440 CHAPTER 8 Hypothesis Testing with Two Samples TRY IT YOURSELF 1 A golf instruction and club fitting company claims that people who play golf can improve (decrease) their average golf scores by using the company’s newly designed golf clubs. The average golf scores of eight randomly selected people who play golf are determined before and after using the company’s newly designed golf clubs. The results are shown in the table. At a = 0.10, is there enough evidence to support the company’s claim? Assume the average golf scores are normally distributed. Answer: Page A41 Note that many advertisements misuse statistical results. For instance, statistical results could be misused by implying a cause-and-effect relationship that has not been substantiated by testing. The t-Test for the Difference Between Means The campaign staff for a state legislator wants to determine whether the legislator’s performance rating (0 –100) has changed from last year to this year. The table below shows the legislator’s performance ratings from the same 16 randomly selected voters for last year and this year. At a = 0.05, is there enough evidence to conclude that the legislator’s performance rating has changed? Assume the performance ratings are normally distributed. Voter 1 2 3 4 5 6 7 8 Rating (last year) 60 54 78 84 91 25 50 65 Rating (this year) 54 46 68 58 83 38 38 53 Voter 9 10111213141516 Rating (last year) 68 81 75 45 62 79 58 63 Rating (this year) 78 72 76 48 48 83 51 58 SOLUTION Because the samples are random and dependent, and the populations are normally distributed, you can use the t@test. If there is a change in the legislator’s rating, then there will be a difference between last year’s ratings and this year’s ratings. Because the legislator wants to determine whether there is a difference, the null and alternative hypotheses are H0: md = 0 and Ha: md ≠ 0. (Claim) Because the test is a two-tailed test, a = 0.05, and d.f. = 16 - 1 = 15, the critical values are -t0 = -2.131 and t0 = 2.131. The rejection regions are t 6 -2.131 and t 7 2.131. EXAMPLE 2 Tech Tip One way to use technology to perform a hypothesis test for the difference between means is to enter the data in two columns and form a third column in which you calculate the difference for each pair. You can now perform a one-sample t@test on the difference column, as shown in Chapter 7. Athlete 1 2 3 4 5 6 7 8 Average golf score (before using newly designed golf clubs) 89 84 96 82 74 92 85 91 Average golf score (after using newly designed golf clubs) 83 83 92 84 76 91 80 91
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