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11_04_09 - STAT 409 Fall 2009 Examples for H0 0 vs 0 1 = H1...

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STAT 409 Examples for 11/04/2009 Fall 2009 H 0 : θ Θ 0 vs. H 1 : θ Θ 1 Θ 0 Θ 1 = . Reject H 0 if Λ = ( ( 29 θ L max θ L max 1 0 θ θ Θ Θ k Λ * = ( ( 29 θ L max θ L max 1 0 0 θ θ Θ Θ Θ k since Λ * = min ( Λ , 1 ) H 0 : θ = θ 0 vs. H 1 : θ θ 0 Reject H 0 if Λ = ( ( 29 θ ˆ L θ L 0 k Example 1 : Let X 1 , X 2 , … , X n be a random sample of size n from N ( μ , σ 2 ) distribution ( σ 2 unknown ) . H 0 : μ = μ 0 vs. H 1 : μ μ 0 μ ˆ = X , σ ˆ = ( 29 = - n i i n 1 2 X X 1 . Under H 0 , 0 μ ˆ = μ 0 , 0 σ ˆ = ( 29 = - n i i n 1 2 0 μ X 1 . Λ = ( ( 29 θ ˆ L θ L 0 = ( 29 ( 29 = = - - - - n i i n i i x x 1 2 2 1 2 0 2 0 0 μ ˆ σ ˆ σ ˆ π μ ˆ σ ˆ σ ˆ π 2 1 exp 2 1 2 1 exp 2 1 = ( 29 ( 29 2 1 2 0 1 2 μ n n i i n i i x x x - - = = = ( 29 ( 29 ( 29 2 1 2 2 0 1 2 μ n n i i n i i x x x n x x - + - - = = .
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Λ k ( ( 29 = - - n i i x x x n 1 2 2 0 μ k 1 μ 0 n s x - c Reject H 0 if μ 0 n s x - c .
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