10_26_09 - STAT 410 H0 θ = θ0 Examples for vs Fall 2009 H...

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Unformatted text preview: STAT 410 H0: θ = θ0 Examples for 10/26/2009 vs. Fall 2009 H 1 : θ = θ 1. Likelihood Ratio: λ ( x 1 , x 2 ,..., x n ) = L ( θ 0 ; x 1 , x 2 ,..., x n ) . L ( θ 1 ; x 1 , x 2 ,..., x n ) Neyman-Pearson Theorem: { ( x 1 , x 2 , … , x n ) : λ ( x 1 , x 2 ,..., x n ) ≤ k } ( “ Reject H 0 if λ ( x 1 , x 2 ,..., x n ) ≤ k ” ) is the best (most powerful) rejection region. 1. Let X 1 , X 2 , … , X n be a random sample of size n from an Exponential distribution with mean 1 / λ. That is, let X 1 , X 2 , … , X n be a random sample of size n from an Exponential distribution with the p.d.f. f ( x ; λ ) = λ e – λ x, x > 0. Consider the test H 0 : λ = 5 vs. H 1 : λ < 5. a) Find form of the best ( uniformly most powerful ) rejection region. b) Suppose n = 20. Find the rejection region with the significance level α = 0.05. Recall: Let X 1 , X 2 , … , X n be a random sample of size Exponential distribution with mean θ. That is, 1 f X ( x ) = e − x θ , x > 0. n from an θ Then n 2 ∑ i =1 X i θ = 2nX θ has a χ2(2n ) n Therefore, 2 λ n X = 2 λ ∑ i =1 X i has a distribution. χ2(2n ) distribution. c) What can be said about the power of the test from part (b) at λ = 2? d) What can be said about the probability of Type II Error of the test from part (b) at λ = 2? e) f) Find the significance level α associated with the rejection region n “Reject H 0 if ∑ X i > 6”. i =1 Find the power of the rejection region from part (e) at λ = 2. ...
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10_26_09 - STAT 410 H0 θ = θ0 Examples for vs Fall 2009 H...

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