When testing H0: σ12 = σ22 vs. Ha: σ12 ≠ σ22 two random samples of sizes 10 and 8, respectively, are used. The calculated value of the test statistic is found to be equal to 2.22. Which of the following statements is true?

a. H0 must be rejected at α =.05.
b. H0 cannot be rejected at α =.05.
c. There is sufficient evidence at α =.05 to conclude that a difference in the population variances exist.
d. H0 cannot be rejected at α =.10.
e. none of these

To determine which statement is true, we need to compare the calculated test statistic (2.22) with the critical value(s) for the given significance level (α).

In this case, the test statistic is calculated as the ratio of the sample variances (s12 / s22). Under the assumption that the population variances are equal (H0: σ12 = σ22), the test statistic follows an F-distribution.

To make a decision, we need to find the critical value(s) from the F-distribution tables or use statistical software. However, without the degrees of freedom for both samples, it is not possible to calculate the critical value(s).

Therefore, we cannot determine which statement is true without additional information.

To determine the correct answer, we need to compare the calculated value of the test statistic (2.22) to the critical value(s) based on the given significance level (α) and degrees of freedom.

To conduct a hypothesis test comparing two population variances, we can use the F-test. The F-test compares the ratio of the variances of the two populations.

Given that the null hypothesis is H0: σ12 = σ22 and the alternative hypothesis is Ha: σ12 ≠ σ22, this is a two-tailed test. We want to determine if there is enough evidence to conclude that the variances are significantly different.

To find the critical value(s) for the F-test, we need to determine the degrees of freedom for the numerator and denominator.

The degrees of freedom for the numerator is equal to the sample size of the first sample minus 1.
The degrees of freedom for the denominator is equal to the sample size of the second sample minus 1.

In this case, the sample sizes are 10 and 8, so the degrees of freedom are 10-1 = 9 and 8-1 = 7.

Using a table or calculator, we can find the critical values for the F-test at the specified significance level (α) and degrees of freedom. For example, at α = 0.05, the critical values for a two-tailed test with 9 and 7 degrees of freedom are approximately 2.70 and 0.37.

Comparing the calculated value of the test statistic (2.22) to the critical values, we can see that it does not exceed either of the critical values.

Therefore, we fail to reject the null hypothesis.

The correct answer is:

b. H0 cannot be rejected at α = 0.05.