e-Statistics

F-Test

In the comparison of population means in two independent groups, say ``Group 1'' and ``Group 2,'' test procedures (pooled t-test and Wilcoxon test) often assume that the two population variances are approximately equal. This assumption itself can be treated as hypothesis test, and F-test is introduced for the plausibility of equal variances. The test is actually concerned with how Group 1 and Group 2 differ in terms of their respective population variance $\sigma_1^2$ and $\sigma_2^2$ .

$H_A:\hspace{0.05in}\sigma_1^2$ $\sigma_2^2$

The test procedure is based upon the sample standard deviations and from Group 1 and 2 with the respective sample size and . The test statistic $F = \frac{S_1^2}{S_2^2}$ is likely observed around, greater than, or less than unity under the respective null hypothesis `` $\sigma_1^2 = \sigma_2^2$ ,'' `` $\sigma_1^2 \ge \sigma_2^2$ ,'' or `` $\sigma_1^2 \le \sigma_2^2$ .'' The opposite of such observation is expressed by p-value < $\alpha$ , suggesting an evidence against the null hypothesis . To check the plausibility of equal variances, we reason that the null hypothesis `` $\sigma_1^2 = \sigma_2^2$ '' can be accepted when we fail to reject it. The F-test is known to be very sensitive to the appropriateness of normality assumption.