552 CHAPTER 10 Chi-Square Tests and the F -Distribution The Two-Sample F@Test for Variances In the remainder of this section, you will learn how to perform a two-sample F@test for comparing two population variances using a sample from each population. A two-sample F@test is used to compare two population variances s 2 1 and s 2 2. To perform this test, these conditions must be met. 1. The samples must be random. 2. The samples must be independent. 3. Each population must have a normal distribution. The test statistic is F = s2 1 s2 2 where s2 1 and s 2 2 represent the sample variances with s 2 1 Ú s 2 2. The numerator has d.f.N = n1 - 1 degrees of freedom and the denominator has d.f.D = n2 - 1 degrees of freedom, where n1 is the size of the sample having variance s2 1 and n2 is the size of the sample having variance s 2 2. Two-Sample F-Test for Variances Using a Two-Sample F@Test to Compare S 2 1 and S 2 2 In Words In Symbols 1. Verify that the samples are random and independent, and the populations have normal distributions. 2. Identify the claim. State the null and State H0 and Ha. alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Identify the degrees of freedom d.f.N = n1 - 1 for the numerator and the denominator. d.f.D = n2 - 1 5. Determine the critical value. Use Table 7 in Appendix B. 6. Determine the rejection region. 7. Find the test statistic and sketch F = s2 1 s2 2 the sampling distribution. 8. Make a decision to reject or fail If F is in the rejection to reject the null hypothesis. region, then reject H0. Otherwise, fail to reject H0. 9. Interpret the decision in the context of the original claim. GUIDELINES In some cases, you will be given the sample standard deviations s1 and s2. Remember to square both standard deviations to calculate the sample variances s2 1 and s 2 2 before using a two-sample F-test to compare variances.
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