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# Let U1, U2 . . . , Un be iid copies of Unif(0, θ). We are interested in testing H0 : θ = 1 and Ha : θ > 1.

Suppose we have two test statistics and two corresponding rules to reject the null hypothesis as follows: • T1 = U1, reject H0 if T1 > 0.95. • T2 = max{U1, . . . , Un}, reject H0 if T2 > 0.951/n . (a) (5 points) What are the Type I error probabilities of those two rules? (b) (5 points) Calculate the number of replications needed to estimate the power of the test with a margin of error of at most 0.01 at the conservative approximate 95% confidence level. ·2· (c) (5 points) Using the number of replications computed above to produce a simulation-based power curve for the two rules using n = 20, θ ∈ {1, 1.04, 1.08, . . . , 2}. Comment and compare the two plots. (d) (5 points) Using the number of replications computed above to produce a simulation-based power curve for the two rules using θ = 1.1, n = {20, 30, 40, . . . 500}. Comment and compare the two plots.

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