2. (25 marks) Let X1, X\" be a random sample from N (6, 0'2), where 0'2 is known. (a) (10 marks) Find the uniformly most powerful(UMP) test for H0 : 9 = 90 versus H, : 9 > 90 at the signicance level 0:. (b) (6 marks) Is the test from part (a) a UMP test for H0 : 9 S 60 versus H, : 9 > 60? Why? Please provide detailed steps for your argument. (c) (9 marks) For the test in part (b), nd an expression for the power function of this test. The experimenter desires a Type I Error probability of 0.05 and a power of 0.8 at 9 = 80 + 0/2 for this test. Find the value of sample size n to achieve this