a 59. The average age in a sample of 90 students at City College is 20. From this sample, it can be concluded that the average age of all the students at City College _____. a. must be more than 20, since the population is always larger than the sample b. must be less than 20, since the sample is only a part of the population c. could not be 20 d. could be larger, smaller, or equal to 20 ANSWER: d 60. _____ analytics consists of analytical techniques that use models constructed from past data to predict the future or to assess the impact of one variable on another. a. Descriptive b. Predictive c. Prescriptive d. Data ANSWER: b 61. The term _____ is used to refer to the process of capturing, storing, and maintaining data. a. data warehousing b. data mining c. data analysis d. data collection ANSWER: a 62. A sample of five Fortune 500 companies showed the following revenues ($ millions): 7505.0, 2904.7, 7208.4, 6819.0, and 19500.0. Based on this information, which of the following statements is correct? a. An estimate of the average revenue for all Fortune 500 companies is 8787.42 ($ millions). b. The average revenue for all Fortune 500 companies is 8787.42 ($ millions). c. Over half of all Fortune 500 companies earn at least 7208.4 ($ millions) in revenues. d. If five other Fortune 500 companies were chosen, the average revenue would be 8787.42 ($ millions). ANSWER: a 63. A sample of five Fortune 500 companies possessed the following industry codes: banking, banking, finance, retail, Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics and banking. Based on this information, which of the following statements is correct? a. Sixty percent of the sample of five companies are banking industries. b. Sixty percent of all Fortune 500 companies are banking industries. c. Banking is the most common type of industry among all Fortune 500 companies. d. If five other Fortune 500 companies were chosen, 60% of them would be banking industries. ANSWER: a 64. The Microsoft Office package used to perform statistical analysis is _____. a. SPSS b. Word c. SAS d. Excel ANSWER: d 65. Dr. Kurt Thearling, a leading practitioner in the field, defines data mining as “the _____ extraction of _____ information from databases." a. thorough, insightful b. timely, accurate c. automated, predictive d. intentional, useful ANSWER: c 66. The major applications of data mining have been made by companies with a strong _____ focus. a. consumer b. manufacturing c. exporting d. research and development ANSWER: a 67. Quantitative data that measure "how many" are ________; quantitative data that measure "how much" are ________. a. interval; ratio b. ratio; interval c. continuous; discrete d. discrete; continuous ANSWER: d 68. Flight time from Cincinnati to Atlanta is an example of a _____ variable and _____ measurement. a. discrete; interval b. discrete; ratio c. continuous; interval d. continuous; ratio ANSWER: d 69. Which of the following is NOT a categorical variable? Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics a. your age on your last birthday b. your cell phone area code c. your accounting class start time d. your high school graduation year ANSWER: a 70. Categorical data use either the ______ or ______ scale of measurement. a. nominal; ordinal b. nominal; interval c. ordinal; interval d. interval; ratio ANSWER: a 71. The term data warehousing is used to refer to the process of doing all of the following, except _____ the data. a. capturing b. storing c. maintaining d. mining ANSWER: d 72. _____ is the process of using procedures from statistics and computer science to extract useful information from extremely large databases. a. Big data b. Analytics c. Data warehousing d. Data mining ANSWER: d Subjective Short Answer 73. After the graduation ceremonies at a university, six graduates were asked whether they were in favor of (identified by 1) or against (identified by 0) abortion. Some characteristics of these graduates are shown below. Graduate 1 2 3 4 5 6 a. b. c. d. e.

Sex F M F M F M

Age 22 21 33 38 25 19

Abortion Issue 1 1 0 0 1 0

Class Rank 3 2 1 20 4 8

How many elements are in the data set? How many variables are in the data set? How many observations are in the data set? Identify the scale of measurement for each of the above (Sex, Age, Abortion Issue, Class Rank). Which of the above (Sex, Age, Abortion Issue, Class Rank) are categorical, and which are

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Chapter 01: Data and Statistics f.

quantitative variables? Are arithmetic operations appropriate for the variable "abortion issue"?

ANSWER: a. b. c. d.

6 4 6 Sex: nominal Age: ratio Abortion Issue: nominal Class Rank: ordinal Sex: categorical Age: quantitative Abortion Issue: categorical Class Rank: categorical No

74. An issue of Fortune Magazine reported the following companies had the lowest sales per employee among the Fortune 500 companies. Company Seagate Technology SSMC Russell Maxxam Dibrell Brothers a. b. c. d. e.

Sales per Employee ($ thousands) 42.20 42.19 41.99 40.88 22.56

Sales Rank 285 414 480 485 470

How many elements are in the above data set? How many variables are in the above data set? How many observations are in the above data set? Name the scale of measurement for each of the variables. Name the variables and indicate whether they are categorical or quantitative.

ANSWER: a. b. c. d. e.

5 2 5 Sales per Employee: ratio; Sales Rank: ordinal Sales per Employee: quantitative; Sales Rank: categorical

75. The following shows the temperatures (high, low) and weather conditions on a given Sunday for seven world cities. For the weather conditions, the following notations are used: c = clear; cl = cloudy; sh = showers; pc = partly cloudy. City Acapulco Bangkok Mexico City Montreal

Hi 99 92 77 72

Lo 77 78 57 56

Condition pc pc sh pc Page 14

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Chapter 01: Data and Statistics Paris Rome Toronto a. b. c. d. e. f.

77 88 78

58 68 61

c cl c

How many elements are in this data set? How many variables are in this data set? How many observations are in this data set? Name the scale of measurement for each of the variables. Name the variables and indicate whether they are categorical or quantitative. For which variables are arithmetic operations appropriate, and for which are they not appropriate?

ANSWER: a. b. c. d. e. f.

7 3 7 Hi: interval, Lo: interval, Condition: nominal Hi: quantitative, Lo: quantitative, Condition: categorical Hi: appropriate, Lo: appropriate, Condition: not appropriate

76. A magazine surveyed a sample of its subscribers. Some of the responses from the survey are shown below. Subscriber ID 0006 4798 2291 4988

Sex F M F M

a. b. c. d. e.

Age 22 21 33 38

Annual Household Income ($1000s) 45 53 82 30

How many elements are in the data set? How many variables are in the data set? How many observations are in the data set? Name the scale of measurement for each of the variables. Which of the above (Sex, Age, Annual Household Income) are categorical, and which are quantitative? Are the data time series or cross-sectional?

f. ANSWER: a. b. c. d. e. f.

4 3 4 Sex: nominal, Age: ratio, Annual Household Income: ratio Sex: categorical, Age: quantitative, Annual Household Income: quantitative cross-sectional

77. A magazine surveys a sample of its subscribers every year. Some of the responses are shown below. Year 1996 1997 1998 1999

Percent Female 5.8 8.6 7.3 9.2

Average Age 35.2 35.8 33.9 35.3

Average Annual Household Income ($1000s) 40 42 41 43 Page 15

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Chapter 01: Data and Statistics 2000 a. b. c. d.

9.8

36.6

How many variables are in the data set? Name the scale of measurement for each variable. Which of the above (Year, Percent Female, Average Age, Average Annual Household Income) are categorical, and which are quantitative? Are the data time series or cross-sectional?

ANSWER: a. b. c. d.

4 Year: interval, Percent Female: ratio, Average Age: ratio, Average Annual Household Income: ratio all are quantitative time series

78. The following data show the yearly income distribution of a sample of 200 employees at MNM, Inc. Yearly Income ($1000s) 20 − 24 25 − 29 30 − 34 35 − 39 40 − 44 a. b. c.

d. e. f. g.

Number of Employees 2 48 60 80 10

What percentage of employees have a yearly incomes of at least $35,000? Is the figure (percentage) that you computed in Part a. an example of statistical inference? If not, what kind of statistics does it represent? Based on this sample, the president of the company said that "45% of all our employees' yearly incomes are at least $35,000." The president's statement represents what kind of statistics? With the statement made in Part c., can we be assured that more than 45% of all employees' yearly incomes are at least $35,000? Explain. What percentage of employees of the sample have a yearly income of less than $30,000? How many variables are presented in the above data set? The above data set represents the results of how many observations?

ANSWER: a. b. c. d. e. f. g.

45% No, it is descriptive statistics. statistical inference No, this is simply an inference and approximation based on the sample information. 25% 2 200

79. A recent issue of a national magazine reported that in a national public opinion survey conducted among 2,000 individuals, 56% were in favor of gun control, 40% opposed gun control, and 4% had no opinion on the subject. a. What is the sample in this survey? b. Based on the sample, what percentage of the population would you think is in favor of gun control? c. Based on the sample, what percentage of the population would you think have no opinion on the subject? Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics ANSWER: a. b. c.

the 2,000 individuals who were approached 56% 4%

80. A pharmaceutical company is performing clinical trials on a new drug that is intended to relieve symptoms for allergy sufferers. Twelve percent of the 300 clinical trial participants experienced dry mouth as a side effect. a. What is the population being studied? b. What is the sample being studied? Based on the sample, what percentage of the population do you think would suffer from dry c. mouth? ANSWER: a. all allergy sufferers b. the 300 participants c. 12%

81. A polling organization conducts a telephone poll of 850 registered voters and asks which candidate they will vote for in the upcoming presidential election. Forty-three percent of the respondents prefer candidate A and 45% prefer candidate B. a. What is the population being studied? b. What is the sample being studied? c. Based on the sample, what percentage of the population do you think would vote for candidate B? ANSWER: a. b. c.

all registered voters the 850 registered voters who were polled 45%

82. The following table shows the starting salaries of a sample of recent business graduates. Income ($1000s) 15 − 19 20 − 24 25 − 29 30 − 34 35 − 39 a. b. c.

Number of Graduates 40 60 80 18 2

What percentage of graduates in the sample had starting salaries of at least $30,000? Of the graduates in the sample, what percentage had starting salaries of less than $25,000? Based on this sample, what percentage of all business graduates do you estimate to have starting salaries of at least $20,000?

ANSWER: a. b. c.

10% 50% 80%

83. Michael, Inc., a manufacturer of electric guitars, is a small firm with 50 employees. The table below shows the hourly wage distribution of the employees. Hourly Wages (In Dollars) 10 − 13.99 Copyright Cengage Learning. Powered by Cognero.

Number of Employees 8 Page 17

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Chapter 01: Data and Statistics 14 − 17.99 18 − 21.99 22 − 25.99 a. b. c.

12 20 10

How many employees receive hourly wages of at least $18? What percentage of the employees has hourly wages of at least $18? What percentage of the employees has hourly wages of less than $14?

ANSWER: a. b. c.

30 60% 16%

84. Laura Naples, manager of Heritage Inn, periodically collects and tabulates information about a sample of the

hotel’s overnight guests. This information aids her in planning and scheduling decisions she must make. The table below lists data on 10 randomly selected hotel registrants, collected as the registrants checked out. The data listed for each registrant are: number of people in the group; birth date of person registering; shuttle service used, yes or no; total telephone charges incurred; and reason for stay, business or personal. Number of Birth Shuttle Telephone ID of People Date Used Charges Registrant in Group (mm/dd/yy) 01 1 05/07/59 yes $ 0.00 02 4 11/23/48 no 12.46 03 2 04/30/73 no 1.20 04 2 12/16/71 no 2.90 05 1 05/09/39 yes 0.00 06 3 09/14/69 yes 4.65 07 2 04/22/66 no 9.35 08 5 10/28/54 yes 2.10 09 1 11/12/49 no 1.85 10 2 01/30/62 no 5.80 a. How many elements are there in the data set? b. How many variables are there in the data set? c. How many observations are there in the data set? d. What are the observations for the second element listed? e. What is the total number of measurements in the data set? f. Which variables are quantitative? g. Which variables are qualitative? h. What is the scale of measurement for each of the variables? i. Does the data set represent cross-sectional or times series data? j. Does the data set represent an experimental or an observational study?

Reason for Stay personal business business business personal business personal personal business business

ANSWER: a. 10 elements b. 5 variables c. 10 observations d. 4, 11/23/48, no, 12.46, business e. 50 f. people in group, telephone charges g. birth date, shuttle used, reason for stay Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics h. people in group – ratio scale, birth date – ordinal scale, shuttle use – nominal, telephone charge – ratio scale, reason for stay – nominal scale i. cross-sectional j. observational 85. Molly Porter owns and operates two convenience stores, one on the East side of the city and the other on the South side. She has workforce-planning decisions to make and has collected some recent sales data that are relevant to her decisions. Listed below are the monthly sales ($1000s) at her two stores for the past six months. Store East South

March 102 72

April 100 74

May 103 81

June 105 86

July 109 92

August 106 93

a. Is the data set cross-sectional or time series data? Explain. b. Comment on any apparent patterns you see in the data. ANSWER: a. Time series data because time series data is collected over a period of time. b. Both stores have been experiencing an overall rise in sales during the past six months. The South store’s increase in sales (as a percentage of sales) has been greater than the East store’s increase. The increases might be temporary, due to the seasonal nature of demand. It is also possible that the increases will continue. 86. The following table shows the starting salaries of a sample of recent VoTech graduates. Income (Rounded to $1000s) 25 − 29 30 − 34 35 − 39 40 − 44 45 − 49

Number of Graduates 40 60 80 18 2

a. What percentage of graduates in the sample had starting salaries of at least $40,000? b. Of the graduates in the sample, what percentage had starting salaries of less than $35,000? c. Based on this sample, what percentage of all VoTech graduates do you estimate to have starting salaries of at least $30,000? ANSWER: a. 10% b. 50% c. 80% 87. Suppose the current weather report for your area contains the following information. Specify the measurement scale for each of the variables. a. b. c. d. e.

Temperature Wind Speed Wind Direction Sky Description Molds Level

84o 10 mph (from the) South Sunny High

ANSWER: a. Temperature – interval b. Wind Speed – ratio Copyright Cengage Learning. Powered by Cognero.

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Chapter 01: Data and Statistics c. Wind Direction – nominal d. Sky Description – nominal e. Molds Level – ordinal

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays Multiple Choice 1. The minimum number of variables represented in a bar chart is _____. a. 1 b. 2 c. 3 d. 4 ANSWER: a 2. The minimum number of variables represented in a histogram is _____. a. 1 b. 2 c. 3 d. 4 ANSWER: a 3. Which of the following graphical methods is most appropriate for categorical data? a. bar chart b. pie chart c. histogram d. scatter diagram ANSWER: b 4. In a stem-and-leaf display, _____. a. a single digit is used to define each stem, and a single digit is used to define each leaf b. a single digit is used to define each stem, and one or more digits are used to define each leaf c. one or more digits are used to define each stem, and a single digit is used to define each leaf d. one or more digits are used to define each stem, and one or more digits are used to define each leaf ANSWER: c 5. A graphical method that can be used to show both the rank order and shape of a data set simultaneously is a _____. a. relative frequency distribution b. pie chart c. stem-and-leaf display d. pivot table ANSWER: c 6. The proper way to construct a stem-and-leaf display for the data set {62, 67, 68, 73, 73, 79, 91, 94, 95, 97} is to _____. a. exclude a stem labeled ‘8’ b. include a stem labeled ‘8’ and enter no leaves on the stem c. include a stem labeled ‘(8)’ and enter no leaves on the stem d. include a stem labeled ‘8’ and enter one leaf value of ‘0’ on the stem ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 7. Data that provide labels or names for groupings of like items are known as _____. a. categorical data b. quantitative data c. label data d. generic data ANSWER: a 8. A researcher is gathering data from four geographical areas designated: South = 1; North = 2; East = 3; West = 4. The designated geographical regions represent _____. a. categorical data b. quantitative data c. directional data d. continuous data ANSWER: a 9. A researcher asked 20 people for their zip code. The respondents zip codes are an example of _____. a. categorical data b. quantitative data c. label data d. category data ANSWER: a 10. The age of employees at a company is an example of _____. a. categorical data b. quantitative data c. label data d. time series data ANSWER: b 11. A frequency distribution is a _____. a. tabular summary of a set of data showing the fraction of items in each of several nonoverlapping classes b. graphical form of representing data c. tabular summary of a set of data showing the number of items in each of several nonoverlapping classes d. graphical device for presenting categorical data ANSWER: c 12. The sum of frequencies for all classes will always equal _____. a. 1 b. the number of elements in a data set c. the number of classes d. a value between 0 and 1 ANSWER: b 13. In constructing a frequency distribution, as the number of classes is decreased, the class width _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays a. decreases b. remains unchanged c. increases d. can increase or decrease depending on the data values ANSWER: c 14. If several frequency distributions are constructed from the same data set, the distribution with the widest class width will have the _____. a. fewest classes b. most classes c. same number of classes as the other distributions since all are constructed from the same data d. None of the answers is correct. ANSWER: a 15. Excel's __________ can be used to construct a frequency distribution for categorical data. a. DISTRIBUTION function b. SUM function c. FREQUENCY function d. PivotTables report ANSWER: d 16. There are 20 boys and 8 girls in a class. What type of graph can be used to display this information? a. bar graph b. stem-and-leaf plot c. histogram d. scatter diagram ANSWER: a 17. The relative frequency of a class is computed by _____. a. dividing the midpoint of the class by the sample size b. dividing the frequency of the class by the midpoint c. dividing the sample size by the frequency of the class d. dividing the frequency of the class by the sample size ANSWER: d 18. The sum of the relative frequencies for all classes will always equal _____. a. the sample size b. the number of classes c. 1 d. 100 ANSWER: c 19. The height and weight are recorded by the school nurse for every student in a school. What type of graph would best display the relationship between height and weight? Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays a. bar graph b. stem-and-leaf plot c. histogram d. scatter diagram ANSWER: d 20. The percent frequency of a class is computed by _____. a. multiplying the relative frequency by 10 b. dividing the relative frequency by 100 c. multiplying the relative frequency by 100 d. adding 100 to the relative frequency ANSWER: c 21. A dot plot can be used to display _____. a. the relationship between two quantitative variables b. the percent a particular category is of the whole c. the distribution of one quantitative variable d. Simpson’s paradox ANSWER: c 22. In a cumulative frequency distribution, the last class will always have a cumulative frequency equal to _____. a. 1 b. 100% c. the total number of elements in the data set d. a value between 0 and 1 ANSWER: c 23. What is the difference between a bar graph and a histogram? a. There is no difference between a bar graph and a histogram. b. A bar graph displays categorical data, while a histogram displays quantitative data. c. A bar graph has no spaces between the bars, while a histogram must have space between the bars. d. A bar graph displays quantitative data, while a histogram displays categorical data. ANSWER: b 24. College students were surveyed to determine how much they planned to spend in various categories during the upcoming academic year. One category is the amount spent on school supplies. The graphs below show the amount of money spent on school supplies by women and men.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

Approximately what percent of women spend more than $105 on school supplies? a. 5% b. 10% c. 15% d. 20% ANSWER: a 25. The difference between the lower class limits of adjacent classes provides the _____. a. number of classes b. class limits c. class midpoint d. class width ANSWER: d Exhibit 2-1 The numbers of hours worked (per week) by 400 statistics students are shown below. Number of Hours 0 x 10

Frequency 20

200

100

26. Refer to Exhibit 2-1. The class width for this distribution _____. a. is 9 b. is 10 c. is 40, which is the largest value minus the smallest value or 40 − 0 =40 d. varies from class to class ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 27. Refer to Exhibit 2-1. The midpoint of the last class is _____. a. 50 b. 34 c. 35 d. 34.5 ANSWER: c 28. Refer to Exhibit 2-1. The number of students working less than 20 hours is _____. a. 80 b. 100 c. 180 d. 300 ANSWER: b 29. Refer to Exhibit 2-1. The relative frequency of students working less than 10 hours is _____. a. 20 b. 100 c. .95 d. .05 ANSWER: d 30. Refer to Exhibit 2-1. The cumulative relative frequency for the class of 20

30 is _____.

a. 300 b. .25 c. .75 d. .5 ANSWER: c 31. Refer to Exhibit 2-1. The percentage of students working between 10 and 20 hours is _____. a. 20% b. 25% c. 75% d. 80% ANSWER: a 32. Refer to Exhibit 2-1. The percentage of students working less than 20 hours is _____. a. 20% b. 25% c. 75% d. 80% ANSWER: b 33. Refer to Exhibit 2-1. The cumulative percent frequency for the class of 30 to 40 is _____. a. 100% Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays b. 75% c. 50% d. 25% ANSWER: a 34. Refer to Exhibit 2-1. The cumulative frequency for the class of 20 to 30 is _____. a. 200 b. 300 c. .75 d. .50 ANSWER: b 35. Refer to Exhibit 2-1. If a cumulative frequency distribution is developed for the above data, the last class will have a cumulative frequency of _____. a. 100 b. 1 c. 30−39 d. 400 ANSWER: d 36. Refer to Exhibit 2-1. The percentage of students who work at least 10 hours per week is _____. a. 50% b. 5% c. 95% d. 100% ANSWER: c Exhibit 2-2 Information on the type of industry is provided for a sample of 50 Fortune 500 companies. Industry Type Banking Consumer Products Electronics Retail

Frequency 7 15 10 18

37. Refer to Exhibit 2-2. The number of industries that are classified as retail is _____. a. 32 b. 18 c. 0.36 d. 36% ANSWER: b 38. Refer to Exhibit 2-2. The relative frequency of industries that are classified as banking is _____. a. 7 b. .07 Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays c. .70 d. .14 ANSWER: d 39. Refer to Exhibit 2-2. The percent frequency of industries that are classified as electronics is _____. a. 10 b. 20 c. .10 d. .20 ANSWER: b Exhibit 2-3 The number of sick days taken (per month) by 200 factory workers is summarized below. Number of Days 0−5 6−10 11−15 16−20

Frequency 120 65 14 1

40. Refer to Exhibit 2-3. The class width for this distribution _____. a. is 5 b. is 6 c. is 20, which is the largest value minus the smallest value or 20 − 0 = 20 d. varies between 5 and 6 ANSWER: d 41. Refer to Exhibit 2-3. The midpoint of the first class is _____. a. 10 b. 2 c. 2.5 d. 3 ANSWER: c 42. Refer to Exhibit 2-3. The number of workers who took less than 11 sick days per month is _____. a. 15 b. 200 c. 185 d. 65 ANSWER: c 43. Refer to Exhibit 2-3. The number of workers who took at most 10 sick days per month is _____. a. 15 b. 200 c. 185 Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays d. 65 ANSWER: c 44. Refer to Exhibit 2-3. The number of workers who took more than 10 sick days per month is _____. a. 15 b. 200 c. 185 d. 65 ANSWER: a 45. Refer to Exhibit 2-3. The number of workers who took at least 11 sick days per month is _____. a. 15 b. 200 c. 185 d. 65 ANSWER: a 46. Refer to Exhibit 2-3. The relative frequency of workers who took 10 or fewer sick days is _____. a. 185 b. .925 c. .075 d. 15 ANSWER: b 47. Refer to Exhibit 2-3. The cumulative relative frequency for the class of 11−15 is _____. a. 199 b. .07 c. 1 d. .995 ANSWER: d 48. Refer to Exhibit 2-3. The percentage of workers who took 0−5 sick days per month is _____. a. 20% b. 120% c. 75% d. 60% ANSWER: d 49. Refer to Exhibit 2-3. The cumulative percent frequency for the class of 16−20 is _____. a. 100% b. 65% c. 92.5% d. 0.5% ANSWER: a Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 50. Refer to Exhibit 2-3. The cumulative frequency for the class of 11−15 is _____. a. 200 b. 14 c. 199 d. 1 ANSWER: c Exhibit 2-4 A survey of 400 college seniors resulted in the following crosstabulation regarding their undergraduate major and whether or not they plan to go to graduate school. Undergraduate Major Graduate School Yes No Total

Business 35 91 126

Engineering 42 104 146

Other 63 65 128

Total 140 260 400

51. Refer to Exhibit 2-4. What percentage of the students does not plan to go to graduate school? a. 280% b. 520% c. 65% d. 32% ANSWER: c 52. Refer to Exhibit 2-4. What percentage of the students' undergraduate major is Engineering? a. 292% b. 520% c. 65% d. 36.5% ANSWER: d 53. Refer to Exhibit 2-4. Of those students who are majoring in Business, what percentage plans to go to graduate school? a. 27.78% b. 8.75% c. 70% d. 72.22% ANSWER: a 54. Refer to Exhibit 2-4. Among the students who plan to go to graduate school, what percentage indicated "Other" majors? Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays a. 15.75% b. 45% c. 54% d. 35% ANSWER: b 55. A graphical device for depicting categorical data that have been summarized in a frequency distribution, relative frequency distribution, or percent frequency distribution is a(n) _____. a. histogram b. stem-and-leaf display c. dot plot d. bar chart ANSWER: d 56. A graphical device for presenting categorical data summaries based on subdivision of a circle into sectors that correspond to the relative frequency for each class is a _____. a. histogram b. stem-and-leaf display c. pie chart d. bar chart ANSWER: c 57. Categorical data can be graphically represented by using a(n) _____. a. histogram b. stem-and-leaf display c. scatter diagram d. bar chart ANSWER: d 58. Fifteen percent of the students in a School of Business Administration are majoring in Economics, 20% in Finance, 35% in Management, and 30% in Accounting. The graphical device(s) that can be used to present these data is(are) _____. a. a line graph b. only a bar chart c. only a pie chart d. both a bar chart and a pie chart ANSWER: d 59. Frequency distributions can be made for _____. a. categorical data only b. quantitative data only c. neither categorical nor quantitative data d. both categorical and quantitative data ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 60. The total number of data items with a value less than or equal to the upper limit for the class is given by the _____. a. frequency distribution b. relative frequency distribution c. cumulative frequency distribution d. cumulative relative frequency distribution ANSWER: c 61. Excel's _____can be used to construct a frequency distribution for quantitative data. a. COUNTIF function b. SUM function c. PivotTable report d. AVERAGE function ANSWER: c 62. A graphical presentation of a frequency distribution, relative frequency distribution, or percent frequency distribution of quantitative data constructed by placing the class intervals on the horizontal axis and the frequencies on the vertical axis is a _____. a. histogram b. bar chart c. stem-and-leaf display d. pie chart ANSWER: a 63. A common graphical presentation of quantitative data is a _____. a. histogram b. bar chart c. relative frequency distribution d. pie chart ANSWER: a 64. When using Excel to create a _____, one must edit the chart to remove the gaps between rectangles. a. scatter diagram b. bar chart c. histogram d. pie chart ANSWER: c 65. A _____can be used to graphically present quantitative data. a. bar chart b. pie chart c. stem-and-leaf display d. stacked bar chart ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 66. A _____ shows the proportion of data items. a. histogram b. cumulative percent frequency distribution c. stem-and-leaf display d. cumulative relative frequency distribution ANSWER: d 67. Excel's Chart Tools can be used to construct a _____. a. dot plot b. pie chart and a dot plot c. histogram d. stem-and-leaf display ANSWER: c 68. To construct a bar chart using Excel's Chart Tools, choose _____ as the chart type. a. column b. pie c. scatter d. line ANSWER: a 69. To construct a pie chart using Excel's Chart Tools, choose _____ as the chart type. a. column b. pie c. scatter d. line ANSWER: b 70. To construct a histogram using Excel's Chart Tools, choose _____ as the chart type. a. column b. pie c. scatter d. line ANSWER: a 71. Excel's Chart Tools does NOT have a chart type for constructing a _____. a. bar chart b. pie chart c. histogram d. stem-and-leaf display ANSWER: d 72. A tabular method that can be used to summarize the data on two variables simultaneously is called _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays a. simultaneous equations b. a crosstabulation c. a histogram d. a dot plot ANSWER: b 73. Excel's _____ can be used to construct a crosstabulation. a. Chart Tools b. SUM function c. PivotTable report d. COUNTIF function ANSWER: c 74. In a crosstabulation, _____. a. both variables must be categorical b. both variables must be quantitative c. one variable must be categorical and the other must be quantitative d. either or both variables can be categorical or quantitative ANSWER: d 75. In a class with 30 students, we ask, “If you could have any super power, what would it be?” Each student could only choose one super power. The resulting pie chart is below. The least popular choice of super power was _____.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

a. ability to fly b. telepathy c. invisibility d. super strength ANSWER: b 76. In Excel, the line of best fit for the points in a scatter diagram is called a _____. a. trendline b. horizontal line c. vertical line d. fit line ANSWER: a 77. When the conclusions based upon the aggregated crosstabulation can be completely reversed if we look at the unaggregated data, the occurrence is known as _____. a. reverse correlation b. inferential statistics c. Simpson's paradox d. disaggregation ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 78. Before drawing any conclusions about the relationship between two variables shown in a crosstabulation, you should _____. a. investigate whether any hidden variables could affect the conclusions b. construct a scatter diagram and find the trendline c. develop a relative frequency distribution d. construct a pie chart for each of the variables ANSWER: a 79. A histogram is NOT appropriate for displaying which of the following types of information? a. frequency b. relative frequency c. cumulative frequency d. percent frequency ANSWER: c 80. For stem-and-leaf displays where the leaf unit is not stated, the leaf unit is assumed to equal _____. a. 0 b. 0.1 c. 1 d. 10 ANSWER: c 81. Which of the following graphical methods is not intended for quantitative data? a. stem-and-leaf display b. dot plot c. scatter diagram d. pie chart ANSWER: d 82. Which of the following is LEAST useful in studying the relationship between two variables? a. trendline b. stem-and-leaf display c. crosstabulation d. scatter diagram ANSWER: b 83. We ask 30 people the following question: “How many people do you live with?” Below are the results in a dot plot.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

What percentage of people surveyed live with 3 or less people? a. 30% b. 40% c. 50% d. 90% ANSWER: a 84. Do males prefer a particular type of smartphone more than females? A survey was conducted to help answer this question. The results are displayed below.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

What type of phone do males prefer? a. Android b. iPhone c. Males prefer Androids and iPhones equally. d. cannot be determined based upon the information given in the graph ANSWER: a Subjective Short Answer 85. Thirty students in the School of Business were asked what their majors were. The following represents their responses (M = Management; A = Accounting; E = Economics; O = Other). A E M a. b.

M E A

M M O

A A A

M O M

M E E

E M E

M A M

O M A

A A M

Construct a frequency distribution. Construct a relative frequency distribution.

ANSWER:

a. and b.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays Major M A E O Total

Frequency 12 9 6 3 30

Relative Frequency 0.4 0.3 0.2 0.1 1.0

86. Twenty employees of ABC Corporation were asked if they liked or disliked the new district manager. Below are their responses. Let L represent liked and D represent disliked. L D D D a. b.

L D L D

D L D D

L L D D

D D L L

Construct a frequency distribution. Construct a relative frequency distribution.

ANSWER:

a. and b. Preferences L D Total

Frequency 8 12 20

Relative Frequency 0.4 0.6 1.0

87. A student has completed 20 courses in the School of Arts and Sciences. Her grades in the 20 courses are shown below. A C B C

B C A B

A B B C

B B B B

C B B A

a. In what percent of her courses did she receive an A? b. In what percent of her courses did she receive a B or better? ANSWER: Grade Frequency Relative Frequency A 4 0.20 B 11 0.55 C 5 0.25 Total 20 1.00 a. 20% b. 75% 88. A sample of 50 TV viewers were asked, "Should TV sponsors pull their sponsorship from programs that draw numerous viewer complaints?" Below are the results of the survey. (Y = Yes; N = No; W = Without Opinion) N N Y W

W Y N W

N N Y N

N N W W

Y N N Y

N N Y W

N N W N

N Y W W

Y N N Y

N N Y W Page 19

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays N a. b.

What percentage of viewers feel that TV sponsors should pull their sponsorship from programs that draw numerous viewer complaints? What percentage of viewers are without opinion?

ANSWER:

Response No Yes Without Opinion Total

Frequency 24 15 11 50

Relative Frequency 0.48 0.30 0.22 1.00

a. 30% b. 22% 89. Forty shoppers were asked if they preferred the weight of a can of soup to be 6 ounces, 8 ounces, or 10 ounces. Below are their responses. 6 10 8 6 a. b.

6 10 8 8

6 8 8 8

10 8 10 8

8 6 8 10

8 6 6 8

10 8 10 10

6 6 8 8

6 6 6 6

Construct a frequency distribution and graphically represent the frequency distribution. Construct a relative frequency distribution and graphically represent the relative frequency distribution.

ANSWER:

a. and b. Preferences 6 ounces 8 ounces 10 ounces Total

Frequency 14 17 9 40

Relative Frequency 0.350 0.425 0.225 1.000

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90. There are 800 students in the School of Business Administration. There are four majors in the school: Accounting, Finance, Management, and Marketing. The following shows the number of students in each major. Major Number of Students Accounting 240 Finance 160 Management 320 Marketing 80 Develop a percent frequency distribution and construct a bar chart and a pie chart. ANSWER: Major Percent Frequency Accounting 30% Finance 20% Management 40% Marketing 10%

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

91. Below are the examination scores of 20 students. 52 63 92 90

99 72 58 75

92 76 65 74

86 95 79 56

84 88 80 99

Construct a frequency distribution for these data. Let the first class be 50–59 and draw a histogram. b. Construct a cumulative frequency distribution. c. Construct a relative frequency distribution. d. Construct a cumulative relative frequency distribution. ANSWER: a. b. c. d. Cumulative Relative Cumulative Score Frequency Frequency Frequency Relative Frequency 50–59 3 3 0.15 0.15 60−69 2 5 0.10 0.25 70−79 5 10 0.25 0.50 80−89 4 14 0.20 0.70 90−99 6 20 0.30 1.00 Total 20 1.00

92. Two hundred members of a fitness center were surveyed. One survey item stated, "The facilities are always clean." The members' responses to the item are summarized below. Fill in the missing value for the frequency distribution. Opinion Strongly Agree Agree Disagree Strongly Disagree No Opinion ANSWER: 16

Frequency 63 92 15 14

93. Fill in the missing value for the following relative frequency distribution. Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays Opinion Strongly Agree Agree Disagree Strongly Disagree No Opinion ANSWER: 0.080

Relative Frequency 0.315 0.460 0.075 0.070

94. Fill in the missing value for the following percent frequency distribution. Annual Salaries Under $30,000 $30,000−$49,999 $50,000−$69,999 $70,000 −$89,999 $90,000 and over

Percent Frequency 10 35 40 5

ANSWER: 10 95. The following is a summary of the number of hours spent per day watching television for a sample of 100 people. What is wrong with the frequency distribution? Hours/Day 0−1 1−3 3−5 5−7 7−9

Frequency 10 45 20 20 5

ANSWER: The classes overlap. 96. A summary of the results of a job satisfaction survey follows. What is wrong with the relative frequency distribution? Rating Poor Fair Good Excellent

Relative Frequency 0.15 0.45 0.25 0.30

ANSWER: The relative frequencies do not sum to 1. 97. The frequency distribution below was constructed from data collected from a group of 25 students. Height (inches) 58−63 64−69 70−75 76−81 82−87 88−93 94−99

Frequency 3 5 2 6 4 3 2

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays a. b. c.

Construct a relative frequency distribution. Construct a cumulative frequency distribution. Construct a cumulative relative frequency distribution.

ANSWER: Height (inches) 58−63 64−69 70−75 76−81 82−87 88−93 94−99

a. Relative Frequency 0.12 0.20 0.08 0.24 0.16 0.12 0.08 1.00

Frequency 3 5 2 6 4 3 2

b. Cumulative Frequency 3 8 10 16 20 23 25

c. Cumulative Relative Frequency 0.12 0.32 0.40 0.64 0.80 0.92 1.00

98. The frequency distribution below was constructed from data collected on the quarts of soft drink consumed per week by 20 students. Quarts of Soft Drink 0−3 4−7 8−11 12−15 16−19 a. b. c.

Frequency 4 5 6 3 2

Construct a relative frequency distribution. Construct a cumulative frequency distribution. Construct a cumulative relative frequency distribution.

ANSWER:

a. Relative Frequency 0.20 0.25 0.30 0.15 0.10 1.00

Quarts of Soft Drink 0−3 4−7 8−11 12−15 16−19 Total

b. Cumulative Frequency 4 9 15 18 20

c. Cumulative Relative Frequency 0.20 0.45 0.75 0.90 1.00

99. The grades of 10 students on their first management test are shown below. 94 68 a. b. c.

61 75

96 85

66 84

92 78

Construct a frequency distribution. Let the first class be 60−69. Construct a cumulative frequency distribution. Construct a relative frequency distribution.

ANSWER:

c. Page 24

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays Class 60−69 70−79 80−89 90−99 Total

Cumulative Frequency 3 5 7 10

Frequency 3 2 2 3 10

Relative Frequency 0.3 0.2 0.2 0.3 1.0

100. You are given the following data on the ages of employees at a company. Construct a stem-and-leaf display. Specify the leaf unit for the display. 26 52 41 42 ANSWER:

32 44 53 44

28 36 55 40

45 42 48 36

Leaf Unit = 1 2|6 7 8 3|2 2 6 4|0 1 2 5|2 3 5

58 27 32 37

6 4 8

7 4

101. Construct a stem-and-leaf display for the following data. Specify the leaf unit for the display. 12 49 ANSWER:

52 43

51 45

37 19

Leaf Unit = 1 1|2 8 9 2|2 6 3|1 2 6 4|0 3 4 5|1 2 7

47 36

7 5

8 7

40 32

38 44

26 48

57 22

31 18

102. You are given the following data on the earnings per share for 10 companies. Construct a stem-and-leaf display. Specify the leaf unit for the display. 2.6 1.1 ANSWER:

1.4 1.1

1.3 0.7

Leaf Unit = 0.1 0|5 7 9 1|1 1 3 2|0 2 6

0.5 0.9

2.2 2.0

103. You are given the following data on the annual salaries for 8 employees. Construct a stem-and-leaf display. Specify the leaf unit for the display. $26,500 $26,890 ANSWER:

$27,850 $25,400

$25,000 $26,150

$27,460 $30,000

Leaf Unit = 100

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 25 | 0 26 | 1 27 | 4 28 | 29 | 30 | 0

4 5 8

104. You are given the following data on the price/earnings (P/E) ratios for 12 companies. Construct a stem-and-leaf display. Specify the leaf unit for the display. 23 8 ANSWER:

25 36

39 48

47 28

Leaf Unit = 1 0|8 1| 2|2 3 5 3|6 7 7 4|7 8

6 9

22 37

37 26

105. You are given the following data on times (in minutes) to complete a race. Construct a stem-and-leaf display. Specify the leaf unit for the display. 15.2 14.7 ANSWER:

15.8 14.8

12.4 11.8

11.9 12.0

15.2 12.1

Leaf Unit = 0.1 11 | 8 9 12 | 0 1 4 13 | 14 | 7 8 15 | 2 2 8

106. The SAT math scores of a sample of business school students and their genders are shown below. SAT Math Scores Gender Female Male Total a. b. c. d. e.

Less than 400 24 40 64

400 up to 600 168 96 264

600 and more 48 24 72

Total 240 160 400

How many students scored less than 400? How many students were female? Of the male students, how many scored 600 or more? Compute row percentages and comment on any relationship that may exist between SAT math scores and gender of the individuals. Compute column percentages.

ANSWER:

a. b.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays c. d.

Gender Female Male

SAT Math Scores Less than 400 10% 25%

400 up to 600 600 and more Total 70% 20% 100% 60% 15% 100% From the above percentages, it can be noted that the largest percentages of both genders' SAT scores are in the 400 to 600 range. However, 70% of females and only 60% of males have SAT scores in this range. Also it can be noted that 10% of females' SAT scores are under 400, whereas 25% of males' SAT scores fall in this category.

e. Gender Female Male Total

SAT Math Scores Less than 400 37.5% 62.5% 100%

400 up to 600 600 and more 63.6% 66.7% 36.4% 33.3% 100% 100%

107. A market research firm has conducted a study to determine consumer preference for a new package design for a particular product. The consumer’s age was also noted. Age Under 25 25−40 Total a. b. c. d. e.

A 18 18 36

Package Design B C Total 18 29 65 12 5 35 30 34 100

Which package design was most preferred overall? What percent of those participating in the study preferred Design A? What percent of those under 25 years of age preferred Design A? What percent of those aged 25 − 40 preferred Design A? Is the preference for Design A the same for both age groups?

ANSWER: a. b. c. d. e.

Design A 36% 27.7% 51.4% No, although both groups have 18 people who prefer Design A, the percentage of those in the "Under 25" age group who prefer Design A is smaller than that of the "25−40" age group (27.7% vs. 51.4%).

108. Partial results of a study follow in a crosstabulation of column percentages. Method of Payment Gender Female

Cash 18%

Credit Card Check 50% 90%

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays Male Total a. b.

82% 100%

50% 100%

10% 100%

Interpret the 18% found in the first row and first column of the crosstabulation. If 50 of those in the study paid by check, how many of the males paid by check?

ANSWER: a. b.

Of those who pay with cash, 18% are female. 5

109. For the following observations, plot a scatter diagram and indicate what kind of relationship (if any) exists between x and y. x y 2 7 6 19 3 9 5 17 4 11 ANSWER: A positive relationship between x and y appears to exist.

110. For the following observations, indicate what kind of relationship (if any) exists between women's height (inches) and annual starting salary ($1000s). Height 64 63 68 65 67 66 65 64

Salary 45 40 39 38 42 45 43 35

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 66 33 ANSWER: No relationship between women's heights and salaries appears to exist. 111. For the following observations, indicate what kind of relationship (if any) exists between the amount of sugar in one serving of cereal (grams) and the amount of fiber in one serving of cereal (grams). Sugar Fiber 1.2 3.2 1.3 3.1 1.5 2.8 1.8 2.4 2.2 1.1 2.8 1.3 3.0 1.0 ANSWER: A negative relationship between amount of sugar and amount of fiber appears to exist. 112. What type of graph is depicted below?

ANSWER: A scatter diagram 113. What type of relationship is depicted in the following scatter diagram?

ANSWER: A positive relationship 114. What type of relationship is depicted in the following scatter diagram?

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

ANSWER: A negative relationship 115. What type of relationship is depicted in the following scatter diagram?

ANSWER: No apparent relationship 116. It is time for Roger Hall, manager of new car sales at the Maxwell Ford dealership, to submit his order for new Mustang coupes. These cars will be parked in the lot, available for immediate sale to buyers who are not special-ordering a car. Roger must decide how many Mustangs of each color he should order. The new color options are very similar to the past year’s options. Roger believes the colors chosen by customers who special-order their cars best reflect most customers’ true color preferences. He has taken a random sample of 40 special orders for Mustang coupes placed in the past year. The color preferences found in the sample are listed below. Blue Black Red Green Blue

Black Red White Black Red

Green White Blue Red Black

White Blue White Black White

Black Blue Red Blue Black

Red Green Red Black Red

Red Red Black White Black

White Black Black Green Blue

a. Prepare a frequency distribution, relative frequency distribution, and percent frequency distribution for the data set. b. Construct a bar chart showing the frequency distribution of the car colors. c. Construct a pie chart showing the percent frequency distribution of the car colors. ANSWER: a. Color Relative Percent of Car Frequency Frequency Frequency Black 12 0.300 30.0 Blue 7 0.175 17.5 Green 4 0.100 10.0 Red 10 0.250 25.0 Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays White Total b.

7 40

0.175 1.000

17.5 100.0

117. Missy Walters owns a mail-order business specializing in clothing, linens, and furniture for children. She is considering offering her customers a discount on shipping charges for furniture based on the dollar amount of the furniture order. Before Missy decides the discount policy, she needs a better understanding of the dollar amount distribution of the furniture orders she receives. Missy had an assistant randomly select 50 recent orders that included furniture. The assistant recorded the value, to the nearest dollar, of the furniture portion of each order. The data collected are listed below. 136 211 194 277 231

281 162 242 348 154

226 212 368 173 166

123 241 258 409 214

178 182 323 264 311

445 290 196 237 141

231 434 183 490 159

389 167 209 222 362

196 246 198 472 189

175 338 212 248 260

a. Prepare a frequency distribution, relative frequency distribution, and percent frequency distribution for the data set using a class width of $50. b. Construct a histogram showing the percent frequency distribution of the furniture-order values in the sample. Copyright Cengage Learning. Powered by Cognero.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays c. Develop a cumulative frequency distribution and a cumulative percent frequency distribution for these data. ANSWER: a. Furniture Relative Percent Order Frequency Frequency Frequency 100−149 3 0.06 6 150−199 15 0.30 30 200−249 14 0.28 28 250−299 6 0.12 12 300−349 4 0.08 8 350−399 3 0.06 6 400−449 3 0.06 6 450−499 2 0.04 4 b.

c. Furniture Order 100−149 150−199 200−249 250−299 300−349 350−399 400−449 450−499

Frequency 3 15 14 6 4 3 3 2

Cumulative Frequency 3 18 32 38 42 45 48 50

Cumulative Percent Frequency 6 36 64 76 84 90 96 100

118. Develop a stretched stem-and-leaf display for the data set below, using a leaf unit of 10. 136 281 211 162 194 242 277 348 231 154 ANSWER:

226 212 368 173 166

123 241 258 409 214

178 182 323 264 311

445 290 196 237 141

231 434 183 490 159

389 196 175 167 246 338 209 198 212 222 472 248 362 189 260 Leaf Unit = 10 Page 32

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays 1 1 2 2 3 3 4 4

2 5 0 5 1 6 0 7

3 5 1 6 2 6 3 9

4 6 1 6 3 8 4

6 1 7 4

6 1 8

7 2 9

7 2

7 3

8 3

8 4

9 4

119. Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The ratings provided by a sample of 20 quests are shown below. Below Average Above Average Below Average Above Average Excellent

Average Above Average Average Average Above Average

Above Average Above Average Poor Above Average Average

Above Average Below Average Poor Average Above Average

a. Provide a frequency distribution showing the number of occurrences of each rating level in the sample. b. Construct relative frequency and percent frequency distributions for the data. c. Display the frequencies graphically with a bar graph. d. Display the percent frequencies graphically with a pie chart. ANSWER: a. Quality Rating Poor Below Average Average Above Average Excellent Total

Frequency 2 3 5 9 1 20

b. Relative Frequency

Quality Rating Poor Below Average Average Above Average Excellent Total

0.10 0.15 0.25 0.45 0.05 1.00

Percent Frequency 10 15 25 45 5 100

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays

120. Ithaca Log Homes manufactures four styles of log houses that are sold in kits. The price ($1000s) and style of homes the company has sold in the past year are shown below. Price ≤99 ≤99 ≥100 ≥100 ≤99 ≤99 ≤99

Style Colonial Ranch Split-Level Split-Level Colonial A-Frame Split-Level

Price ≥100 ≥100 ≤99 ≥100 ≥100 ≤99 ≤99

Style A-Frame Split-Level Colonial Ranch Colonial A-Frame Split-Level

Price ≥100 ≤99 ≤99 ≥100 ≥100 ≤99 ≥100

Style Colonial Colonial A-Frame Split-Level Ranch Split-Level Split-Level Page 34

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays ≤99 ≥100 ≥100 ≤99 ≤99 ≥100 ≤99

≤99 ≤99 ≤99 ≥100 ≥100 ≤99

A-Frame Ranch Split-Level A-Frame Colonial Ranch Colonial

Split-Level Colonial Ranch Split-Level Colonial Split-Level

≥100 ≥100 ≥100 ≤99 ≥100 ≤99

Colonial Ranch Split-Level Colonial Colonial Split-Level

A-Frame 5 1 6

Grand Total 21 19 40

Prepare a crosstabulation for the variables price and style. ANSWER: Count of Home Price ($1000s) ≤99 ≥100 Grand Total

Style Colonial 8 5 13

Ranch 2 5 7

Split-Level 6 8 14

121. Tony Zamora, a real estate investor, has just moved to Clarksville and wants to learn about the local real estate market. He wants to understand, for example, the relationship between geographical segment of the city and selling price of a house, the relationship between selling price and number of bedrooms, and so on. Tony has randomly selected 25 house-for-sale listings from the Sunday newspaper and collected the data listed below.

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Chapter 02: Descriptive Statistics: Tabular and Graphical Displays a. Construct a crosstabulation for the variables segment of city and number of bedrooms. b. Compute the row percentages for your crosstabulation in part (a). c. Comment on any apparent relationship between the variables. ANSWER: a. CROSSTABULATION Count of Home Segment of City Northeast Northwest South West Grand Total

Number of Bedrooms 2 3 0 1 0 0 2 2 0 1 2 4

4 4 4 2 3 13

5 0 3 0 3 6

4 80.0 57.1 33.3 42.9

5 0.0 42.9 0.0 42.9

Grand Total 5 7 6 7 25

b. ROW PERCENTAGES Percent of Home Segment of City Northeast Northwest South West

Number of Bedrooms 2 3 0.0 20.0 0.0 0.0 33.3 33.3 0.0 14.3

Grand Total 100.0 100.0 100.0 100.1

c. We see that fewest bedrooms are associated with the South, and the most bedrooms are associated with the West and particularly the Northwest.

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Chapter 03: Descriptive Statistics: Numerical Measures Multiple Choice 1. The interquartile range is the difference between the _____. a. first and second quartiles b. first and third quartiles c. second and third quartiles d. second and fourth quartiles ANSWER: b 2. The variance is equal to the _____. a. absolute value of the standard deviation b. squared value of the standard deviation c. square root of the standard deviation d. inverse value of the standard deviation ANSWER: b 3. Generally, which of the following is the least appropriate measure of central tendency for a data set that contains outliers? a. mean b. median c. mode d. midrange ANSWER: a 4. An important measure of location for categorical data is the _____. a. mean b. median c. mode d. range ANSWER: c 5. The measure of variability easiest to compute, but seldom used as the only measure, is the _____. a. range b. interquartile range c. standard deviation d. variance ANSWER: a 6. When dividing a data set into four parts, the division points are referred to as the _____. a. class lower limits b. quartiles c. midpoints d. percentiles ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 7. Which of the following is NOT a measure of variability of a single variable? a. range b. covariance c. standard deviation d. coefficient of variation ANSWER: b 8. The empirical rule states that, for data having a bell-shaped distribution, the portion of data values being within one standard deviation of the mean is approximately _____. a. 33% b. 50% c. 68% d. 95% ANSWER: c 9. A boxplot is a graphical representation of data that is based on _____. a. the empirical rule b. z-scores c. a histogram d. a five-number summary ANSWER: d 10. The coefficient of variation indicates how large the standard deviation is relative to the _____. a. mean b. median c. range d. variance ANSWER: a 11. Which of the following descriptive statistics is NOT measured in the same units as the data? a. 35th percentile b. standard deviation c. variance d. interquartile range ANSWER: c 12. A numerical measure computed from a sample, such as sample mean, is known as a _____. a. population parameter b. sample parameter c. sample statistic d. population statistic ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 13. A numerical measure, such as a mean, computed from a population is known as a _____. a. population parameter b. sample parameter c. sample statistic d. population statistic ANSWER: a 14. Which of the following depicts a moderately left-skewed distribution? a.

ANSWER: a 15. A graph with skewness –1.8 would be which of the following? a. moderately skewed left b. highly skewed left c. moderately skewed right d. highly skewed right ANSWER: b 16. The mean of a sample is _____. a. always equal to the mean of the population b. always smaller than the mean of the population c. computed by summing the data values and dividing the sum by (n − 1) Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures d. computed by summing all the data values and dividing the sum by the number of items ANSWER: d 17. The mean of the sample _____. a. is always larger than the mean of the population from which the sample was taken b. is always smaller than the mean of the population from which the sample was taken c. can never be zero d. is affected by outliers ANSWER: d 18. In statistics, what does IQR mean? a. individual quantity range b. inter-quantity relativity c. individual quartile relativity d. interquartile range ANSWER: d 19. After the data have been arranged from smallest value to largest value, the value in the middle is called the _____. a. range b. median c. mean d. interquartile range ANSWER: b 20. If a data set has an even number of observations, the median _____. a. cannot be determined b. is the observation recorded most often c. must be equal to the mean d. is the average value of the two middle items when all items are arranged in ascending order ANSWER: d 21. The median of the data set is _____. a. always very close in value to the mean b. larger than the mean c. smaller than the mean d. preferred over the mean when the data set contains outliers ANSWER: d 22. The most frequently occurring value of a data set is called the _____. a. range b. mode c. mean d. median ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 23. Since the mode is the most frequently occurring data value, _____. a. it can never be larger than the mean b. it is always larger than the median c. it is always larger than the mean d. more than one mode can exist ANSWER: d 24. Excel's _____ function can be used to compute the mean. a. MAX b. AVERAGE c. MEDIAN d. MODE ANSWER: b 25. Excel's _____ function can be used to compute the middle value of an ordered data set. a. MAX b. AVERAGE c. MEDIAN d. MODE ANSWER: c 26. Excel's _____ function can be used to compute the data value occurring most frequently. a. MAX b. AVERAGE c. MEDIAN d. MODE.SNGL ANSWER: d 27. The five-number summary consists of what five statistical measures? a. min, Q1, median, Q3, max b. Q1, Q2, median, Q3, Q4 c. min, Q1, mean, Q3, max d. Q1, Q2, mean, Q3, Q4 ANSWER: a 28. In computing the pth percentile, if the index i is an integer the pth percentile is the _____. a. data value in position i b. data value in position i + 1 c. average of data values in positions i and i + 1 d. average of data values in positions i and i – 1 ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 29. The 50th percentile is the _____. a. mode b. median c. mean d. third quartile ANSWER: b 30. The 75th percentile is also the _____. a. first quartile b. second quartile c. third quartile d. fourth quartile ANSWER: c 31. The first quartile _____. a. contains at least one third of the data elements b. is the same as the 25th percentile c. is the same as the 50th percentile d. is the same as the 75th percentile ANSWER: b 32. Which of the following is NOT a measure of location? a. mean b. median c. variance d. mode ANSWER: c 33. The median of a sample will always equal the _____. a. mode b. mean c. 50th percentile d. 75th percentile ANSWER: c 34. The measure of location most likely to be influenced by extreme values in the data set is the _____. a. range b. median c. mode d. mean ANSWER: d Exhibit 3-1 A researcher has collected the following sample data. Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 5 6

12 7

6 5

8 12

5 4

35. Refer to Exhibit 3-1. The median is _____. a. 5 b. 6 c. 7 d. 8 ANSWER: b 36. Refer to Exhibit 3-1. The mode is _____. a. 5 b. 6 c. 7 d. 8 ANSWER: a 37. Refer to Exhibit 3-1. The mean is _____. a. 5 b. 6 c. 7 d. 8 ANSWER: c 38. Refer to Exhibit 3-1. The 75th percentile is _____. a. 5 b. 6 c. 7 d. 8 ANSWER: d Exhibit 3-2 A researcher has collected the following sample data. The mean of the sample is 5. 3

39. Refer to Exhibit 3-2. The variance is _____. a. 80 b. 4.062 c. 13.2 d. 16.5 ANSWER: d 40. Refer to Exhibit 3-2. The standard deviation is _____. a. 8.944 Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures b. 4.062 c. 13.2 d. 16.5 ANSWER: b 41. Refer to Exhibit 3-2. The coefficient of variation is _____. a. 72.66% b. 81.24% c. 264% d. 330% ANSWER: b 42. Refer to Exhibit 3-2. The range is _____. a. 1 b. 2 c. 10 d. 12 ANSWER: c 43. Refer to Exhibit 3-2. The interquartile range is _____. a. 1 b. 2 c. 10 d. 12 ANSWER: b Exhibit 3-3 Suppose annual salaries for sales associates from Hayley's Heirlooms have a bell-shaped distribution with a mean of $32,500 and a standard deviation of $2,500. 44. Refer to Exhibit 3-3. The z-score for a sales associate from this store who earns $37,500 is _____. a. 37.5 b. 2 c. –2 d. .92 ANSWER: b 45. Refer to Exhibit 3-3. The z-score for a sales associate from this store who earns $28,000 is _____. a. 28 b. 1.8 c. –1.8 d. .78 ANSWER: c Exhibit 3-4 Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures The following is the frequency distribution for the speeds of a sample of automobiles traveling on an interstate highway. Speed (mph) 50−54 55−59 60−64 65−69 70−74 75−79

Frequency 2 4 5 10 9 5 35

46. Refer to Exhibit 3-4. The mean is _____. a. 35 b. 670 c. 10 d. 67 ANSWER: d 47. Refer to Exhibit 3-4. The variance is _____. a. 6.969 b. 7.071 c. 48.570 d. 50.000 ANSWER: d 48. Refer to Exhibit 3-4. The standard deviation is _____. a. 6.969 b. 7.071 c. 48.570 d. 50.000 ANSWER: b 49. The difference between the largest and smallest data values is the _____. a. variance b. interquartile range c. range d. coefficient of variation ANSWER: c 50. The interquartile range is _____. a. the 50th percentile b. another name for the variance c. the difference between the largest and smallest values d. the difference between the third and first quartiles Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures ANSWER: d 51. The interquartile range is used as a measure of variability to overcome what difficulty of the range? a. The sum of the range variances is zero. b. The range is difficult to compute. c. The range is influenced too much by extreme values. d. The range is negative. ANSWER: c 52. The sample variance _____. a. is always smaller than the true value of the population variance b. is always larger than the true value of the population variance c. could be smaller, equal to, or larger than the true value of the population variance d. can never be zero ANSWER: c 53. The variance of a sample or a population cannot be _____. a. negative b. calculated c. zero d. less than1 ANSWER: a 54. The symbol for _____ is rxy. a. sample covariance b. population covariance c. sample correlation d. population correlation ANSWER: c 55. The sum of deviations of the individual data elements from their mean is _____. a. always greater than zero b. always less than zero c. sometimes greater than and sometimes less than zero, depending on the data elements d. always equal to zero ANSWER: d 56. The value of the sum of the squared deviations from the mean, i.e., must always be _____. a. less than the mean b. negative c. either positive or negative, depending on whether the mean is negative or positive d. at least zero ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 57. If the variance of a data set is correctly computed with the formula using n − 1 in the denominator, which of the following is true? a. The data set is a sample. b. The data set is a population. c. The data set could be either a sample or a population. d. The data set is from a census. ANSWER: a 58. During a cold winter, the temperature stayed below zero for 10 days (ranging from –20 to –5). The variance of the temperatures of the 10 day period _____. a. is negative since all the numbers are negative b. must be at least zero c. cannot be computed since all the numbers are negative d. can be either negative or positive ANSWER: b 59. The variance of a sample of 81 observations is 64. The standard deviation of the sample is which of the following? a. 9 b. 4,096 c. 8 d. 6,561 ANSWER: c 60. The standard deviation of a sample of 100 observations is 64. The variance of the sample is which of the following? a. 8 b. 10 c. 6,400 d. 4,096 ANSWER: d 61. The numerical value of the standard deviation can never be _____. a. larger than the variance b. zero c. negative d. equal to the mean ANSWER: c 62. Excel's _____ function can be used to compute the sample variance. a. VAR b. VAR.C c. VAR.S d. STDEV.SQ ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 63. Which of Excel's functions can be used to compute the population variance? a. VAR.S b. VAR.C c. VAR d. Population variance cannot be computed using an Excel function. ANSWER: d 64. Excel's _____ function can be used to compute the sample standard deviation. a. STDEV b. VAR.SQRT c. STDEV d. STDEV.S ANSWER: d 65. Which of Excel's functions can be used to compute the population standard deviation? a. STDEV.S b. VAR.SQRT c. STDEV d. Population standard deviation cannot be computed using an Excel function. ANSWER: d 66. The coefficient of variation is _____. a. the same as the variance b. the square root of the variance c. the square of the standard deviation d. usually expressed as a percentage ANSWER: d 67. The weight (in pounds) of a sample of 36 individuals was recorded, and the following statistics were calculated. mean = 160 range = 60 mode = 165 variance = 324 median = 170 The coefficient of variation is _____. a. 0.1125% b. 11.25% c. 203.12% d. 0.20312% ANSWER: b 68. Which of the following is a measure of dispersion? a. percentiles b. quartiles c. interquartile range d. geometric mean Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures ANSWER: c 69. Which of the following is not a measure of dispersion? a. range b. 50th percentile c. standard deviation d. interquartile range ANSWER: b 70. The measure of dispersion that is influenced most by extreme values is the _____. a. variance b. standard deviation c. range d. interquartile range ANSWER: c 71. The descriptive measure of dispersion that is based on the concept of a deviation about the mean is the _____. a. range b. interquartile range c. weighted mean d. standard deviation ANSWER: d 72. The symbol for _____ is xy a. sample covariance b. population covariance c. sample correlation d. population correlation ANSWER: d 73. The descriptive measure NOT measured in the same units as the original data is the _____. a. median b. standard deviation c. mode d. variance ANSWER: d 74. Which of the following symbols represents the size of a population? a. σ 2 b. σ c. μ d. N ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 75. Which of the following symbols represents the size of a sample? a. σ 2 b. σ c. N d. n ANSWER: d 76. Which of the following symbols represents the mean of a population? a. σ 2 b. σ c. μ d. ANSWER: c 77. Which of the following symbols represents the mean of a sample? a. σ 2 b. σ c. μ d. ANSWER: d 78. Which of the following symbols represents the variance of a population? a. σ 2 b. σ c. μ d. ANSWER: a 79. The symbol σ 2 is used to represent the _____. a. variance of a population b. standard deviation of a sample c. standard deviation of a population d. variance of a sample ANSWER: a 80. Which of the following symbols represents the standard deviation of a population? a. σ 2 b. σ c. μ d. ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 81. The symbol σ is used to represent the _____. a. variance of a population b. standard deviation of a sample c. standard deviation of a population d. variance of a sample ANSWER: c 82. The _____ denotes the number of standard deviations xi is from the mean . a. variance b. median c. z-score d. normal deviation ANSWER: c 83. A(n) _____ is an unusually small or unusually large data value. a. sample statistic b. median c. z-score d. outlier ANSWER: d 84. _____ can be used to make statements about the proportion of data values that must be within a specified number of standard deviations of the mean, regardless of the shape of the distribution. a. Chebyshev's theorem b. The empirical rule c. A five-number summary d. A boxplot ANSWER: a 85. _____ can be used to determine the percentage of data values that must be within one, two, and three standard deviations of the mean for data having a bell-shaped distribution. a. Chebyshev's theorem b. The empirical rule c. A five-number summary d. A boxplot ANSWER: b 86. In a five-number summary, which of the following is NOT used for data summarization? a. smallest value b. largest value c. median d. mean ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 87. A graphical summary of data that is based on a five-number summary is a _____. a. histogram b. stem-and-leaf display c. scatter diagram d. boxplot ANSWER: d 88. A numerical measure of linear association between two variables is the _____. a. variance b. covariance c. standard deviation d. coefficient of variation ANSWER: b 89. Positive values of covariance indicate _____. a. a positive variance of the x values b. a positive variance of the y values c. the standard deviation is positive d. a positive relation between the x and y variables ANSWER: d 90. Excel's _____ function can be used to compute the sample covariance. a. COVARIANCE b. COVARIANCE.S c. COVAR d. COVAR.S ANSWER: b 91. A numerical measure of linear association between two variables is the _____. a. variance b. z-score c. correlation coefficient d. standard deviation ANSWER: c 92. The correlation coefficient ranges from which two values? a. 0 and 1 b. −1 and +1 c. minus infinity and plus infinity d. 1 and 100 ANSWER: b 93. The correlation coefficient _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures a. is the same as the covariance b. can be larger than 1 c. cannot be larger than 1 d. cannot be negative ANSWER: c 94. Excel's _____ function can be used to compute the sample correlation coefficient. a. CORR.S b. CORR c. CORREL.S d. CORREL ANSWER: d 95. A mean computed in such a way that each data value is given a weight reflecting its importance is referred to as a(n) _____. a. important mean b. trimmed mean c. weighted mean d. heavy mean ANSWER: c 96. If r = 0.48 for data set A and r = –0.48 for data set B, which of the following is true? a. The variables in A are more strongly correlated than the variables in B. b. The data values in A are all positive, while the data values in B are all negative. c. The two data sets have the same level of correlation. d. The two data sets have an almost perfect linear relationship between their respective variables. ANSWER: c 97. Which of the following values of r indicates the strongest correlation? a. .82 b. .361 c. 0 d. –.9 ANSWER: d 98. An important numerical measure related to the shape of a distribution is the _____. a. correlation coefficient b. variance c. skewness d. relative location ANSWER: c 99. If the data distribution is symmetric, the skewness is _____. a. 0 Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures b. .5 c. 1 d. –.5 ANSWER: a 100. For data skewed to the left, the skewness is _____. a. between 0 and .5 b. less than 1 c. positive d. negative ANSWER: d 101. When the data are positively skewed, the mean will usually be _____. a. less than the median b. greater than the median c. less than the mode d. greater than the mode ANSWER: b 102. The measure of location often used in analyzing growth rates in financial data is the _____. a. arithmetic mean b. weighted mean c. geometric mean d. hyperbolic mean ANSWER: c 103. The measure of central location most often reported for annual income and property value data is the _____. a. median b. mode c. weighted mean d. aggregate mean ANSWER: a 104. Chebyshev’s theorem requires that z be _____. a. an integer b. greater than 1 c. less than or equal to 3 d. between 0 and 4 ANSWER: b 105. Chebyshev’s theorem is applicable _____. a. only to large (n > 30) data sets b. only to data sets with no outliers c. only to bell-shaped data sets Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures d. to any data set ANSWER: d 106. A set of visual displays organizing and presenting information used to monitor the performance of a company or organization in a manner that is easy to read, understand, and interpret is called a _____. a. stem-and-leaf display b. stacked bar chart c. data dashboard d. crosstabulation ANSWER: c Subjective Short Answer 107. The hourly wages of a sample of eight individuals is given below. Individual Hourly Wage ($) A 27 B 25 C 20 D 10 E 12 F 14 G 17 H 19 For the above sample, determine the following measures: a. Mean b. Standard deviation c. 25th percentile ANSWER: a. b. c.

18 6 12.5

108. In 1998, the average age of students at UTC was 22 with a standard deviation of 3.96. In 1999, the average age was 24 with a standard deviation of 4.08. In which year do the ages show a more dispersed distribution? Show your complete work and support your answer. ANSWER: Coefficient of Variation for 1998 = 18% Coefficient of Variation for 1999 = 17% Since 18% > 17%, 1998 shows a more dispersed distribution. 109. Compute the measures below for the following data: 5 7 9 11 Compute the following measures: a. Mean b. Variance c. Standard deviation d. Coefficient of variation

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Chapter 03: Descriptive Statistics: Numerical Measures e. f. g.

25th percentile Median 75th percentile

ANSWER: a. b. c. d. e. f. g.

11 27.2 5.22 47.41 6.5 10 16

110. Compute the measures below for the following data: 20 a. b. c. d. e. f. g.

Mean Variance Standard deviation Coefficient of variation 25th percentile Median 75th percentile

ANSWER: a. b. c. d. e. f. g.

20 5.75 2.4 11.99 17.5 20 22.5

111. A private research organization studying families in various countries reported the following data for the time that 4year-old children spent alone with their fathers each day. Country Belgium Canada China Finland Germany Nigeria Sweden United States

Time with Dad (minutes) 30 44 54 50 36 42 46 42

For the above sample, determine the following measures: a. Mean b. Standard deviation c. Mode d. 75th percentile ANSWER: a.

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Chapter 03: Descriptive Statistics: Numerical Measures b. c. d.

7.56 42 49

112. The following data show the yearly salaries of football coaches at some state-supported universities. University A B C D E F G H

Salary ($1,000s) 53 44 68 47 62 59 53 94

For the above sample, determine the following measures. a. Mean yearly salary b. Standard deviation c. Mode d. Median e. 70th percentile ANSWER: a. b. c. d. e.

60 15.8 53 56 63.8

113. The amount of time that a sample of students spends watching television per day is given below. Student 1 2 3 4 5 6 7 8 a. b. c. d.

Time (minutes) 40 28 71 48 49 35 40 57

Compute the mean. Compute the median. Compute the standard deviation. Compute the 75th percentile.

ANSWER: a. b. c. d.

46 44 13.5 55

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Chapter 03: Descriptive Statistics: Numerical Measures 114. The number of hours worked per week for a sample of 10 students is shown below. Student 1 2 3 4 5 6 7 8 9 10 a. b. c.

Hours 20 0 18 16 22 40 8 6 30 40

Determine the median and explain its meaning. Compute the 70th percentile and explain its meaning. What is the mode of the above data? What does it signify?

ANSWER: a. b. c.

19; approximately 50% of the students work at least 19 hours 27.6; at least 70% of the students work less than or equal to 26 hours per week 40; the most frequent data element

115. A researcher has obtained the number of hours worked per week during the summer for a sample of 15 students. 40 25 35 30 20 40 30 Using this data set, compute the following: a. Median b. Mean c. Mode d. 40th percentile e. Range f. Sample variance g. Standard deviation ANSWER: a. b. c. d. e. f. g.

25 25 20 20 35 128.6 11.3

116. A sample of 12 families was asked how many times per week they dine in restaurants. Their responses are given below. 2 1 0 2 0 2 Using this data set, compute the following: a. Mode b. Median c. Mean d. Range Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures e. f. g. h.

Interquartile range Variance Standard deviation Coefficient of variation

ANSWER: a. b. c. d. e. f. g. h.

2 1.5 1.25 2 1.5 0.75 0.866 69.28%

117. A sample of 9 mothers was asked the age of their oldest child. Their responses are given below. 3 a. b. c. d. e. f. g. h.

Compute the mean. Compute the variance. Compute the standard deviation. Compute the coefficient of variation. Determine the 25th percentile. Determine the median. Determine the 75th percentile. Determine the range.

ANSWER: a. b. c. d. e. f. g. h.

7.56 17.78 4.22 55.8 3.5 7.0 11.5 12

118. A sample of 11 individuals shows the following monthly incomes. Individual 1 2 3 4 5 6 7 8 9 10 11 a.

Income ($) 1,500 2,000 2,500 4,000 4,000 2,500 2,000 4,000 3,500 3,000 43,000

What would be a representative measure of central location for the above data? Explain.

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Chapter 03: Descriptive Statistics: Numerical Measures b. c. d. e.

Determine the mode. Determine the median. Determine the 60th percentile. Remove the income of individual number 11 and compute the standard deviation for the first 10 individuals.

ANSWER: a. b. c. d. e.

Median, because the income of individual 11 is unusually high. 4,000 3,000 3,500 936.90

119. Suppose annual salaries for sales associates from Geoff's Computer Shack have a mean of $32,500 and a standard deviation of $2,500. a. Calculate and interpret the z-score for a sales associate who makes $36,000. Use Chebyshev's theorem to calculate the percentage of sales associates with salaries between b. $26,250 and $38,750. Suppose that the distribution of annual salaries for sales associates at this store is bell-shaped. c. Use the empirical rule to calculate the percentage of sales associates with salaries between $27,500 and $37,500. Use the empirical rule to determine the percentage of sales associates with salaries less than d. $27,500. Still suppose that the distribution of annual salaries for sales associates at this store is belle. shaped. A sales associate makes $42,000. Should this salary be considered an outlier? Explain. ANSWER: a. 1.4. This sales associate's annual salary is 1.4 standard deviations higher than the mean annual salary for sales associates from this store. b. 84% c. 95% d. 2.5% e. Yes, because this salary is more than three standard deviations from the mean. It has a z-score of 3.8.

120. Provide a five-number summary for the following data. 115 191 153 184 216 185 ANSWER: Min = 115, Q1 = 183, Median = 188, Q3 = 202, Max = 236

194 183

236 202

121. The following observations are given for two variables. x 5 8

y 2 12

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Chapter 03: Descriptive Statistics: Numerical Measures 18 20 22 30 10 7 a. b.

3 6 11 19 18 9

Compute and interpret the sample covariance for the above data. Compute and interpret the sample correlation coefficient.

ANSWER: a. b.

19.3. Since the covariance is positive, a positive relationship between x and y is indicated. 0.345. There is a positive relationship between x and y. The relationship is not very strong.

122. The following data represent the daily demand (y in thousands of units) and the unit price (x in dollars) for a product. Daily Demand (y) 47 39 35 44 34 20 15 30 a. b.

Unit Price (x) 1 3 5 3 6 8 16 6

Compute and interpret the sample covariance for the above data. Compute and interpret the sample correlation coefficient. –47.00 (rounded). Since the covariance is negative, a negative relationship between x and y is indicated. –0.922. There is a strong negative relationship between daily demand and unit price.

ANSWER: a. b.

123. Compute the weighted mean for the following data. xi 9 8 5 3 2

Weight (wi) 10 12 4 5 3

ANSWER: 6.68 124. Compute the weighted mean for the following data. xi 19 17 14

Weight (wi) 12 30 28

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Chapter 03: Descriptive Statistics: Numerical Measures 13 18

10 10

ANSWER: 16 125. Paul, a freshman at a local college, just completed 15 credit hours. His grade report is presented below. Course Credit Hours Grade Calculus 5 C Biology 4 A English 3 D Music 2 B P.E. 1 A The local university uses a 4-point grading system, i.e., A = 4, B = 3, C = 2, D = 1, F = 0. Compute Paul's semester grade point average. ANSWER: 2.6 126. Suppose the mean GMAT score is 550 with a standard deviation of 100. Hayden takes the GMAT and is told his zscore is 2.5. Interpret the meaning of a z-score in this context. ANSWER: Since Hayden’s z-score is positive, he scored above the mean of 550. The value, 2.5, indicates he scored 2.5 standard deviations from the mean. One standard deviation is 100, so 2.5 standard deviations is 250. Therefore, Hayden scored 250 points above the mean of 550, or 800 on the GMAT. 127. Explain why Chebyshev’s theorem cannot be used for one standard deviation from the mean. ANSWER: Chebyshev’s Theorem requires an input value greater than 1, since an input of 1 would cause a divide by 0 error. 128. Use this data set to answer the following questions. 15 68

11 26

53 15

14 13

36 20

a. Find the five-number summary. b. Find the IQR. c. Use IQR to determine whether there are any outliers. Explain. ANSWER: a. 11, 14, 17.5, 36, 68 b. 22 c. 68 is a suspected outlier, since it seems to be much higher than most other data values. 1.5(IQR) = 33, and 33 + 36 = 69. Since 68 < 69, it is not an outlier. 129. Use Excel to make a box plot of the following data: 15 68

11 26

53 15

14 13

36 20

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Chapter 03: Descriptive Statistics: Numerical Measures ANSWER:

130. Use Excel to make a boxplot of the following data: 115 191 184 216 ANSWER:

153 185

194 183

236 202

131. Use this data set to answer the following questions. 115 184 a.

191 216

153 185

194 183

236 202

Find the five-number summary.

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Chapter 03: Descriptive Statistics: Numerical Measures b. c.

Find the IQR. Use IQR to determine whether there are any outliers. Explain.

ANSWER: a. b. c.

115, 183, 188, 202, 236 19 1.5(IQR) = 28.5. 183 – 28.5 = 154.5 and 202 + 28.5 = 230.5 115 and 153 are both less than 154.5, and 236 is greater than 230.5, so 115, 153, and 236 are all outliers.

132. Describe how a comparative analysis using boxplots of men’s vs. women’s salaries at ACME, Inc. could be used. ANSWER: Two boxplots, one depicting men’s and one depicting women’s salaries, side by side would visually show differences between minimum, maximum, and median values. Differences or similarities of variability could also be easily recognized, as would any outliers in either direction. ACME, Inc. could use this depiction to correct gender bias, if it existed, with regard to salary, or the company could publish the comparative analysis using boxplots to demonstrate no gender bias exists, if it doesn’t. 133. Del Michaels had a successful morning, or so he thinks, selling 1,300 surplus notebook computers over the telephone to three commercial customers. The three customers were not equally skillful at negotiating a low unit price. Customer A bought 600 computers for $1,252 each, B bought 300 units at $,1,310 each, and C bought 400 at $1,375 each. a. What is the average unit price at which Del sold the 1,300 computers? b. Del’s manager told Del he expected him to sell, by the end of the day, a total of 2,500 surplus computers at an average price of $1,312 each. What is the average unit price at which Del must sell the remaining 1,200 computers? ANSWER: a. $1,303.23 b. $1,321.50 134. Missy Walters owns a mail-order business specializing in baby clothes. She is considering offering her customers a discount on shipping charges based on the dollar amount of the mail order. Before Missy decides the discount policy, she needs a better understanding of the dollar amount distribution of the mail orders she receives. Missy had an assistant randomly select 50 recent orders and record the value, to the nearest dollar, of each order as shown below. 136 211 194 277 231

281 162 242 348 154

226 212 368 173 166

123 241 258 409 214

178 182 323 264 311

445 290 196 237 141

231 434 183 490 159

389 167 209 222 362

196 246 198 472 189

175 338 212 248 260

a. Determine the mean, median, and mode for this data set. b. Determine the 80th percentile. c. Determine the first quartile. d. Determine the range and interquartile range. e. Determine the sample variance, sample standard deviation, and coefficient of variation. f. Determine the z-scores for the minimum and maximum values in the data set. ANSWER: a. mean = 251.46, median = 228.5, modes = 196 and 231 b. 331.5 c. 183 d. range = 367, interquartile range = 107 e. variance = 8398.5, standard deviation = 91.64, coefficient of variation = 36.44 f. minimum’s z-score = –.40, maximum’s z-score = 2.60 Copyright Cengage Learning. Powered by Cognero.

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Chapter 03: Descriptive Statistics: Numerical Measures 135. Ron Butler, a custom home builder, is looking over the expenses he incurred for a house he just finished constructing. To price future construction projects, he would like to know the average wage ($/hr.) he paid the workers he employed. Listed below are the categories of worker he employed, along with their respective wage and total hours worked. What is the average wage ($/hr.) he paid the workers? Worker Carpenter Electrician Laborer Painter Plumber

Wage ($/hr.) 21.60 28.72 11.80 19.75 24.16

Total Hours 520 230 410 270 160

ANSWER: $20.05 136. Interpret the scatter plot in terms of correlation.

ANSWER: There is a moderate, positive linear correlation between the ages of spouses at first marriage. 137. Interpret the scatter plot in terms of correlation.

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Chapter 03: Descriptive Statistics: Numerical Measures

ANSWER: There is a very strong, negative linear correlation between the age of a car and its value. 138. Angela Lopez, a golf instructor, is interested in investigating the relationship between a golfer’s average driving distance and 18-hole score. She recently observed the performance of six golfers during one round of a tournament and measured, as accurately as possible, the distances (yards) of their drives and noted their final scores. She then computed each golfer’s average drive distance for 18 holes. The results of her sample are shown below. Golfer 1 2 3 4 5 6

Avg. Drive (yards) 277.6 259.5 269.1 267.0 255.6 272.9

18-Hole Score 69 71 70 70 71 69

Compute and interpret both the sample covariance and the sample correlation coefficient. ANSWER: The sample covariance value of –7.08 indicates a negative relationship between driving distance and final score. This is not surprising. Unless a golfer’s putting performance is relatively poor, we would expect longer driving distances to lead to lower final scores. The correlation coefficient value of –.96 indicates (because it is so close to –1.00) an exceptionally strong negative correlation between driving distance and final score. 139. Reed Auto periodically has a special week-long sale. As part of the advertising campaign, Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of five previous sales are shown below. Week 1 2

TV Ads 1 3

Cars Sold 14 24 Page 30

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Chapter 03: Descriptive Statistics: Numerical Measures 3 2 18 4 1 17 5 3 27 Compute and interpret both the sample covariance and the sample correlation coefficient. ANSWER: The sample covariance value of 5.0 indicates a positive relationship between the number of TV ads and the number of cars sold. This is expected, as advertisements often bring in more business. The sample correlation coefficient value of .937 indicates an exceptionally strong positive correlation between the number of TV ads and the number of cars sold.

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Chapter 04: Introduction to Probability Multiple Choice 1. The probability of at least one head in two flips of a coin is _____. a. .33 b. .50 c. .75 d. 1 ANSWER: c 2. Revised probabilities of events based on additional information are _____. a. joint probabilities b. posterior probabilities c. marginal probabilities d. complementary probabilities ANSWER: b 3. Posterior probabilities are computed using _____. a. the classical method b. Chebyshev’s theorem c. the empirical rule d. Bayes’ theorem ANSWER: d 4. The complement of P(A | B) is _____. a. P(Ac | B) b. P(A | Bc) c. P(B | A) d. P(A  B) ANSWER: a 5. An element of the sample space is a(n) _____. a. event b. estimator c. sample point d. outlier ANSWER: c 6. The probability of an intersection of two events is computed using the _____. a. addition law b. subtraction law c. multiplication law d. division law ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability 7. If A and B are mutually exclusive, then _____. a. P(A) + P(B) = 0 b. P(A) + P(B) = 1 c. P(A  B) = 0 d. P(A  B) = 1 ANSWER: c 8. Posterior probabilities are _____. a. simple probabilities b. marginal probabilities c. joint probabilities d. conditional probabilities ANSWER: d 9. The range of probability is _____, a. any value larger than 0 b. any value between minus infinity to plus infinity c. 0 to 1, inclusive d. any value between –1 to 1 ANSWER: c 10. Since the sun MUST rise tomorrow, then the probability of the sun rising tomorrow is _____. a. much larger than 1 b. 0 c. infinity d. 1 ANSWER: d 11. Any process that generates well-defined outcomes is _____. a. an event b. an experiment c. a sample point d. a probability ANSWER: b 12. Suppose we flip a fair coin five times and each time it lands heads up. The probability of landing heads up on the next flip is _____. a. .5 b. 1 c. 0 d. .75 ANSWER: a 13. There is a 60% chance of getting stuck in traffic when leaving the city. On two separate days, what is the probability Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability that you get stuck in traffic both days? a. .36 b. .60 c. 1.20 d. .30 ANSWER: a 14. A sample point refers to a(n) _____. a. numerical measure of the likelihood of the occurrence of an event b. set of all possible experimental outcomes c. individual outcome of an experiment d. initial estimate of the probabilities of an event ANSWER: c 15. The collection of all possible sample points in an experiment is _____. a. the sample space b. a sample point c. an experiment d. the population ANSWER: a 16. Twenty percent of people at a company picnic got food poisoning. What percent of the people at the picnic did NOT get food poisoning? a. 20% b. 40% c. 60% d. 80% ANSWER: d 17. The sample space refers to _____. a. any particular experimental outcome b. the sample size minus 1 c. the set of all possible experimental outcomes d. a revised probability of an event based on additional information ANSWER: c 18. An experiment consists of three steps. There are four possible results on the first step, three possible results on the second step, and two possible results on the third step. The total number of experimental outcomes is _____. a. 9 b. 14 c. 24 d. 36 ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability 19. An experiment consists of tossing four coins successively. The number of sample points in this experiment is _____. a. 16 b. 8 c. 4 d. 2 ANSWER: a 20. A lottery is conducted using three urns. Each urn contains chips numbered from 0 to 9. One chip is selected at random from each urn. The total number of sample points in the sample space is _____. a. 30 b. 100 c. 729 d. 1,000 ANSWER: d 21. Three applications for admission to a local university are checked to determine whether each applicant is male or female. The number of sample points in this experiment is _____. a. 2 b. 4 c. 6 d. 8 ANSWER: d 22. Assume your favorite football team has two games left to finish the season. The outcome of each game can be win, lose, or tie. The number of possible outcomes is _____. a. 2 b. 4 c. 6 d. 9 ANSWER: d 23. Each customer entering a department store will either buy or not buy some merchandise. An experiment consists of following three customers and determining whether or not they purchase any merchandise. The number of sample points in this experiment is _____. a. 2 b. 4 c. 6 d. 8 ANSWER: d 24. A graphical device used for enumerating sample points in a multiple-step experiment is a _____. a. bar chart b. tree diagram c. histogram Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability d. Venn diagram ANSWER: b 25. Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible selections are there? a. 20 b. 7 c. 5 d. 10 ANSWER: d 26. The "Top Three" at a racetrack consists of picking the correct order of the first three horses in a race. If there are 10 horses in a particular race, how many "Top Three" outcomes are there? a. 302,400 b. 720 c. 1,814,400 d. 10 ANSWER: b 27. When the assumption of equally likely outcomes is used to assign probability values, the method used to assign probabilities is referred to as the _____. a. relative frequency method b. subjective method c. probability method d. classical method ANSWER: d 28. A magician holds a standard deck of cards and draws one card. The probability of drawing the ace of diamonds is 1/52. What method of assigning probabilities was used? a. objective method b. classical method c. subjective method d. experimental method ANSWER: b 29. When the results of experimentation or historical data are used to assign probability values, the method used to assign probabilities is referred to as the _____. a. relative frequency method b. subjective method c. classical method d. posterior method ANSWER: a 30. A method of assigning probabilities based upon judgment is referred to as the _____. a. relative frequency method Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability b. probability method c. classical method d. subjective method ANSWER: d 31. Of the last 100 customers entering a computer shop, 25 have purchased a computer. If the relative frequency method for computing probability is used, the probability that the next customer will purchase a computer is _____. a. .5 b. .67 c. .3 d. .167 ANSWER: a 32. A professor rolls a fair, six-sided die. Using the classical method of probability, what is the probability that at least three spots will be showing up on the die? a. .5 b. .67 c. .3 d. .167 ANSWER: b 33. An experiment consists of four outcomes with P(E1) = .2, P(E2) = .3, and P(E3) = .4. The probability of outcome E4 is _____. a. .500 b. .024 c. .100 d. .900 ANSWER: c 34. A graphical method of representing the sample points of a multiple-step experiment is a(n) _____. a. frequency polygon b. histogram c. ogive d. tree diagram ANSWER: d 35. A __________ is a graphical representation in which the sample space is represented by a rectangle and events are represented as circles. a. frequency polygon b. histogram c. Venn diagram d. tree diagram ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability 36. A(n) __________ is a collection of sample points. a. probability b. permutation c. experiment d. event ANSWER: d 37. Given that event E has a probability of .25, the probability of the complement of event E _____. a. cannot be determined with the above information b. can have any value between 0 and 1 c. must be .75 d. is .25 ANSWER: c 38. The symbol ∪ indicates the _____. a. union of events b. intersection of events c. sum of the probabilities of events d. sample space ANSWER: a 39. The union of events A and B is the event containing _____. a. all the sample points common to both A and B b. all the sample points belonging to A or B c. all the sample points belonging to A or B or both d. all the sample points belonging to A or B, but not both ANSWER: c 40. The probability of the union of two events with nonzero probabilities cannot be _____. a. less than 1 b. 1 c. less than 1 and cannot be 1 d. more than 1 ANSWER: d 41. The symbol ∩ shows the _____. a. union of events b. intersection of events c. sum of the probabilities of events d. sample space of events ANSWER: b 42. The addition law is potentially helpful when we are interested in computing the probability of _____. a. independent events Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability b. the intersection of two events c. the union of two events d. conditional events ANSWER: c 43. If P(A) = .38, P(B) = .83, and P(A ∩ B) = .24; then P(A ∪ B) = a. 1.21 b. .97 c. .76 d. 1.45 ANSWER: b 44. If P(A) = .62, P(B) = .47, and P(A ∪ B) = .88; then P(A ∩ B) = a. .291 b. 1.970 c. .670 d. .210 ANSWER: d 45. If P(A) = .75, P(A ∪ B) = .86, and P(A ∩ B) = .56, then P(B) = a. .25 b. .67 c. .56 d. .11 ANSWER: b 46. Two events are mutually exclusive if _____. a. the probability of their intersection is 1 b. they have no sample points in common c. the probability of their intersection is .5 d. the probability of their intersection is 1 and they have no sample points in common ANSWER: b 47. You roll a fair six-sided die with the hopes of rolling a 5 or a 6. These two events are ___________ because they have no sample points in common. a. independent events b. posterior events c. mutually exclusive events d. complements ANSWER: c 48. The probability of the intersection of two mutually exclusive events _____. a. can be any value between 0 and 1 b. must always be equal to 1 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability c. must always be equal to 0 d. can be any positive value ANSWER: c 49. If two events are mutually exclusive, then the probability of their intersection _____. a. will be equal to 0 b. can have any value larger than 0 c. must be larger than 0, but less than 1 d. will be 1 ANSWER: a 50. Two events, A and B, are mutually exclusive and each has a nonzero probability. If event A is known to occur, the probability of the occurrence of event B is _____. a. 1 b. any positive value c. 0 d. any value between 0 and 1 ANSWER: c 51. If A and B are mutually exclusive events with P(A) = .3 and P(B) = 0.5, then P(A ∩ B) = a. .30 b. .15 c. 0 d. .20 ANSWER: c 52. If A and B are mutually exclusive events with P(A) = .3 and P(B) = .5, then P(A ∪ B) = a. 0 b. .15 c. .8 d. .2 ANSWER: c 53. In an experiment, events A and B are mutually exclusive. If P(A) = .6, then the probability of B _____. a. cannot be larger than .4 b. can be any value greater than .6 c. can be any value between 0 and 1 d. must also be .6 ANSWER: a 54. Which of the following statements is always true? a. −1 ≤ P(Ei) ≤ 1 b. P(A) = 1 − P(Ac) Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability c. P(A) + P(B) = 1 d. P(A) = 1 + P(Ac) ANSWER: b 55. One of the basic requirements of probability is _____. a. for each experimental outcome Ei, we must have P(Ei) ≥ 1 b. P(A) = P(Ac) – 1 c. if there are k experimental outcomes, then P(E1) + P(E2) + ... + P(Ek) = 1 d. P(A) = P(Ac) ANSWER: c 56. Events A and B are mutually exclusive with P(A) = .3 and P(B) = .2. The probability of the complement of event B equals _____. a. 0 b. .06 c. .70 d. .80 ANSWER: d 57. The multiplication law is potentially helpful when we are interested in computing the probability of _____. a. mutually exclusive events b. the intersection of two events c. the union of two events d. the complement of an event ANSWER: b 58. If P(A) = .62, P(B) = .56, and P(A ∪ B) = .70, then P(B | A) = _____. a. .4800 b. .7742 c. .9032 d. .6340 ANSWER: b 59. If two events are independent, then _____. a. they must be mutually exclusive b. the sum of their probabilities must be equal to 1 c. the probability of their intersection must be 0 d. the events have no influence on each other ANSWER: d 60. If A and B are independent events with P(A) = .38 and P(B) = .55, then P(A | B) = a. .209 b. 0 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability c. .550 d. .38 ANSWER: d 61. If X and Y are mutually exclusive events with P(X) = .295, P(Y) = .32, then P(X ∩ Y) = _____. a. .094 b. .615 c. 1 d. 0 ANSWER: d 62. Two events with nonzero probabilities _____. a. can be both mutually exclusive and independent b. cannot be both mutually exclusive and independent c. are always mutually exclusive d. are always independent ANSWER: b 63. If P(A) = .50, P(B) = .60, and P(A ∩ B) = .30, then events A and B are _____. a. mutually exclusive events b. dependent events c. independent events d. posterior probabilities ANSWER: c 64. On a December day, the probability of snow is .30. The probability of a "cold" day is .50. The probability of snow and a "cold" day is .15. Thus, snow and "cold" weather are _____. a. mutually exclusive events b. independent events c. dependent events d. discrete events ANSWER: b 65. If A and B are mutually exclusive events with P(A) = .5 and P(B) = .5, then P(A ∩ B) is _____. a. 0 b. .25 c. 1 d. .5 ANSWER: a 66. If A and B are independent events with P(A) = .4 and P(B) = .6, then P(A ∩ B) = _____. a. .76 b. 1 c. .24 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability d. .2 ANSWER: c 67. If A and B are independent events with P(A) = .2 and P(B) = .6, then P(A ∪ B) = _____. a. .62 b. .12 c. .60 d. .68 ANSWER: d 68. If A and B are independent events with P(A) = .4 and P(B) = .25, then P(A ∪ B) = _____. a. .65 b. .55 c. .10 d. .70 ANSWER: b 69. Events A and B are mutually exclusive. Which of the following statements is also true? a. A and B are also independent. b. P(A ∪ B) = P(A)P(B) c. P(A ∪ B) = P(A) + P(B) d. P(A ∩ B) = P(A) + P(B) ANSWER: c 70. If A and B are independent events with P(A) = .05 and P(B) = .65, then P(A | B) = _____. a. .05 b. .0325 c. .65 d. .8 ANSWER: a 71. A six-sided die is rolled three times. The probability of observing a 1 three times in a row is _____. a. b. c. d. ANSWER: d 72. If a coin is tossed three times, the likelihood of obtaining three heads in a row is _____. a. 0 b. .500 c. .875 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability d. .125 ANSWER: d 73. If a fair penny is tossed four times and comes up heads all four times, the probability of heads on the fifth trial is _____. a. 0 b. .03125 c. .50 d. .20 ANSWER: c 74. If a fair penny is tossed three times and comes up heads all three times, the probability of heads on the fourth trial is _____. a. smaller than the probability of tails b. larger than the probability of tails c. .0625 d. .50 ANSWER: d 75. A couple has three children. Assuming each child has an equal chance of being a boy or a girl, what is the probability that they have at least one girl? a. .125 b. .5 c. .875 d. 1 ANSWER: c 76. The probability of the occurrence of event A in an experiment is 1/3. If the experiment is performed two times and event A did not occur, then on the third trial event A _____. a. must occur b. may occur c. could not occur d. has a 2/3 probability of occurring ANSWER: b 77. Bayes' theorem is used to compute _____. a. the prior probabilities b. the union of events c. the intersection of events d. the posterior probabilities ANSWER: d 78. Initial estimates of the probabilities of events are known as _____. a. sets Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability b. posterior probabilities c. conditional probabilities d. prior probabilities ANSWER: d 79. A B C D E 1 Prior Conditional Joint 2 Event Probability Probability Probability 3 A1 0.25 0.31 For the Excel worksheet above, which of the following formulas would correctly calculate the joint probability for cell D3? a. =SUM(B3:C3) b. =B3+C3 c. =B3/C3 d. =B3*C3 ANSWER: d 80. A B C D E 1 Prior Conditional Joint Posterior 2 Event Probability Probability Probability Probability 3 A1 0.45 0.22 0.099 4 A2 0.55 0.16 0.088 5 0.187 For the Excel worksheet above, which of the following formulas would correctly calculate the posterior probability for cell E3? a. =SUM(B3:D3) b. =D3/$D$5 c. =D5/$D$3 d. =B3/C3+D3 ANSWER: b 81. If P(A ∩ B) = 0, _____. a. P(A) + P(B) = 1 b. either P(A) = 0 or P(B) = 0 c. A and B are mutually exclusive events d. A and B are independent events ANSWER: c 82. The probability of an event is _____. a. the sum of the probabilities of the sample points in the event b. the product of the probabilities of the sample points in the event c. the minimum of the probabilities of the sample points in the event d. the maximum of the probabilities of the sample points in the event Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability ANSWER: a 83. If P(A | B) = 0.3, a. P(B | A) = 0.7 b. P(Ac | B) = 0.7 c. P(A | Bc) = 0.7 d. P(Ac | Bc) = 0.7 ANSWER: b 84. If A and B are independent events with P(A) = 0.1 and P(B) = .4, then _____. a. P(A ∩ B) = 0 b. P(A ∩ B) = .04 c. P(A ∩ B) = .5 d. P(A ∩ B) = .25 ANSWER: b 85. If P(A | B) = .3 and P(B) = .8, then _____. a. P(A) = .24 b. P(B | A) = .7 c. P(A ∩ B) = .5 d. P(A ∩ B) = .24 ANSWER: d 86. If P(A) = .6, P(B) = .3, and P(A ∩ B) = .2, then P(B | A) =_____. a. .33 b. .5 c. .67 d. .9 ANSWER: a 87. If P(A) = .85, P(B) = .76, and P(A ∩ B) = .72, then P(A | B) = _____. a. .15 b. .53 c. .25 d. .95 ANSWER: d 88. If P(A) = .80, P(B) = .65, and P(A ∩ B) = .78, then P(B  A) = _____. a. .6700 b. .8375 c. .9750 d. .5420 ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability Subjective Short Answer 89. All the employees of ABC Company are assigned ID numbers. The ID number consists of the first letter of an employee's last name, followed by four digits ranging from 0 to 9. a. How many possible different ID numbers are there? How many possible different ID numbers are there for employees whose last name starts with b. an "A"? ANSWER: a. 260,000 b. 10,000 90. A company plans to interview 10 recent graduates for possible employment. The company has three positions open. How many groups of three can the company select? ANSWER: 120 91. A student has to take seven more courses before she can graduate. If none of the courses are prerequisites to others, how many groups of three courses can she select for the next semester? ANSWER: 35 92. A committee of four is to be selected from a group of 12 people. How many possible committees can be selected? ANSWER: 495 93. The sales records of a real estate agency show the following sales over the past 200 days: Number of Houses Sold 0 1 2 3 4

Number of Days 60 80 40 16 4

a. How many sample points are there? b. Assign probabilities to the sample points and show their values. c. What is the probability that the agency will not sell any houses in a given day? d. What is the probability of selling at least two houses? e. What is the probability of selling one or two houses? f. What is the probability of selling less than three houses? ANSWER: a. 5 b. Number of Houses Sold Probability 0 .30 1 .40 2 .20 3 .08 4 .02 c. d. e.

.3 .3 .6

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Chapter 04: Introduction to Probability f.

94. The results of a survey of 800 married couples and the number of children they had is shown below. Number of Children Probability 0 .050 1 .125 2 .600 3 .150 4 .050 5 .025 If a couple is selected at random, what is the probability that the couple will have a. less than four children? b. more than two children? c. either two or three children? ANSWER: a. b. c.

.925 .225 .75

95. An experiment consists of rolling two six-sided dice and observing the number of spots on the upper faces. Determine the probability that a. the sum of the spots is 3. b. each die shows four or more spots. c. the sum of the spots is not 3. d. neither a 1 nor a 6 appears on each die. e. a pair of sixes appears. f. the sum of the spots is 7. ANSWER: a. b. c.

2/36 or 9/36 or .25 34/36 or

16/36 or

1/36 or

6/36 or

96. Assume that in your hand you hold an ordinary six-sided die and a dime. You toss both the die and the dime on a table. a. What is the probability that a head appears on the dime and a 6 on the die? b. What is the probability that a tail appears on the dime and any number more than 3 on the die? c. What is the probability that a number larger than 2 appears on the die? ANSWER: a. b.

1/12 or 3/12 or .25

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Chapter 04: Introduction to Probability c.

8/12 or

97. A very short quiz has one multiple-choice question with five possible choices (a, b, c, d, e) and one true or false question. Assume you are taking the quiz but do not have any idea what the correct answer is to either question, but you mark an answer anyway. a. What is the probability that you have given the correct answer to both questions? b. What is the probability that only one of the two answers is correct? c. What is the probability that neither answer is correct? d. What is the probability that only your answer to the multiple-choice question is correct? e. What is the probability that you have only answered the true or false question correctly? ANSWER: a. 1/10 or .1 b. 5/10 or .5 c. 4/10 or .4 d. 1/10 or .1 e. 4/10 or .4 98. Two of the cylinders in an eight-cylinder car are defective and need to be replaced. If two cylinders are selected at random, what is the probability that a. both defective cylinders are selected? b. no defective cylinder is selected? c. at least one defective cylinder is selected? ANSWER: a. 2/56 or .0357 (rounded) b. 30/56 or .5357 (rounded) c. 26/56 or .4643 (rounded) 99. Assume two events A and B are mutually exclusive and, furthermore, P(A) = .2 and P(B) = .4. a. Find P(A ∩ B). b. Find P(A ∪ B). c. Find P(A | B). ANSWER: a. b. c.

0 .6 0

100. You are given the following information on events A, B, C, and D. P(A) = .4 P(B) = .2 P(C) = .1 a. b. c. d. e. f. g. h.

P(A ∪ D) = .6 P(A⏐B) = .3

P(A ∩ C) = .04 P(A ∩ D) = .03

Compute P(D). Compute P(A ∩ B). Compute P(A⏐C). Compute the probability of the complement of C. Are A and D mutually exclusive? Explain your answer. Are A and B independent? Explain your answer. Are A and C mutually exclusive? Explain your answer. Are A and C independent? Explain your answer.

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Chapter 04: Introduction to Probability ANSWER: a. b. c. d. e. f. g. h.

0.23 .06 0.4 .9 No, P(A ∩ D) ≠ 0 No, P(A ⏐ B) ≠ P(A) No, P(A ∩ C) ≠ 0 Yes, P(A ⏐ C) = P(A)

101. A government agency has 6,000 employees. The employees were asked whether they preferred a four-day work week (10 hours per day), a five-day work week (8 hours per day), or flexible hours. You are given information on the employees' responses broken down by gender. Four days Five days Flexible Total a. b. c. d. e. f.

Male 300 1,200 300 1,800

Female 600 1,500 2,100 4,200

Total 900 2,700 2,400 6,000

What is the probability that a randomly selected employee is a man and is in favor of a fourday work week? What is the probability that a randomly selected employee is female? A randomly selected employee turns out to be female. Compute the probability that she is in favor of flexible hours. What percentage of employees is in favor of a five-day work week? Given that a person is in favor of flexible time, what is the probability that the person is female? What percentage of employees is male and in favor of a five-day work week?

ANSWER: a. b. c. d. e. f.

.05 .7 .5 45% .875 20%

102. A bank has the following data on the gender and marital status of 200 customers. Male 20 100

Single Married a. b. c. d. e. f. g.

Female 30 50

What is the probability of finding a single female customer? What is the probability of finding a married male customer? If a customer is female, what is the probability that she is single? What percentage of customers is male? If a customer is male, what is the probability that he is married? Are gender and marital status mutually exclusive? Is marital status independent of gender? Explain using probabilities.

ANSWER: a.

.15

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Chapter 04: Introduction to Probability b. c. d. e. f. g.

.5 .375 60% .833 No, the probability of intersection is not 0. They are not independent because P(male) = .6 and P(male⏐single) = .4

103. A survey of a sample of business students resulted in the following information regarding the genders of the individuals and their major.

a. b. c. d.

Major Gender Management Marketing Others Male 40 10 30 Female 30 20 70 Total 70 30 100 What is the probability of selecting an individual who is majoring in Marketing? What is the probability of selecting an individual who is majoring in Management, given that the person is female? Given that a person is male, what is the probability that he is majoring in Management? What is the probability of selecting a male individual?

ANSWER: a. b. c. d.

Total 80 120 200

.15 .25 .50 .40

104. The following table shows the number of students in three different degree programs and whether they are graduate or undergraduate students: Degree Program Business Engineering Arts and Sciences Total a. b. c. d. e.

Undergraduate 150 150 100 400

Graduate 50 25 25 100

Total 200 175 125 500

What is the probability that a randomly selected student is an undergraduate? What percentage of students is Engineering majors? If we know that a selected student is an undergraduate, what is the probability that he or she is a Business major? A student is enrolled in the Arts and Sciences school. What is the probability that the student is an undergraduate student? What is the probability that a randomly selected student is a graduate Business major?

ANSWER: a. b. c. d. e.

.8 35% .375 .8 .1

105. A small town has 5,600 residents. The residents in the town were asked whether or not they favored building a new bridge across the river. You are given the following information on the residents' responses, broken down by gender. Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability Men 1,400 840 2,240

In favor Opposed Total

Women 280 3,080 3,360

Let:

a. b. c. d. e. f. g.

Total 1,680 3,920 5,600 M be the event a resident is a man W be the event a resident is a woman F be the event a resident is in favor P be the event a resident is opposed

Find the joint probability table. Find the marginal probabilities. What is the probability that a randomly selected resident is a man and is in favor of building the bridge? What is the probability that a randomly selected resident is a man? What is the probability that a randomly selected resident is in favor of building the bridge? What is the probability that a randomly selected resident is a man or in favor of building the bridge or both? A randomly selected resident turns out to be male. Compute the probability that he is in favor of building the bridge.

ANSWER: a. and b.

In favor Opposed Total c. d. e. f. g.

Men .25 .15 .40

Women .05 .55 .60

Total .3 .7 1.0

.25 .4 .3 .45 .625

106. On a recent holiday evening, a sample of 500 drivers was stopped by the police. Three hundred were under 30 years of age. A total of 250 were under the influence of alcohol. Of the drivers under 30 years of age, 200 were under the influence of alcohol. Let A be the event that a driver is under the influence of alcohol. Let Y be the event that a driver is less than 30 years old. a. b. c. d. e. f.

Determine P(A) and P(Y). What is the probability that a driver is under 30 and NOT under the influence of alcohol? Given that a driver is NOT under 30, what is the probability that he/she is under the influence of alcohol? What is the probability that a driver is under the influence of alcohol if we know the driver is under 30? Show the joint probability table. Are A and Y mutually exclusive events? Explain.

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Chapter 04: Introduction to Probability g.

Are A and Y independent events? Explain.

ANSWER: a. b. c. d. e.

f. g.

P(Y) = .6 .2 .25 .667

P(A) = .5

A Y .4 Yc .1

Ac .2 .3

No P(A∩Y) ≠ 0 No, P(A ⏐ Y) ≠P(A)

107. Six vitamin and three sugar tablets identical in appearance are in a box. One tablet is taken at random and given to Person A. A tablet is then selected and given to Person B. What is the probability that a. Person A was given a vitamin tablet? b. Person B was given a sugar tablet given that Person A was given a vitamin tablet? c. neither was given vitamin tablets? d. both were given vitamin tablets? e. exactly one person was given a vitamin tablet? f. Person A was given a sugar tablet and Person B was given a vitamin tablet? g. Person A was given a vitamin tablet and Person B was given a sugar tablet? ANSWER: a. 6/9 or b.

3/8 or

1/12 or

d. e. f. g.

5/12 or 1/2 or .5 1/4 or .25 1/4 or .25

108. In a random sample of UTC students, 50% indicated they are Business majors, 40% Engineering majors, and 10% Other majors. Of the Business majors, 60% were females; whereas 30% of Engineering majors were females. Finally, 20% of the Other majors were female. a. What percentage of students in this sample was female? b. Given that a person is female, what is the probability that she is an engineering major? ANSWER: a. 44% b. .2727 109. Sixty percent of the student body at UTC is from the state of Tennessee (T), 30% is from other states (O), and the remainder is international students (I). Twenty percent of students from Tennessee lives in the dormitories, whereas 50% of students from other states lives in the dormitories. Finally, 80% of the international students live in the dormitories. a. What percentage of UTC students lives in the dormitories? b. Given that a student lives in the dormitory, what is the probability that she/he is an Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability international student? Given that a student does not live in the dormitory, what is the probability that she/he is an c. international student? ANSWER: a. 35% b. .2286 (rounded) c. .0308 (rounded) 110. Tammy is a general contractor and has submitted two bids for two projects (A and B). The probability of getting project A is .65. The probability of getting project B is .77. The probability of getting at least one of the projects is .90. a. What is the probability that she will get both projects? b. Are the events of getting the two projects mutually exclusive? Explain, using probabilities. c. Are the two events independent? Explain, using probabilities. ANSWER: a. .52 b. No, the probability of their intersection is not zero. c. No, P(A ⏐ B) = .6753 ≠ P(A) 111. Assume you are taking two courses this semester (A and B). Based on your opinion, you believe the probability that you will pass course A is .835; the probability that you will pass both courses is .276. You further believe the probability that you will pass at least one of the courses is .981. a. What is the probability that you will pass course B? Would the passing of the two courses be independent events? Use probability information to b. justify your answer. c. Are the events of passing the courses mutually exclusive? Explain. d. What method of assigning probabilities did you use? ANSWER: a. b. c. d.

.422 No, P(A ⏐ B) = .654 ≠ P(A) No, the probability of their intersection is not 0. the subjective method

112. Assume you have applied to two different universities (let's refer to them as Universities A and B) for your graduate work. In the past, 25% of students (with similar credentials as yours) who applied to University A were accepted, while University B accepted 35% of the applicants. Assume events are independent of each other. a. What is the probability that you will be accepted to both universities? b. What is the probability that you will be accepted to at least one graduate program? c. What is the probability that one and only one of the universities will accept you? d. What is the probability that neither university will accept you? ANSWER: a. b. c. d.

.0875 .5125 .425 .4875

113. Assume you have applied for two scholarships, a Merit scholarship (M) and an Athletic scholarship (A). The probability that you receive an Athletic scholarship is .18. The probability of receiving both scholarships is .11. The probability of getting at least one of the scholarships is .3. a. What is the probability that you will receive a Merit scholarship? b. Are events A and M mutually exclusive? Why or why not? Explain. Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability c. d. e.

Are the two events, A and M, independent? Explain, using probabilities. What is the probability of receiving the Athletic scholarship given that you have been awarded the Merit scholarship? What is the probability of receiving the Merit scholarship given that you have been awarded the Athletic scholarship?

ANSWER: a. b. c. d. e.

.23 No, because P(A ∩ M) ≠ 0 No, because P(A ∩ M) ≠ P(A) P(M) .4783 .6111

114. In the two upcoming basketball games, the probability that UTC will defeat Marshall is .63, and the probability that UTC will defeat Furman is .55. The probability that UTC will defeat both opponents is .3465. a. What is the probability that UTC will defeat Furman given that they defeat Marshall? b. What is the probability that UTC will win at least one of the games? c. What is the probability of UTC winning both games? d. Are the outcomes of the games independent? Explain your answer using probabilities. ANSWER: a. .55 b. .8335 c. .3465 Yes, the probability of defeating Furman (.55) is equal to the probability of defeating Furman d. given that they have defeated Marshall (.55). 115. The probability of an economic decline in the year 20Y1 is 0.23. There is a probability of 0.64 that we will elect a Republican president in the year 20Y0. If we elect a Republican president, there is a .35 probability of an economic decline. Let D represent the event of an economic decline, and R represent the event of election of a Republican president a. Are R and D independent events? What is the probability of electing a Republican president in 20Y0 and an economic decline in b. the year 20Y1? If we experience an economic decline in the year 20Y1, what is the probability that a c. Republican president will have been elected in the year 20Y0? What is the probability of economic decline in 20Y1 or a Republican president elected in the d. year 20Y0 or both? ANSWER: a. b. c. d.

No, because P(D) ≠ P(D ⏐ R) .224 .9739 .646

116. As a company manager for Claimstat Corporation there is a .40 probability that you will be promoted this year. There is a .72 probability that you will get a promotion or a raise. The probability of getting a promotion and a raise is .25. a. If you get a promotion, what is the probability that you will also get a raise? b. What is the probability of getting a raise? c. Are getting a raise and being promoted independent events? Explain using probabilities. d. Are these two events mutually exclusive? Explain using probabilities. ANSWER: a. .625 b. .57 c. No, because P(R) ≠ P(R ⏐ P) Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability d.

No, because P(R ∩ P) ≠ 0

117. An applicant has applied for positions at Company A and Company B. The probability of getting an offer from Company A is .4, and the probability of getting an offer from Company B is .3. Assuming that the two job offers are independent of each other, what is the probability that a. the applicant gets an offer from both companies? b. the applicant will get at least one offer? c. the applicant will not be given an offer from either company? d. Company A does not offer the applicant a job, but Company B does? ANSWER: a. .12 b. .58 c. .42 d. .18 118. A corporation has 15,000 employees. Sixty-two percent of the employees are male. Twenty-three percent of the employees earn more than $30,000 a year. Eighteen percent of the employees are male and earn more than $30,000 a year. a. If an employee is taken at random, what is the probability that the employee is male? If an employee is taken at random, what is the probability that the employee earns more than b. $30,000 a year? If an employee is taken at random, what is the probability that the employee is male and earns c. more than $30,000 a year? If an employee is taken at random, what is the probability that the employee is male or earns d. more than $30,000 a year or both? The employee taken at random turns out to be male. Compute the probability that he earns e. more than $30,000 a year. f. Are being male and earning more than $30,000 a year independent? ANSWER: a. .62 b. .23 c. .18 d. .67 e. .2903 f. No 119. A statistics professor has noted from past experience that a student who follows a program of studying two hours for each hour in class has a probability of .9 of getting a grade of C or better, while a student who does not follow a regular study program has a probability of .2 of getting a C or better. It is known that 70% of the students follow the study program. Find the probability that if a student who has earned a C or better grade followed the program. ANSWER: .9130 120. A survey of business students who have taken the Graduate Management Admission Test (GMAT) indicated that students who have spent at least five hours studying GMAT review guides have a probability of .85 of scoring above 400. Students who do not spend at least five hours reviewing have a probability of .65 of scoring above 400. It has been determined that 70% of the business students spent at least five hours reviewing for the test. a. Find the probability of scoring above 400. Find the probability that given a student scored above 400, he/she spent at least five hours b. reviewing for the test. ANSWER: a. .79 b. .7532 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability 121. A machine is used in a production process. From past data, it is known that 97% of the time the machine is set up correctly. Furthermore, it is known that if the machine is set up correctly, it produces 95% acceptable (non-defective) items. However, when it is set up incorrectly, it produces only 40% acceptable items. An item from the production line is selected. What is the probability that the selected item is a. non-defective? Given that the selected item is non-defective, what is the probability that the machine is set up b. correctly? c. What method of assigning probabilities was used here? ANSWER: a. .9335 b. .9871 c. the relative frequency method 122. In a recent survey in a statistics class, it was determined that only 60% of the students attend class on Fridays. From past data, it was noted that 98% of those who went to class on Fridays pass the course, while only 20% of those who did not go to class on Fridays passed the course. a. What percentage of students is expected to pass the course? Given that a person passes the course, what is the probability that he/she attended classes on b. Fridays? ANSWER: a. 66.8% b. .88 123. Thirty-five percent of the students who enroll in a statistics course go to the statistics laboratory on a regular basis. Past data indicates that 40% of those students who use the lab on a regular basis make a grade of B or better. On the other hand, 10% of students who do not go to the lab on a regular basis make a grade of B or better. If a particular student made an A, determine the probability that she or he used the lab on a regular basis. ANSWER: .6829 124. In a city, 60% of the residents live in houses and 40% of the residents live in apartments. Of the people who live in houses, 20% own their own business. Of the people who live in apartments, 10% own their own business. If a person owns his or her own business, find the probability that he or she lives in a house. ANSWER: .75 125. A market study taken at a local sporting goods store showed that of 20 people questioned, 6 owned tents, 10 owned sleeping bags, 8 owned camping stoves, 4 owned both tents and camping stoves, and 4 owned both sleeping bags and camping stoves. Let event A = owns a tent, event B = owns a sleeping bag, event C = owns a camping stove, and sample space = 20 people questioned. a. Find P(A), P(B), P(C), P(A∩C), P(B∩C). b. Are events A and C mutually exclusive? Explain briefly. c. Are events B and C independent events? Explain briefly. d. If a person questioned owns a tent, what is the probability he also owns a camping stove? e. If two people questioned own a tent, a sleeping bag, and a camping stove, how many own only a camping stove? f. Is it possible for three people to own both a tent and a sleeping bag, but not a camping stove? ANSWER: a. P(A) = .3, P(B) = .5, P(C) = .4, P(A∩C) = .2, P(B∩C) = .2 b. not mutually exclusive; there are four people who own both a tent and stove c. independent: P(B∩C) = P(B) · P(C) d. P(C | A) = .667 e. two people own only a stove f. no 126. The board of directors of Bidwell Valve Company has made the following estimates for the upcoming year's annual Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability earnings: P(earnings lower than this year) = .30 P(earnings about the same as this year) = .50 P(earnings higher than this year) = .20 After talking with union leaders, the human resources department has drawn the following conclusions: P(union will request wage increase | lower earnings next year) = .25 P(union will request wage increase | same earnings next year) = .40 P(union will request wage increase | higher earnings next year) = .90 a. Calculate the probability that the company earns the same as this year and the union requests a wage increase. b. Calculate the probability that the company has higher earnings next year and the union does not request a wage increase. c. Calculate the probability that the union requests a wage increase. ANSWER: a. .20 b. .02 c. .455 127. An accounting firm noticed that of the companies it audits, 85% show no inventory shortages, 10% show small inventory shortages, and 5% show large inventory shortages. The firm has devised a new accounting test for which it believes the following probabilities hold: P(company will pass test | no shortage) = .90 P(company will pass test | small shortage) = .50 P(company will pass test | large shortage) = .20 a. If a company being audited fails this test, what is the probability of a large or small inventory shortage? b. If a company being audited passes this test, what is the probability of no inventory shortage? ANSWER: a. .515 b. .927 128. Global Airlines operates two types of jet planes: jumbo and ordinary. On jumbo jets, 25% of the passengers are on business, while on ordinary jets 30% of the passengers are on business. Of Global's air fleet, 40% of its capacity is provided on jumbo jets. (Hint: You have been given two conditional probabilities.) a. What is the probability a randomly chosen business customer flying with Global is on a jumbo jet? b. What is the probability a randomly chosen non-business customer flying with Global is on an ordinary jet? ANSWER: a. .357 b. .583 129. Safety Insurance Company has compiled the following statistics. For any one-year period: P(accident | male driver under 25) = .22 P(accident | male driver over 25) = .15 P(accident | female driver under 25) = .16 P(accident | female driver over 25) = .14 The percentage of Safety's policyholders in each category is: Male under 25: 20% Male over 25: 40% Female under 25: 10% Female over 25: 30% a. What is the probability that a randomly selected policyholder will have an accident within the next year? b. Given that a driver has an accident, what is the probability the driver is a male over 25? c. Given that a driver has no accident, what is the probability the driver is a female? ANSWER: a. .162 b. .370 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability c. .408 130. Super Cola sales break down as 80% regular soda and 20% diet soda. Men purchase 60% of the regular soda, but only 30% of the diet soda. If a woman purchases Super Cola, what is the probability that it is a diet soda? ANSWER: .304 131. An investment advisor recommends the purchase of shares in Infogenics, Inc. He has made the following predictions: P(stock goes up 20% | rise in GDP) = .6 P(stock goes up 20% | level GDP) = .5 P(stock goes up 20% | fall in GDP) = .4 An economist has predicted that the probability of a rise in the GDP is 30%, whereas the probability of a fall in the GDP is 40%. a. Draw a tree diagram to represent this multiple-step experiment. b. What is the probability that the stock will go up 20%? c. We have been informed that the stock has gone up 20%. What is the probability of a rise or fall in the GDP? ANSWER: a.

b. .490 c. .694

132. The following probability model describes the number of snowstorms for Washington, D.C. for a given year: Number of Snowstorms Probability

.25

.33

.24

.11

.04

.02

.01

The probability of seven or more snowstorms in a year is 0. a. What is the probability of more than two but less than five snowstorms? b. Given this is a particularly cold year (in which two snowstorms have already been observed), what is the conditional probability that four or more snowstorms will be observed? c. If at the beginning of winter there is a snowfall, what is the probability of at least one more snowstorm before winter is over? ANSWER: a. .15 Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability b. .167 c. .56 133. Ambell Company uses batteries from two different manufacturers. Historically, 60% of the batteries are from manufacturer 1, and 90% of these batteries last for over 40 hours. Only 75% of the batteries from manufacturer 2 last for over 40 hours. A battery in a critical tool fails at 32 hours. What is the probability it was from manufacturer 2? ANSWER: .625 134. It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance performance. The test for this drug is 90% accurate. What is the probability that an athlete who tests positive is actually a user? ANSWER: .2177 135. Through a telephone survey, a low-interest bank credit card is offered to 400 households. The responses are as follows: Income ≤ $50,000 Income > $50,000 Accept offer 40 30 Reject offer 210 120 a. Develop a joint probability table and show the marginal probabilities. b. What is the probability of a household whose income exceeds $50,000 and who rejects the offer? c. If income is ≤ $50,000, what is the probability the offer will be accepted? d. If the offer is accepted, what is the probability that income exceeds $50,000?

ANSWER: a. Accept offer Reject offer Total

Income  $50,000 .100 .525 .625

Income > $50,000 .075 .300 .375

Total .175 .825 1.000

b. .3 c. .16 d. .4286 136. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an A for the semester. Your subjective assessment of your performance is Event A on paper and A on exam A on paper only A on exam only A on neither

Probability .25 .10 .30 .35

a. What is the probability of getting an A on the paper? b. What is the probability of getting an A on the exam? c. What is the probability of getting an A in the course? Copyright Cengage Learning. Powered by Cognero.

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Chapter 04: Introduction to Probability d. Are the grades on the assignments independent? ANSWER: a. .35 b. .55 c. .65 d. no 137. A marina has two party boats available for customers to rent. Historically, demand for party boats has followed this distribution shown below. The revenue per rental is $400. If a customer wants a party boat and none is available, the store gives a $150 coupon for jet ski rental. Demand 0 1 2 3 4

Relative Frequency .35 .30 .20 .10 .05

Revenue Cost 0 0 400 0 800 0 800 150 800 300

a. What is the expected demand? b. What is the expected revenue? c. What is the expected cost? d. What is the expected profit? ANSWER: a. 1.2 b. 400 c. 30 d. 370

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Chapter 05: Discrete Probability Distributions Multiple Choice 1. A numerical description of the outcome of an experiment is called a _____. a. descriptive statistic b. probability function c. variance d. random variable ANSWER: d 2. A random variable that can assume only a finite number of values is referred to as a(n) _____. a. infinite sequence b. finite sequence c. discrete random variable d. discrete probability function ANSWER: c 3. A continuous random variable may assume _____. a. any value in an interval or collection of intervals b. only integer values in an interval or collection of intervals c. only fractional values in an interval or collection of intervals d. only the positive integer values in an interval ANSWER: a 4. A marketing manager instructs his team to make 80 telephone calls to attempt to sell an insurance policy. The random variable in this experiment is the number of sales made. This random variable is a _____. a. discrete random variable b. continuous random variable c. complex random variable d. binomial random variable ANSWER: a 5. The number of customers who enter a store during one day is an example of _____. a. a continuous random variable b. a discrete random variable c. either a continuous or a discrete random variable, depending on the number of the customers d. either a continuous or a discrete random variable, depending on the gender of the customers ANSWER: b 6. Highway patrol officers measure the speed of automobiles on a highway using radar equipment. The random variable in this experiment is speed, measured in miles per hour. This random variable is a _____. a. discrete random variable b. continuous random variable c. complex random variable d. uniform random variable Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions ANSWER: b 7. The weight of an object, measured in grams, is an example of _____. a. a continuous random variable b. a discrete random variable c. either a continuous or a discrete random variable, depending on the weight of the object d. either a continuous or a discrete random variable, depending on the units of measurement ANSWER: a 8. The weight of an object, measured to the nearest gram, is an example of _____. a. a continuous random variable b. a discrete random variable c. either a continuous or a discrete random variable, depending on the weight of the object d. either a continuous or a discrete random variable, depending on the units of measurement ANSWER: b 9. A description of how the probabilities are distributed over the values the random variable can assume is called a(n) _____. a. probability distribution b. probability function c. random variable d. expected value ANSWER: a 10. Which of the following is a required condition for a discrete probability function? a. ∑f(x) = 0 b. f(x) ≥ 1 for all values of x c. f(x) < 0 d. ∑f(x) = 1 ANSWER: d 11. Which of the following is NOT a required condition for a discrete probability function? a. f(x) ≥ 0 for all values of x b. ∑f(x) = 1 c. ∑f(x) = 0 d. There are no required conditions for a discrete probability function. ANSWER: c 12. Which of the following statements about a discrete random variable and its probability distribution is true? a. Values of the random variable can never be negative. b. Negative values of f(x) are allowed if ∑f(x) = 1. c. Values of f(x) must be greater than or equal to zero. d. The values of f(x) increase to a maximum point and then decrease. ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 13. A measure of the average value of a random variable is called a(n) _____. a. variance b. standard deviation c. expected value d. median ANSWER: c 14. A weighted average of the value of a random variable, where the probability function provides weights, is known as _____. a. a probability function b. a random variable c. the expected value d. the standard deviation ANSWER: c 15. The expected value of a random variable is the _____. a. value of the random variable that should be observed on the next repeat of the experiment b. value of the random variable that occurs most frequently c. square root of the variance d. measure of the central location of a random variable ANSWER: d 16. The expected value of a discrete random variable _____. a. is the most likely or highest probability value for the random variable b. will always be one of the values x can take on, although it may not be the highest probability value for the random variable c. is the average value for the random variable over many repeats of the experiment d. cannot be calculated using Excel ANSWER: c 17. Excel's _____ function can be used to compute the expected value of a discrete random variable. a. SUMPRODUCT b. AVERAGE c. MEDIAN d. VAR ANSWER: a 18. Variance is _____. a. a measure of the average, or central value of a random variable b. a measure of the dispersion of a random variable c. the square root of the standard deviation d. the sum of the deviation of data elements from the mean ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 19. The variance is a weighted average of the _____. a. square root of the deviations from the mean b. square root of the deviations from the median c. squared deviations from the median d. squared deviations from the mean ANSWER: d 20. Excel's _____ function can be used to compute the variance of a discrete random variable. a. SUMPRODUCT b. AVERAGE c. MEDIAN d. VAR ANSWER: a 21. The standard deviation is the _____. a. variance squared b. square root of the sum of the deviations from the mean c. same as the expected value d. positive square root of the variance ANSWER: d 22. x is a random variable with the probability function: f(x) = x/6 for x = 1, 2, or 3. The expected value of x is _____. a. 0.333 b. 0.500 c. 2.000 d. 2.333 ANSWER: d 23. The number of electrical outages in a city varies from day to day. Assume that the number of electrical outages (x) in the city has the following probability distribution. x f(x) 0 0.80 1 0.15 2 0.04 3 0.01 The mean and the standard deviation for the number of electrical outages (respectively) are _____. a. 2.6 and 5.77 b. 0.26 and .577 c. 3 and .01 d. 0 and .8 ANSWER: b Exhibit 5-1 The following represents the probability distribution for the daily demand of microcomputers at a local store. Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions Demand 0 1 2 3 4

Probability .1 .2 .3 .2 .2

24. Refer to Exhibit 5-1. The expected daily demand is _____. a. 1.0 b. 2.2 c. 2 d. 4 ANSWER: b 25. Refer to Exhibit 5-1. The probability of having a demand for at least two microcomputers is _____. a. .7 b. .3 c. .4 d. 1.0 ANSWER: a Exhibit 5-2 The probability distribution for the daily sales at Michael's Co. is given below. Daily Sales ($1000s) 40 50 60 70

Probability .1 .4 .3 .2

26. Refer to Exhibit 5-2. The expected daily sales are _____. a. $55,000 b. $56,000 c. $50,000 d. $70,000 ANSWER: b 27. Refer to Exhibit 5-2. The probability of having sales of at least $50,000 is _____. a. .5 b. .10 c. .30 d. .90 ANSWER: d Exhibit 5-3 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions The probability distribution for the number of goals the Lions soccer team makes per game is given below. Number of Goals 0 1 2 3 4

Probability .05 .15 .35 .30 .15

28. Refer to Exhibit 5-3. The expected number of goals per game is _____. a. 0 b. 1 c. 2 d. 2.35 ANSWER: d 29. Refer to Exhibit 5-3. What is the probability that in a given game the Lions will score at least 1 goal? a. .20 b. .55 c. 1.0 d. .95 ANSWER: d 30. Refer to Exhibit 5-3. What is the probability that in a given game the Lions will score less than 3 goals? a. .85 b. .55 c. .45 d. .80 ANSWER: b 31. Refer to Exhibit 5-3. What is the probability that in a given game the Lions will score no goals? a. .95 b. .85 c. .75 d. .05 ANSWER: d Exhibit 5-4 A local bottling company has determined the number of machine breakdowns per month and their respective probabilities as shown below. Number of Breakdowns 0 1 2

Probability .12 .38 .25

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Chapter 05: Discrete Probability Distributions 3 4

.18 .07

32. Refer to Exhibit 5-4. The expected number of machine breakdowns per month is _____. a. 2 b. 1.70 c. 1 d. 2.50 ANSWER: b 33. Refer to Exhibit 5-4. The probability of at least 3 breakdowns in a month is _____. a. .5 b. .10 c. .30 d. .25 ANSWER: d 34. Refer to Exhibit 5-4. The probability of no breakdowns in a month is _____. a. .88 b. .00 c. .50 d. .12 ANSWER: d Exhibit 5-5 AMR is a computer-consulting firm. The number of new clients that it has obtained each month has ranged from 0 to 6. The number of new clients has the probability distribution that is shown below. Number of New Clients 0 1 2 3 4 5 6

Probability .05 .10 .15 .35 .20 .10 .05

35. Refer to Exhibit 5-5. The expected number of new clients per month is _____. a. 6 b. 0 c. 3.05 d. 21 ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 36. Refer to Exhibit 5-5. The variance is _____. a. 1.431 b. 2.0475 c. 3.05 d. 21 ANSWER: b 37. Refer to Exhibit 5-5. The standard deviation is _____. a. 1.431 b. 2.047 c. 3.05 d. 21 ANSWER: a Exhibit 5-6 Probability Distribution x 10 20 30 40

f(x) .2 .3 .4 .1

38. Refer to Exhibit 5-6. The expected value of x equals _____. a. 24 b. 25 c. 30 d. 100 ANSWER: a 39. Refer to Exhibit 5-6. The variance of x equals _____. a. 9.165 b. 84 c. 85 d. 93.33 ANSWER: b Exhibit 5-7 A sample of 2,500 people was asked how many cups of coffee they drink in the morning. You are given the following sample information. Cups of Coffee 0 1 2 3

Frequency 700 900 600 300 2,500

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Chapter 05: Discrete Probability Distributions 40. Refer to Exhibit 5-7. The expected number of cups of coffee is _____. a. 1 b. 1.2 c. 1.5 d. 1.7 ANSWER: b 41. Refer to Exhibit 5-7. The variance of the number of cups of coffee is _____. a. .96 b. .9798 c. 1 d. 2.4 ANSWER: a 42. Which of the following is a characteristic of a binomial experiment? a. At least two outcomes are possible. b. The probability of success changes from trial to trial. c. The trials are independent. d. The experiment consists of a sequence of different trials. ANSWER: c 43. In a binomial experiment, the _____. a. probability of success does not change from trial to trial b. probability of success does change from trial to trial c. probability of success could change from trial to trial, depending on the situation under consideration d. probability of success is always the same as the probability of failure ANSWER: a 44. Which of the following is NOT a characteristic of an experiment where the binomial probability distribution is applicable? a. The experiment has a sequence of n identical trials. b. Exactly two outcomes are possible on each trial. c. The trials are dependent. d. The probabilities of the outcomes do not change from one trial to another. ANSWER: c 45. Which of the following is NOT a property of a binomial experiment? a. The experiment consists of a sequence of n identical trials. b. Each outcome can be referred to as a success or a failure. c. The probabilities of the two outcomes can change from one trial to the next. d. The trials are independent. ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 46. A probability distribution showing the probability of x successes in n trials, where the probability of success does not change from trial to trial, is termed a _____. a. uniform probability distribution b. binomial probability distribution c. hypergeometric probability distribution d. normal probability distribution ANSWER: b 47. The binomial probability distribution is used with _____. a. a continuous random variable b. a discrete random variable c. any distribution, as long as it is not bell shaped d. any random variable ANSWER: b 48. If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability of two successes in 15 trials, the correct probability function to use is the _____. a. standard normal probability density function b. normal probability density function c. Poisson probability function d. binomial probability function ANSWER: d 49. In a binomial experiment, the probability of success is .06. What is the probability of two successes in seven trials? a. .0036 b. .06 c. .0554 d. .28 ANSWER: c 50. Four percent of the customers of a mortgage company default on their payments. A sample of five customers is selected. What is the probability that exactly two customers in the sample will default on their payments? a. .2592 b. .0142 c. .9588 d. .7408 ANSWER: b 51. A production process produces 2% defective parts. A sample of five parts from the production process is selected. What is the probability that the sample contains exactly two defective parts? a. .0004 b. .0038 c. .10 d. .02 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions ANSWER: b 52. Excel's BINOM.DIST function can be used to compute _____. a. bin width for histograms b. binomial probabilities c. cumulative binomial probabilities d. binomial probabilities and cumulative binomial probabilities ANSWER: d 53. Excel's BINOM.DIST function has how many inputs? a. 2 b. 3 c. 4 d. 5 ANSWER: c 54. When using Excel's BINOM.DIST function, one should choose TRUE for the fourth input if _____. a. a probability is desired b. a cumulative probability is desired c. the expected value is desired d. the correct answer is desired ANSWER: b 55. The expected value for a binomial probability distribution is _____. a. E(x) = pn(1 − n) b. E(x) = p(1 − p) c. E(x) = np d. E(x) = np(1 − p) ANSWER: c 56. The variance for the binomial probability distribution is _____. a. Var(x) = p(1 − p) b. Var(x) = np c. Var(x) = n(1 − p) d. Var(x) = np(1 − p) ANSWER: d 57. The standard deviation of a binomial distribution is _____. a. E(x) = pn(1 − n) b. E(x) = np(1 − p) c. E(x) = np d. the positive square root of the variance ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 58. Assume that you have a binomial experiment with p = 0.5 and a sample size of 100. The expected value of this distribution is _____. a. 0.50 b. 0.30 c. 50 d. .25 ANSWER: c 59. Assume that you have a binomial experiment with p = 0.4 and a sample size of 50. The variance of this distribution is _____. a. 20 b. 12 c. 3.46 d. 12.5 ANSWER: b 60. Twenty percent of the students in a class of 100 are planning to go to graduate school. The standard deviation of this binomial distribution is _____. a. 20 b. 16 c. 4 d. 2 ANSWER: c Exhibit 5-8 The student body of a large university consists of 60% female students. A random sample of 8 students is selected. 61. Refer to Exhibit 5-8. What is the random variable in this experiment? a. the 60% of female students b. the random sample of 8 students c. the number of female students out of 8 d. the student body size ANSWER: c 62. Refer to Exhibit 5-8. What is the probability that among the students in the sample exactly two are female? a. .0896 b. .2936 c. .0413 d. .0007 ANSWER: c 63. Refer to Exhibit 5-8. What is the probability that among the students in the sample at least 7 are female? a. .1064 b. .0896 c. .0168 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions d. .8936 ANSWER: a 64. Refer to Exhibit 5-8. What is the probability that among the students in the sample at least 6 are male? a. .0413 b. .0079 c. .0007 d. .0499 ANSWER: d Exhibit 5-9 Forty percent of all registered voters in a national election are female. A random sample of 5 voters is selected. 65. Refer to Exhibit 5-9. What is the random variable in this experiment? a. the 40% of female registered voters b. the random sample of 5 voters c. the number of female voters out of 5 d. the number of registered voters in the nation ANSWER: c 66. Refer to Exhibit 5-9. The probability that the sample contains 2 female voters is _____. a. .0778 b. .7780 c. .5000 d. .3456 ANSWER: d 67. Refer to Exhibit 5-9. The probability that there are no females in the sample is _____. a. .0778 b. .7780 c. .5000 d. .3456 ANSWER: a Exhibit 5-10 The probability Pete will catch fish when he goes fishing is .8. Pete is going fishing 3 days next week. 68. Refer to Exhibit 5-10. What is the random variable in this experiment? a. the .8 probability of catching fish b. the 3 days c. the number of days out of 3 that Pete catches fish d. the number of fish in the body of water ANSWER: c 69. Refer to Exhibit 5-10. The probability that Pete will catch fish on exactly 1 day is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions a. .008 b. .096 c. .104 d. .8 ANSWER: b 70. Refer to Exhibit 5-10. The probability that Pete will catch fish on 1 or fewer days is _____. a. .008 b. .096 c. .104 d. .8 ANSWER: c 71. Refer to Exhibit 5-10. The expected number of days Pete will catch fish is _____. a. .6 b. .8 c. 2.4 d. 3 ANSWER: c 72. Refer to Exhibit 5-10. The variance of the number of days Pete will catch fish is _____. a. .16 b. .48 c. .8 d. 2.4 ANSWER: b Exhibit 5-11 The random variable x is the number of occurrences of an event over an interval of 10 minutes. It can be assumed the probability of an occurrence is the same in any two time periods of an equal length. It is known that the mean number of occurrences in 10 minutes is 5.3. 73. Refer to Exhibit 5-11. The random variable x satisfies which of the following probability distributions? a. bell shaped b. Poisson c. binomial d. hypergeometric ANSWER: b 74. Refer to Exhibit 5-11. The appropriate probability distribution for the random variable is _____. a. discrete b. continuous c. either discrete or continuous, depending on how the interval is defined d. binomial Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions ANSWER: a 75. Refer to Exhibit 5-11. The expected value of the random variable x is _____. a. 2 b. 5.3 c. 10 d. 2.30 ANSWER: b 76. Refer to Exhibit 5-11. The probability there are 8 occurrences in 10 minutes is _____. a. .0241 b. .0771 c. .1126 d. .9107 ANSWER: b 77. Refer to Exhibit 5-11. The probability there are less than 3 occurrences is _____. a. .0659 b. .0948 c. .1016 d. .1239 ANSWER: c 78. The Poisson probability distribution is a _____. a. continuous probability distribution b. discrete probability distribution c. uniform probability distribution d. normal probability distribution ANSWER: b 79. The Poisson probability distribution is used with _____. a. a continuous random variable b. a discrete random variable c. either a continuous or discrete random variable d. any random variable ANSWER: b 80. When dealing with the number of occurrences of an event over a specified interval of time or space and when the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval, the appropriate probability distribution is a _____. a. binomial distribution b. Poisson distribution c. normal distribution d. hypergeometric probability distribution Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions ANSWER: b 81. Excel's POISSON.DIST function can be used to compute _____. a. bin width for histograms b. only Poisson probabilities c. only cumulative Poisson probabilities d. both Poisson probabilities and cumulative Poisson probabilities ANSWER: d 82. Excel's POISSON.DIST function has how many inputs? a. 2 b. 3 c. 4 d. 5 ANSWER: b 83. When using Excel's POISSON.DIST function, one should choose TRUE for the third input if _____. a. a probability is desired b. a cumulative probability is desired c. the expected value is desired d. the correct answer is desired ANSWER: b 84. In the textile industry, a manufacturer is interested in the number of blemishes or flaws occurring in each 100 feet of material. The probability distribution that has the greatest chance of applying to this situation is the _____. a. normal distribution b. binomial distribution c. Poisson distribution d. uniform distribution ANSWER: c 85. When sampling without replacement, the probability of obtaining a certain sample is best given by a _____. a. hypergeometric distribution b. binomial distribution c. Poisson distribution d. normal distribution ANSWER: a 86. The key difference between binomial and hypergeometric distributions is that with the hypergeometric distribution the _____. a. probability of success must be less than .5 b. probability of success changes from trial to trial c. trials are independent of each other d. random variable is continuous Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions ANSWER: b 87. Excel's HYPGEOM.DIST function can be used to compute _____. a. bin width for histograms b. only hypergeometric probabilities c. only cumulative hypergeometric probabilities d. both hypergeometric probabilities and cumulative hypergeometric probabilities ANSWER: b 88. Excel's HYPGEOM.DIST function has how many inputs? a. 2 b. 3 c. 4 d. 5 ANSWER: d 89. When using Excel's HYPGEOM.DIST function, one should choose TRUE for the fifth input if _____. a. a probability is desired b. a cumulative probability is desired c. the expected value is desired d. the correct answer is desired ANSWER: b 90. The binomial probability distribution is most symmetric when _____. a. n is 30 or greater b. n equals p c. p approaches 1 d. p equals 0.5 ANSWER: d 91. If one wanted to find the probability of 10 customer arrivals in an hour at a service station, one would generally use the _____. a. binomial probability distribution b. Poisson probability distribution c. hypergeometric probability distribution d. exponential probability distribution ANSWER: b 92. The _____ probability function is based in part on the counting rule for combinations. a. uniform b. Poisson c. hypergeometric d. exponential ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 93. To compute the probability that in a random sample of n elements, selected without replacement, we will obtain x successes, we would use the _____. a. binomial probability distribution b. Poisson probability distribution c. hypergeometric probability distribution d. exponential probability distribution ANSWER: c 94. Experimental outcomes that are based on measurement scales such as time, weight, and distance can be described by _____ random variables. a. discrete b. continuous c. uniform d. intermittent ANSWER: b 95. Which of the following properties of a binomial experiment is called the stationarity assumption? a. The experiment consists of n identical trials. b. Two outcomes are possible on each trial. c. The probability of success is the same for each trial. d. The trials are independent. ANSWER: c 96. The function used to compute the probability of x successes in n trials, when the trials are dependent, is the _____. a. binomial probability function b. Poisson probability function c. hypergeometric probability function d. exponential probability function ANSWER: c 97. The expected value of a random variable is the _____. a. most probable value b. most occurring value c. median value d. mean value ANSWER: d 98. In a binomial experiment consisting of five trials, the number of different values that x (the number of successes) can assume is _____. a. 2 b. 5 c. 6 d. 10 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions ANSWER: c 99. A binomial probability distribution with p = 0.3 is _____. a. negatively skewed b. symmetrical c. positively skewed d. bimodal ANSWER: a 100. An example of a bivariate experiment is _____. a. tossing a coin once b. rolling a pair of dice c. winning or losing a football game d. passing or failing a course ANSWER: b 101. Bivariate probabilities are often called _____. a. union probabilities b. conditional probabilities c. marginal probabilities d. joint probabilities ANSWER: d 102. To compute a binomial probability. we must know all of the following except the _____. a. probability of success b. number of elements in the population c. number of trials d. value of the random variable ANSWER: b 103. A property of the Poisson distribution is that the mean equals the _____. a. mode b. median c. variance d. standard deviation ANSWER: c Subjective Short Answer 104. The probability distribution for the rate of return on an investment is given below. Rate of Return (%) 9.5 9.8 10.0

Probability .1 .2 .3

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Chapter 05: Discrete Probability Distributions 10.2 10.6 a. b. c.

.3 .1

What is the probability that the rate of return will be at least 10%? What is the expected rate of return? What is the variance of the rate of return?

ANSWER: a. b. c.

.7 10.03 .0801

105. A random variable x has the following probability distribution: x 0 1 2 3 4 a. b.

f(x) .08 .17 .45 .25 .05

Determine the expected value of x. Determine the variance.

ANSWER: a. b.

2.02 .9396

106. For the following probability distribution: x 0 1 2 3 4 5 6 7 8 9 10 a. b. c.

f(x) .01 .02 .10 .35 .20 .11 .08 .05 .04 .03 .01

Determine E(x). Determine the variance. Determine the standard deviation.

ANSWER: a. b. c.

4.14 3.7 1.924

107. A company sells its products to wholesalers in batches of 1,000 units only. The daily demand for its product and the respective probabilities are given below. Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions Demand (Units) 0 1000 2000 3000 4000 a. b.

Probability .2 .2 .3 .2 .1

Determine the expected daily demand. Assume that the company sells its product for $3.75 per unit. What is the expected daily revenue?

ANSWER: a. b.

1800 $6,750

108. The demand for a product varies from month to month. Based on the past year's data, the following probability distribution shows MNM company's monthly demand. x Unit Demand 0 1,000 2,000 3,000 4,000 a. b.

f(x) Probability .10 .10 .30 .40 .10

Determine the expected number of units demanded per month. Each unit produced costs the company $8.00 and is sold for $10.00. How much will the company gain or lose in a month if it stocks the expected number of units demanded, but sells 2000 units?

ANSWER: a. b.

2300 Profit = $1,600

109. The probability distribution of the daily demand for a product is shown below. Demand 0 1 2 3 4 5 6 a. b.

Probability .05 .10 .15 .35 .20 .10 .05

What is the expected number of units demanded per day? Determine the variance and the standard deviation.

ANSWER: a.

3.05

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Chapter 05: Discrete Probability Distributions b.

variance = 2.0475

std. dev. = 1.431

110. The random variable x has the following probability distribution: x 0 1 2 3 4 a. b. c. d.

f(x) .25 .20 .15 .30 .10

Is this probability distribution valid? Explain and list the requirements for a valid probability distribution. Calculate the expected value of x. Calculate the variance of x. Calculate the standard deviation of x.

ANSWER: a. b. c. d.

Yes it is valid because it meets the requirements of: f(x) ≥ 0 and ∑f(x) = 1 1.8 1.86 1.364

111. The probability function for the number of insurance policies John will sell to a customer is given by f(x) = .5 − (x/6) for x = 0, 1, or 2 a. Is this a valid probability function? Explain your answer. b. What is the probability that John will sell exactly 2 policies to a customer? c. What is the probability that John will sell at least 2 policies to a customer? d. What is the expected number of policies John will sell? e. What is the variance of the number of policies John will sell? ANSWER: a. Yes it is valid because it meets the requirements of: f(x) ≥ 0 and ∑f(x) = 1 b. .167 c. .167 d. .667 e. .556

112. Thirty-two percent of the students in a management class are graduate students. A random sample of 5 students is selected. Using the binomial probability function, determine the probability that the sample contains exactly 2 graduate students. ANSWER: .322 (rounded) 113. A production process produces 2% defective parts. A sample of 5 parts from the production is selected. What is the probability that the sample contains exactly 2 defective parts? ANSWER: .0037648 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions 114. When a machine is functioning properly, 80% of the items produced are non-defective. If 3 items are examined, what is the probability that 1 is defective? Use the binomial probability function to answer this question. ANSWER: .384 115. The records of a department store show that 20% of its customers who make a purchase return the merchandise to exchange it. In the next 6 purchases, a. what is the probability that 3 customers will return the merchandise for exchange? b. what is the probability that 4 customers will return the merchandise for exchange? c. what is the probability that none of the customers will return the merchandise for exchange? ANSWER: a. b. c.

.0819 .0154 .2621

116. Ten percent of the items produced by a machine are defective. Out of 15 items chosen at random, a. what is the probability that exactly 3 items will be defective? b. what is the probability that less than 3 items will be defective? c. what is the probability that exactly 11 items will be non-defective? ANSWER: a. b. c.

.1285 .816 .0428

117. In a large university, 15% of the students are female. If a random sample of 20 students is selected, a. what is the probability that the sample contains exactly 4 female students? b. what is the probability that the sample will contain no female students? c. what is the probability that the sample will contain exactly 20 female students? d. what is the probability that the sample will contain more than 9 female students? e. what is the probability that the sample will contain fewer than 5 female students? f. what is the expected number of female students? ANSWER: a. b. c. d. e. f.

.1821 .0388 .0000 .0002 .8298 3

118. Seventy percent of the students applying to a university are accepted. What is the probability that among the next 18 applicants, a. at least 6 will be accepted? b. exactly 10 will be accepted? c. exactly 5 will be rejected? d. fifteen or more will be accepted? e. Determine the expected number of acceptances. f. Compute the standard deviation. ANSWER: a. .9988 b. .0811 c. .2017 d. .1646 e. 12.6 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions f.

1.9442

119. Twenty percent of the applications received for a position are rejected. What is the probability that among the next 14 applications, a. none will be rejected? b. all will be rejected? c. less than 2 will be rejected? d. more than 4 will be rejected? e. Determine the expected number of rejected applications and its variance. ANSWER: a. b. c. d. e.

.0440 .0000 .1979 .1298 2.8, 2.24

120. Fifty-five percent of the applications received for a credit card are accepted. Among the next 12 applications, a. what is the probability that all will be rejected? b. what is the probability that all will be accepted? c. what is the probability that exactly 4 will be accepted? d. what is the probability that fewer than 3 will be accepted? e. Determine the expected number and the variance of the accepted applications. ANSWER: a. .0001 b. .0008 c. .0762 d. .0079 e. 6.60, 2.9700 121. According to company records, 5% of all automobiles brought to Geoff’s Garage last year for a state-mandated annual inspection did not pass. Of the next 10 automobiles entering the inspection station, a. what is the probability that none will pass inspection? b. what is the probability that all will pass inspection? c. what is the probability that exactly 2 will not pass inspection? d. what is the probability that more than 3 will not pass inspection? e. what is the probability that fewer than 2will not pass inspection? f. Find the expected number of automobiles not passing inspection. g. Determine the standard deviation for the number of cars not passing inspection. ANSWER: a. b. c. d. e. f. g.

.0000 .5987 .0746 .0010 .9139 .5 .6892

122. Only 0.02% of credit card holders of a company report the loss or theft of their credit cards each month. The company has 15,000 credit cards in the city of Memphis. What is the probability that during the next month in the city of Memphis a. no one reports the loss or theft of their credit cards? b. every credit card is lost or stolen? Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions c. d. e. f.

6 people report the loss or theft of their cards? at least 9 people report the loss or theft of their cards? Determine the expected number of reported lost or stolen credit cards. Determine the standard deviation for the number of reported lost or stolen cards.

ANSWER: a. b. c. d. e. f.

.0498 .0000 .0504 .0038 3 1.73

123. Two percent of the parts produced by a machine are defective. Forty parts are selected. Define the random variable x to be the number of defective parts. a. What is the probability that exactly 3 parts will be defective? b. What is the probability that the number of defective parts will be more than 2 but fewer than 6? c. What is the probability that fewer than 4 parts will be defective? d. What is the expected number of defective parts? e. What is the variance for the number of defective parts? ANSWER: a. .0374 b. .0455 c. .9918 d. .8 e. .784 124. A manufacturing company has 5 identical machines that produce nails. The probability that a machine will break down on any given day is .1. Define a random variable x to be the number of machines that will break down in a day. a. What is the appropriate probability distribution for x? Explain how x satisfies the properties of the distribution. b. Compute the probability that 4 machines will break down. c. Compute the probability that at least 4 machines will break down. d. What is the expected number of machines that will break down in a day? e. What is the variance of the number of machines that will break down in a day? ANSWER: a.

b. c. d. e.

Binomial. x satisfies the properties since 1) there are an identical number of trials; 2) two outcomes are possible: breaks down/doesn’t break down; 3) the probability of success does not change from trial to trial; 4) the trials are independent of one another. .0004 .0004 .5 .45

125. In a large corporation, 65% of the employees are male. A random sample of 5 employees is selected. a. Define the random variable in words for this experiment. b. What is the probability that the sample contains exactly 3 male employees? c. What is the probability that the sample contains no male employees? d. What is the probability that the sample contains more than 3 female employees? e. What is the expected number of female employees in the sample? ANSWER: a.

x = the number of male employees out of 5 or y = the number of female employees out of 5

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Chapter 05: Discrete Probability Distributions b. c. d. e.

.3364 .0053 .0541 1.75

126. In a large university, 75% of students live in the dormitories. A random sample of 5 students is selected. a. Define the random variable in words for this experiment. b. What is the probability that the sample contains exactly 3 students who live in the dormitories? c. What is the probability that the sample contains no students who live in the dormitories? d. What is the probability that the sample contains more than 3 students who do not live in the dormitories? e. What is the expected number of students (in the sample) who do not live in the dormitories? ANSWER: a. b. c. d. e.

x = the number of students out of 5 who live in the dormitories or y = the number of students out of 5 who do not live in the dormitories .2637 .001 .0156 1.25

127. A production process produces 90% non-defective parts. A sample of 10 parts from the production process is selected. a. Define the random variable in words for this experiment. b. What is the probability that the sample will contain 7 non-defective parts? c. What is the probability that the sample will contain at least 4 defective parts? d. What is the probability that the sample will contain less than 5 non-defective parts? e. What is the probability that the sample will contain no defective parts? ANSWER: a. x = the number of non-defective parts out of 10 or y = the number of defective parts out of 10 b. .0574 c. .0128 d. .0001 e. .3487

128. The student body of a large university consists of 30% Business majors. A random sample of 20 students is selected. a. Define the random variable in words for this experiment. b. What is the probability that among the students in the sample at least 10 are Business majors? c. What is the probability that at least 16 are not Business majors? d. What is the probability that exactly 10 are Business majors? e. What is the probability that exactly 12 are not Business majors? ANSWER: a. x = the number of students out of 20 who are Business majors or y = the number of students out of 20 who are not Business majors b. .0480 c. .2375 d. .0308 e. .1144 Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions

129. A local university reports that 3% of its students take their general education courses on a pass/fail basis. Assume that 50 students are registered for a general education course. a. Define the random variable in words for this experiment. b. What is the expected number of students who have registered on a pass/fail basis? c. What is the probability that exactly 5 are registered on a pass/fail basis? d. What is the probability that more than 3 are registered on a pass/fail basis? e. What is the probability that less than 4 are registered on a pass/fail basis? ANSWER: a.

x = the number of students out of 50 who are registered for a general education course on a pass/fail basis or

b. c. d. e.

y = the number of students out of 50 who are not registered for a general education course on a pass/fail basis. 1.5 .0131 .0628 .9372

130. Twenty-five percent of the employees of a large company are minorities. A random sample of 7 employees is selected. a. Define the random variable in words for this experiment. b. What is the probability that the sample contains exactly 4 minorities? c. What is the probability that the sample contains fewer than 2 minorities? d. What is the probability that the sample contains exactly 1 non-minority? e. What is the expected number of minorities in the sample? f. What is the variance of the minorities? ANSWER: x = the number of minority employees out of 7 or a. y = the number of non-minority employees out of 7 b. .0577 c. .4449 d. .0013 e. 1.75 f. 1.3125

131. Twenty-five percent of all résumés received by a corporation for a management position are from females. Fifteen résumés will be received tomorrow. a. Define the random variable in words for this experiment. b. What is the probability that exactly 5 of the résumés will be from females? c. What is the probability that fewer than 3 of the résumés will be from females? d. What is the expected number of résumés from women? e. What is the variance of the number of résumés from women? ANSWER: a.

x = the number of résumés out of 15 that are from females

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Chapter 05: Discrete Probability Distributions

b. c. d. e.

or y = the number of résumés out of 15 that are from males .1651 .2361 3.75 2.8125

132. A salesperson contacts 8 potential customers per day. From past experience, we know that the probability of a potential customer making a purchase is .10. a. Define the random variable in words for this experiment. b. What is the probability the salesperson will make exactly 2 sales in a day? c. What is the probability the salesperson will make at least 2 sales in a day? d. What percentage of days will the salesperson not make a sale? e. What is the expected number of sales per day? ANSWER: a. b. c. d. e.

x = the number of sales made out of 8 contacts .1488 .1869 43.05% .8

133. An insurance company has determined that each week an average of 9 claims are filed in its Atlanta branch. What is the probability that during the next week a. exactly 7 claims will be filed? b. no claims will be filed? c. less than 4 claims will be filed? d. at least 18 claims will be filed? ANSWER: a. b. c. d.

.1171 .0001 .0212 .0053

134. John parks cars at a hotel. On average, 6.7 cars will arrive in an hour. Assume a driver's decision on whether to let John park the car does not depend upon any other person's decision. Define the random variable x to be the number of cars arriving in any hour period. a. What is the appropriate probability distribution for x? Explain how x satisfies the properties of the distribution. b. Compute the probability that exactly 5 cars will arrive in the next hour. c. Compute the probability that no more than 5 cars will arrive in the next hour. ANSWER: a. b. c.

The appropriate probability distribution for x is the Poisson probability distribution since we are interested in the number of cars arriving during a specified interval. .1385 .3406

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Chapter 05: Discrete Probability Distributions 135. The average number of calls received by a switchboard in a 30-minute period is 15. a. Define the random variable in words for this experiment. b. What is the probability that between 10:00 and 10:30 the switchboard will receive exactly 10 calls? c. What is the probability that between 10:00 and 10:30 the switchboard will receive more than 9 calls but fewer than 15 calls? d. What is the probability that between 10:00 and 10:30 the switchboard will receive fewer than 7 calls? ANSWER: a. b. c. d.

x = the number of calls received in a 30-minute period .0486 .3958 .0076

136. A life insurance company has determined that each week an average of 7 claims is filed in its Nashville branch. a. Define the random variable in words for this experiment. b. What is the probability that during the next week exactly 7 claims will be filed? c. What is the probability that during the next week no claims will be filed? d. What is the probability that during the next week fewer than 4 claims will be filed? e. What is the probability that during the next week at least 17 claims will be filed? ANSWER: a. b. c. d. e.

x = the number of claims filed in a one-week period .1490 .0009 .0818 .0010

137. General Hospital has noted that it admits an average of 8 patients per hour. a. Define the random variable in words for this experiment. b. What is the probability that during the next hour fewer than 3 patients will be admitted? c. What is the probability that during the next 2 hours exactly 8 patients will be admitted? ANSWER: a. b. c.

x = the number of patients admitted per hour .0137 .0120

138. Shoppers enter Hamilton Place Mall at an average of 120 per hour. a. Define the random variable in words for this experiment. b. What is the probability that exactly 5 shoppers will enter the mall between noon and 1:00 P.M.? What is the probability that exactly 5 shoppers will enter the mall between noon and 12:05 c. P.M.? d. What is the probability that at least 35 shoppers will enter the mall between 5:00 and 5:10 P.M.? ANSWER: a. b. c. d.

x = the number of shoppers entering the mall in a one-hour period .0000 .0378 .0015

139. Compute the hypergeometric probabilities for the following values of n and x. Assume N = 8 and r = 5. Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions a. n = 5, x = 2 b. n = 6, x = 4 c. n = 3, x = 0 d. n = 3, x = 3 ANSWER: a. .1786 b. .5357 c. .01786 d. .1786 140. A retailer of electronic equipment received 6 HDTVs from the manufacturer. Three of the HDTVs were damaged in the shipment. The retailer sold 2 HDTVs to 2 customers. a Can a binomial formula be used for the solution of the above problem? b. What kind of probability distribution does the above satisfy? c. What is the probability that both customers received damaged HDTVs? d. What is the probability that 1 of the 2 customers received a defective HDTV? ANSWER: a. b. c. d.

No, in a binomial experiment, trials are independent of each other. Hypergeometric probability distribution .2 .6

141. Waters’ Edge is a clothing retailer that promotes its products via catalog and accepts customer orders by all of the conventional ways including the Internet. The company has gained a competitive advantage by collecting data about its operations and the customer each time an order is processed. Among the data collected with each order are: a. number of items ordered, b. total shipping weight of the order, c. whether or not all items ordered were available in inventory, d. time taken to process the order, e. customer’s number of prior orders in the last 12 months, and f. method of payment. For each of the six aforementioned variables, identify which of the variables are discrete and which are continuous. ANSWER: Discrete: number of items ordered, whether or not all items ordered were available in inventory,

customer’s number of prior orders in the last 12 months, method of payment Continuous: total shipping weight of the order, time taken to process the order 142. June's Specialty Shop sells designer original dresses. On 10% of her dresses, June makes a profit of $10, on 20% of her dresses she makes a profit of $20, on 30% of her dresses she makes a profit of $30, and on 40% of her dresses she makes a profit of $40. On a given day, the probability of June having no customers is .05, of 1 customer is .10, of 2 customers is .20, of 3 customers is .35, of 4 customers is .20, and of 5 customers is .10. a. What is the expected profit June earns on the sale of a dress? b. June's daily operating cost is $40 per day. Find the expected net profit June earns per day. (Hint: To find the expected daily gross profit, multiply the expected profit per dress by the expected number of customers per day.) c. June is considering moving to a larger store. She estimates that doing so will double the expected number of customers. If the larger store will increase her operating costs to $100 per day, should she make the move? ANSWER: a. $30 b. $45.50 c. Yes; new daily net profit expected is $71. 143. The salespeople at Gold Key Realty sell up to 9 houses per month. The probability distribution of a salesperson selling x houses in a month is as follows: Sales (x)

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Chapter 05: Discrete Probability Distributions Probability f (x) .05 .10 .15 .20 .15 .10 .10 .05 .05 .05 a. What is the mean for the number of houses sold by a salesperson per month? b. What is the standard deviation for the number of houses sold by a salesperson per month? c. Any salesperson selling more houses than the amount equal to the mean plus two standard deviations receives a bonus. How many houses per month must a salesperson sell to receive a bonus? ANSWER: a. mean = 3.9 b. standard deviation = 2.34 c. 8.58, or 9 houses 144. Sandy's Pet Center grooms large and small dogs. It takes Sandy 40 minutes to groom a small dog and 70 minutes to groom a large dog. Large dogs account for 20% of Sandy's business. Sandy has 5 appointments tomorrow. a. What is the probability that all 5 appointments tomorrow are for small dogs? b. What is the probability that 2 of the appointments tomorrow are for large dogs? c. What is the expected amount of time to finish all 5 dogs tomorrow? ANSWER: a. .3277 b. .2048 c. 230 minutes 145. Ralph's Gas Station is running a giveaway promotion. With every fill-up of gasoline, Ralph gives out a lottery ticket that has a 25% chance of being a winning ticket. Customers who collect 4 winning lottery tickets are eligible for the "BIG SPIN" for large payoffs. What is the probability of qualifying for the big spin if a customer fills up: (a) 3 times; (b) 4 times; (c) 7 times? ANSWER: a. 0 b. .0039 c. .0705 146. The number of customers at Winkies Donuts between 8:00 A.M. and 9:00 A.M. is believed to follow a Poisson distribution with a mean of 2 customers per minute. a. During a randomly selected one-minute interval during this time frame, what is the probability of 6 customers arriving to Winkies? b. What is the probability at least 2 minutes elapse between customer arrivals? ANSWER: a. .0120 b. .0183 147. During lunchtime, customers arrive at Bob's Drugs according to a Poisson distribution with μ = 4 per minute. During a one-minute interval, determine the following probabilities: a. no arrivals; b. 1 arrival; c. 2 arrivals; and, d. 3 or more arrivals. e. What is the probability of 2 arrivals in a two-minute period? ANSWER: a. .0183 b. .0733 c. .1465 d. .7619 e. .0107 148. Telephone calls arrive at the Global Airline reservation office in Louisville according to a Poisson distribution with a mean of 1.2 calls per minute. a. What is the probability of receiving exactly 1 call during a one-minute interval? b. What is the probability of receiving at most 2 calls during a one-minute interval? c. What is the probability of receiving at least 2 calls during a one-minute interval? d. What is the probability of receiving exactly 4 calls during a five-minute interval? Copyright Cengage Learning. Powered by Cognero.

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Chapter 05: Discrete Probability Distributions e. What is the probability that at most 2 minutes elapse between 1 call and the next? ANSWER: a. .36 b. .88 c. .34 d. .135 e. .9093 149. Before dawn Josh hurriedly packed some clothes for a job interview trip while his roommate was still sleeping. He reached in his disorganized sock drawer where there were 5 black socks and 5 navy blue socks, although they appeared to be the same color in the dimly lighted room. Josh grabbed 6 socks, hoping that at least 2, and preferably 4, of them were black to match the gray suit he had packed. With no time to spare, he then raced to the airport to catch his plane. a. What is the probability Josh packed at least 2 black socks so that he will be dressed appropriately the day of his interview? b. What is the probability Josh packed at least 4 black socks so that he will be dressed appropriately the latter day of his trip as well? ANSWER: a. .976 b. .262 150. Consider a Poisson probability distribution in a process with an average of 3 flaws every 100 feet. Find the probability of a. no flaws in 100 feet. b. 2 flaws in 100 feet. c. 1 flaw in 150 feet. d. 3 or 4 flaws in 150 feet.

ANSWER: a. .0498 b. .2240 c. .0500 d. .3585

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Chapter 06: Continuous Probability Distributions Multiple Choice 1. If arrivals follow a Poisson probability distribution, the time between successive arrivals must follow a(n) _____ probability distribution. a. Poisson b. normal c. uniform d. exponential ANSWER: d 2. Whenever the probability is proportional to the length of the interval in which the random variable can assume a value, the random variable follows a(n) _____ distribution. a. uniform b. normal c. exponential d. Poisson ANSWER: a 3. There is a lower limit but no upper limit for a random variable that follows the _____ probability distribution. a. uniform b. normal c. exponential d. binomial ANSWER: c 4. The form of the continuous uniform probability distribution is _____. a. triangular b. rectangular c. bell-shaped d. a series of vertical lines ANSWER: b 5. The mean, median, and mode have the same value for which of the following probability distributions? a. uniform b. normal c. exponential d. Poisson ANSWER: b 6. The probability distribution that can be described by just one parameter is the _____ distribution. a. uniform b. normal c. exponential d. continuous Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions ANSWER: c 7. A continuous random variable may assume _____. a. all values in an interval or collection of intervals b. only integer values in an interval or collection of intervals c. only fractional values in an interval or collection of intervals d. all the positive integer values in an interval ANSWER: a 8. For a continuous random variable x, the probability density function f(x) represents _____. a. the probability at a given value of x b. the area under the curve at x c. the distribution of a given value of x d. the height of the function at x ANSWER: d 9. What type of function defines the probability distribution of ANY continuous random variable? a. Normal distribution function b. Uniform distribution function c. Exponential distribution function d. Probability density function ANSWER: d 10. For any continuous random variable, the probability that the random variable takes on exactly a specific value is _____. a. 1 b. .50 c. any value between 0 and 1 d. 0 ANSWER: d 11. The uniform probability distribution is used with _____. a. a continuous random variable b. a discrete random variable c. a normally distributed random variable d. any random variable ANSWER: a 12. A uniform probability distribution is a continuous probability distribution where the probability that the random variable assumes a value in any interval of equal length is _____. a. different for each interval b. the same for each interval c. either different or the same depending on the magnitude of the standard deviation d. always 0 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions ANSWER: b 13. For a uniform probability density function, the height of the function _____. a. cannot be larger than1 b. is the same for each value of x c. is different for various values of x d. decreases as x increases ANSWER: b 14. A continuous random variable is uniformly distributed between a and b. The probability density function between a and b is _____. a. 0 b. (a − b) c. (b − a) d. 1/(b − a) ANSWER: d 15. The probability density function for a uniform distribution ranging between 2 and 6 is _____. a. 4 b. undefined c. any positive value d. .25 ANSWER: d 16. The random variable x is known to be uniformly distributed between 70 and 90. The probability of x having a value between 80 and 95 is _____. a. .75 b. .5 c. .05 d. 1 ANSWER: b 17. The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability density function has what value in the interval between 6 and 10? a. .25 b. 4.00 c. 5.00 d. 0 ANSWER: a 18. The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability of assembling the product in 7 to 9 minutes is _____. a. 0 b. .50 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions c. .20 d. 1 ANSWER: b 19. The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability of assembling the product in less than 6 minutes is _____. a. 0 b. .50 c. .15 d. 1 ANSWER: a 20. The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability of assembling the product in 7 minutes or more is _____. a. .25 b. .75 c. 0 d. 1 ANSWER: b 21. The assembly time for a product is uniformly distributed between 6 and 10 minutes. The expected assembly time (in minutes) is _____. a. 16 b. 2 c. 8 d. 4 ANSWER: c 22. The assembly time for a product is uniformly distributed between 6 and 10 minutes. The standard deviation of assembly time (in minutes) is approximately _____. a. .33 b. .13 c. 16 d. 1.15 ANSWER: d 23. A normal probability distribution _____. a. is a continuous probability distribution b. is a discrete probability distribution c. can be either continuous or discrete d. always has a standard deviation of 1 ANSWER: a 24. Which of the following is NOT a characteristic of the normal probability distribution? Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. The mean, median, and mode are equal. b. The mean of the distribution can be negative, zero, or positive. c. The distribution is symmetrical. d. The standard deviation must be 1. ANSWER: d 25. Which of the following is NOT a characteristic of the normal probability distribution? a. The graph of the curve is the shape of a rectangle. b. The total area under the curve is always equal to 1. c. The random variable assumes a value within plus or minus three standard deviations of its mean 99.72% of the time. d. The mean is equal to the median, which is also equal to the mode. ANSWER: a 26. The highest point of a normal curve occurs at _____. a. one standard deviation to the right of the mean b. two standard deviations to the right of the mean c. approximately three standard deviations to the right of the mean d. the mean ANSWER: d 27. If the mean of a normal distribution is negative, _____. a. the standard deviation must also be negative b. the variance must also be negative c. a mistake has been made in the computations, because the mean of a normal distribution cannot be negative d. the median and mode must also be negative ANSWER: d 28. Larger values of the standard deviation result in a normal curve that is _____. a. shifted to the right b. shifted to the left c. narrower and more peaked d. wider and flatter ANSWER: d 29. A standard normal distribution is a normal distribution with _____. a. a mean of 1 and a standard deviation of 0 b. a mean of 0 and a standard deviation of 1 c. any mean and a standard deviation of 1 d. any mean and any standard deviation ANSWER: b 30. In a standard normal distribution, the range of values of z is from _____. a. minus infinity to infinity Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions b. –1 to 1 c. 0 to 1 d. –3.09 to 3.09 ANSWER: a 31. For a standard normal distribution, a negative value of z indicates _____. a. a mistake has been made in computations, because z is always positive b. the area corresponding to the z is negative c. the z is to the left of the mean d. the z is to the right of the mean ANSWER: c 32. For the standard normal probability distribution, the area to the left of the mean is _____. a. –0.5 b. 0.5 c. any value between 0 and 1 d. 1 ANSWER: b 33. The mean of a standard normal probability distribution _____. a. is always equal to 1 b. can be any value as long as it is positive c. can be any value d. is always equal to 0 ANSWER: d 34. The standard deviation of a standard normal distribution _____. a. is always equal to 0 b. is always equal to 1 c. can be any positive value d. can be any value ANSWER: b 35. For a standard normal distribution, the probability of z ≤ 0 is _____. a. 0 b. –.5 c. .5 d. 1 ANSWER: c 36. Assume z is a standard normal random variable. Then P(1.20 ≤ z ≤ 1.85) equals _____. a. .4678 b. .3849 c. .8527 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions d. .0829 ANSWER: d 37. Assume z is a standard normal random variable. Then P(1.05 ≤ z ≤ 2.13) equals _____. a. .8365 b. .1303 c. .4834 d. .0687 ANSWER: b 38. Assume z is a standard normal random variable. Then P(1.41 < z < 2.85) equals _____. a. .4772 b. .3413 c. .8285 d. .0771 ANSWER: d 39. Assume z is a standard normal random variable. Then P(–1.96 ≤ z ≤ –1.4) equals _____. a. .8942 b. .0558 c. .475 d. .4192 ANSWER: b 40. Assume z is a standard normal random variable. Then P(–1.20 ≤ z ≤ 1.50) equals _____. a. .0483 b. .3849 c. .4332 d. .8181 ANSWER: d 41. Assume z is a standard normal random variable. Then P(–1.5 ≤ z ≤ 1.09) equals _____. a. .4322 b. .3621 c. .7953 d. .0711 ANSWER: c 42. Assume z is a standard normal random variable. Then P(z ≥ 2.11) equals _____. a. .4821 b. .9821 c. .5 d. .0174 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions 43. Given that z is a standard normal random variable, what is the value of z if the area to the right of z is .1112? a. .3888 b. 1.22 c. 2.22 d. 3.22 ANSWER: b 44. Given that z is a standard normal random variable, what is the value of z if the area to the right of z is .1401? a. 1.08 b. .1401 c. 2.16 d. –1.08 ANSWER: a 45. Given that z is a standard normal random variable, what is the value of z if the area to the left of z is .9382? a. 1.8 b. 1.54 c. 2.1 d. 1.77 ANSWER: b 46. Assume z is a standard normal random variable. What is the value of z if the area between –z and z is .754? a. .377 b. .123 c. 2.16 d. 1.16 ANSWER: d 47. Assume z is a standard normal random variable. What is the value of z if the area to the right of z is .9803? a. –2.06 b. .4803 c. .0997 d. 3.06 ANSWER: a 48. For a standard normal distribution, the probability of obtaining a z value between –2.4 and –2.0 is _____. a. .4000 b. .0146 c. .0400 d. .5000 ANSWER: b 49. For a standard normal distribution, the probability of obtaining a z value of less than 1.6 is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. .1600 b. .0160 c. .0016 d. .9452 ANSWER: d 50. For a standard normal distribution, the probability of obtaining a z value between –1.9 and 1.7 is _____. a. .9267 b. .4267 c. 1.4267 d. .5000 ANSWER: a 51. Suppose x is a normally distributed random variable with a mean of 8 and a standard deviation of 4. The probability that x is between 1.48 and 15.56 is _____. a. .0222 b. .4190 c. .5222 d. .9190 ANSWER: d 52. Suppose x is a normally distributed random variable with a mean of 5 and a variance of 4. The probability that x is greater than 10.52 is _____. a. .0029 b. .0838 c. .4971 d. .9971 ANSWER: a 53. Suppose x is a normally distributed random variable with a mean of 12 and a standard deviation of 3. The probability that x equals 19.62 is _____. a. 0 b. .0055 c. .4945 d. .9945 ANSWER: a 54. Suppose x is a normally distributed random variable with a mean of 22 and a standard deviation of 5. The probability that x is less than 9.7 is _____. a. 0 b. .4931 c. .0069 d. .9931 ANSWER: c Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions 55. The ages of students at a university are normally distributed with a mean of 21. What percentage of the student body is at least 21 years old? a. It could be any value, depending on the magnitude of the standard deviation. b. 50% c. 21% d. 1.96% ANSWER: b Exhibit 6-1 Consider the continuous random variable x, which has a uniform distribution over the interval from 20 to 28. 56. Refer to Exhibit 6-1. The probability density function has what value in the interval between 20 and 28? a. 0 b. .050 c. .125 d. 1 ANSWER: c 57. Refer to Exhibit 6-1. The probability that x will take on a value between 21 and 25 is_____. a. .125 b. .25 c. .5 d. 1 ANSWER: c 58. Refer to Exhibit 6-1. The probability that x will take on a value of at least 26 is _____. a. 0 b. .125 c. .250 d. 1 ANSWER: c 59. Refer to Exhibit 6-1. The mean of x is _____. a. 0 b. .125 c. 23 d. 24 ANSWER: d 60. Refer to Exhibit 6-1. The variance of x is approximately _____. a. 2.309 b. 5.333 c. 32 d. .667 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions ANSWER: b Exhibit 6-2 The time it takes for a college student to travel between her home and her college is uniformly distributed between 40 and 90 minutes. 61. Refer to Exhibit 6-2. What is the random variable in this experiment? a. Distance from home to college b. 40 minutes c. 90 minutes d. Travel time ANSWER: d 62. Refer to Exhibit 6-2. The probability that she will finish her trip in 80 minutes or less is _____. a. .02 b. .8 c. .2 d. 1 ANSWER: b 63. Refer to Exhibit 6-2. The probability that her trip will take longer than 60 minutes is _____. a. 1 b. .40 c. .02 d. .60 ANSWER: d 64. Refer to Exhibit 6-2. The probability that her trip will take exactly 50 minutes is _____. a. 0 b. .02 c. .06 d. .20 ANSWER: a Exhibit 6-3 The weight of football players is normally distributed with a mean of 200 pounds and a standard deviation of 25 pounds. 65. Refer to Exhibit 6-3. What is the random variable in this experiment? a. Weight of football players b. 200 pounds c. 25 pounds d. Height of football players ANSWER: a 66. Refer to Exhibit 6-3. The probability of a player weighing more than 241.25 pounds is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. .4505 b. .0495 c. .9505 d. .9010 ANSWER: b 67. Refer to Exhibit 6-3. The probability of a player weighing less than 250 pounds is _____. a. .4772 b. .9772 c. .0528 d. .5000 ANSWER: b 68. Refer to Exhibit 6-3. What percent of players weigh between 180 and 220 pounds? a. 34.13% b. 68.26% c. 0.3413% d. 57.62% ANSWER: d 69. Refer to Exhibit 6-3. What is the minimum weight of the middle 95% of the players? a. 196 b. 151 c. 249 d. 205 ANSWER: b Exhibit 6-4 The starting salaries of individuals with an MBA degree are normally distributed with a mean of $40,000 and a standard deviation of $5,000. 70. Refer to Exhibit 6-4. What is the random variable in this experiment? a. Starting salaries b. Type of degree c. $40,000 d. $5,000 ANSWER: a 71. Refer to Exhibit 6-4. What is the probability that a randomly selected individual with an MBA degree will get a starting salary of at least $30,000? a. .4772 b. .9772 c. .0228 d. .5000 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions ANSWER: b 72. Refer to Exhibit 6-4. What is the probability that a randomly selected individual with an MBA degree will get a starting salary of at least $47,500? a. .4332 b. .9332 c. .0668 d. .5000 ANSWER: c 73. Refer to Exhibit 6-4. What percentage of MBAs will have starting salaries of $34,000 to $46,000? a. 38.49% b. 38.59% c. 50% d. 76.98% ANSWER: d Exhibit 6-5 The weight of items produced by a machine is normally distributed with a mean of 8 ounces and a standard deviation of 2 ounces. 74. Refer to Exhibit 6-5. What is the random variable in this experiment? a. Weight of items produced by a machine b. 8 ounces c. 2 ounces d. Accuracy of the machine ANSWER: a 75. Refer to Exhibit 6-5. What is the probability that a randomly selected item will weigh more than 10 ounces? a. .3413 b. .8413 c. .1587 d. .5000 ANSWER: c 76. Refer to Exhibit 6-5. What is the probability that a randomly selected item will weigh between 11 and 12 ounces? a. .4772 b. .4332 c. .9104 d. .0440 ANSWER: d 77. Refer to Exhibit 6-5. What percentage of items will weigh at least 11.7 ounces? a. 46.78% b. 96.78% Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions c. 3.22% d. 53.22% ANSWER: c 78. Refer to Exhibit 6-5. What percentage of items will weigh between 6.4 and 8.9 ounces? a. .1145 b. .2881 c. .1736 d. .4617 ANSWER: d 79. Refer to Exhibit 6-5. What is the probability that a randomly selected item weighs exactly 8 ounces? a. .5 b. 1.0 c. .3413 d. 0 ANSWER: d Exhibit 6-6 The life expectancy of a particular brand of tire is normally distributed with a mean of 40,000 and a standard deviation of 5,000 miles. 80. Refer to Exhibit 6-6. What is the random variable in this experiment? a. Life expectancy of this brand of tire b. 5,000 miles c. 40,000 miles d. Brand of tire ANSWER: a 81. Refer to Exhibit 6-6. What is the probability that a randomly selected tire will have a life of at least 30,000 miles? a. .4772 b. .9772 c. .0228 d. .7761 ANSWER: b 82. Refer to Exhibit 6-6. What is the probability that a randomly selected tire will have a life of at least 47,500 miles? a. .4332 b. .9332 c. .0668 d. .1459 ANSWER: c 83. Refer to Exhibit 6-6. What percentage of tires will have a life of 34,000 to 46,000 miles? a. 38.49% Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions b. 76.98% c. 50% d. 56.3% ANSWER: b 84. Refer to Exhibit 6-6. What is the probability that a randomly selected tire will have a life of exactly 47,500 miles? a. .4332 b. .9332 c. .0668 d. 0 ANSWER: d Exhibit 6-7 f(x) = (1/10) e-x/10

x≥0

85. Refer to Exhibit 6-7. The mean of x is _____. a. .10 b. 10 c. 100 d. 1,000 ANSWER: b 86. Refer to Exhibit 6-7. The probability that x is less than 5 is _____. a. .6065 b. .0606 c. .3935 d. .9393 ANSWER: c 87. Refer to Exhibit 6-7. The probability that x is between 3 and 6 is _____. a. .4512 b. .1920 c. .2592 d. .6065 ANSWER: b 88. Excel's NORM.S.DIST function can be used to compute _____. a. cumulative probabilities for a standard normal z value b. the standard normal z value given a cumulative probability c. cumulative probabilities for a normally distributed x value d. the normally distributed x value given a cumulative probability ANSWER: a 89. Excel's NORM.S.INV function can be used to compute _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. cumulative probabilities for a standard normal z value b. the standard normal z value given a cumulative probability c. cumulative probabilities for a normally distributed x value d. the normally distributed x value given a cumulative probability ANSWER: b 90. Excel's NORM.DIST function can be used to compute _____. a. cumulative probabilities for a standard normal z value b. the standard normal z value given a cumulative probability c. cumulative probabilities for a normally distributed x value d. the normally distributed x value given a cumulative probability ANSWER: c 91. Excel's NORM.INV function can be used to compute _____. a. cumulative probabilities for a standard normal z value b. the standard normal z value given a cumulative probability c. cumulative probabilities for a normally distributed x value d. the normally distributed x value given a cumulative probability ANSWER: d 92. A continuous probability distribution that is useful in describing the time, or space, between occurrences of an event is a(n) _____ probability distribution. a. normal b. uniform c. exponential d. Poisson ANSWER: c 93. The exponential probability distribution is used with _____. a. a discrete random variable b. a continuous random variable c. any probability distribution with an exponential term d. an approximation of the binomial probability distribution ANSWER: b 94. An exponential probability distribution _____. a. is a continuous distribution b. is a discrete distribution c. must be uniformly distributed d. must be normally distributed ANSWER: a 95. Excel's EXPON.DIST function can be used to compute _____. a. exponents Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions b. exponential probabilities only c. cumulative exponential probabilities only d. both exponential and cumulative exponential probabilities ANSWER: d 96. Excel's EXPON.DIST function has how many inputs? a. 2 b. 3 c. 4 d. 5 ANSWER: b 97. When using Excel's EXPON.DIST function, one should choose TRUE for the third input if _____ is desired. a. a probability b. a cumulative probability c. the expected value d. the correct answer ANSWER: b 98. A property of the exponential distribution is that the mean equals the _____. a. mode b. median c. variance d. standard deviation ANSWER: d 99. The skewness measure for exponential distributions is _____. a. 0 b. 1 c. 2 d. 3 ANSWER: c 100. Exponential distributions _____. a. are skewed to the left b. are skewed to the right c. can be skewed to the left or right d. are not skewed ANSWER: b 101. About 95.4% of the values of a normal random variable are within approximately how many standard deviations of its mean? a. ±1.7 b. ±2 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions c. ±2.5 d. ±3 ANSWER: b 102. All of the following distributions are symmetric EXCEPT the _____ distribution. a. uniform b. normal c. exponential d. standard normal ANSWER: c Subjective Short Answer 103. A random variable x is uniformly distributed between 45 and 150. a. Determine the probability of x = 48. b. What is the probability of x ≤ 60? c. What is the probability of x ≥ 50? d. Determine the expected vale of x and its standard deviation. ANSWER: a. b. c. d.

0 .1429 .9524 97.5, 30.31

104. The price of a bond is uniformly distributed between $80 and $85. a. What is the probability that the bond price will be at least $83? b. What is the probability that the bond price will be between $81 and $90? c. Determine the expected price of the bond. d. Compute the standard deviation for the bond price. ANSWER: a. .4 b. .8 c. $82.50 d. $1.44 105. The price of a stock is uniformly distributed between $30 and $40. a. What is the probability that the stock price will be more than $37? b. What is the probability that the stock price will be less than or equal to $32? c. What is the probability that the stock price will be between $34 and $38? d. Determine the expected price of the stock. e. Determine the standard deviation for the stock price. ANSWER: a. .3 b. .2 c. .4 d. $35 e. $2.89 106. The time it takes to hand carve a guitar neck is uniformly distributed between 110 and 190 minutes. Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. What is the probability that a guitar neck can be carved between 95 and 165 minutes? b. What is the probability that the guitar neck can be carved between 120 and 200 minutes? c. Determine the expected completion time for carving the guitar neck. d. Compute the standard deviation. ANSWER: a. .6875 b. .875 c. 150 minutes d. 23.09 minutes 107. The length of time it takes students to complete a statistics examination is uniformly distributed and varies between 40 and 60 minutes. a. Find the mathematical expression for the probability density function. b. Compute the probability that a student will take between 45 and 50 minutes to complete the examination. c. Compute the probability that a student will take no more than 40 minutes to complete the examination. d. What is the expected amount of time it takes a student to complete the examination? e. What is the variance for the amount of time it takes a student to complete the examination? ANSWER: a. b. c. d. e.

f(x) = .05 for 40 ≤ x ≤ 60; 0 elsewhere .25 0 50 minutes 33.33 (minutes)2

108. The advertised weight on a can of soup is 10 ounces. The actual weight in the can follows a uniform distribution and varies between 9.3 and 10.3 ounces. a. Give the mathematical expression for the probability density function. b. What is the probability that a can of soup will weigh between 9.4 and 10.3 ounces? c. What is the mean weight of a can of soup? d. What is the standard deviation of the weight? ANSWER: a. b. c. d.

f(x) = 1.000 for 9.3 ≤ x ≤ 10.3; 0 elsewhere .90 9.8 ounces .289 ounce

109. The length of time patients must wait to see a doctor in a local clinic is uniformly distributed between 15 minutes and 2 1/2 hours. a. Define the random variable in words. b. What is the probability of a patient waiting exactly 50 minutes? c. What is the probability that a patient would have to wait between 45 minutes and 2 hours? d. Compute the probability that a patient would have to wait over 2 hours. e. Determine the expected waiting time and its standard deviation. ANSWER: a. the length of time patients must wait b. 0 c. .556 d. .222 e. 82.5 minutes, 38.97 minutes 110. For the standard normal distribution, determine the probability of obtaining a z value _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. b. c. d. e.

greater than0 between –2.34 and –2.55 less than 1.86 between –1.95 and 2.7 between 1.5 and 2.75

ANSWER: a. b. c. d. e.

.5000 .0042 .9686 .9709 .0638

111. Assume z is a standard normal random variable. Compute the following probabilities. a. P(–1.33 ≤ z ≤ 1.67) b. P(1.23 ≤ z ≤ 1.55) c. P(z ≥ 2.32) d. P(z ≥ –2.08) e. P(z ≥ –1.08) ANSWER: a. b. c. d. e.

.8607 .0487 .0102 .9812 .8599

112. Suppose z is a standard normal random variable. Compute the following probabilities. a. P(–1.23 ≤ z ≤ 2.58) b. P(1.83 ≤ z ≤ 1.96) c. P(z ≥ 1.32) d. P(z ≤ 2.52) e. P(z ≥ –1.63) f. P(z ≤ –1.38) g. P(–2.37 ≤ z ≤ –1.54) h. P(z = 2.56) ANSWER: a. b. c. d. e. f. g. h.

.8858 .0086 .0934 .9941 .9484 .0838 .0529 0

113. Suppose z is a standard normal variable. Find the value of z in the following. a. The area between 0 and z is .4678. b. The area to the right of z is .1112. c. The area to the left of z is .8554 Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions d. e. f.

The area between –z and z is .754. The area to the left of –z is .0681. The area to the right of –z is .9803.

ANSWER: a. b. c. d. e. f.

1.85 1.22 1.06 1.16 1.49 2.06

114. The miles per gallon obtained by the 1995 model Q cars is normally distributed with a mean of 22 miles per gallon and a standard deviation of 5 miles per gallon. a. What is the probability that a car will get between 13.35 and 35.1 miles per gallon? b. What is the probability that a car will get more than 29.6 miles per gallon? c. What is the probability that a car will get less than 21 miles per gallon? d. What is the probability that a car will get exactly 22 miles per gallon? ANSWER: a. b. c. d.

.9538 .0643 .4207 0

115. The salaries at a corporation are normally distributed with an average salary of $19,000 and a standard deviation of $4,000. a. What is the probability that an employee will have a salary between $12,520 and $13,480? b. What is the probability that an employee will have a salary more than $11,880? c. What is the probability that an employee will have a salary less than $28,440? ANSWER: a. .0312 b. .9625 c. .9909 116. A major department store has determined that its customers charge an average of $500 per month, with a standard deviation of $80. Assume the amounts of charges are normally distributed. a. What percentage of customers charges more than $380 per month? b. What percentage of customers charges less than $340 per month? c. What percentage of customers charges between $644 and $700 per month? ANSWER: a. 93.32% b. 2.28% c. 2.97% 117. The contents of soft drink bottles are normally distributed with a mean of 12 ounces and a standard deviation of 1 ounce. a. What is the probability that a randomly selected bottle will contain more than 10 ounces of soft drink? b. What is the probability that a randomly selected bottle will contain between 9.5 and 11 ounces? c. What percentage of the bottles will contain less than 10.5 ounces of soft drink? ANSWER: a.

.9772

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Chapter 06: Continuous Probability Distributions b. c.

.1525 6.68%

118. The life expectancy of computer terminals is normally distributed with a mean of 4 years and a standard deviation of 10 months. a. What is the probability that a randomly selected terminal will last more than 5 years? b. What percentage of terminals will last between 5 and 6 years? c. What percentage of terminals will last less than 4 years? d. What percentage of terminals will last between 2.5 and 4.5 years? e. If the manufacturer guarantees the terminals for 3 years (and will replace them if they malfunction), what percentage of terminals will be replaced? ANSWER: a. b. c. d. e.

.1151 10.69% 50% 68.98% 11.51%

119. Scores on a recent national statistics exam were normally distributed with a mean of 80 and a standard deviation of 6. a. What is the probability that a randomly selected exam will have a score of at least 71? b. What percentage of exams will have scores between 89 and 92? c. If the top 2.5% of test scores receive merit awards, what is the lowest score eligible for an award? d. If there were 334 exams with scores of at least 89, how many students took the exam? ANSWER: a. b. c. d.

.9332 4% 91.76 5,000

120. The average starting salary for this year's graduates at a large university (LU) is $30,000 with a standard deviation of $8,000. Furthermore, it is known that the starting salaries are normally distributed. a. What is the probability that a randomly selected LU graduate will have a starting salary of at least $30,400? b. Individuals with starting salaries of less than $15,600 receive a low income tax break. What percentage of the graduates will receive the tax break? c. What are the minimum and the maximum starting salaries of the middle 95% of the LU graduates? d. If 303 of the recent graduates have salaries of at least $43,120, how many students graduated this year from this university? ANSWER: a. b. c. d.

.4801 3.59% minimum = $14,320; maximum = $45,680 6,000

121. The weights of items produced by a company are normally distributed with a mean of 4.5 ounces and a standard deviation of 0.3 ounces. Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions a. b. c. d.

What is the probability that a randomly selected item from the production will weigh at least 4.14 ounces? What percentage of the items weighs between 4.8 and 5.04 ounces? Determine the minimum weight of the heaviest 5% of all items produced. If 27,875 items of the entire production weigh at least 5.01 ounces, how many items have been produced?

ANSWER: a. b. c. d.

.8849 12.28% 4.99 ounces 625,000

122. The life expectancy of Timely brand watches is normally distributed with a mean of four years and a standard deviation of eight months. a. What is the probability that a randomly selected watch will be in working condition for more than five years? b. The company has a three-year warranty period on its watches. What percentage of its watches will be in operating condition after the warranty period? c. What are the minimum and maximum life expectancies of the middle 95% of the watches? d. Ninety-five percent of the watches will have a life expectancy of at least how many months? ANSWER: a. b. c. d.

.0668 93.32% minimum = 32.32 months; maximum = 63.68 months 34.84 months

123. Cans of tomato paste produced by a company are normally distributed with a mean of 6 ounces and a standard deviation of 0.3 ounces. a. What percentage of all cans produced contains more than 6.51 ounces of tomato paste? b. What percentage of all cans produced contains less than 5.415 ounces of tomato paste? c. What percentage of cans contains between 5.46 and 6.495 ounces of tomato paste? d. Ninety-five percent of cans will contain at least how many ounces of tomato paste? e. What percentage of cans contains between 6.3 and 6.6 ounces of tomato paste? ANSWER: a. 4.46% b. 2.56% c. 91.46% d. 5.5067 ounces e. 13.59% 124. A professor at a local university noted that the grades of her students were normally distributed with a mean of 78 and a standard deviation of 10. a. The professor has informed us that 16.6% of her students received grades of A. What is the minimum score needed to receive a grade of A? b. If 12.1% of her students failed the course and received an F, what was the maximum score among those who received an F? c. If 33% of the students received grades of B or better (i.e., an A or B), what is the minimum score of those who received a B? Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions ANSWER: a. b. c.

87.7 66.3 82.4

125. DRUGS R US is a large manufacturer of various kinds of liquid vitamins. The quality control department has noted that the bottles of vitamins marked 6 ounces vary in content with a standard deviation of 0.3 ounce. Assume the contents of the bottles are normally distributed. a. What percentage of all bottles produced contains more than 6.51 ounces of vitamins? b. What percentage of all bottles produced contains less than 5.415 ounces of vitamins? c. What percentage of bottles produced contains between 5.46 and 6.495 ounces of vitamins? d. Ninety-five percent of the bottles will contain at least how many ounces of vitamins? e. What percentage of the bottles contains between 6.3 and 6.6 ounces of vitamins? ANSWER: a. 4.46% b. 2.56% c. 91.46% d. 5.508 ounces e. 13.59% 126. The daily dinner bills in a local restaurant are normally distributed with a mean of $28 and a standard deviation of $6. a. Define the random variable in words. b. What is the probability that a randomly selected bill will be at least $39.10? c. What percentage of the bills will be less than $16.90? d. What are the minimum and maximum of the middle 95% of the bills? If 12 of one day's bills had a value of at least $43.06, how many bills did the restaurant collect e. on that day? ANSWER: a. b. c. d. e.

daily dinner bills .0322 3.22% minimum = $16.24; maximum = $39.76 2,000

127. The monthly income of residents of Daisy City is normally distributed with a mean of $3,000 and a standard deviation of $500. a. Define the random variable in words. b. The mayor of Daisy City makes $2,250 a month. What percentage of Daisy City's residents has incomes that are more than the mayor's? c. Individuals with incomes of less than $1,985 per month are exempt from city taxes. What percentage of residents is exempt from city taxes? d. What are the minimum and maximum incomes of the middle 95% of the residents? e. Two hundred residents have incomes of at least $4,440 per month. What is the population of Daisy City? ANSWER: a. b. c.

monthly income of residents of Daisy City 93.32% 2.12%

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Chapter 06: Continuous Probability Distributions d. e.

minimum = $2,020; maximum = $3,980 100,000

128. The average starting salary of this year's MBA students is $35,000 with a standard deviation of $5,000. Furthermore, it is known that the starting salaries are normally distributed. What are the minimum and maximum starting salaries of the middle 95% of MBA graduates? ANSWER: minimum = $25,200; maximum = $44,800 129. A local bank has determined that the daily balances of the checking accounts of its customers are normally distributed with an average of $280 and a standard deviation of $20. a. What percentage of its customers has daily balances of more than $275? b. What percentage of its customers has daily balances less than $243? c. What percentage of its customers' balances is between $241 and $301.60? ANSWER: a. 59.87% b. 3.22% c. 83.43% 130. The weekly earnings of bus drivers are normally distributed with a mean of $395. If only 1.1% of the bus drivers have a weekly income of more than $429.35, what is the standard deviation of the weekly earnings of the bus drivers? ANSWER: $15 131. The monthly earnings of computer programmers are normally distributed with a mean of $4,000. If only 1.7% of programmers have monthly incomes of less than $2,834, what is the standard deviation of the monthly earnings of the computer programmers? ANSWER: $550 132. The Globe Fishery packs shrimp that weigh more than 1.91 ounces each in packages marked" large" and shrimp that weigh less than 0.47 ounces each into packages marked "small"; the remainder are packed in "medium" size packages. If a day's catch showed that 19.77% of the shrimp were large and 6.06% were small, determine the mean and standard deviation for the shrimp weights. Assume that the shrimps' weights are normally distributed. ANSWER: mean = 1.4; standard deviation = 0.6 133. In grading eggs into small, medium, and large, Linda Farms packs the eggs that weigh more than 3.6 ounces in packages marked "large" and the eggs that weigh less than 2.4 ounces into packages marked "small"; the remainder are packed in packages marked "medium." If a day's packaging contained 10.2% large and 4.18% small eggs, determine the mean and standard deviation for the eggs' weights. Assume that the distribution of the weights is normal. ANSWER: mean = 3.092; standard deviation = 0.4 134. A manufacturing process produces items whose weights are normally distributed. It is known that 22.57% of all the items produced weigh between 100 grams up to the mean and 49.18% weigh from the mean up to 190 grams. Determine the mean and standard deviation. ANSWER: mean = 113; standard deviation = 30 135. Suppose z is the standard normal random variable. Use Excel to calculate the following: a. P(z ≤ 2.5) b. P(0 ≤ z ≤ 2.5) c. P(–2 ≤ z ≤ 2) d. P(z ≤ –0.38) e. P(z ≥ 1.62) Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions f. g.

z value with 0.05 in the lower tail z value with 0.05 in the upper tail

ANSWER: a. b. c. d. e. f. g.

P(z ≤ 2.5) P(0 ≤ z ≤ 2.5) P(–2 ≤ z ≤ 2) P(z ≤ –0.38) P(z ≥ 1.62) z value with 0.05 in the lower tail z value with 0.05 in the upper tail

=NORM.S.DIST(2.5,TRUE) =NORM.S.DIST(2.5,TRUE)-NORM.S.DIST(0,TRUE) =NORM.S.DIST(2,TRUE)-NORM.S.DIST(-2,TRUE) =NORM.S.DIST(-0.38,TRUE) =1-NORM.S.DIST(1.62,TRUE) =NORM.S.INV(0.05) =NORM.S.INV(0.95)

136. Suppose x is a normally distributed random variable with a mean of 50 and a standard deviation of 5. Use Excel to calculate the following: a. P(x ≤ 45) b. P(45 ≤ x ≤ 55) c. P(x ≥ 55) d. x value with 0.20 in the lower tail e. x value with 0.01 in the upper tail ANSWER: a. b. c. d. e.

P(x ≤ 45) P(45 ≤ x ≤ 55) P(x ≥ 55) x value with 0.20 in the lower tail x value with 0.01 in the upper tail

=NORM.DIST(45,50,5,TRUE) =NORM.DIST(55,50,5,TRUE)-NORM.DIST(45,50,5,TRUE) =1-NORM.DIST(55,50,5,TRUE) =NORM.INV(0.2,50,5) =NORM.INV(0.99,50,5)

137. The time it takes a mechanic to change the oil in a car is exponentially distributed with a mean of 5 minutes. a. What is the probability density function for the time it takes to change the oil? b. What is the probability that it will take a mechanic less than 6 minutes to change the oil? c. What is the probability that it will take a mechanic between 3 and 5 minutes to change the oil? ANSWER: a. f(x) = (1/5) e-x/5 for x ≥ 0 b. .6988 c. .1809

138. The time it takes a worker on an assembly line to complete a task is exponentially distributed with a mean of 8 minutes. a. What is the probability density function for the time it takes to complete the task? b. What is the probability that it will take a worker less than 4 minutes to complete the task? c. What is the probability that it will take a worker between 6 and 10 minutes to complete the task? ANSWER: a. b. c.

f(x) = (1/8 ) e-x/8 for x ≥ 0 .3935 .1859

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Chapter 06: Continuous Probability Distributions 139. The time between arrivals of customers at the drive-up window of a bank follows an exponential probability distribution with a mean of 10 minutes. a. What is the probability that the arrival time between customers will be 7 minutes or less? b. What is the probability that the arrival time between customers will be between 3 and 7 minutes? ANSWER: a. b.

.5034 .2442

140. The time required to assemble a part of a machine follows an exponential probability distribution with a mean of 14 minutes. a. What is the probability that the part can be assembled in 7 minutes or less? b. What is the probability that the part can be assembled between 3.5 and 7 minutes? ANSWER: a. .3935 b. .1723 141. The time it takes to completely tune an engine of an automobile follows an exponential distribution with a mean of 40 minutes. a. Define the random variable in words. b. What is the probability of tuning an engine in 30 minutes or less? c. What is the probability of tuning an engine between 30 and 35 minutes? ANSWER: a. time it takes to completely tune an engine b. .5276 c. .0555

142. Suppose x is an exponentially distributed random variable with a mean of 10. Use Excel to calculate the following: a. P(x ≤ 15) b. P(8 ≤ x ≤ 12) c. P(x ≥ 8) ANSWER: a. b. c.

P(x ≤ 15) P(8 ≤ x ≤ 12) P(x ≥ 8)

=EXPON.DIST(15,1/10,TRUE) =EXPON.DIST(12,1/10,TRUE)-EXPON.DIST(8,1/10,TRUE) =1-EXPON.DIST(8,1/10,TRUE)

143. The Harbour Island Ferry leaves on the hour and at 15-minute intervals. The time, x, it takes John to drive from his house to the ferry has a uniform distribution with x between 10 and 20 minutes. One morning John leaves his house at precisely 8:00 A.M. a. What is the probability John will wait less than 5 minutes for the ferry? b. What is the probability John will wait less than 10 minutes for the ferry? c. What is the probability John will wait less than 15 minutes for the ferry? d. What is the probability John will not have to wait for the ferry? e. Suppose John leaves at 8:05 A.M. What is the probability John will wait (1) less than 5 minutes for the ferry; (2) less than 10 minutes for the ferry? f. Suppose John leaves at 8:10 A.M. What is the probability John will wait (1) less than 5 minutes for the ferry; (2) less than 10 minutes for the ferry? g. What appears to be the best time for John to leave home if he wishes to maximize the probability of waiting less than 10 minutes for the ferry? Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions ANSWER: a. .5 b. .5 c. 1 d. 0 e. (1) 0, (2) .5 f. (1) .5, (2) 1 g. 8:10 144. Delicious Candy markets a two-pound box of assorted chocolates. Because of imperfections in the candy making equipment, the actual weight of the chocolate has a uniform distribution ranging from 31.8 to 32.6 ounces. a. Define a probability density function for the weight of the box of chocolate. b. What is the probability that a box weighs (1) exactly 32 ounces; (2) more than 32.3 ounces; (3) less than 31.8 ounces? c. The government requires that at least 60% of all products sold weigh at least as much as the stated weight. Is Delicious violating government regulations? ANSWER: a. f(x) = 1.25 for 31.8 ≤ x ≤ 32.6, and 0 otherwise b. (1) 0, (2) .375, (3) 0 c. no; 75% are 32 oz. or more 145. The time at which the mailman delivers the mail to Ace Bike Shop follows a normal distribution with mean 2:00 P.M. and standard deviation of 15 minutes. a. What is the probability the mail will arrive after 2:30 P.M.? b. What is the probability the mail will arrive before 1:36 P.M.? c. What is the probability the mail will arrive between 1:48 P.M. and 2:09 P.M.? ANSWER: a. .0228 b. .0548 c. .5138 146. The township of Middleton sets the speed limit on its roads by conducting a traffic study and determining the speed (to the nearest 5 miles per hour) at which 80% of the drivers travel at or below. A study was done on Brown's Dock Road that indicated drivers' speeds follow a normal distribution with a mean of 36.25 miles per hour and a variance of 6.25. a. What should the speed limit be? b. What percent of the drivers travel below that speed? ANSWER: a. 40 miles per hour b. 93.32% 147. A light bulb manufacturer claims its light bulbs will last 500 hours on average. The lifetime of a light bulb is assumed to follow an exponential distribution. a. What is the probability that the light bulb will have to be replaced within 500 hours? b. What is the probability that the light bulb will last more than 1,000 hours? c. What is the probability that the light bulb will last between 200 and 800 hours? ANSWER: a. .632 b. .135 c. .468 148. The weight of a 0.5-cubic-yard bag of landscape mulch is uniformly distributed over the interval from 38.5 to 41.5 pounds. a. Give a mathematical expression for the probability density function. b. What is the probability that a bag will weigh more than 40 pounds? c. What is the probability that a bag will weigh less than 39 pounds? Copyright Cengage Learning. Powered by Cognero.

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Chapter 06: Continuous Probability Distributions d. What is the probability that a bag will weigh between 39 and 40 pounds? ANSWER: a. f(x) = 1/3 for 38.5  x  41.5 f(x) = 0, otherwise b. .5000 c. .1667 d. .3333

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Chapter 07: Sampling and Sampling Distributions Multiple Choice 1. The expected value of

equals the mean of the population from which the sample is drawn _____.

a. only if the sample size is 30 or greater b. only if the sample size is 50 or greater c. only if the sample size is 100 or greater d. for any sample size ANSWER: d 2. The basis for using a normal probability distribution to approximate the sampling distribution of

and

is _____.

a. Chebyshev’s theorem b. the empirical rule c. the central limit theorem d. Bayes’ theorem ANSWER: c 3. The standard deviation of

is referred to as the _____.

a. standard proportion b. sample proportion c. average proportion d. standard error of the proportion ANSWER: d 4. The standard deviation of is referred to as the _____. a. standard x b. standard error of the mean c. sample standard mean d. sample mean deviation ANSWER: b 5. The value of the _____ is used to estimate the value of the population parameter. a. population statistic b. sample parameter c. population estimate d. sample statistic ANSWER: d 6. The population being studied is usually considered ______ if it involves an ongoing process that makes listing or counting every element in the population impossible. a. finite b. infinite c. skewed d. symmetric Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions ANSWER: b 7. A probability sampling method in which we randomly select one of the first k elements and then select every kth element thereafter is _____. a. stratified random sampling b. cluster sampling c. systematic sampling d. convenience sampling ANSWER: c 8. The standard deviation of a point estimator is the _____. a. standard error b. sample statistic c. point estimate d. sampling error ANSWER: a 9. The finite correction factor should be used in the computation of

when n/N is greater than _____.

a. .01 b. .025 c. .05 d. .10 ANSWER: c 10. The set of all elements of interest in a study is _____. a. set notation b. a set of interest c. a sample d. a population ANSWER: d 11. A subset of a population selected to represent the population is a _____. a. subset b. sample c. small population d. parameter ANSWER: b 12. The purpose of statistical inference is to provide information about the _____. a. sample based upon information contained in the population b. population based upon information contained in the sample c. population based upon information contained in the population d. mean of the sample based upon the mean of the population Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions ANSWER: b 13. A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size _____. a. N has the same probability of being selected b. n has a probability of .5 of being selected c. n has a probability of .1 of being selected d. n has the same probability of being selected ANSWER: d 14. The number of random samples (without replacement) of size 3 that can be drawn from a population of size 5 is _____. a. 15 b. 10 c. 20 d. 125 ANSWER: b 15. There are 6 children in a family. The number of children defines a population. The number of simple random samples of size 2 (without replacement) that are possible equals _____. a. 12 b. 15 c. 3 d. 16 ANSWER: b 16. How many different samples of size 3 (without replacement) can be taken from a finite population of size 10? a. 30 b. 1,000 c. 720 d. 120 ANSWER: d 17. A population consists of 8 items. The number of different simple random samples of size 3 (without replacement) that can be selected from this population is _____. a. 24 b. 56 c. 512 d. 128 ANSWER: b 18. A population consists of 500 elements. We want to draw a simple random sample of 50 elements from this population. On the first selection, the probability of an element being selected is _____. a. .100 b. .010 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions c. .001 d. .002 ANSWER: d 19. Excel's RAND function _____. a. determines sample size b. selects a simple random sample c. randomizes a population d. generates random numbers ANSWER: d 20. A simple random sample of size n from a finite population of size N is to be selected. Each possible sample should have _____. a. the same probability of being selected b. a probability of 1/n of being selected c. a probability of 1/N of being selected d. a probability of N/n of being selected ANSWER: a 21. A simple random sample from an infinite population is a sample selected such that _____. a. the probability of each element being selected is 1/n b. each element selected comes from a different population c. each element selected comes from the same population and each element is selected independently d. the probability of being selected changes ANSWER: c 22. A numerical measure from a population, such as a population mean, is called _____. a. a statistic b. a parameter c. a sample d. the mean deviation ANSWER: b 23. A numerical measure from a sample, such as a sample mean, is known as _____. a. a statistic b. a parameter c. the mean deviation d. the central limit theorem ANSWER: a 24. A sample statistic, such as , that estimates the value of the corresponding population parameter is known as a _____. a. point estimator b. parameter Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions c. population parameter d. parameter and a population parameter ANSWER: a 25. A single numerical value used as an estimate of a population parameter is known as a _____. a. parameter b. population parameter c. parameter or a population parameter d. point estimate ANSWER: d 26. In point estimation, data from the _____. a. population are used to estimate the population parameter b. sample are used to estimate the population parameter c. sample are used to estimate the sample statistic d. population are used to estimate the sample statistic ANSWER: b 27. The sample mean is the point estimator of _____. a. μ b. σ c. d. ANSWER: a 28. The sample statistic s is the point estimator of _____. a. μ b. σ c. d. ANSWER: b 29. Which of the following is a point estimator? a. σ b. p c. s d. t ANSWER: c 30. A simple random sample of 5 observations from a population containing 400 elements was taken, and the following values were obtained. 12 18 19 20 21 A point estimate of the population mean is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions a. 5 b. 18 c. 19 d. 20 ANSWER: b 31. A probability distribution for all possible values of a sample statistic is known as a _____. a. sample statistic b. parameter c. simple random sample d. sampling distribution ANSWER: d 32. A simple random sample of 28 observations was taken from a large population. The sample mean equaled 50. Fifty is a _____. a. population parameter b. point estimator c. sample parameter d. point estimate ANSWER: d 33. If we consider the simple random sampling process as an experiment, the sample mean is _____. a. always zero b. always smaller than the population mean c. a random variable d. exactly equal to the population mean ANSWER: c 34. The probability distribution of all possible values of the sample mean is called the ____. a. central probability distribution b. sampling distribution of the sample mean c. random variation d. standard error ANSWER: b 35. The sampling distribution of the sample mean _____. a. is the probability distribution showing all possible values of the sample mean b. is used as a point estimator of the population mean μ c. is an unbiased estimator d. shows the distribution of all possible values of μ ANSWER: a 36. The difference between the value of the sample statistic and the value of the corresponding population parameter is called the _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions a. statistical error b. standard error c. proportion error d. sampling error ANSWER: d 37. The expected value of the random variable

a. σ b. the standard error c. the sample size d. μ ANSWER: d 38. The standard deviation of all possible

values is called the _____.

a. standard error of proportion b. standard error of the mean c. mean deviation d. central variation ANSWER: b 39. In computing the standard error of the mean, the finite population correction factor is NOT used when _____. a. n/N > 0.05 b. N/n ≤ 0.05 c. n/N ≤ 0.05 d. n ≥ 30 ANSWER: c 40. A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means _____. a. whenever the population is infinite b. whenever the sample size is more than 5% of the population size c. whenever the sample size is less than 5% of the population size d. The correction factor is not necessary if the population has a normal distribution. ANSWER: b 41. From a population of 200 elements, the standard deviation is known to be 14. A sample of 49 elements is selected. It is determined that the sample mean is 56. The standard error of the mean is _____. a. 3 b. 2 c. greater than 2 d. less than 2 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions 42. From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately _____. a. 1.1022 b. 2 c. 30 d. 1.4847 ANSWER: d 43. A simple random sample of 64 observations was taken from a large population. The population standard deviation is 120. The sample mean was determined to be 320. The standard error of the mean is _____. a. 1.875 b. 40 c. 5 d. 15 ANSWER: d 44. As the sample size increases, the _____. a. standard deviation of the population decreases b. population mean increases c. standard error of the mean decreases d. standard error of the mean increases ANSWER: c 45. As the sample size increases, the variability among the sample means _____. a. increases b. decreases c. remains the same d. depends upon the specific population being sampled ANSWER: b 46. Doubling the size of the sample will _____. a. reduce the standard error of the mean to one-half its current value b. reduce the standard error of the mean to approximately 70% of its current value c. have no effect on the standard error of the mean d. double the standard error of the mean ANSWER: b 47. Random samples of size 49 are taken from a population that has 200 elements, a mean of 180, and a variance of 196. The distribution of the population is unknown. The mean and the standard error of the distribution of sample means are _____. a. 180 and 24.39 b. 180 and 28 c. 180 and 1.74 d. 180 and 2 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions ANSWER: c 48. Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 200 and 18, respectively. The distribution of the population is unknown. The mean and the standard error of the distribution of sample means are _____. a. 200 and 18 b. 81 and 18 c. 9 and 2 d. 200 and 2 ANSWER: d 49. Random samples of size 36 are taken from an infinite population whose mean and standard deviation are 20 and 15, respectively. The distribution of the population is unknown. The mean and the standard error of the distribution of sample mean are _____. a. 36 and 15 b. 20 and 15 c. 20 and 0.417 d. 20 and 2.5 ANSWER: d 50. A theorem that allows us to use the normal probability distribution to approximate the sampling distribution of sample means and sample proportions whenever the sample size is large is known as the _____. a. approximation theorem b. normal probability theorem c. central limit theorem d. central normality theorem ANSWER: c 51. The fact that the sampling distribution of the sample mean can be approximated by a normal probability distribution whenever the sample size is large is based on the _____. a. central limit theorem b. fact that there are tables of areas for the normal distribution c. assumption that the population has a normal distribution d. normal probability theorem ANSWER: a 52. As the sample size becomes larger, the sampling distribution of the sample mean approaches a _____. a. binomial distribution b. Poisson distribution c. hypergeometric distribution d. normal probability distribution ANSWER: d 53. Whenever the population has a normal probability distribution, the sampling distribution of is a normal probability distribution for _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions a. only large sample sizes b. only small sample sizes c. any sample size d. only samples of size 30 or greater ANSWER: c 54. For a population with an unknown distribution, the form of the sampling distribution of the sample mean is _____. a. approximately normal for all sample sizes b. exactly normal for large sample sizes c. exactly normal for all sample sizes d. approximately normal for large sample sizes ANSWER: d 55. A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of is _____. a. approximately normal because is always approximately normally distributed b. approximately normal because the sample size is large in comparison to the population size c. approximately normal because of the central limit theorem d. normal if the population is normally distributed ANSWER: d 56. A sample of 92 observations is taken from an infinite population. The sampling distribution of is approximately normal because _____. a. is always approximately normally distributed b. the sample size is small in comparison to the population size c. of the central limit theorem d. the sample is greater than 50 ANSWER: c 57. A population has a mean of 80 and a standard deviation of 7. A sample of 49 observations will be taken. The probability that the mean from that sample will be larger than 82 is _____. a. .5228 b. .9772 c. .4772 d. .0228 ANSWER: d 58. A population has a mean of 180 and a standard deviation of 24. A sample of 64 observations will be taken. The probability that the mean from that sample will be between 183 and 186 is _____. a. 0.1359 b. 0.8185 c. 0.3413 d. 0.4772 ANSWER: a Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions 59. A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is _____. a. .0347 b. .7200 c. .9511 d. .5645 ANSWER: c 60. A population has a mean of 53 and a standard deviation of 21. A sample of 49 observations will be taken. The probability that the sample mean will be greater than 57.95 is _____. a. 0 b. .0495 c. .4505 d. .6721 ANSWER: b Exhibit 7-1 The following data were collected from a simple random sample from an infinite population. 13

61. Refer to Exhibit 7-1. The point estimate of the population mean _____. a. is 5 b. is 14 c. is 4 d. cannot be determined because the population is infinite ANSWER: b 62. Refer to Exhibit 7-1. The point estimate of the population standard deviation is _____. a. 2.500 b. 1.581 c. 2.000 d. 1.414 ANSWER: b 63. Refer to Exhibit 7-1. The mean of the population _____. a. is 14 b. is 15 c. is 15.1581 d. could be any value ANSWER: d Exhibit 7-2 Four hundred registered voters were randomly selected and asked whether gun laws should be changed. Three hundred said "yes," and 100 said "no." Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions 64. Refer to Exhibit 7-2. The point estimate of the proportion in the population who will respond "yes" is _____. a. 300 b. approximately 300 c. .75 d. .25 ANSWER: c 65. Refer to Exhibit 7-2. The point estimate of the proportion in the population who will respond "no" is _____. a. 75 b. .25 c. .75 d. .50 ANSWER: b Exhibit 7-3 The following information was collected from a simple random sample of a population. 16

66. Refer to Exhibit 7-3. The point estimate of the mean of the population is _____. a. 18.0 b. 19.6 c. 108 d. 16 ANSWER: a 67. Refer to Exhibit 7-3. The point estimate of the population standard deviation is _____. a. 2.000 b. 1.291 c. 1.414 d. 1.667 ANSWER: c Exhibit 7-4 A random sample of 121 bottles of cologne showed an average content of 4 ounces. It is known that the standard deviation of the contents (i.e., of the population) is 0.22 ounces. 68. Refer to Exhibit 7-4. The standard error of the mean equals _____. a. .3636 b. .0331 c. .0200 d. 4.000 ANSWER: c 69. Refer to Exhibit 7-4. The point estimate of the mean content of all bottles is _____. a. .22 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions b. 4 c. 121 d. .02 ANSWER: b 70. Refer to Exhibit 7-4. In this problem, the .22 is _____. a. a parameter b. a statistic c. the standard error of the mean d. the average content of colognes in the long run ANSWER: a Exhibit 7-5 Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. 71. Refer to Exhibit 7-5. The mean and the standard deviation of the sampling distribution of the sample means are _____. a. 8.7 and 1.94 b. 36 and 1.94 c. 36 and 1.86 d. 36 and 8 ANSWER: c 72. Refer to Exhibit 7-5. The value 36 is _____. a. the standard error of the proportion b. the standard error of the mean c. a statistic d. a parameter ANSWER: d 73. The probability distribution of all possible values of the sample proportion

is the _____.

a. probability density function of b. sampling distribution of c. same as

, since it considers all possible values of the sample proportion

d. sampling distribution of ANSWER: d 74. Random samples of size 525 are taken from an infinite population whose population proportion is .3. The standard deviation of the sample proportions (i.e., the standard error of the proportion) is _____. a. .0004 b. .2100 c. .3000 d. .0200 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions ANSWER: d 75. A random sample of 150 people was taken from a very large population. Ninety of the people in the sample were females. The standard error of the proportion of females is _____. a. .0016 b. .2400 c. .1600 d. .0400 ANSWER: d 76. A population of size 1,000 has a proportion of .5. Therefore, the proportion and the standard deviation of the sample proportion for samples of size 100 are _____. a. 500 and .047 b. 500 and .050 c. .5 and .047 d. .5 and .050 ANSWER: c 77. Random samples of size 100 are taken from a process (an infinite population) whose population proportion is .2. The mean and standard deviation of the distribution of sample proportions are _____. a. .2 and .04 b. .2 and .2 c. 20 and .04 d. 20 and .2 ANSWER: a 78. As a general rule, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever _____. a. np ≥ 5 b. n(1 − p) ≥ 5 c. n ≥ 30 d. np ≥ 5 and n(1 − p) ≥ 5 ANSWER: d 79. A sample of 25 observations is taken from an infinite population. The sampling distribution of a. not normal since n < 30 b. approximately normal because

is _____.

is always normally distributed

c. approximately normal if np ≥ 5 and n(1 – p) ≥ 5 d. approximately normal if np > 30 and n(1 – p) > 30 ANSWER: c 80. A sample of 400 observations will be taken from an infinite population. The population proportion equals .8. The probability that the sample proportion will be greater than 0.83 is _____. a. .4332 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions b. .9332 c. .0668 d. .5668 ANSWER: c 81. A sample of 66 observations will be taken from an infinite population. The population proportion equals .12. The probability that the sample proportion will be less than .1768 is _____. a. .0568 b. .0778 c. .4222 d. .9222 ANSWER: d 82. A sample of 51 observations will be taken from an infinite population. The population proportion equals .85. The probability that the sample proportion will be between .9115 and .946 is _____. a. .8633 b. .6900 c. .0819 d. .0345 ANSWER: c 83. Stratified random sampling is a method of selecting a sample in which _____. a. the sample is first divided into groups, and then random samples are taken from each group b. various strata are selected from the sample c. the population is first divided into groups, and then random samples are drawn from each group d. the elements are selected on the basis of convenience ANSWER: c 84. Cluster sampling is _____. a. a nonprobability sampling method b. the same as convenience sampling c. a probability sampling method d. based on judgment ANSWER: c 85. Convenience sampling is an example of _____. a. probabilistic sampling b. stratified sampling c. a nonprobability sampling technique d. cluster sampling ANSWER: c 86. Which of the following is an example of a nonprobability sampling technique? a. simple random sampling Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions b. stratified random sampling c. cluster sampling d. judgment sampling ANSWER: d 87. Which of the following sampling methods does NOT lead to probability samples? a. stratified sampling b. cluster sampling c. systematic sampling d. convenience sampling ANSWER: d 88. The population we want to make inferences about is the _____. a. sampled population b. frame c. target population d. finite population ANSWER: c 89. When the population has a normal distribution, the sampling distribution of

is normally distributed _____.

a. for any sample size b. for any sample size of 30 or more c. for any sample size of 50 or more d. for any sample from a finite population ANSWER: a 90. It is impossible to construct a frame for a(n) _____. a. finite population b. infinite population c. target population d. sampled population ANSWER: b 91. The standard error of the proportion will become larger as _____. a. n increases b. p approaches 0 c. p approaches .5 d. p approaches 1 ANSWER: c 92. All of the following are true about the standard error of the mean EXCEPT _____. a. it is larger than the standard deviation of the population b. it decreases as the sample size increases Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions c. its value is influenced by the standard deviation of the population d. it measures the variability in sample means ANSWER: a Subjective Short Answer 93. A simple random sample of 8 employees of a corporation provided the following information: Employee Age Gender

1 25 M

2 32 M

3 26 M

4 40 M

5 50 F

6 54 M

7 22 M

8 23 F

a. Determine the point estimate for the average age of all employees. b. What is the point estimate for the standard deviation of the population? c. Determine a point estimate for the proportion of all employees who are female. ANSWER: a. 34 b. 12.57 c. .25 94. Starting salaries of a sample of 5 management majors along with their genders are shown below. Employee 1 2 3 4 5

Salary ($1000s) 30 28 22 26 19

Gender F M F F M

a. What is the point estimate for the starting salaries of all management majors? b. Determine the point estimate for the variance of the population. c. Determine the point estimate for the proportion of male employees. ANSWER: a. 25 ($1000s) b. 20 ($1000s) c. .4 95. A sample of 8 new models of automobiles provides the following data on highway miles per gallon. Use Excel to answer the questions that follow the data. Model 1 2 3 4 5 6 7 8 a. b.

Highway Miles per Gallon 33.6 26.8 20.2 38.7 35.1 28.0 26.2 27.6

What is the point estimate for the average highway miles per gallon for all new models of autos? Determine the point estimate for the standard deviation of the population.

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Chapter 07: Sampling and Sampling Distributions ANSWER:

A 1 2 3 4 5 6 7 8 9 10 11

a. b.

B Model 1 2 3 4 5 6 7 8 Average Std. Dev.

C Hwy mpg 33.6 26.8 20.2 38.7 35.1 28.0 26.2 27.6 29.53 5.9

=AVERAGE(C2:C9) =STDEV.S(C2:C9)

96. A sample of 10 members of a video club provides the following data on number of videos they own. Use Excel to answer the questions that follow the data. Member 1 2 3 4 5 6 7 8 9 10

Number Owned 200 26 158 75 52 352 17 276 488 129

a. What is the point estimate for the mean number of videos owned by all video club members? b. Determine the point estimate for the standard deviation of the population. ANSWER: A B C D 1 Member Number Owned 2 1 200 3 2 26 4 3 158 5 4 75 6 5 52 7 6 352 8 7 17 9 8 276 10 9 488 11 10 129 12 a. Average 177.3 =AVERAGE(C2:C11) 13 b. Std. Dev. 154.5 =STDEV.S(C2:C11) 97. Consider a population of 5 weights identical in appearance but weighing 1, 3, 5, 7, and 9 ounces. a. Determine the mean and the variance of the population. Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions b.

Sampling without replacement from the above population with a sample size of 2 produces 10 possible samples. Using the 10 sample mean values, determine the mean of the population and the variance of .

Compute the standard error of the mean.

ANSWER: a. b. c.

5 and 8 5 and 3 1.732

98. Consider a population of 5 families with the following data representing the number of pets in each family. Family A B C D E a. b. c.

Number of Pets 2 6 4 3 1

There are 10 possible samples of size 2 (sampling without replacement). List the 10 possible samples of size 2, and determine the mean of each sample. Determine the mean and the variance of the population. Using the 10 sample mean values, compute the mean and the standard error of the mean.

ANSWER: a.

Possible Samples AB AC AD AE BC BD BE CD CE DE

b. c.

Sample Means 4 3 2.5 1.5 5 4.5 3.5 3.5 2.5 2 3.2 and 2.96 3.2 and 1.11

99. The following information gives the number of days absent from work for a population of 5 workers at a small factory. Worker A B C D E a.

Number of Days Absent 5 7 1 4 8

Find the mean and the standard deviation for the population.

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Chapter 07: Sampling and Sampling Distributions b.

Samples of size 2 will be drawn from the population. Use the answers in Part a. to calculate the expected value and the standard deviation of the sampling distribution of the sample mean. Find all the samples of 2 workers that can be extracted from this population. Choose the samples without replacement. Compute the sample mean for each of the samples in Part c.

d. ANSWER: a. b. c. d.

5; 2.449 5; 1.5 AB, AC, AD, AE, BC, BD, BE, CD, CE, DE 6, 3, 4.5, 6.5, 4, 5.5, 7.5, 2.5, 4.5, 6

100. The average weekly earnings of bus drivers in a city are $950 (that is μ) with a standard deviation of $45 (that is σ). Assume that we select a random sample of 81 bus drivers. a. Assume the number of bus drivers in the city is large compared to the sample size. Compute the standard error of the mean. b. What is the probability that the sample mean will be greater than $960? If the population of bus drivers consisted of 400 drivers, what would be the standard error of c. the mean? ANSWER: a. b. c.

5 .0228 4.47

101. An automotive repair shop has determined that the average service time on an automobile is 2 hours with a standard deviation of 32 minutes. A random sample of 64 services is selected. a. What is the probability that the sample of 64 will have a mean service time greater than 114 minutes? b. Assume the population consists of 400 services. Determine the standard error of the mean. ANSWER: a. .9332 b. 3.67 102. A population of 1,000 students spends an average of $10.50 a day on dinner. The standard deviation of the expenditure is $3. A simple random sample of 64 students is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? b. What is the probability that these 64 students will spend a combined total of more than $715.21? c. What is the probability that these 64 students will spend a combined total between $703.59 and $728.45? ANSWER: a. b. c.

10.5; .363; normal .0314 .0794

103. There are 8,000 students at the University of Tennessee at Chattanooga. The average age of all the students is 24 years with a standard deviation of 9 years. A random sample of 36 students is selected. a. Determine the standard error of the mean. b. What is the probability that the sample mean will be larger than 19.5? c. What is the probability that the sample mean will be between 25.5 and 27 years? ANSWER: a. 1.5 Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions b. c.

0.9986 0.1359

104. The life expectancy in the United States is 75 with a standard deviation of 7 years. A random sample of 49 individuals is selected. a. What is the probability that the sample mean will be larger than 77 years? b. What is the probability that the sample mean will be less than 72.7 years? c. What is the probability that the sample mean will be between 73.5 and 76 years? d. What is the probability that the sample mean will be between 72 and 74 years? e. What is the probability that the sample mean will be larger than 73.46 years? ANSWER: a. .0228 b. .0107 c. .7745 d. .1573 e. .9382 105. SAT scores have an average of 1200 with a standard deviation of 60. A sample of 36 scores is selected. a. What is the probability that the sample mean will be larger than 1224? b. What is the probability that the sample mean will be less than 1230? c. What is the probability that the sample mean will be between 1200 and 1214? d. What is the probability that the sample mean will be greater than 1200? e. What is the probability that the sample mean will be larger than 73.46? ANSWER: a. .0082 b. .9986 c. .4192 d. .5 e. 1.0

106. A bank has kept records of the checking balances of its customers and determined that the average daily balance of its customers is $300 with a standard deviation of $48. A random sample of 144 checking accounts is selected. a. What is the probability that the sample mean will be more than $306.60? b. What is the probability that the sample mean will be less than $308? c. What is the probability that the sample mean will be between $302 and $308? d. What is the probability that the sample mean will be at least $296? ANSWER: a. .0495 b. .9772 c. .2857 d. .8413 107. Students of a large university spend an average of $5 a day on lunch. The standard deviation of the expenditure is $3. A simple random sample of 36 students is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? b. What is the probability that the sample mean will be at least $4? c. What is the probability that the sample mean will be at least $5.90? ANSWER: a. b. c.

5.0; .5; normal .9772 .0359

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Chapter 07: Sampling and Sampling Distributions 108. The average lifetime of a light bulb is 3,000 hours with a standard deviation of 696 hours. A simple random sample of 36 bulbs is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of ? b. What is the random variable in this problem? Define it in words. c. What is the probability that the average life in the sample will be between 2,670.56 and 2,809.76 hours? d. What is the probability that the average life in the sample will be greater than 3,219.24 hours? e. What is the probability that the average life in the sample will be less than 3,180.96 hours? ANSWER: a. b. c. d. e.

3,000; 116; normal , the average life in a sample of 36 bulbs .0482 .0294 .9406

109. MNM Corporation gives each of its employees an aptitude test. The scores on the test are normally distributed with a mean of 75 and a standard deviation of 15. A simple random sample of 25 is taken from a population of 500. a. What are the expected value, the standard deviation, and the shape of the sampling distribution of ? b. What is the random variable in this problem? Define it in words. c. What is the probability that the average aptitude test score in the sample will be between 70.14 and 82.14? d. What is the probability that the average aptitude test score in the sample will be greater than 82.68? e. What is the probability that the average aptitude test score in the sample will be less than 78.69? ) = .015. f. Find a value, C, such that P( ANSWER: a. b. c. d. e. f.

75; 3; normal , the average aptitude test score in a sample of 25 employees .9387 .0052 .8907 81.51

110. The price of a particular brand of jeans has a mean of $37.99 and a standard deviation of $7. A sample of 49 pairs of jeans is selected. Use Excel to answer the following questions: a. What is the probability that the sample of jeans will have a mean price less than $40? b. What is the probability that the sample of jeans will have a mean price between $38 and $39? c. What is the probability that the sample of jeans will have a mean price within $3 of the population mean? ANSWER: a. b. c.

.97778 =NORM.DIST(40,37.99,1,TRUE) .339763 =NORM.DIST(39,37.99,1,TRUE) – NORM.DIST(38,37,99,1,TRUE) .9973 =NORM.DIST(40.99,37.99,1,TRUE) – NORM.DIST(34.99,37.99,1,TRUE)

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Chapter 07: Sampling and Sampling Distributions 111. The mean diameter of a ball bearing produced by a certain manufacturer is 0.80 cm with a standard deviation of 0.03 cm. A sample of 36 ball bearings is randomly selected from a production run. Use Excel to answer the following questions: a. What is the probability that the sample of ball bearings will have a mean less than 0.798 cm? b. What is the probability that the sample of ball bearings will have a mean of at least 0.815 cm? c. What is the probability that the sample of ball bearings will have a mean between 0.798 and 0.815 cm? d. For samples of size 36, 15% of all sample means are at most what diameter? ANSWER: a. b. c.

.344578 =NORM.DIST(0.798,0.8,0.005,TRUE) .00135 =1 – NORM.DIST(0.815,0.8,0.005,TRUE) .654072 =NORM.DIST(0.815,0.8,0.005,TRUE) – NORM.DIST(0.798,0.8,0.005,TRUE) .794818 =NORM.INV(0.15,0.8,0.005)

112. There are 500 employees in a firm, and 45% are female. A sample of 60 employees is selected randomly. a. Determine the standard error of the proportion. b. What is the probability that the sample proportion of females is between .40 and .55? ANSWER: a. b.

.0603 .7482

113. Ten percent of the items produced by a machine are defective. A random sample of 100 items is selected and checked for defects. a. Determine the standard error of the proportion. b. What is the probability that the sample will contain more than 2.5% defective units? c. What is the probability that the sample will contain more than 13% defective units? ANSWER: a. .03 b. .9938 c. .1587 114. A new soft drink is being market tested. It is estimated that 60% of consumers will like the new drink. A sample of 96 taste-tested the new drink. a. Determine the standard error of the proportion b. What is the probability that more than 70.4% of consumers will indicate they like the drink? c. What is the probability that more than 30% of consumers will indicate they do NOT like the drink? ANSWER: a. b. c.

.05 .0188 .9772

115. In a large university, 20% of the students are Business majors. A random sample of 100 students is selected, and their majors are recorded. a. Compute the standard error of the proportion. Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions b. c. d.

What is the probability that the sample contains at least 12 Business majors? What is the probability that the sample contains less than 15 Business majors? What is the probability that the sample contains between 12 and 14 Business majors?

ANSWER: a. b. c. d.

.04 .9772 .1056 .044

116. In a local university, 10% of the students live in the dormitories. A random sample of 100 students is selected for a study. a. What is the probability that the sample proportion of students living in the dormitories is between .172 and .178? b. What is the probability that the sample proportion of students living in the dormitories is greater than .025? ANSWER: a. b.

.0035 .9938

117. A department store has determined that 25% of all its sales are credit sales. A random sample of 75 sales is selected. a What is the probability that the sample proportion will be greater than .34? b. What is the probability that the sample proportion will be between .196 and .354? c. What is the probability that the sample proportion will be less than .25? d. What is the probability that the sample proportion will be less than .10? ANSWER: a. b. c. d.

.0359 .8411 .5 .0014

118. Candidate A is running for president of the student government at a large university. The proportion of voters who favor the candidate is .8. A simple random sample of 100 voters is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of ? b. What is the probability that the number of voters in the sample who will not favor Candidate A will be between 26 and 30? c. What is the probability that the number of voters in the sample who will NOT favor Candidate A will be more than 16? ANSWER: a. b. c.

.8; .04; normal .0606 .8413

119. In a restaurant, the proportion of people who order coffee with their dinner is .9. A simple random sample of 144 patrons of the restaurant is taken. a. What are the expected value, standard deviation, and shape of the sampling distribution of ? b. c.

What is the random variable in this problem? Define it in words. What is the probability that the proportion of people who will order coffee with their meal is between .85 and .875?

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Chapter 07: Sampling and Sampling Distributions d.

What is the probability that the proportion of people who will order coffee with their meal is at least .945?

ANSWER: a. b. c. d.

.9; .025; normal the sample proportion of people who order coffee with their dinners out of 144 patrons .1359 .0359

120. Thirty percent of a magazine's subscribers are female. A random sample of 50 subscribers is taken. Answer the following questions using Excel: a. What is the probability that the proportion of females from this sample is at most .25? b. What is the probability that the proportion of females from this sample is between .22 and .28? c. What is the probability that the proportion of females from this sample is within .03 of the population proportion? ANSWER: a. b. c.

.220 =NORM.DIST(0.25,0.3,0.0648,TRUE) .270 =NORM.DIST(0.28,0.3,0.0648,TRUE) – NORM.DIST(0.22,0.3,0.0648,TRUE) .357 =NORM.DIST(0.33,0.3,0.0648,TRUE) – NORM.DIST(0.27,0.3,0.0648,TRUE)

121. The proportion of Americans who support the death penalty is .53. A sample of 1000 randomly selected Americans is surveyed by telephone interview. Use Excel to answer the following questions: a. What is the probability that the sample proportion of those supporting the death penalty will be less than .50? b. What is the probability that the sample proportion of those supporting the death penalty will be at least .55? c. What is the probability that the sample proportion of those supporting the death penalty will be between .50 and .55? d. For samples of size 1000, 15% of all sample proportions are at most what value? ANSWER: a. b. c. d.

.028 =NORM.DIST(0.5,0.53,0.0157,TRUE) .101 =1-NORM.DIST(0.55,0.53,0.0157,TRUE) .871 =NORM.DIST(0.55,0.53,0.0157,TRUE) – NORM.DIST(0.5,0.53,0.0157,TRUE) .514 =NORM.INV(0.15,0.53,0.0157)

122. A random sample of 9 telephone calls in an office provided the following information: Call Number 1 2 3 4 5 6

Duration (minutes) 3 8 4 3 5 6

Type of Call local long distance local local long distance local Page 25

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Chapter 07: Sampling and Sampling Distributions 7 8 9

3 5 8

local local local

a. Determine the point estimate for the average duration of all calls. b. What is the point estimate for the standard deviation of the population? c. What is the point estimate for the proportion of all calls that were long distance? ANSWER: a. 5 b. 1.97 c. .222 123. A random sample of 10 examination papers in a course that was given on a pass or fail basis showed the following scores: Paper Number 1 2 3 4 5 6 7 8 9 10 a. b. c.

Grade 65 87 92 35 79 100 48 74 79 91

Status pass pass pass fail pass pass fail pass pass pass

What is the point estimate for the mean of the population? What is the point estimate for the standard deviation of the population? What is the point estimate for the proportion of all students who passed the course?

ANSWER: a. b. c.

75 20.48 .8

124. Roger, who oversees 6 Ford dealerships, believes that the colors chosen by customers who special-order their cars best reflect most customers’ true color preferences. For that reason, he has tabulated the color requests specified in a sample of 56 Mustang coupe special orders placed this year. The sample data are listed below. Black Red Green Blue Blue Red Green

Red White Black Red Black Red Red

White Blue Red Black Green Blue Black

Blue White Black White White Black White

Blue Red Blue Black Black Red Black

Green Red Black Red Red Black Red

Red Black White Black Red Green Black

Black Black Green Blue White Black White

a. What is the point estimate of the proportion of all Mustang coupe special orders that specify a color preference of black? b. Describe the sampling distribution of , where is the proportion of Mustang coupe special orders that specify a color preference of black. Assume that the proportion of all Mustang coupe special orders having a color preference of black is .36. c. What is the probability that a simple random sample of 56 special orders will provide an estimate of the population Copyright Cengage Learning. Powered by Cognero.

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Chapter 07: Sampling and Sampling Distributions proportion of special orders specifying the color black that is within plus or minus .05 of the actual population proportion, assuming p = .36? In other words, what is the probability that will be between .31 and .41? ANSWER: a. .32143 b. normally distributed with c. .5646

= .36 and

= .064

125. Lily owns a mail-order business specializing in baby clothes. Lily is confident the dollar amounts of ALL her orders are normally distributed or nearly so. Assume she knows the mean and standard deviation are $249 and $46, respectively, for ALL orders she receives. a. Describe the sampling distribution of , where is the mean dollar amount of an order for a sample of 10 orders. b. What is the probability that a simple random sample of 30 orders will provide an estimate of the population mean dollar amount of an order that is within plus or minus $10 of the actual population mean? c. What happens to the sampling distribution of when the sample size is increased from 30 to 90? With a sample size of 90, what is the probability that will be between $239 and $259? ANSWER: a. normally distributed with E(x) = $249 and = $14.5465 b. .7660 c. the sampling distribution is narrower; .9606

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Chapter 08: Interval Estimation Multiple Choice 1. As the degrees of freedom increase, the t distribution approaches the _____ distribution. a. uniform b. normal c. exponential d. p ANSWER: b 2. If the margin of error in an interval estimate of μ is 4.6, the interval estimate equals _____. a. b. c. d. ANSWER: b 3. The t distribution is a family of similar probability distributions, with each individual distribution depending on a parameter known as the _____. a. finite correction factor b. sample size c. degrees of freedom d. standard deviation ANSWER: c 4. The probability that the interval estimation procedure will generate an interval that contains the actual value of the population parameter being estimated is the _____. a. level of significance b. confidence level c. confidence coefficient d. error factor ANSWER: c 5. To compute the minimum sample size for an interval estimate of μ when the population standard deviation is known, we must first determine all of the following EXCEPT _____. a. desired margin of error b. confidence level c. population standard deviation d. degrees of freedom ANSWER: d 6. The use of the normal probability distribution as an approximation of the sampling distribution of condition that both np and n(1 – p) equal or exceed _____. a. .05 b. 5 Copyright Cengage Learning. Powered by Cognero.

is based on the

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Chapter 08: Interval Estimation c. 15 d. 30 ANSWER: b 7. The sample size that guarantees all estimates of proportions will meet the margin of error requirements is computed using a planning value of p equal to _____. a. .01 b. .50 c. .51 d. .99 ANSWER: b 8. We can reduce the margin of error in an interval estimate of p by doing any of the following EXCEPT _____. a. increasing the sample size b. using a planning value p* closer to .5 c. increasing the level of significance d. reducing the confidence coefficient ANSWER: b 9. In determining an interval estimate of a population mean when σ is unknown, we use a t distribution with _____ degrees of freedom. a. b. c. n − 1 d. n ANSWER: c 10. The expression used to compute an interval estimate of μ may depend on any of the following factors EXCEPT _____. a. the sample size b. whether the population standard deviation is known c. whether the population has an approximately normal distribution d. whether there is sampling error ANSWER: d 11. The mean of the t distribution is _____. a. 0 b. .5 c. 1 d. dependent upon the sample size ANSWER: a 12. An interval estimate is used to estimate _____. a. the shape of the population's distribution Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation b. the sampling distribution c. a sample statistic d. a population parameter ANSWER: d 13. An estimate of a population parameter that provides an interval believed to contain the value of the parameter is known as the _____. a. confidence level b. interval estimate c. parameter value d. population estimate ANSWER: b 14. As the sample size increases, the margin of error _____. a. increases b. decreases c. stays the same d. becomes negative ANSWER: b 15. The confidence associated with an interval estimate is called the _____. a. level of significance b. degree of association c. confidence level d. precision ANSWER: c 16. The ability of an interval estimate to contain the value of the population parameter is described by the _____. a. confidence level b. degrees of freedom c. precise value of the population mean μ d. sample statistic ANSWER: a 17. If an interval estimate is said to be constructed at the 90% confidence level, the confidence coefficient would be _____. a. .1 b. .95 c. .9 d. .05 ANSWER: c 18. If we want to provide a 95% confidence interval for the mean of a population, the confidence coefficient is _____. a. .485 Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation b. 1.96 c. .95 d. 1.645 ANSWER: c 19. For the interval estimation of μ when σ is assumed known, the proper distribution to use is the _____. a. standard normal distribution b. t distribution with n degrees of freedom c. t distribution with n − 1 degrees of freedom d. t distribution with n − 2 degrees of freedom ANSWER: a 20. The z value for a 97.8% confidence interval estimation is _____. a. 2.02 b. 1.96 c. 2.00 d. 2.29 ANSWER: d 21. It is known that the variance of a population equals 1,936. A random sample of 121 has been selected from the population. There is a .95 probability that the sample mean will provide a margin of error of _____. a. 7.84 or less b. 31.36 or less c. 344.96 or less d. 1,936 or less ANSWER: a 22. A random sample of 144 observations has a mean of 20, a median of 21, and a mode of 22. The population standard deviation is known to equal 4.8. The 95.44% confidence interval for the population mean is _____. a. 15.2 to 24.8 b. 19.2 to 20.8 c. 19.216 to 20.784 d. 21.2 to 22.8 ANSWER: b Exhibit 8-1 In order to estimate the average time spent on the computer terminals per student at a local university, data were collected from a sample of 81 business students over a one-week period. Assume the population standard deviation is 1.2 hours. 23. Refer to Exhibit 8-1. The standard error of the mean is _____. a. 7.5 b. .014 c. .160 d. .133 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation 24. Refer to Exhibit 8-1. With a .95 probability, the margin of error is approximately _____. a. .26 b. 1.96 c. .21 d. 1.64 ANSWER: a 25. Refer to Exhibit 8-1. If the sample mean is 9 hours, then the 95% confidence interval is approximately _____. a. 7.04 to 110.96 hours b. 7.36 to 10.64 hours c. 7.80 to 10.20 hours d. 8.74 to 9.26 hours ANSWER: d Exhibit 8-2 The manager of a grocery store has selected a random sample of 100 customers. The average length of time it took these 100 customers to check out was 3.0 minutes. It is known that the standard deviation of the checkout time is 1 minute. 26. Refer to Exhibit 8-2. The standard error of the mean equals _____. a. .001 b. .01 c. .1 d. 1 ANSWER: c 27. Refer to Exhibit 8-2. With a .95 probability, the sample mean will provide a margin of error of _____. a. .95 b. .10 c. .196 d. 1.96 ANSWER: c 28. Refer to Exhibit 8-2. If the confidence coefficient is reduced to .80, the standard error of the mean _____. a. will increase b. will decrease c. remains unchanged d. becomes negative ANSWER: c 29. Refer to Exhibit 8-2. The 95% confidence interval for the average checkout time for all customers is _____. a. 3 to 5 b. 1.36 to 4.64 c. 2.804 to 3.196 d. 1.04 to 4.96 Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation ANSWER: c Exhibit 8-3 A random sample of 81 automobiles traveling on a section of an interstate showed an average speed of 60 mph. The distribution of speeds of all cars on this section of highway is normally distributed, with a standard deviation of 13.5 mph. 30. Refer to Exhibit 8-3. If we are interested in determining an interval estimate for μ at 86.9% confidence, the z value to use is _____. a. 1.96 b. 1.31 c. 1.51 d. 2.00 ANSWER: c 31. Refer to Exhibit 8-3. The value to use for the standard error of the mean is _____. a. 13.5 b. 9 c. 2.26 d. 1.5 ANSWER: d 32. Refer to Exhibit 8-3. The 86.9% confidence interval for μ is _____. a. 46.500 to 73.500 b. 57.735 to 62.265 c. 59.131 to 60.869 d. 50 to 70 ANSWER: b 33. Refer to Exhibit 8-3. If the sample size was 25 (other factors remain unchanged), the interval for μ would _____. a. not change b. become narrower c. become wider d. become zero ANSWER: c 34. In general, higher confidence levels provide _____. a. wider confidence intervals b. narrower confidence intervals c. a smaller standard error d. unbiased estimates ANSWER: a 35. When the level of confidence increases, the confidence interval _____. a. stays the same b. becomes wider Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation c. becomes narrower d. becomes negative ANSWER: b 36. A 95% confidence interval for a population mean is determined to be 100 to 120. If the confidence coefficient is reduced to .90, the interval for μ _____. a. becomes narrower b. becomes wider c. does not change d. becomes .1 ANSWER: a 37. If we change a 95% confidence interval estimate to a 99% confidence interval estimate, we can expect the _____. a. width of the confidence interval to increase b. width of the confidence interval to decrease c. width of the confidence interval to remain the same d. sample size to increase ANSWER: a 38. In developing an interval estimate of the population mean, if the population standard deviation is unknown _____. a. it is impossible to develop an interval estimate b. a sample proportion can be used c. the sample standard deviation and t distribution can be used d. a normal distribution must be used ANSWER: c 39. A bank manager wishes to estimate the average waiting time for customers in line for tellers. A random sample of 50 times is measured and the average waiting time is 5.7 minutes. The population standard deviation of waiting time is 2 minutes. Which Excel function would be used to construct a confidence interval estimate? a. CONFIDENCE.NORM b. NORM.INV c. T.INV d. INT ANSWER: a 40. An auto manufacturer wants to estimate the annual income of owners of a particular model of automobile. A random sample of 200 current owners is selected. The population standard deviation is known. Which Excel function would NOT be appropriate to use to construct a confidence interval estimate? a. NORM.S.INV b. COUNTIF c. AVERAGE d. STDEV ANSWER: b 41. Whenever the population standard deviation is unknown, which distribution is used in developing an interval estimate Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation for a population mean? a. standard distribution b. z distribution c. binomial distribution d. t distribution ANSWER: d 42. The t distribution should be used whenever _____. a. the sample size is less than 30 b. the sample standard deviation is used to estimate the population standard deviation c. the population is not normally distributed d. the sample mean is unknown ANSWER: b 43. Whenever using the t distribution in interval estimation, we must assume that _____. a. the sample size is less than 30 b. a random sample was selected c. the population is approximately normal d. the finite population correction factor is necessary ANSWER: b 44. From a population that is normally distributed with an unknown standard deviation, a sample of 25 elements is selected. For the interval estimation of μ, the proper distribution to use is the _____. a. standard normal distribution b. z distribution c. t distribution with 26 degrees of freedom d. t distribution with 24 degrees of freedom ANSWER: d 45. From a population that is not normally distributed and whose standard deviation is not known, a sample of 50 items is selected to develop an interval estimate for μ. Which of the following statements is true? a. The standard normal distribution can be used. b. The t distribution with 50 degrees of freedom must be used. c. The t distribution with 49 degrees of freedom must be used. d. The sample size must be increased in order to develop an interval estimate. ANSWER: c 46. As the number of degrees of freedom for a t distribution increases, the difference between the t distribution and the standard normal distribution _____. a. becomes larger b. becomes smaller c. stays the same d. becomes larger or smaller, depending on the sample size ANSWER: b Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation 47. The t value with a 95% confidence and 24 degrees of freedom is _____. a. 1.711 b. 2.064 c. 2.492 d. 2.069 ANSWER: b 48. A sample of 26 elements from a normally distributed population is selected. The sample mean is 10 with a standard deviation of 4. The 95% confidence interval for μ is _____. a. 6.000 to 14.000 b. 9.846 to 10.154 c. 8.384 to 11.616 d. 8.462 to 11.538 ANSWER: c 49. A random sample of 36 students at a community college showed an average age of 25 years. Assume the ages of all students at the college are normally distributed with a standard deviation of 1.8 years. The 98% confidence interval for the average age of all students at this college is _____. a. 24.301 to 25.699 b. 24.385 to 25.615 c. 23.200 to 26.800 d. 23.236 to 26.764 ANSWER: a 50. A random sample of 25 statistics examinations was selected. The average score in the sample was 76 with a variance of 144. Assuming the scores are normally distributed, the 99% confidence interval for the population average examination score is _____. a. 70.02 to 81.98 b. 69.82 to 82.18 c. 70.06 to 81.94 d. 69.29 to 82.71 ANSWER: d 51. A random sample of 25 employees of a local company has been measured. A 95% confidence interval estimate for the mean systolic blood pressure for all company employees is 123 to 139. Which of the following statements is valid? a. 95% of the sample of employees has a systolic blood pressure between 123 and 139. b. If the sampling procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure. c. 95% of the population of employees has a systolic blood pressure between 123 and 139. d. If the sampling procedure were repeated many times, 95% of the sample means would be between 123 and 139. ANSWER: b 52. To estimate a population mean, the sample size needed to provide a margin of error of 2 or less with a .95 probability Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation when the population standard deviation equals 11 is _____. a. 10 b. 11 c. 116 d. 117 ANSWER: d 53. It is known that the population variance equals 484. With a .95 probability, the sample size that needs to be taken to estimate the population mean if the desired margin of error is 5 or less is _____. a. 25 b. 74 c. 189 d. 75 ANSWER: d 54. We can use the normal distribution to make confidence interval estimates for the population proportion, p, when _____. a. np ≥ 5 b. n(1 − p) ≥ 5 and np ≤ 5 c. p has a normal distribution d. np ≥ 5 and n(1 − p) ≥ 5 ANSWER: d 55. Using α = .04, a confidence interval for a population proportion is determined to be .65 to .75. If the level of significance is decreased, the interval for the population proportion _____. a. becomes narrower b. becomes wider c. does not change d. is reduced to ½ of the size ANSWER: b 56. In determining the sample size necessary to estimate a population proportion, which of the following is NOT needed? a. the maximum margin of error that can be tolerated b. the confidence level required c. a preliminary estimate of the true population proportion p d. the mean of the population ANSWER: d 57. For which of the following values of p is the value of P(1 − p) maximized? a. p = .99 b. p = .90 c. p = 1.0 d. p = .50 ANSWER: d Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation 58. A manufacturer wants to estimate the proportion of defective items that are produced by a certain machine. A random sample of 50 items is selected. Which Excel function would NOT be appropriate to construct a confidence interval estimate? a. NORM.S.INV b. COUNTIF c. STDEV d. COUNTA ANSWER: c 59. A newspaper wants to estimate the proportion of Americans who will vote for Candidate A. A random sample of 1000 voters is selected. Of the 1000 respondents, 526 say that they will vote for Candidate A. Which Excel function would be used to construct a confidence interval estimate? a. NORM.S.INV b. NORM.INV c. T.INV d. INT ANSWER: a 60. The general form of an interval estimate of a population mean or population proportion is the _____ plus or minus the _____. a. population mean, standard error b. level of significance, degrees of freedom c. point estimate, margin of error d. planning value, confidence coefficient ANSWER: c 61. The degrees of freedom associated with a t distribution are a function of the _____. a. area in the upper tail b. sample standard deviation c. confidence coefficient d. sample size ANSWER: d 62. The margin of error in an interval estimate of the population mean is a function of all of the following EXCEPT _____. a. level of significance b. sample mean c. sample size d. variability of the population ANSWER: b 63. Computing the necessary sample size for an interval estimate of a population proportion requires a planning value for . In case of any uncertainty about an appropriate planning value, we know the value that will provide the largest sample size for a given level of confidence and a given margin of error is _____. Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation a. .10 b. .50 c. .90 d. 1 ANSWER: b Subjective Short Answer 64. In order to estimate the average electric usage per month, a sample of 196 houses was selected and the electric usage determined. a. Assume a population standard deviation of 350 kilowatt-hours. Determine the standard error of the mean. b. With a .95 probability, determine the margin of error. c. If the sample mean is 2,000 KWH, what is the 95% confidence interval estimate of the population mean?

ANSWER: a. b. c.

25 49 1951 to 2049

65. A random sample of 100 credit sales in a department store showed an average sale of $120.00. From past data, it is known that the standard deviation of the population is $40.00. a. Determine the standard error of the mean. b. With a .95 probability, determine the margin of error. c. What is the 95% confidence interval of the population mean? ANSWER: a. b. c.

4.00 7.84 112.16 to 127.84

66. A random sample of 49 lunch customers was selected at a restaurant. The average amount of time the customers in the sample stayed in the restaurant was 33 minutes. From past experience, it is known that the population standard deviation equals 10 minutes. a. Compute the standard error of the mean. b. What can be said about the sampling distribution for the average amount of time customers spent in the restaurant? Be sure to explain your answer. c. With a .95 probability, what statement can be made about the size of the margin of error? d. Construct a 95% confidence interval for the true average amount of time customers spent in the restaurant. e. With a .95 probability, what sample size would have to be selected to provide a margin of error of 2.5 minutes or less? ANSWER: a. b. c. d. e.

1.4286 Normal by the central limit theorem There is a .95 probability that the sample mean will provide a margin of error of 2.80 or less. 30.20 to 35.80 62

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Chapter 08: Interval Estimation 67. In order to determine the average weight of carry-on luggage by passengers in airplanes, a sample of 36 pieces of carry-on luggage was weighed. The average weight was 20 pounds. Assume that we know the standard deviation of the population to be 8 pounds. a. Determine a 97% confidence interval estimate for the mean weight of the carry-on luggage. b. Determine a 95% confidence interval estimate for the mean weight of the carry-on luggage. ANSWER: a. 17.11 to 22.89 b. 17.39 to 22.61 68. A small stock brokerage firm wants to determine the average daily sales (in dollars) of stocks to their clients. A sample of the sales for 36 days revealed average daily sales of $200,000. Assume that the standard deviation of the population is known to be $18,000. a. Provide a 95% confidence interval estimate for the average daily sale. b. Provide a 97% confidence interval estimate for the average daily sale. ANSWER: a. $194,120 to $205,880 b. $193,490 to $206,510 69. A random sample of 121 checking accounts at a bank showed an average daily balance of $280. The population standard deviation is known to be $60. a. Is it necessary to know anything about the shape of the distribution of the account balances in order to make an interval estimate of the mean of all the account balances? Explain. b. Find the standard error of the mean. c. Give a point estimate of the population mean. d. Construct a 95% confidence interval estimate for the population mean. e. Interpret the confidence interval estimate that you constructed in part (d). ANSWER: a. b. c. d. e.

No, since the sample means will be normally distributed by the central limit theorem. 5.4545 280 269.31 to 290.69 With a 95% level of confidence, we can state that the average daily balance of all checking accounts at this bank is between $269.31 and $290.69.

70. A simple random sample of 144 items resulted in a sample mean of 1080. The population standard deviation is known to be 240. Develop a 95% confidence interval for the population mean. ANSWER: 1040.8 to 1119.2 71. A random sample of 26 checking accounts at a bank showed an average daily balance of $300 and a standard deviation of $45. The balances of all checking accounts at the bank are normally distributed. Develop a 95% confidence interval estimate for the mean of the population. ANSWER: $281.82 to $318.18 72. A random sample of 81 students at a local university showed that they work an average of 100 hours per month. The population standard deviation is known to be 27 hours. Compute a 95% confidence interval for the mean hours per month all students at the university work. ANSWER: 94.12 to 105.88 73. A random sample of 81 children with working mothers showed that they were absent from school an average of 6 days per term. The population standard deviation is known to be 1.8 days. Provide a 90% confidence interval for the average number of days absent per term for all children with working mothers. Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation ANSWER: 5.671 to 6.329 74. The Highway Safety Department wants to study the driving habits of individuals. A sample of 41 cars traveling on the highway revealed an average speed of 60 miles per hour and a standard deviation of 7 miles per hour. The population of car speeds is approximately normally distributed. Determine a 90% confidence interval estimate for the speed of all cars. ANSWER: 58.16 to 61.84 75. Computer Services, Inc. wants to determine a confidence interval for the average CPU time of their teleprocessing transactions. A sample of 196 transactions yielded a mean of 5 seconds. The population standard deviation is 1.4 seconds. Determine a 97% confidence interval for the average CPU time. ANSWER: 4.783 to 5.217 76. The average monthly electric bill of a random sample of 256 residents of a city is $90. The population standard deviation is assumed to be $24. a. Construct a 90% confidence interval for the mean monthly electric bills of all residents. b. Construct a 95% confidence interval for the mean monthly electric bills of all residents. ANSWER: a. 87.5325 to 92.4675 b. 87.06 to 92.94 77. A sample of 100 cans of coffee showed an average weight of 13 ounces. The population standard deviation is 0.8 ounces. a. Construct a 95% confidence interval for the mean of the population. b. Construct a 95.44% confidence interval for the mean of the population. c. Discuss why the answers in parts (a) and (b) are different. ANSWER: a. b. c.

12.8432 to 13.1568 12.84 to 13.16 As the level of confidence increases, the confidence interval becomes wider.

78. In order to determine how many hours per week freshmen college students watch television, a random sample of 256 students was selected. It was determined that the students in the sample spent an average of 14 hours per week watching television. The standard deviation is 3.2 hours per week for all freshmen college students. a. Provide a 95% confidence interval estimate for the average number of hours that all college freshmen spend watching TV per week. b. Suppose the sample mean came from a sample of 25 students. Provide a 95% confidence interval estimate for the average number of hours that all college freshmen spend watching TV per week. Assume that the hours are normally distributed. ANSWER: a. 13.608 to 14.392 b. 12.679 to 15.321 79. A random sample of 36 magazine subscribers is selected to estimate the mean age of all subscribers. The data follow. Use Excel to construct a 90% confidence interval estimate of the mean age of all of this magazine's subscribers. Subscriber 1 2 3 4 5 6

Age 39 27 38 33 40 35

Subscriber 13 14 15 16 17 18

Age 40 35 35 41 34 46

Subscriber 25 26 27 28 29 30

Age 38 51 26 39 35 37 Page 14

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Chapter 08: Interval Estimation 7 8 9 10 11 12 ANSWER:

51 36 47 28 33 35

19 20 21 22 23 24

A 1 Subscriber 2 1 3 2 4 3 5 4 6 5 7 6 8 7 9 8 10 9 11 10 12 11 13 12 14 13 15 14 16 15 35.6905 to 39.3095

44 44 43 32 29 33 B Age 39 27 38 33 40 35 51 36 50 28 33 35 40 35 35

31 32 33 34 35 36

C Sample Size Sample Mean

33 41 36 33 46 37 D =COUNT(B2:B37) =AVERAGE(B2:B37)

Popul. Std. Dev. 6.6 Confid. Coeffic. 0.9 Level of Signif. =1-D5

Value for D 36 37.5 6.6 0.9 0.1

Margin of Error

=CONFIDENCE.NORM(D6,D4,D1) 1.8095

Point Estimate Lower Limit Upper Limit

=D2 =D12-D8 =D12+D8

37.5 35.6905 39.3095

80. A simple random sample of 25 items from a normally distributed population resulted in a sample mean of 28 and a standard deviation of 7.5. Construct a 95% confidence interval for the population mean. ANSWER: 24.904 to 31.096 81. A sample of 25 patients in a doctor's office showed that they had to wait an average of 35 minutes with a standard deviation of 10 minutes before they could see the doctor. Provide a 98% confidence interval estimate for the average waiting time of all the patients who visit this doctor. Assume the population of waiting times is normally distributed. ANSWER: 30.016 to 39.984 82. A sample of 16 students from a large university is selected. The average age in the sample was 22 years with a standard deviation of 6 years. Construct a 95% confidence interval for the average age of the population. Assume the population of student ages is normally distributed. ANSWER: 18.8035 to 25.1965 83. The proprietor of a boutique in New York wanted to determine the average age of his customers. A random sample of 25 customers revealed an average age of 28 years with a standard deviation of 10 years. Determine a 95% confidence interval estimate for the average age of all his customers. Assume the population of customer ages is normally distributed. ANSWER: 23.872 to 32.128 84. A statistician selected a sample of 16 accounts receivable and determined the mean of the sample to be $5,000 with a standard deviation of $400. She reported that the sample information indicated the mean of the population ranges from $4,739.80 to $5,260.20. She did not report what confidence coefficient she had used. Based on the above information, determine the confidence coefficient that was used. Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation ANSWER: .98 85. The makers of a soft drink want to identify the average age of its consumers. A sample of 16 consumers is selected. The average age in the sample was 22.5 years with a standard deviation of 5 years. Assume the population of consumer ages is normally distributed. a. Construct a 95% confidence interval for the average age of all the consumers. b. Construct an 80% confidence interval for the average age of all the consumers. c. Discuss why the 95% and 80% confidence intervals are different. ANSWER: a. 19.836 to 25.164 b. 20.824 to 24.176 c. As the level of confidence increases, the confidence interval gets wider. 86. A random sample of 25 observations was selected from a normally distributed population. The average in the sample was 84.6 with a variance of 400. a. Construct a 90% confidence interval for μ. b. Construct a 99% confidence interval for μ. c. Discuss why the 90% and 99% confidence intervals are different. d. What would you expect to happen to the confidence interval in part (a) if the sample size was increased? Be sure to explain your answer.

ANSWER: a. b. c. d.

77.756 to 91.444 73.412 to 95.788 As the level of confidence increases, the confidence interval gets wider. Decrease in width since the margin of error decreased.

87. You are given the following information obtained from a random sample of four observations selected from a large, normally distributed population. 25 47 32 56 Construct a 95% confidence interval for the mean of the population. ANSWER: 17.613 to 62.387 88. You are given the following information obtained from a random sample of four observations selected from a large, normally distributed population. 25

a. What is the point estimate of μ? b. Construct a 95% confidence interval for μ. c. Construct a 90% confidence interval for μ. d. Discuss why the 90% and 95% confidence intervals are different. ANSWER: a. 40 b. 17.613 to 62.387 c. 23.445 to 56.555 d. As the level of confidence increases, the confidence interval gets wider. 89. The monthly incomes from a random sample of faculty at a university are shown below. Monthly Income ($1000s) 3.0 Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation 4.0 6.0 3.0 5.0 5.0 6.0 8.0 Compute a 90% confidence interval for the mean of the population. The population of all faculty incomes is known to be normally distributed. Give your answer in dollars. ANSWER: $3,867.52 to $6,132.48 90. Fifty students are enrolled in an Economics class. After the first examination, a random sample of five papers was selected. The grades were 60, 75, 80, 70, and 90. a. Calculate the estimate of the standard error of the mean. b. What assumption must be made before we can determine an interval for the mean grade of all the students in the class? Explain why. c. Assume the assumption of part (b) is met. Provide a 90% confidence interval for the mean grade of all the students in the class. d. If there were 200 students in the class, what would be the 90% confidence interval for the mean grade of all the students in the class? ANSWER: a. b. c. d.

4.79 Since the sample is small (n < 30) and σ is estimated from s, we must assume the distribution of all the grades is normal. 64.783 to 85.217 64.34 to 85.66

91. A local university administers a comprehensive examination to the recipients of a B.S. degree in Business Administration. A sample of five examinations is selected at random and scored. The scores are shown below. Grade 56 85 65 86 93 Use Excel to determine an interval estimate for the mean of the population at a 98% confidence level. Interpret your results. ANSWER: A B C D Value for D 1 Grade Mean 77 2 56 Standard Error 7.021396 3 85 Median 85 4 65 Mode #N/A 5 86 Standard Deviation 15.70032 6 93 Sample Variance 246.5 7 Kurtosis -2.00512 8 Skewness -0.608507 9 Range 37 Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation 10 Minimum 56 11 Maximum 93 12 Sum 385 13 Count 5 14 Confidence Level (98.0%) 26.308723 15 16 Point Estimate =D1 77.0000 17 Lower Limit =D1-D14 50.6913 18 Upper Limit =D1+D14 103.3087 Interpretation of Interval Estimate: With a 98% level of confidence, we can state that the mean comprehensive examination score of all recipients of the B.S. degree in Business Administration is between 50.6913 and 103.3087. 92. Below you are given ages that were obtained by taking a random sample of nine undergraduate students. 19 22 23 19 21 22 19 23 21 Use Excel to determine an interval estimate for the mean of the population at a 99% confidence level. Interpret your results. ANSWER: A B C D Value for D 1 Grade Mean 21 2 19 Standard Error 0.552771 3 22 Median 21 4 23 Mode 19 5 19 Standard Deviation 1.658312 6 21 Sample Variance 2.75 7 22 Kurtosis -1.667060 8 19 Skewness -0.211450 9 23 Range 4 10 21 Minimum 19 11 Maximum 23 12 Sum 189 13 Count 9 14 Confidence Level (98.0%) 1.854756 15 16 Point Estimate =D1 21.0000 17 Lower Limit =D1-D14 19.1452 18 Upper Limit =D1+D14 22.8548 Interpretation of Interval Estimate: With a 99% level of confidence, we can state that the mean age of undergraduate students is between 19.1452 and 22.8548. 93. The monthly starting salaries of students who receive an MBA degree have a standard deviation of $110. What size sample should be selected to obtain a .95 probability of estimating the mean monthly income within $20 or less? ANSWER: 117 94. A coal company wants to determine a 95% confidence interval estimate for the average daily tonnage of coal that it mines. Assuming the company reports that the standard deviation of daily output is 200 tons, how many days should it sample so that the margin of error will be 39.2 tons or less? ANSWER: 100 Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation 95. If the standard deviation of the lifetimes of vacuum cleaners is estimated to be 300 hours, what sample size must be selected in order to be 97% confident that the margin of error will not exceed 40 hours? ANSWER: 265 96. A researcher is interested in determining the average number of years employees of a company stay with the company. If past information shows a standard deviation of 7 months, what size sample should be selected so that at 95% confidence the margin of error will be 2 months or less? ANSWER: 48 97. The standard deviation for the lifetimes of washing machines is estimated to be 800 hours. What sample size must be selected in order to be 97% confident that the margin of error will not exceed 50 hours? ANSWER: 1206 98. A real estate agent wants to estimate the mean selling price of two-bedroom homes in a particular area. She wants to estimate the mean selling price to within $10,000 with an 89.9% level of confidence. The standard deviation of selling prices is unknown but the agent estimates that the highest selling price is $1,000,000 and the lowest is $50,000. How many homes should be sampled? ANSWER: 1518 99. For inventory purposes, a grocery store manager wants to estimate the mean number of pounds of cat food sold per month. The estimate is desired to be within 10 pounds with a 95% level of confidence. A pilot study provided a standard deviation of 27.6 pounds. How many months should be sampled? ANSWER: 30 100. It is known that the variance of a population equals 484. A random sample of 81 observations is going to be selected from the population. a. With an .80 probability, what statement can be made about the size of the margin of error? b. With an .80 probability, what sample size would have to be selected to provide a margin of error of 3 or less? ANSWER: a. b.

There is an .80 probability that the sample mean will provide a margin of error of 3.129 or less. 89

101. In a random sample of 400 registered voters, 120 indicated they plan to vote for Candidate A. Determine a 95% confidence interval for the proportion of all the registered voters who will vote for Candidate A. ANSWER: .255 to .345 102. In a random sample of 200 registered voters, 120 indicated they are Democrats. Develop a 95% confidence interval for the proportion of registered voters in the population who are Democrats. ANSWER: .5321 to .6679 103. In a random sample of 500 college students, 23% say that they read or watch the news every day. Develop a 90% confidence interval for the population proportion. Interpret your results. ANSWER: .199 to .261 With a 90% level of confidence we can state that the proportion of all college students who read or watch the Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation news every day is between .199 and .261. 104. Six hundred consumers were asked whether they would like to purchase a domestic or a foreign automobile. Their responses are given below. Preference Frequency Domestic 240 Foreign 360 Develop a 95% confidence interval for the proportion of all consumers who prefer to purchase domestic automobiles. ANSWER: .3608 to .4392 105. A university planner wants to determine the proportion of spring semester students who will attend summer school. She surveys 32 current students and discovers that 12 will return for summer school. a. Construct a 90% confidence interval estimate for the proportion of current spring students who will return for summer school. b. With a .95 probability, what sample size would have to be selected to provide a margin of error of 3% or less? ANSWER: a. b.

.234 to .516 1001

106. A new brand of breakfast cereal is being market tested. One hundred boxes of the cereal were given to consumers to try. The consumers were asked whether they liked or disliked the cereal. You are given their responses below. Response Liked Disliked a. b. c. d.

Frequency 60 40 100

What is the point estimate of the proportion of people who will like the cereal? Construct a 95% confidence interval for the proportion of all consumers who will like the cereal. What is the margin of error for the 95% confidence interval that you constructed in part (b)? With a .95 probability, what sample size needs to be selected to provide a margin of error of .09 or less?

ANSWER: a. b. c. d.

.6 .504 to .696 .096 114

107. A marketing firm is developing a new television advertisement for a large discount retail chain. A sample of 30 people is shown two potential ads and asked their preference. The results for ad #1 follow. Use Excel to develop a 95% confidence interval estimate of the proportion of people in the population who will prefer ad #1. yes no no

no no no

Prefer Advertisement #1 no yes no yes yes yes

yes no yes

no yes no Page 20

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Chapter 08: Interval Estimation yes yes ANSWER:

yes no

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A Prefer Ad 1 yes no no yes yes no no no no yes no yes no no

no yes

no no

yes no

C Sample size Response of Interest Count for Response Sample Proportion

D =COUNTA(A2:A31) yes =COUNTIF(A2:A31,"yes") =D3/D1

Value for D 30 yes 14 0.466667

Confidence Coefficient Level of Significance z value

0.95 =1-D5 =NORM.S.INV(1-D7/2)

0.95 0.05 1.959961

Standard Error Margin of Error

=SQRT((D4*(1-D4)/D1)) =D8*D10

0.091084 0.178521

Point Estimate Lower Limit Upper Limit

=D4 =D13-D11 =D13+D11

0.466667 0.28815 0.64519

108. A survey of 40 students at a local college asks, "Where do you buy the majority of your books?" The responses fell into three categories: "at the campus bookstore," "on the Internet," and "other." The results follow. Use Excel to develop a 95% confidence interval estimate of the proportion of college students who buy their books on the Internet. Where Most Books Bought bookstore bookstore Internet other bookstore bookstore bookstore bookstore bookstore other other other other other Internet bookstore Internet Internet

other bookstore Internet other other other

Internet bookstore Internet other bookstore bookstore

other bookstore other Internet bookstore

bookstore other other bookstore other

ANSWER: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A where bought bookstore bookstore Internet other Internet other bookstore bookstore bookstore bookstore bookstore bookstore bookstore Internet

C Sample size Response of Interest Count for Response Sample Proportion

D =COUNTA(A2:A41) Internet =COUNTIF(A2:A41,"internet") =D2/D1

Value for D 40 Internet 8 0.2

Confidence Coefficient Level of Significance z value

0.95 =1-D6 =NORM.S.INV(1-D6/2)

0.95 0.05 1.95996108

Standard Error Margin of Error

=SQRT((D4*(1-D4)/D1)) =D8*D10

0.06324555 0.12395882

Point Estimate Lower Limit Upper Limit

=D4 =D13-D11 =D13+D11

0.2 0.07604 0.32396

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Chapter 08: Interval Estimation 109. A health club annually surveys its members. Last year, 33% of the members said they use the treadmill at least four times a week. What size sample should be selected this year to estimate the percentage of members who use the treadmill at least four times a week? The estimate is desired to have a margin of error of 5% with a 95% level of confidence. ANSWER: 340 110. A local hotel wants to estimate the proportion of its guests that are from out of state. Preliminary estimates are that 45% of the hotel guests are from out-of-state.What sample size should be selected to estimate the proportion of out of state guests with a margin of error no larger than 5% and with a 95% level of confidence? ANSWER: 381 111. The manager of a department store wants to determine what proportion of people who enter the store use the store's credit card for their purchases. What size sample should he take so that at 99% confidence the error will not be more than 8%? ANSWER: 260 112. The manager of Hudson Auto Repair wants to advertise one price for an engine tune-up, with parts included. Before he decides the price to advertise, he needs a good estimate of the average cost of tune-up parts. A sample of 20 customer invoices for tune-ups has been selected and the costs of parts, rounded to the nearest dollar, are listed below. 91 104

78 74

93 62

57 68

75 97

52 73

99 77

80 65

105 80

62 109

Provide a 90% confidence interval estimate of the mean cost of parts per tune-up for all of the tune-ups performed at Hudson Auto Repair. ANSWER: 73.51 to 86.59 113. The manager of University Credit Union (UCU) is concerned about checking account transaction discrepancies. Customers are bringing transaction errors to the attention of the bank’s staff several months after they occur. The manager would like to know what proportion of his customers balance their checking accounts within 30 days of receiving a transaction statement from the bank. Using random sampling, 400 checking account customers are contacted by telephone and asked if they routinely balance their accounts within 30 days of receiving a statement. 271 of the 400 customers respond Yes. a. Develop a 95% confidence interval estimate for the proportion of the population of checking account customers at UCU who routinely balance their accounts in a timely manner. b. Suppose UCU wants a 95% confidence interval estimate of the population proportion with a margin of error of E = .025. What sample size is needed? ANSWER: a. .6317 to .7233 b. 1343 114. National Discount has 260 retail outlets throughout the United States. National evaluates each potential location for a new retail outlet in part on the mean annual income of the households in the marketing area of the new location. National develops an interval estimate of the mean annual income in a potential marketing area after taking a random sample of households. For a marketing area being studied, a sample of 36 households was selected. The sample mean income was $21,100.39. Based on past experience, National Discount assumes a known value of $4500 for the population standard deviation of incomes. a. Develop a 95% confidence interval for the mean annual income of households in this marketing area. Interpret the interval. Copyright Cengage Learning. Powered by Cognero.

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Chapter 08: Interval Estimation b. Suppose that National’s management team wants a 95% confidence interval estimate of the population mean with a margin of error of E = $500. What sample size is needed to meet these requirements? ANSWER: a. $19,630.39 to $22,570.39 We are 95% confident that the average annual income for all households in the market area being studied falls in the interval $19,630.39 to $22,570.39. b. We need to sample 312 households to reach a desired margin of error of $500 at 95% confidence.

115. A reporter for a student newspaper is writing an article on the cost of off-campus housing. A sample was selected of 10 one-bedroom units within a half-mile of campus and the rents paid. The sample mean is $550 and the sample standard deviation is $60.05. Assume this population is normally distributed. Develop a 95% confidence interval estimate of the mean rent per month for the population of one-bedroom units within a half-mile of campus. Interpret the interval. ANSWER: $507.05 to $592.95 We are 95% confident that the mean rent per month for the population of one-bedroom units within a halfmile of campus is between $507.05 and $592.95. 116. Political Science, Inc. (PSI) specializes in voter polls and surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, interviewers ask registered voters who they would vote for if the election were held that day. In a recent election campaign, PSI found that 220 registered voters, out of 500 contacted, favored a particular candidate. a. Develop a 95% confidence interval estimate for the proportion of the population of registered voters that favors the candidate. Interpret the interval. b. Suppose that PSI would like 99% confidence that the sample proportion is within ± .03 of the population proportion. What sample size is needed to provide the desired margin of error? ANSWER: a. .3965 to .4835 We are 95% confident that the proportion of the population of registered voters that favors the candidate is between .3965 and .4835. b. The required sample size is 1816. 117. An apartment complex developer is considering building apartments in College Town, but first wants to do a market study. A sample was selected of monthly rent values for 70 studio apartments in College Town. The sample mean is $490.80. Based on past experience, the developer assumes a known value of  = $55 for the population standard deviation. a. Develop a 98% confidence interval for the mean monthly rent for all studio apartments in this city. b. Suppose the apartment developer wants a 98% confidence interval estimate of the population mean with a margin of error of E = $10. What sample size is needed? ANSWER: a. 475.48 to 506.12 b. 162

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