(a) State and explain the N-P lemma, paying particular attention to the framing of the null and alternative hypotheses.

(b) State and explain the two conditions needed to extend the N-P lemma to more realistic cases for the existence of these α-UMP tests.

(c) In the case of the simple Normal model the two formulations of the null and alternative hypotheses: (i) H0: μ ≤ μ0, vs. H1: μ > μ0, (ii) H0: μ = μ0, vs. H1:
μ > μ0, give rise to the same optimal test. Explain why the t-test is UMP in this case.

(d) Explain why in testing the hypotheses: (iii) H0: μ = μ0, vs. H1: μ = μ0, the optimal t-test for (i)-(ii) in

(c) is biased when used to test (iii). Suggest an unbiased test for (iii).