484 CHAPTER 9 Inferences from Two Samples Critical Value Method As with the P-value method, we find that the critical value is 3.5879. The test statistic F = 1.5638 is not greater than the critical value, so the test statistic does not fall in the critical region. See Figure 9-5. Step7: Figure 9-5 shows that the test statistic F = 1.5638 does not fall within the critical region, so we fail to reject the null hypothesis of equal variances. There is not sufficient evidence to warrant rejection of the claim of equal standard deviations. 0 Lower Critical Value F5 0.2556 Upper Critical Value F5 3.5879 a/2 5 0.025 Test Statistic: F5 1.5638 a/2 5 0.025 FIGURE 9-5 F-Test of Equal Standard Deviations INTERPRETATION Step8: There is not sufficient evidence to warrant rejection of the claim that the two standard deviations are equal. It appears that the variation among weights of male U.S. Army personnel did not change from 1988 to 2012. YOUR TURN. Do Exercise 5 “Minting of Quarters.” Caution: Part 2 of Section 9-2 includes methods for testing claims about two population means, and one of those methods has a requirement that s1 = s2. Using the F test is not recommended as a way to decide whether this requirement is met. For Section 9-2, using the F test runs the risk of using differences that are too small to have an effect on the t test for two independent samples. That approach is often described as being analogous to sending someone out to sea in a rowboat (the preliminary F test) to determine whether the sea is safe for an ocean liner (the t test). PART 2 Alternative Methods Part 1 of this section presents the F test for testing claims made about the standard deviations (or variances) of two independent populations. Because that test is so sensitive to departures from normality, we now briefly describe alternative methods that are not so sensitive to departures from normality. Count Five The count five method is a relatively simple alternative to the F test, and it does not require normally distributed populations. (See “A Quick, Compact, Two-Sample Dispersion Test: Count Five,” by McGrath and Yeh, American Statistician, Vol. 59, No. 1.) If the two sample sizes are equal, and if one sample has at least five of the largest mean absolute deviations (MAD), then we conclude that its population has a larger variance. See Exercise 17 “Count Five Test” for the specific procedure. Go Figure 12,386,344: Number of different possible plays in every baseball game. 9: Number of different possible ways that a batter can get to first base. 5 to 7: Number of pitches that a typical baseball is used in a major league game.
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