2. (25 marks) Let X1, .., Xn he a random sample from N (6, 02). where 0'2 is known. (a) (10 marks) Find the uniformly most powerful(UMP) test for H0 : 6 = 90 versus H1 : 6 > 60 at the signicance level or. (b) (6 marks) Is the test from part (a) a UMP test for H0 : 6 s 60 versus H1 : 9 > 90'? Why? Please provide detailed steps for your argument. (0) (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 = 60 + 0/2 for this test. Find the value of sample size n to achieve this