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A. Suppose that data are observed under the model X1, X2, Xn - Fo. Consider testing the hypothesis H0 : 0 = 00 vs H1 : 0 of 00 using test statistic S(x) with significance regions of the form Ta = {x : S(x) c1-a}, as described in the lectures. Suppose that the null hypothesis is true. Derive the sampling distribution of the p- value for this test. B. Continuing from part a, show that calling a test significant whenever p-value < a corresponds to a Type I error rate of a. C. Suppose that you have data X1, XN iid - Normal(u, o X1, XN ild - Normal(u,o²) where both parameters are unknown. Derive the generalized likelihood ratio statistic for Ho: = VS H1 : po. Demonstrate whether the set of significance regions for this test is identical to the set of significance regions for the one- sample two-sided t-test. D. For a given significance region, argue whether it is true or not that one test will compute a larger Type I error rate even though the significance region is the same for the two tests.

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