Fat-free or Regular Pringles: Can Tasters Tell the Difference?

of the United States 2009, p. 178. 10.49 According to the Federal Reserve Board, the mean net worth of U.S. households headed by persons 75 years or older is $640,000. Suppose a simple random sample of 50 households in this age group is obtained from a certain E X E R C I S E S 5217X_10_ch10_p311-363.indd 336 04/02/10 8:40 PM Chapter 10: Hypothesis Tests Involving a Sample Mean or Proportion 337 region of the United States and is found to have a mean net worth of $615,000, with a standard deviation of $120,000. From these sample results, and using the 0.05 level of significance in a two-tail test, comment on whether the mean net worth for all the region’s households in this age category might not be the same as the mean value reported for their counterparts across the nation. Source: Federal Reserve Board, Changes in U.S. Family Finances from 2004 to 2007, p. A11. 10.50 Using the sample results in Exercise 10.49, construct and interpret the 95% confidence interval for the population mean. Is the hypothesized population mean ($640,000) within the interval? Given the presence or absence of the $640,000 value within the interval, is this consistent with the findings of the hypothesis test conducted in Exercise 10.49? 10.51 It has been reported that the average life for halogen lightbulbs is 4000 hours. Learning of this figure, a plant manager would like to find out whether the vibration and temperature conditions that the facility’s bulbs encounter might be having an adverse effect on the service life of bulbs in her plant. In a test involving 15 halogen bulbs installed in various locations around the plant, she finds the average life for bulbs in the sample is 3882 hours, with a standard deviation of 200 hours. Assuming the population of halogen bulb lifetimes to be approximately normally distributed, and using the 0.025 level of significance, do the test results tend to support the manager’s suspicion that adverse conditions might be detrimental to the operating lifespan of halogen lightbulbs used in her plant? Source: Cindy Hall and Gary Visgaitis, “Bulbs Lasting Longer,” USA Today, March 9, 2000, p. 1D. 10.52 In response to an inquiry from its national office, the manager of a local bank has stated that her bank’s average service time for a drive-through customer is 93 seconds. A student intern working at the bank happens to be taking a statistics course and is curious as to whether the true average might be some value other than 93 seconds. The intern observes a simple random sample of 50 drive-through customers whose average service time is 89.5 seconds, with a standard deviation of 11.3 seconds. From these sample results, and using the 0.05 level of significance, what conclusion would the student reach with regard to the bank manager’s claim? 10.53 Using the sample results in Exercise 10.52, construct and interpret the 95% confidence interval for the population mean. Is the hypothesized population mean (93 seconds) within the interval? Given the presence or absence of the 93 seconds value within the interval, is this consistent with the findings of the hypothesis test conducted in Exercise 10.52? 10.54 The U.S. Census Bureau says the 52-question “long form” received by 1 in 6 households during the 2000 census takes a mean of 38 minutes to complete. Suppose a simple random sample of 35 persons is given the form, and their mean time to complete it is 36.8 minutes, with a standard deviation of 4.0 minutes. From these sample results, and using the 0.10 level of significance, would it seem that the actual population mean time for completion might be some value other than 38 minutes? Source: Haya El Nasser, “Census Forms Can Be Filed by Computer,” USA Today, February 10, 2000, p. 4A. 10.55 Using the sample results in Exercise 10.54, construct and interpret the 90% confidence interval for the population mean. Is the hypothesized population mean (38 minutes) within the interval? Given the presence or absence of the 38 minutes value within the interval, is this consistent with the findings of the hypothesis test conducted in Exercise 10.54? ( DATA SET ) Note: Exercises 10.56–10.58 require a computer and statistical software. 10.56 The International Council of Shopping Centers reports that the average teenager spends $57 during a shopping trip to the mall. The promotions director of a local mall has used a variety of strategies to attract area teens to his mall, including live bands and “teenappreciation days” that feature special bargains for this age group. He believes teen shoppers at his mall respond to his promotional efforts by shopping there more often and spending more when they do. Mall management decides to evaluate the promotions director’s success by surveying a simple random sample of 45 local teens and finding out how much they spent on their most recent shopping visit to the mall. The results are listed in data file XR10056. Use a suitable hypothesis test in examining whether the mean mall shopping expenditure for teens in this area might be higher than for U.S. teens as a whole. Identify and interpret the p-value for the test. Using the 0.025 level of significance, what conclusion do you reach? Source: icsc.org, July 23, 2009. 10.57 According to the Insurance Information Institute, the mean annual expenditure for automobile insurance for U.S. motorists is $817. Suppose that a government official in North Carolina has surveyed a simple random sample of 80 residents of her state, and that their auto insurance expenditures for the most recent year are in data file XR10057. Based on these data, examine whether the mean annual auto insurance expenditure for motorists in North Carolina might be different from the $817 for the country as a whole. Identify and interpret the p-value for the test. Using the 0.05 level of significance, what conclusion do you reach? Source: iii.org, July 23, 2009. 10.58 Using the sample data in Exercise 10.57, construct and interpret the 95% confidence interval for the popula tion mean. Is the hypothesized population mean ($817) within the interval? Given the presence or absence of the $817 value within the interval, is this consistent with the findings of the hypothesis test conducted in Exercise 10.57? 5217X_10_ch10_p311-363.indd 337 04/02/10 8:40 PM 338 Part 4: Hypothesis Testing TESTING A PROPORTION Occasions may arise when we wish to compare a sample proportion, p, with a value that has been hypothesized for the population proportion, . As we noted in Figure 10.2, the theoretically correct distribution for dealing with proportions is the binomial distribution. However, the normal distribution is a good approximation when n $ 5 and n(1 2 ) $ 5. The larger the sample size, the better this approximation becomes, and for most practical settings, this condition is satisfied. When using the normal distribution for hypothesis tests of a sample proportion, the test statistic is as follows: Test statistic, z-test for a sample proportion: