In a two-sample Hypothesis Test for Variance, what does an $F$-statistic near 1 indicate?

• A. Both populations have zero variance.
• B. We reject the null hypothesis $H_0$.
• C. The numerator variance is smaller than the denominator variance.
• D. The two sample variances are approximately equal.

Answer: (D)

Verified step by step guidance

1. Step 1: Identify the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)) for the test. Typically, for testing two variances, the hypotheses are \(H_0: \sigma_1^2 = \sigma_2^2\) (the variances are equal) versus \(H_a: \sigma_1^2 \neq \sigma_2^2\) (the variances are not equal), or one-sided alternatives depending on the problem context.
2. Step 2: Collect the sample variances (\(s_1^2\) and \(s_2^2\)) and sample sizes (\(n_1\) and \(n_2\)) from the two independent samples. These values are necessary to compute the test statistic.
3. Step 3: Calculate the test statistic using the formula for the F-test for equality of variances: \[F = \frac{s_1^2}{s_2^2}\] where \(s_1^2\) is the larger sample variance to ensure \(F \geq 1\).
4. Step 4: Determine the degrees of freedom for the numerator and denominator: \[df_1 = n_1 - 1\] \[df_2 = n_2 - 1\] These are used to find the critical value or p-value from the F-distribution.
5. Step 5: Compare the calculated \(F\) statistic to the critical value from the F-distribution table at the chosen significance level (e.g., \(\alpha = 0.05\)). If \(F\) is greater than the critical value, reject the null hypothesis; otherwise, do not reject it. Alternatively, calculate the p-value and compare it to \(\alpha\).