Chapter 9: Introduction to Hypothesis Testing

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416 CHAPTER 9 • INTRODUCTION TO HYPOTHESIS TESTING 9-3: Exercises Skill Development 9-46. You are given the following null and alternative hypotheses: H 0 : 1.20 H A : 1.20 0.10 a. If the true population mean is 1.25, determine the value of beta. Assume the population standard deviation is known to be 0.50 and the sample size is 60. b. Referring to part a, calculate the power of the test. c. Referring to parts a and b, what could be done to increase power and reduce beta when the true population mean is 1.25? Discuss. d. Indicate clearly the decision rule that would be used to test the null hypothesis, and determine what decision should be made if the sample mean were 1.23. 9-47. You are given the following null and alternative hypotheses: H 0 : 4,350 H A : 4,350 0.05 a. If the true population mean is 4,345, determine the value of beta. Assume the population standard deviation is known to be 200 and the sample size is 100. b. Referring to part a, calculate the power of the test. c. Referring to parts a and b, what could be done to increase power and reduce beta when the true population mean is 4,345? Discuss. d. Indicate clearly the decision rule that would be used to test the null hypothesis, and determine what decision should be made if the sample mean were 4,337.50. 9-48. You are given the following null and alternative hypotheses: H 0 : 500 H A : 500 0.01 Calculate the probability of committing a Type II error when the population mean is 505, the sample size is 64, and the population standard deviation is known to be 36. 9-49. You are given the following null and alternative hypotheses: H 0 : 200 H A : 200 0.10 Calculate the probability of committing a Type II error when the population mean is 197, the sample size is 36, and the population standard deviation is known to be 24. 9-50. For each of the following situations, indicate what the general impact on the Type II error probability will be: a. The alpha level is increased. b. The “true” population mean is moved farther from the hypothesized population mean. c. The alpha level is decreased. d. The sample size is increased. 9-51. Solve for beta when the “true” population mean is 103 and the following information is given: H 0 : 100 H A : 100 0.05 10 n 49 9-52. Consider the following hypotheses: H 0 : 103 H A : 103 A sample of size 20 is to be taken from a population with a mean of 100 and a standard deviation of 4. Determine the probability of committing a Type II error for each of the following significance levels: a. α 0.01 b. α 0.025 c. α 0.05 9-53. Consider the following hypotheses: H 0 : 30 H A : 30 A sample of size 50 is to be taken from a population with a standard deviation of 13. The hypothesis is to be conducted using a significance level of 0.05. Determine the probability of committing a Type II error when a. 22 b. 25 c. 29 9-54. Consider the following hypotheses: H 0 : 201 H A : 201 A sample is to be taken from a population with a mean of 203 and a standard deviation of 3. The hypothesis is to be conducted using a significance
CHAPTER 9 • INTRODUCTION TO HYPOTHESIS TESTING 417 level of 0.05. Determine the probability of committing a Type II error when a. n 10 b. n 20 c. n 50 9-55. The following hypotheses are to be tested: H 0 : 0.35 H A : 0.35 A random sample of 400 is taken. Using each set of information following, compute the power of the test. a. 0.01, true 0.32 b. 0.025, true 0.33 c. 0.05, true 0.34 9-56. The following hypotheses are to be tested: H 0 : 0.65 H A : 0.65 A random sample of 500 is taken. Using each set of information following, compute the power of the test. a. 0.01, true 0.68 b. 0.025, true 0.67 c. 0.05, true 0.66 Business Applications 9-57. Waiters at Finegold’s Restaurant and Lounge earn most of their income from tips. Each waiter is required to “tip-out” a portion of tips to the table bussers and hostesses. The manager has based the “tip-out” rate on the assumption that the mean tip is at least 15% of the customer bill. To make sure that this is the correct assumption, he has decided to conduct a test by randomly sampling 60 bills and recording the actual tips. a. State the appropriate null and alternative hypotheses. b. Calculate the probability of a Type II error if the true mean is 14%. Assume that the population standard deviation is known to be 2% and a significance level equal of 0.01 will be used to conduct the hypothesis test. 9-58. According to data from the Environmental Protection Agency, the average daily water consumption for a household of four people in the United States is approximately 243 gallons. (Source: http://www.catskillcenter.org/programs/ csp/H20/Lesson3/house3.htm) Suppose a state agency plans to test this claim using an alpha level equal to 0.05 and a random sample of 100 households with four people. a. State the appropriate null and alternative hypotheses. b. Calculate the probability of committing a Type II error if the true population mean is 230 gallons. Assume that the population standard deviation is known to be 40 gallons. 9-59. According to JobWeb ® , a Web site of career development and job-search information for college students and new college graduates, owned and sponsored by the National Association of Colleges and Employers (NACE), the average starting salary for accounting graduates in 2004 exceeded $41,000. Suppose that the American Society for Certified Public Accountants planned to test this claim by randomly sampling 200 accountants who graduated in 2004. a. State the appropriate null and alternative hypotheses. b. Compute the power of the hypothesis test to reject the null hypothesis if the true population mean starting salary is only $40,000. Assume that the population standard deviation is known to be $4,600 and the test is to be conducted using an alpha level equal to 0.01. 9-60. Swift is the holding company for Swift Transportation Co., Inc., a truckload carrier headquartered in Phoenix, Arizona. Swift operates the largest truckload fleet in the United States. Before Swift switched to its current computer-based billing system, the average payment time from customers was approximately 40 days. Suppose before purchasing the present billing system, it performed a test by examining a random sample of 24 invoices to see if the system would reduce the average billing time. The sample indicates that the average payment time is 38.7 days. a. The company that created the billing system indicates that it would reduce the average billing time to less than 40 days. Conduct a hypothesis test to determine if the new computer-based billing system would reduce the average billing time to less than 40 days. Assume the standard deviation is known to be 6 days. Use a significance level of 0.025. b. If the billing system actually reduced the average billing time to 36 days, determine the probability that a wrong decision was made in part a. 9-61. Nationwide Mutual Insurance, based in Columbus, Ohio, is one of the largest diversified insurance and financial services organizations in the world, with more than $157 billion in assets. Nationwide ranked #99 on the Fortune 100 list in 2005. The company provides a full range of insurance and financial services. In a July 26, 2005 news release (“ID Theft Victims Struggle to Achieve Resolution”), Nationwide reported the results of a new survey of 1,097 identity theft victims. The survey shows victims spend an average of 81 hours trying to resolve their case. If the true average time spent was 81 hours, determine the probability that a test of hypothesis designed to test that the average was less than 85 hours would select the research
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Chapter 9: Introduction to Hypothesis Testing

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