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# 10_28_09 - STAT 409 Fall 2009 Examples for 1 Let X 1 X 2 X...

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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 }. b) If n = 10, find the value of c such that α = 0.10. Hint: ( 29 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 . c) If n = 10 and c from part (b), find the probability of Type II Error. 2. Consider a family of probability distributions with a p.d.f. of the form f ( x ; θ ) = θ x θ – 1 , 0 < x < 1, zero elsewhere, θ 1. To test the simple hypothesis H 0 : θ = 1 against the alternative simple hypothesis H 1 : θ = θ 1 , where θ 1 > 1, we will use a random sample X 1 , X 2 of size n = 2. a)

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10_28_09 - STAT 409 Fall 2009 Examples for 1 Let X 1 X 2 X...

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