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# 09 - April/08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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April /08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1302 PROBABILITY AND STATISTICS II Example Class 9 1. (a) It is known that an observation made on a certain system will yield a result falling into one of k categories, C 1 , C 2 , · · · , C k . To each value of the real parameter θ in a given interval corresponds a probability distribution p i ( θ )( i = 1 , 2 , · · · , k ). The null hypothesis is made that, for some unknown value of θ , the probability of a result falling into category C i ( i = 1 , 2 , · · · , k ). Show carefully how to use a χ 2 distribution in testing the above hypothesis, and describe how you would carry out the test, using a statistic of the form n i =1 ( n i - e i ) 2 /e 2 , where e i is the expected number of observations in category C i under (an appropriate case of) the null hypothesis. ( E2, Assignment 4 ) (b) A scientist gets observations in three categories, across which he suspects a linear trend of probability, in which case p 1 ( θ ) = 1 3 - θ, p 2 ( θ ) = 1 3 , p 3 ( θ ) = 1 3 + θ,

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09 - April/08 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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