1. Let X1,X2 be Mid observations from Uniform(l9,6 + 1), 19 > 0. For testing H0 :6 = 0 versus H1 : I9 > 0, we have two competing tests: _ 1(X1) : Reject a! if X1 > 0.95 2(X1,X2) : Reject so if X1 +X2 > C (a) Find the value of 0 so that (:52 has the same size as gb. (b) Calculate the power function of ql and d2 obtained from part (a). (c) Show that $2 from part (a) is not uniformly more powerful test than $1. ((1) Find a test that is always equally or more powerful than (,252 and ql with the same size. Justify your answer. { Hint: Consider the test Reject H0 if X1 +X2 > Cor X1 > 1 orX2 > 1 where O is as in part (a). } (e) (Optional) Consider now a random sample of size n, n > 2 from the same population for testing the same hypothesis. Demonsate that the test Reject H0 if X\") 2 1 or X\") 2 k, where k is a constant, is UMP of its size. Determine k such that the test hassizea, U