# q9 - are observed as follows J F M A M J J A S O N D 39 28...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II EXAMPLE CLASS 9 1. 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. (b) Calculate the (approximate) p -value of the generalized likelihood ratio test with the help of Wilk’s Theorem. 2. The numbers of members of a learned society born in the different months are as- sumed to be Poisson random variables with probably different means. The numbers
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Unformatted text preview: are observed as follows: J F M A M J J A S O N D 39 28 51 49 48 48 41 39 33 60 48 44 It is claimed that members are born at a constant rate throughout the whole year. Test H : a constant rate against H 1 : the rate may vary from month to month. (a) Evaluate the likelihood ratio for the speciﬁed hypotheses. (b) Calculate the (approximate) p-value of the generalized likelihood ratio test with the help of Wilk’s Theorem. 3. From each of six batches of seed a random sample of 100 seeds was selected for sowing. The numbers of seeds that failed to germinate in the six samples of 100 seeds were 12, 20, 9, 17, 24, 16. Calculate the p-value for the Pearson chi-squared test of the hypothesis that the proportion of non-germinating seeds was the same for all batches. Would you reject this hypothesis at 0.05 signiﬁcance level? 1...
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