Statistics2009aug - Statistics Qualifying Exam for MAT...

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Statistics Qualifying Exam for MAT 651/652 August, 2009 1. Let X 1 , X 2 , . . . , X 10 be a random sample of size 10 from a continuous uniform distribution on (0 , 2) . Let W = X (10) - X (1) where X (10) is the largest order statistic and X (1) the smallest order statistic of the sample. (a) Find P ( X 1 X 2 < 3) . (b) Find P ( W > 1) . 2. Let X 1 , X 2 , . . . , X 20 be a random sample from a discrete uniform distribution on { 1 , 2 , . . . , θ } where θ is unknown. We are interested in testing the null hypothesis H 0 : θ = 4 versus H 1 : θ = 5 . (a) Assume that we consider only nonrandomized tests. For what α levels does there exists a MP test? (b) Find a randomized MP test with P(type I error) = .1. (c) What is the power of the MP test in (b)? 3. Let X 1 , . . . , X n be i.i.d. with density function proportional to exp - x 2 2 - θx 4 · . (a) Find the uniformly most powerful test of H 0 : θ = 0 versus H 1 : θ > 0 . (b) Find an appropriate transformation of your test statistic in (a) for which you can find the critical value or compute the P-value, at least asymptotically. For α = 0 . 05 , find the critical value of your asymptotic test.
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