482 CHAPTER 9 Inferences from Two Samples F Distribution For two normally distributed populations with equal variances 1s 2 1 = s 2 22, the sampling distribution of the test statistic F = s2 1>s 2 2 is the F distribution shown in Figure 9-4 (provided that we have not yet imposed the stipulation that the larger sample variance is s2 1). If you repeat the process of selecting samples from two normally distributed populations with equal variances, the distribution of the ratio s2 1>s 2 2 is the F distribution. 0 Not symmetric (skewed to the right) F Nonnegative values only a Value of F 5 s1 2 2s 2 FIGURE 9-4 F Distribution There is a different F distribution for each different pair of degrees of freedom for the numerator and denominator. See Figure 9-4 and note these properties of the F distribution: ■ The F distribution is not symmetric. ■ Values of the F distribution cannot be negative. ■ The exact shape of the F distribution depends on the two different degrees of freedom. Interpreting the Value of the F Test Statistic If the two populations have equal variances, then the ratio s2 1>s 2 2 will tend to be close to 1. Because we are stipulating that s2 1 is the larger sample variance, the ratio s 2 1>s 2 2 will be a large number whenever s2 1 and s 2 2 are far apart in value. Consequently, a value of F near 1 will be evidence in favor of s 2 1 = s 2 2, but a large value of F will be evidence against s 2 1 = s 2 2. Large values of F are evidence against s 2 1 = s 2 2. Weights of Male Army Personnel EXAMPLE 1 Listed below are weights (kg) of randomly selected male U.S. Army personnel from Data Set 2 “ANSUR I 1988” and Data Set 3 “ANSUR II 2012.” Use a 0.05 significance level to test the claim that the variation among weights did not change from the ANSUR I study in 1988 to the ANSUR II study in 2012. ANSUR I 1988 63.088.9 71.183.684.276.369.574.4 81.472.085.5111.1 ANSUR II 2012 90.8 86.1 101.1 76.9 63.0 98.4 83.5 65.1 111.5 78.0
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