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# 07_31 - STAT 410 Examples for vs Summer 2009 H0 = 0...

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STAT 410 Examples for 07/31/2009 Summer 2009 H 0 : θ = θ 0 vs. H 1 : θ = θ 1 . Likelihood Ratio: ( 29 ( ( 29 ,..., , ; ,..., , ; ,..., , 2 1 1 2 1 0 2 1 L L λ n n n x x x x x x x x x θ θ = . Neyman-Pearson Theorem : C = { ( x 1 , x 2 , … , x n ) : ( k x x x n ,..., , 2 1 λ } . ( Reject H 0 if ( k x x x n ,..., , 2 1 λ ) 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 ( μ , σ 2 ) distribution ( σ 2 is known ) . Use the likelihood ratio to find the best rejection region for the test H 0 : μ = μ 0 vs. H 1 : μ = μ 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 0 2 2 1 1 2 1 0 2 1 μ σ σ π μ σ σ π μ L μ L λ 2 1 exp 2 1 2 1 exp 2 1 ,..., , ; ,..., , ; ,..., , = ( 29 ( 29 [ ] - - - = 2 1 exp 1 2 0 2 1 2 μ μ σ n i i i x x = ( ( 29 - + - 2 exp 2 1 0 2 2 0 2 1 σ μ μ σ μ μ x n n ( k x x x n ,..., , 2 1 λ ( x - 1 0 μ μ k 1 < 0 1 0 1 μ μ μ μ if if c x c x

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2. Let X 1 , X 2 , … , X n be a random sample of size n from a Poisson distribution with mean λ . That is, P ( X 1 = k ) = !
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07_31 - STAT 410 Examples for vs Summer 2009 H0 = 0...

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