Question: Problem 3. Let X1, . .., Xn be i.i.d U(0, 0 + 1). To test Ho : 0 = 0 versus H1 : 0 >

Problem 3. Let X1, . .., Xn be i.i.d U(0, 0 + 1). To test Ho : 0 = 0 versus H1 : 0 > 0, use the test with a rejection region R = {X(1) > k or X(n) 2 1}, in which k is constant to be determined by a pre-specified significance level. (1) Determine k so that the test will have significance level a; (2) Derive the power function of the test in (1); (3) Prove that the test is a UMP test of size a. (4) Determine n and k such that the test in (1) has significance level a = 0.1 and its power function is at least 0.8 of 0 > 1

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