Suppose X1, ..., Xn N(u, 1), where u E R is unknown. (a) Find the best critical region of size a for a test of Ho : u = 3 vs. HA : M = 4. Name the theorem you used to get your result. (b) Suppose for n = 43, we observe x = 3.21. Find the p-value, and state your conclusion at the 5% level of significance. (c) Assume still that n = 43. Find the value of c such that the critical region C = {x : x > c} has size a = 0.05. (d) Suppose the rejection region is C = {x : x 2 3.21}. Find the sample size n necessary to reject Ho at the 5% significance level. (e) Suppose n = 67 and the critical region is C = {x : x 2 3.21}. Find the statistical power K (u) for the hypothesis test in part (a), when A = 3.6. Give one way we can increase the power of this test