Suppose that X1, . . . , Xn form a random sample from the normal distribution with unknown mean μ and known variance σ2. In this problem, you will prove the missing steps from the proof that there is no UMP level α0 test for the hypotheses in (9.3.22). Let δ1 be the test procedure with level α0 defined in Example 9.3.8.
a. Let A be a set of possible values for the random vector X = (X1 . . . , Xn). Let μ1 ≠ μ0. Prove that Pr(X ∈ A|μ = μ0) > 0 if and only if Pr(X ∈ A|μ = μ1) > 0.
b. Let δ be a size α0 test for the hypotheses in (9.3.22) that differs from δ1 in the following sense: There is a set A for which δ rejects its null hypothesis when X ∈ A, δ1 does not reject its null hypothesis when X ∈ A, and Pr(X ∈ A|μ = μ0) > 0. Prove that π(μ|δ) < π(μ|δ1) for all μ > μ0.