Could you help giving me explanations on this problem ? Thank you.

Let X be a normal random variable with unknown mean ,u and known variance 02 = 1. . The null hypothesis is that no = 2; . The alternative hypothesis is that m = 3. Let X1, . . . Xn be H = 20 independent samples of X , and let Z be Z := Zoning- : X1 + 2X2 3X3 + 4)::1 + (1)nan. i=1 (a) I I Compute the expectation and variance of Z. (b) __ F4 Let C be the critical region C := {(X1,...,Xn) | Z 2 19}. Find the value It so that the type I error of this critical region is o: = 0.05. \f(d) Consider another critical region C" given by C' ( X 1 , . . . , Xn ) | X Z Hot Z0.05 := In the type I error for this test is a = 0.05. Compute the type II error for this test.(e) I 11 I Prove that the critical region 0 in part (b) is not a uni- fornily most powerful critical region of size (.1: = 0.05. (Hint: use previous parts of the question)