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11_02_11_2 - STAT 409 1 Let X 1 X 2 X n be a random sample...

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STAT 409 Examples for 11/02/2011 (2) Fall 2011 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) θ 2 X X ln 1 θ θ ; x x x f x f - = = , x > 1, θ > 1. a) Find the form of the uniformly most powerful rejection region for testing H 0 : θ = 2 vs. H 1 : θ > 2. b) Suppose n = 5. Find the rejection region with the significance level α = 0.05. c) Find the power function of the test from part (b). d) Suppose n = 5. Find the significance level α for the rejection region “Reject H 0 if = n i i x 1 ln 5 .” e) Find the power of the test from part (d) when θ = 3 and θ = 4. f) Suppose n = 5. Find the significance level α for the rejection region “Reject H 0 if = n i i x 1 ln 6 .” g) Find the power of the test from part (f) when θ = 3 and θ = 4. h) Suppose n = 5, and x 1 = 5, x 2 = 1.2, x 3 = 2, x 4 = 12, x 5 = 1.5. Find the p-value. State your decision ( Reject H 0 or Do NOT Reject H 0 ) at α = 0.05.
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Answers: 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) θ 2 X X ln 1
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