Chapter 9: Introduction to Hypothesis Testing

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422 CHAPTER 9 • INTRODUCTION TO HYPOTHESIS TESTING A random sample of 50 accounts revealed an average additional payment of $114 with a sample standard deviation of $50. a. Testing at a significance level of 0.10, do the sample data refute the director’s claim? b. Determine the largest sample mean (with the same sample size and standard deviation) that would fail to refute the director’s claim. 9-80. Bowman Electronics sells electronic components for car stereos. It claims that the average life of a component exceeds 4,000 hours. To test this claim, it has selected a random sample of n 12 of the components and traced the life between installation and failure. The following data were obtained: 1,973 4,838 3,805 4,494 4,738 5,249 4,459 4,098 4,722 5,894 3,322 4,800 a. State the appropriate null and alternative hypotheses. b. Assuming that the test is to be conducted using a 0.05 level of significance, what conclusion should be reached based on these sample data? Be sure to examine the required normality assumption. 9-81. The college of business at a state university has a computer literacy requirement for all graduates: Students must show proficiency with a computer spreadsheet software package and with a wordprocessing software package. To assess whether students are computer literate, a test is given at the end of each semester. The test is designed so that at least 70% of all students who have taken a special microcomputer course will pass. Suppose that, in a random sample of 100 students who have recently finished the microcomputer course, 63 pass the proficiency test. a. Using a significance level of 0.05, what conclusions should the administrators make regarding the difficulty of the test? b. Describe a Type II error in the context of this problem. 9-82. In a March 2002 article, Golf Digest reported on a survey conducted involving 300 golfers in which the respondents were asked their views about the impact of new technologies on the game of golf. Before the study, a group of United States Golf Association officials believed that less than 50% of golfers believed that professional golfers should have different equipment rules than amateurs. The survey conducted by the Golf Digest found 67% did not favor different equipment rules. a. If the claim made by the USGA is to be tested, what should the null and alternative hypotheses be? b. Based on the sample data, and an alpha level equal to 0.05, use the p-value approach to conduct the hypothesis test. 9-83. USA Today reports (Darryl Haralson, “It’s all about Overstock.com,” July 25, 2005) on an ad for Overstock.com which sells discounted merchandize on its Web site. To evaluate the effectiveness of the ads, Harris Interactive conducted a nationwide poll of 883 adults. Of the 883 adults, 168 thought the ads were very effective. This was compared to the Harris Ad Track average of 21%. a. Determine if the sample size is large enough for the test in part b to warrant approximating the sample proportion’s distribution with a normal distribution. b. Does the Harris poll provide evidence to contend that the proportion of adults who find Overstock.com’s ads to be very effective is smaller than the Harris Ad Track average? Use a significance level of 0.05. 9-84. ACNielsen is a New York-based leading global provider of marketing research information services, analytical systems and tools, and professional client service. A recent issue of its magazine (Todd Hale, “Winning Retail Strategies Start with High Value Consumers,” Consumer Insight, Spring 2005) addressed, in part, consumers attitudes to self-checkout lines. Of the 17,346 EDLP (every day low price) shoppers, only 3,470 indicated an interest in this service. If Wal-Mart’s CEO, Lee Scott, had decided not to install self-checkout lines unless consumer interest was more than 17.5%, would he order the installation? a. Determine if the sample size for the test indicated is large enough to warrant approximating the sample proportion’s distribution with a normal distribution. b. Use a significance level of 0.05 and the p-value approach to answer the question put forward above. 9-85. The Arrow Tire and Rubber Company plans to warranty its new mountain bike tire for 12 months. However, before it does this, the company wants to be sure that the mean lifetime of the tires is at least 18 months under normal operations. It will put the warranty in place unless the sample data strongly suggest that the mean lifetime of the tires is less than 18 months. The company plans to test this statistically using a random sample of tires. The test will be conducted using an alpha level of 0.03. a. If the population mean is actually 16.5 months, determine the probability the hypothesis test will lead to incorrectly accepting the null hypothesis. Assume that the population standard deviation is known to be 2.4 months and the sample size is 60.
CHAPTER 9 • INTRODUCTION TO HYPOTHESIS TESTING 423 b. If the population mean is actually 17.3, calculate the chance of committing a Type II error. This is a specific example of a generalization relating the probability of committing a Type II error and the parameter being tested. State this generalization. c. Without calculating the probability, state whether the probability of a Type II error would be larger or smaller than that calculated in part b if you were to calculate it for a hypothesized mean of 15 months. Justify your answer. d. Suppose the company decides to increase the sample size from 60 to 100 tires. What can you expect to happen to the probabilities calculated in part a? 9-86. The makers of Mini-Oats Cereal have an automated packaging machine that can be set at any targeted fill level between 12 and 32 ounces. At the end of every shift (eight hours), 16 boxes are selected at random and the mean and standard deviation of the sample are computed. Based on these sample results, the production control manager determines whether the filling machine needs to be readjusted or whether it remains all right to operate. Previous data suggest the fill level has a normal distribution with a standard deviation of 0.65 ounces. Use α 0.05. The test is a two-sided test to determine if the mean fill level was equal to 24 ounces. a. Calculate the probability that the test procedure will detect that the average fill level is not equal to 24 ounces when in fact it equals 24.5 ounces. b. On the basis of your calculation in part a, would you suggest a change in the test procedure? Explain what change you would make and the reasons you would make this change. 9-87. The Lazer Company has a contract to produce a part for Boeing Corporation that must have an average diameter of 6 inches and a standard deviation of 0.10 inch. The Lazer Company has developed the process that will meet the specifications with respect to the standard deviation, but it is still trying to meet the mean specifications. A test run (considered a random sample) of parts was produced and the company wishes to determine whether this latest process that produced the sample will produce parts meeting the requirement of an average diameter equal to 6 inches. a. Specify the appropriate research, null, and alternative hypotheses. b. Develop the decision rule assuming that the sample size is 200 parts and the significance level is 0.01. c. What should the Lazer Company conclude if the sample mean diameter for the 200 parts is 6.03 inches? Discuss. 9-88. The personnel manager for a large airline has claimed that, on average, workers are asked to work no more than 3 hours overtime per week. Past studies show the standard deviation in overtime hours per worker to be 1.2 hours. Suppose the union negotiators wish to test this claim by sampling payroll records for 250 employees. They believe that the personnel manager’s claim is untrue but want to base their conclusion on the sample results. a. State the research, null, and alternative hypotheses and discuss the meaning of Type I and Type II errors in the context of this case. b. Establish the appropriate decision rule if the union wishes to have no more than a 0.01 chance of a Type I error. c. The payroll records produced a sample mean of 3.15 hours. Do the union negotiators have a basis for a grievance against the airline? Support your answer with a relevant statistical procedure. 9-89. About 74% of Freddie Mac-owned loans were refinanced in the second quarter of 2005 (USA Today, Lifeline, August 3, 2005), resulting in new mortgages carrying loan amounts at least 5% above the original mortgage balance. The median loan refinanced in the second quarter was 2.6 years old. If a sample size of 2,500 was used to obtain this information, a. determine if the sample size for the test in part b is large enough to warrant approximating the sample proportion’s distribution with a normal distribution. b. use this information to determine if less than 75% of new mortgages had a loan amount at least 5% above the original mortgage balance. Use a test statistic approach with 0.025. 9-90. Cisco Systems, Inc. is the leading maker of networking gear that connects computers to the Internet. Company managers are concerned with the productivity of their workers as well as their job satisfaction. Kate D’Camp is the senior vice president for human resources. She often initiates surveys concerning Cisco’s personnel. A typical survey asked, “Do you feel it’s OK for your company to monitor your Internet use?” Two hundred and twenty-three of the 405 respondents chose “Only after informing me.” Cisco would consider monitoring if more than 50% of its workers wouldn’t mind if informed beforehand that the company was going to monitor their Internet usage. a. D’Camp may have read the USA Today issue (Snapshots, August 2, 2005) that indicated 55% of American workers wouldn’t object after being informed. So she might desire that the test indicate with a high probability that more than 50% of Cisco workers wouldn’t object when in fact her workers reflect the opinion of all
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Chapter 9: Introduction to Hypothesis Testing

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