07_29ans - STAT 410 Examples for 07/29/2009 Summer 2009 H 0...

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STAT 410 Examples for 07/29/2009 Summer 2009 H 0 true H 0 false Accept H 0 ( Do NOT Reject H 0 ) Type II Error Reject H 0 Type I Error α = significance level = P ( Type I Error ) = P ( Reject H 0 | H 0 is true ) β = P ( Type II Error ) = P ( Do Not Reject H 0 | H 0 is NOT true ) Power = 1 – P ( Type II Error ) = P ( Reject H 0 | H 0 is NOT true ) 1. 5.5.11 Let Y 1 < Y 2 < Y 3 < Y 4 be the order statistics of a random sample of size n = 4 from a distribution with a p.d.f. f ( x ; θ ) = 1 / θ , for 0 x θ , zero elsewhere, where 0 < θ . The hypothesis H 0 : θ = 1 is rejected and H 1 : θ > 1 is accepted if the observed Y 4 c . a) Find the constant c so that the significance level is α = 0.05. Recall: Let X 1 , X 2 , … , X n be i.i.d. Uniform ( 0 , θ ) . Y n = max X i . F max X i ( x ) = n x θ , f max X i ( x ) = n n x n θ 1 - , 0 < x < θ . F Y 4 ( x ) = P ( Y 4 x ) = P ( X 1 x , X 2 x , X 3 x , X 4 x ) = P ( X 1 x ) P ( X 2 x ) P ( X 3 x ) P ( X 4 x ) = 4 θ x , 0 < x < θ . Want 0.05 = α = P ( Y 4 c | θ = 1 ) = 1 – c 4 . c 4 = 0.95. c = 0.95 ¼ 0.98726.
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b) Determine the power function of the test. Power ( θ ) = P ( Y 4 c | θ ) = 4 θ 1 - c = 4 θ 95 . 0 1 - , θ > 1. c)* The hypothesis H 0 : θ = 1 is rejected and H 1 : θ > 1 is accepted if the observed Y 4 1. Find the significance level α of this test. α = P ( Y 4 1 | θ = 1 ) = 0 . d)* Determine the power function of the test in part (c). Power ( θ ) = P ( Y 4 1 | θ ) = 4 θ 1 1 - , θ > 1. In general, c = ( 29 n 1 α 1 - , α 0. α = 1 – c n , c 1. Power ( θ ) = n θ 1 1 α - - , θ > 1.
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2. Let X 1 , X 2 , … , X 16 be a random sample of size n = 16 from a N ( μ , σ 2 ) distribution. We are interested in testing H 0 : σ = 39 vs. H 1 : σ > 39. Recall: If X 1 , X 2 , … , X n are i.i.d. N ( μ , σ 2 ), then ( 29 2 2 σ S 1 n - is χ 2 ( n – 1 ). a) Find the “best” critical ( rejection ) region with the significance level α = 0.05. Test Statistic: ( 29 2 2 2 0 2 2 39 15 1 s σ s χ = = - n . Reject H 0 if 2 2 α χ χ ( n – 1 ) = 2 0.05 χ ( 15 ) = 25.00.
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This note was uploaded on 10/29/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Summer '08 term at University of Illinois at Urbana–Champaign.

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07_29ans - STAT 410 Examples for 07/29/2009 Summer 2009 H 0...

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