Identifying the Appropriate Variance Ratio Test

The question asks for the appropriate hypothesis test to determine if two samples, assumed to be from normal populations, have the same variance. This involves comparing the variances of these two populations.

Understanding Hypothesis Testing for Variances

When comparing the variances of two populations, the standard approach uses the ratio of their sample variances. The goal is to test the null hypothesis that the population variances are equal.

Appropriate Test: F-test

• The F-test is specifically designed to compare the variances of two populations.
• It utilizes the ratio of the sample variances ($s_1^2$ and $s_2^2$) from the two samples.
• The test statistic follows an F-distribution under the null hypothesis ($H_0: \sigma_1^2 = \sigma_2^2$).
• This directly addresses the question of whether the two samples can be regarded as having the same variance.

Why Other Tests Are Inappropriate

• Z-test: Primarily used for comparing means when population variance is known or for large sample sizes. Not suitable for comparing variances directly.
• t-test: Used for comparing means when population variance is unknown. It does not directly test for equality of variances.
• Chi-square test: Can be used to test if a *single* sample variance matches a hypothesized population variance, but it's not the standard test for comparing the variances of *two* samples using their ratio.

Therefore, the F-test is the correct choice for testing the equality of variances between two populations using the variance ratio.