04_07 - STAT 410 Examples for 04/07/2008 Spring 2008 H 0...

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STAT 410 Examples for 04/07/2008 Spring 2008 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. 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 ( ) 2 2 S 1 n - is χ 2 ( n – 1 ). a) Find the “best” critical ( rejection ) region with the significance level α = 0.05. b) Find the power of the test from part (a) at σ = 66.7. c) What is the probability of Type II Error if σ = 66.7? The Chi-Square Distribution P ( X x ) 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 r ( ) r 2 99 . 0 ± ( ) r 2 975 . 0 ± ( ) r 2 95 . 0 ± ( ) r 2 90 . 0 ± ( ) r 2 10 . 0 ± ( ) r 2 05 . 0 ± ( ) r 2 025 . 0 ± ( ) r 2 01 . 0 ± 15 5.229 6.262 7.261 8.547 22.31 25.00 27.49 30.58
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2. 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,
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04_07 - STAT 410 Examples for 04/07/2008 Spring 2008 H 0...

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