12_02ans - STAT 410 Examples for Fall 2009 1 Let X 1 X 2 X...

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STAT 410 Examples for 12/02/2009 Fall 2009 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 ( 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. 2 2 39 15 s > 25.00 s 2 > 2535 . b) Find the power of the test from part (a) at σ = 66.7. Power = P ( Reject H 0 | H 0 is not true ) = P ( S 2 > 2535 | σ = 66.7 ) = P ( ( 29 2 2 σ S 1 n - > 2 7 . 66 2535 15 | σ = 66.7 ) = P ( χ 2 ( 15 ) > 8.547 ) = 0.90 . c) What is the probability of Type II Error if σ = 66.7? P ( Type II Error ) = 1 – Power = 0.10 . The Chi-Square Distribution P ( X x ) 0.010 0.025 0.050 0.100 0.900 0.950 0.975 0.990 r ( 29 r 2 99 . 0 χ
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This note was uploaded on 04/15/2010 for the course STAT 410 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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12_02ans - STAT 410 Examples for Fall 2009 1 Let X 1 X 2 X...

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