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Unformatted text preview: Chapter 8 Hypothesis Testing 8.1 Let X = # of heads out of 1000. If the coin is fair, then X binomial(1000 , 1 / 2). So P ( X 560) = 1000 X x =560 1000 x 1 2 x 1 2 n x . 0000825 , where a computer was used to do the calculation. For this binomial, E X = 1000 p = 500 and Var X = 1000 p (1 p ) = 250. A normal approximation is also very good for this calculation. P { X 560 } = P X 500 250 559 . 5 500 250 P { Z 3 . 763 } . 0000839 . Thus, if the coin is fair, the probability of observing 560 or more heads out of 1000 is very small. We might tend to believe that the coin is not fair, and p > 1 / 2. 8.2 Let X Poisson( ), and we observed X = 10. To assess if the accident rate has dropped, we could calculate P ( X 10  = 15) = 10 X i =0 e 15 15 i i ! = e 15 1+15+ 15 2 2! + + 15 10 10! . 11846 . This is a fairly large value, not overwhelming evidence that the accident rate has dropped. (A normal approximation with continuity correction gives a value of .12264.) 8.3 The LRT statistic is ( y ) = sup L (  y 1 ,...,y m ) sup L (  y 1 ,...,y m ) . Let y = m i =1 y i , and note that the MLE in the numerator is min { y/m, } (see Exercise 7.12) while the denominator has y/m as the MLE (see Example 7.2.7). Thus ( y ) = 1 if y/m ( ) y (1 ) m y ( y/m ) y (1 y/m ) m y if y/m > , and we reject H if ( ) y (1 ) m y ( y/m ) y (1 y/m ) m y < c. To show that this is equivalent to rejecting if y > b , we could show ( y ) is decreasing in y so that ( y ) < c occurs for y > b > m . It is easier to work with log ( y ), and we have log ( y ) = y log + ( m y ) log (1 ) y log y m ( m y ) log m y m , 82 Solutions Manual for Statistical Inference and d dy log ( y ) = log  log(1 ) log y m y 1 y + log m y m + ( m y ) 1 m y = log y/m ( m y m ) 1 ! . For y/m > , 1 y/m = ( m y ) /m < 1 , so each fraction above is less than 1, and the log is less than 0. Thus d dy log < 0 which shows that is decreasing in y and ( y ) < c if and only if y > b . 8.4 For discrete random variables, L (  x ) = f ( x  ) = P ( X = x  ). So the numerator and denomi nator of ( x ) are the supremum of this probability over the indicated sets. 8.5 a. The loglikelihood is log L ( ,  x ) = n log + n log  ( + 1)log Y i x i ! , x (1) , where x (1) = min i x i . For any value of , this is an increasing function of for x (1) . So both the restricted and unrestricted MLEs of are = x (1) . To find the MLE of , set log L ( ,x (1)  x ) = n + n log x (1) log Y i x i !...
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 Spring '08
 Peruggia,M
 Binomial

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