10_23_09 - STAT 409 Examples for Fall 2009 H θ = θ vs H 1...

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Unformatted text preview: STAT 409 Examples for 10/23/2009 Fall 2009 H : θ = θ vs. H 1 : θ = θ 1 . Likelihood Ratio: ( 29 ( 29 ( 29 ,..., , ; ,..., , ; ,..., , 2 1 1 2 1 2 1 L L λ n n n x x x x x x x x x θ θ = . Neyman-Pearson Lemma : C = { ( x 1 , x 2 , … , x n ) : ( 29 k x x x n ≤ ,..., , 2 1 λ }. ( “ Reject H if ( 29 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 ). Find the best rejection region for the test H : μ = μ 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 2 2 1 1 2 1 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 2 1 2 μ μ σ n i i i x x = ( 29 ( 29 - +- ⋅ 2 exp...
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

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10_23_09 - STAT 409 Examples for Fall 2009 H θ = θ vs H 1...

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