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**Unformatted text preview: **MA2B HOMEWORK 6 SOLUTIONS Problem 1. We have that X 1 ,...,X n are i.i.d. with normal distribution N ( , 2 ) where = 10. We want to test the null hypothesis H : = against K : = , where = 0. From example A on page 308309 of Rice, we know that such a test re- jects the null hypothesis when | X n- | n z ( 2 ), where z ( ) is the inverse of as defined on page 305, namely P ( Z > z ( )) = . For n = 25, = . 1, = 10, and = 0, this gives z ( . 05) = 1 . 64485 (by table or mathematica) and so | X n | 10 5 (1 . 64485) = 3 . 28971 For n = 100, this gives | X n | 10 10 (1 . 64485) = 1 . 64485 The power curve with n = 100 should depart more rapidly away than the power curve for n = 25. Problem 2. (a) As defined in 9.1, the likelihood ratio is given by ( x ) = P ( x | H ) P ( x | H A ) . For each x j we compute: ( x 1 ) = . 2 . 1 = 2 ( x 2 ) = . 3 . 4 = 0 . 75 ( x 3 ) = . 3 . 1 = 3 ( x 4 ) = . 2 . 4 = 0 . 5 . We relabel the ordering of x i according to by y 1 = x 3 ,y 2 = x 1 ,y 3 = x 2 ,y 4 = x 4 and we let the random variable j = J...

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