Problem 8.6 Suppose we use a sample of size n = 9 taken from a normal distribution with = 4 and the rejection region

Problem 8.6 Suppose we use a sample of size n = 9 taken from a normal distribution with σ = 4 and the rejection region C = {x : x < 28} to test the hypothesis H0: µ ≥ 30 against H1 : µ < 30.

(a) What is the significance level of this test? Compute the probability of a type II error at the alternative: µ = 25.

(b) Suppose the cutoff value is changed to c = 27.5, so the rejection region is the set C = {x : x < 27.5}. Compute the significance level of this test; next, compute the probability of a type II error at the alternative µ = 25. Compare the type I and type II errors of this test with the test of part (a). Interpret your results.