Please the answer the following question manually

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Let X1, ..., Xn be a random sample from a N(0, o2) population with pdf f (x 10, 02) = 1 V27102 expl- (x - 0)2/ (202) ]. Consider testing Ho : 0 00. (a) If o2 known, show that the test that rejects Ho when X > 0 + ZaVo2, is a test of size a. (b) Show that the test in part (a) can be derived as a likelihood ratio test (LRT). (c) Show that the test in part (a) is a UMP test. (d) If o2 unknown, show that the test that rejects Ho when X > 0 + tn-1,a \\ S2, is a test of size a. (e) Assume o2 = 1 is known. Find the most powerful level a = 0.05 test of Ho : 0 = 0 versus H1 : 0 = 1. (Hint: Use the Neyman-Pearson Lemma)