Return to the situations described in Example 9.1.17 and Exercise 19.We wish to compare what might happen if we switch the null and alternative hypotheses. That is, we want to compare the results of testing the hypotheses in (9.1.22) at level α0 to the results of testing the hypotheses in (9.1.27) at level α0.
a. Let α0 < 0.5. Prove that there are no possible data sets such that we would reject both of the null hypotheses simultaneously. That is, for every possible n and σ', we must fail to reject at least one of the two null hypotheses.
b. Let α0 < 0.5. Prove that there are data sets that would lead to failing to reject both null hypotheses. Also prove that there are data sets that would lead to rejecting each of the null hypotheses while failing to reject the other.
c. Let α0 > 0.5. Prove that there are data sets that would lead to rejecting both null hypotheses.