Tutorial 8

Tutorial 8 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2009-10) EXAMPLE CLASS 8 1. Let X 1 and X 2 be i.i.d. exponential random variables with mean θ . It is well known that S = X 1 + X 2 has a gamma distribution with the density function f ( s ) = s exp(- s/θ ) /θ 2 , s > . Suppose the test with critical region C k = { ( x 1 ,x 2 ) : x 1 + x 2 > k } for some positive k is used to test H : θ ≤ 1 against H 1 : θ > 1 . (a) Given the power function of the test ω ( θ ) = (1 + k/θ )exp(- k/θ ) , show that it is an increasing function in θ . (b) Deduce from (a) the size of the test. 2. Let X 1 ,...,X n be i.i.d. Poisson random variables with mean θ . (a) Find the critical region for the most powerful size α test of H : θ = 1 vs H 1 : θ = 1 . 21 (b) Use the Central Limit Theorem to show that the smallest value of n required to make α = 0 . 05 and β ≤ . 1 ( where α and β are the type I and II error probabilities respectively) is somewhere near 212.probabilities respectively) is somewhere near 212....
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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

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Tutorial 8 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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