Math Stat Final-2009 - Mathematical Statistics-2009 Final...

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Mathematical Statistics-2009 6/25/2009 Final Exam 10:00-12:30 1. State the definitions of the following terms. (1) Type I error and type II error. (2) Power function and the relationship with (1) (3) Size of a test. (4) The most powerful size α test. (5)The likelihood ratio (LR) test. 2. Suppose that you are interested in conducting the statistical test of H : µ ൌ 255 vs H : µ ൐ 255 , and you have decided to use following decision rule: Reject H if the sample mean of a random sample of 81 items is more than 270. Assume that the population standard deviation is 63. (1) Find the probability of making a type I error. (2) Find the probability of making a type I error when µ ൌ 260 . What is the shape of the power function would you expect? Draw the picture and explain in words. 3. Let X , X , … , X be a random sample of size n. (1) State and prove the Neyman-Pearson (NP) Lemma. (2) Suppose that the random sample is from ܰሺߤ, 1ሻ . Find the most powerful (MP) test for testing H : µ ൌ 0 vs H : µ ൌ 2 using NP lemma.
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