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hw6 - APPM 4/5520 Problem Set Six(Due Wednesday October 19...

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Unformatted text preview: APPM 4/5520 Problem Set Six (Due Wednesday, October 19) 1. Let X 1 , X 2 be a random sample from the N (0 ,σ 2 ) distribution. Find the distribution of X 1 | X 2 | ? Name it! ( Hint: | X 2 | = radicalBig X 2 2 . ) 2. Let X (1) ,X (2) ,... ,X ( n ) denote the order statistics of a random sample of size n from a dis- tribution that has pdf f ( x ) = 3 θ 3 x 2 I (0 ,θ ) ( x ) . (a) Show that P ( c < X ( n ) /θ < 1) = 1- c 3 n where 0 < c < 1. (b) If n = 4 and the observed value of X (4) is 2 . 3, find a 95% confidence interval for θ based on X (4) . 3. Independent random samples of size n 1 = 16 and n 2 = 8 were drawn from two normal populations. The samples yielded the estimates x 1 = 4 . 31, x 2 = 5 . 22, s 2 1 = 0 . 15, and s 2 2 = 0 . 10. A 90% confidence interval constructed (we don’t know how to do this yet) for the ratio of variances σ 2 1 /σ 2 2 is (0 . 358 , 1 . 83). Since this interval contains the value 1, there is not strong evidence that the variances are unequal, hence it is not unreasonable to assume that...
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