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# Example Class9 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 9 1. Let X 1 , . . . , X n be an i.i.d. sample from a Poisson distribution with mean λ . i.e. f ( x | λ ) = e - λ λ x /x !, x = 0 , 1 , . . . . For n = 30 and x i = 327, with the help of Wilk’s Theorem, construct the generalized likelihood ratio test for H 0 : λ = 10 versus H 1 : λ 6 = 10 at α = 0 . 05. 2. A survey of 223 phone boxes found 53 of them to be out of order. A similar survey of 202 phone boxes found 45 of them out of order one month later. It is assumed that each phone box has a fixed probability p of being out of order during the first survey, and a fixed probability q of being out of order during the second survey. The telephone company reports that the condition of the phone boxes is improving. Test H 0 : no change in condition against H 1 : the condition may have changed. (a) Evaluate the likelihood ratio for the specified hypotheses.

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