Final answer:
The question involves constructing a likelihood ratio test using an F-test to compare two variances from normal distributions, determining if they share a common variance or not.
Step-by-step explanation:
The question relates to constructing a likelihood ratio test for comparing two variances from independent samples drawn from normal distributions with the assumption of a common variance. When testing for the equality of variances, the F-test is used, where the F-statistic is the ratio of the two sample variances.
For this scenario, where we have S1² and S2² denoting the variances of independent samples of sizes n and m, we can ascertain whether the common variance is equal to σ0² (the null hypothesis, H0) or σa² (the alternative hypothesis, Ha), assuming σa² > σ0².
Under the null hypothesis, the F-statistic should be close to 1 since both sample variances would be estimates of the same population variance. If the observed F is much larger than 1, it suggests that the common variance is greater than σ0², which supports the alternative hypothesis. This test relies on both samples coming from normal distributions to produce reliable results.
Note that this test is generally right-tailed due to the assumption that σa² is greater than σ0², and degrees of freedom for the F distribution are n - 1 and m - 1, respectively.