Multiple Choice
Which statement about the -distribution is always true?
10. Hypothesis Testing for Two Samples
Two Variances and F Distribution
Multiple Choice
In a two-sample Hypothesis Test for Variance, what does an -statistic near 1 indicate?
A
Both populations have zero variance.
B
We reject the null hypothesis .
C
The numerator variance is smaller than the denominator variance.
D
The two sample variances are approximately equal.
Verified step by step guidance1
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.
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.
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 \(F \geq 1\).
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.
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 \(\alpha\).
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