Suppose that X1, . . . , xm form a random sample from the normal distribution with mean 1 and variance 21, and Y1, .

Suppose that X1, . . . , xm form a random sample from the normal distribution with mean μ1 and variance σ21, and Y1, . . . , yn form a random sample from the normal distribution with mean μ2 and variance σ22 . Suppose that we use the usual improper prior and that we wish to test the hypotheses
H0: σ21 ≤ σ22,
H1: σ21 > σ22.
a. Prove that the level α0 F test is the same as the test in (9.8.7) when α0 = w1/(w0 + w1).
b. Prove that the p-value for the F test is the posterior probability that H0 is true.