10_28_09ans - STAT 409 Fall 2009 Examples for 1 Let X 1 X 2...

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STAT 409 Examples for 10/28/2009 Fall 2009 1. Let X 1 , X 2 , … , X n be a random sample of size n from a N ( 0 , σ 2 ) distribution. We are interested in testing H 0 : σ = 2 vs. H 1 : σ = 5. a) Use the likelihood ratio to show that the best rejection region is C = { ( x 1 , x 2 , … , x n ) : = n i i x 1 2 > c } . ( 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 2 1 2 1 2 1 50 5 2 1 8 2 2 1 ,..., , ; 5 ,..., , ; 2 ,..., , exp π exp π σ L σ L λ = - = - = = 200 21 2 5 8 1 50 1 2 5 1 2 1 2 exp exp n i i n n i i n x x . ( k x x x n < ,..., , 2 1 λ k x n n i i ln ln 1 2 200 21 2 5 < - = c x n i i = 1 2 . b) If n = 10, find the value of c such that α = 0.10. Hint: ( 2 2 σ μ X i - is χ 2 ( n ) ; here μ = 0. ( 29 = = 2 2 1 2 σ χ P X P c n c n i i . Want 0.10 = α = P ( Reject H 0 | H 0 is true ) = P ( = n i i 1 2 X > c | σ = 2 ) = P ( 4 X 4 1 1 2 c n i i = | σ = 2 ) = P ( χ 2 ( 10 ) > 4 c ) .
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χ 2 Table ( Table IV, p. 685 ) : χ 2 0.10 ( 10 ) = 15.99 . c = 4 χ 2 0.10 ( 10 ) = 4 × 15.96 = 63.96 . c) If n = 10 and c from part (b), find the probability of Type II Error. P ( Type II Error ) = P ( Accept H 0 | H 0 is not true ) = P ( = n i i 1 2 X c | σ = 5 ) = P ( = n i i 1 2 X 63.96 | σ = 5 ) = P ( 25 96 . 64 X 25 1 1 2 = n i i | σ = 5 ) = P ( χ 2 ( 10 ) 2.5584 ) = 1 – P ( χ 2 ( 10 ) > 2.5584 ) .
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