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...
View
Full Document
 Fall '11
 Manuel
 Normal Distribution, Variance

Click to edit the document details