Hi, It says on page 485 of the CFAI test 1 point #2 that a greater than or less than alternative hypothesis: rject the null hypothesis … if the test statistic is greater than the upper alpha point of the F-distribution… Isn’t the F formula the same for the two cases of H0 Variance1 <= Variance 2 and H0: Variance1 >= Variance 2? Then if the F is rejected for H0 of Variance1 <= Variance 2, then it is rejected for the 2nd case as well? This seems counter-intuitive and I am missing some idea. Any comments or corrections? Thanks.

Suppose that you start out with H0: s1^2 <= s2^2 If you do your sampling and s1 < s2, even much less. Your test is over - fail to reject H0 because it looks to be true. If s1 > s2 then go ahead with the F-test with level of significance alpha and make your decision. You are doing a 1-sided test here. Now suppose that you start out with H0: s1^2 <> s2^2. If s1 < s2 we can still maybe reject H0, but we need to look up the 1-alpha/2 quantile of the F-distribution (i.e., a bigger number than we looked up above). Ditto with the s1 > s2. The deal is that if you start with the 1-sided hypotheses it is easier to reject H0 if your assumption is correct but impossible if your assumption is incorrect (which of course might lead you to ask why anybody would want their scientific knowledge based on your assumptions of what you “know” before collecting data and thus you might decide that 1-tail tests of differences in variances are bogus).