07_31 - STAT 410 Examples for 07/31/2009 Summer 2009 H : =...

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Unformatted text preview: STAT 410 Examples for 07/31/2009 Summer 2009 H : = vs. H 1 : = 1 . Likelihood Ratio: ( 29 ( 29 ( 29 ,..., , ; ,..., , ; ,..., , 2 1 1 2 1 2 1 L L n n n x x x x x x x x x = . Neyman-Pearson Theorem : C = { ( x 1 , x 2 , , x n ) : ( 29 k x x x n ,..., , 2 1 }. ( Reject H if ( 29 k x x x n ,..., , 2 1 ) is the best (most powerful) rejection region. 1. Let X 1 , X 2 , , X n be a random sample of size n from a N ( , 2 ) distribution ( 2 is known ). Use the likelihood ratio to find the best rejection region for the test H : = vs. H 1 : = 1 . ( 29 ( 29 ( 29 ( 29 ( 29 = = -- -- = = n i i n i i n n n x x x x x x x x x x x 1 2 1 2 1 2 2 2 1 1 2 1 2 1 L L 2 1 exp 2 1 2 1 exp 2 1 ,..., , ; ,..., , ; ,..., , = ( 29 ( 29 [ ] --- = 2 1 exp 1 2 2 1 2 n i i i x x = ( 29 ( 29 - +- 2 exp 2 1 2 2 2 1 x n n ( 29 k x x x n ,..., , 2 1 ( 29 x - 1 k 1 < 1 1 if if c x c x 2. Let X 1 , X...
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07_31 - STAT 410 Examples for 07/31/2009 Summer 2009 H : =...

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