Comparing Two Variances 10.3 SECTION 10.3 Comparing Two Variances 549 What You Should Learn How to interpret the F-distribution and use an F-table to find critical values How to perform a two-sample F-test to compare two variances The F-Distribution The Two-Sample F-Test for Variances The F@Distribution In Chapter 8, you learned how to perform hypothesis tests to compare population means and population proportions. Recall from Section 8.2 that the t@test for the difference between two population means depends on whether the population variances are equal. To determine whether the population variances are equal, you can perform a two-sample F@test. In this section, you will learn about the F@distribution and how it can be used to compare two variances. As you read the next definition, recall that the sample variance s2 is the square of the sample standard deviation s. Let s2 1 and s 2 2 represent the sample variances of two different populations. If both populations are normal and the population variances s 2 1 and s 2 2 are equal, then the sampling distribution of F = s2 1 s2 2 is an F@distribution. Here are several properties of the F@distribution. 1. The F@distribution is a family of curves, each of which is determined by two types of degrees of freedom: the degrees of freedom corresponding to the variance in the numerator, denoted by d.f.N, and the degrees of freedom corresponding to the variance in the denominator, denoted by d.f.D. 2. The F@distribution is positively skewed and therefore the distribution is not symmetric (see figure below). 3. The total area under each F@distribution curve is equal to 1. 4. All values of F are greater than or equal to 0. 5. For all F@distributions, the mean value of F is approximately equal to 1. F d.f.N = 1 and d.f.D = 8 d.f.N = 3 and d.f.D = 11 d.f.N = 8 and d.f.D = 26 d.f.N = 16 and d.f.D = 7 1 2 3 4 F-Distribution for Different Degrees of Freedom DEFINITION For unequal variances, designate the greater sample variance as s1 2. So, in the sampling distribution of F = s2 1 s 2 2, the variance in the numerator is greater than or equal to the variance in the denominator. This means that F is always greater than or equal to 1. As such, all one-tailed tests are right-tailed tests, and for all two-tailed tests, you need only to find the right-tailed critical value.
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