10_19_11ans - STAT 409 Fall 2011 Examples for 10/19/2011...

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STAT 409 Examples for 10/19/2011 Fall 2011 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 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: ( ) 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 ( ( ) 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 ( ) r 2 99 . 0 χ ( ) r 2 975 . 0 χ ( ) r 2 95 . 0 χ ( ) r 2 90 . 0 χ ( ) r 2 10 . 0 χ ( ) r 2 05 . 0 χ ( ) r 2 025 . 0 χ ( ) r 2 01 . 0 χ 15 5.229 6.262 7.261 8.547 22.31 25.00 27.49 30.58
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2. A scientist wishes to test if a new treatment has a better cure rate than the traditional treatment which cures only 60% of the patients. In order to test whether the new treatment is more effective or not, a test group of 20 patients were given the new treatment. Assume that each personal result is independent of the others. Let X denote the number of patients in the test group who were cured. H 0 : p = 0.60 vs. H 1 : p > 0.60. Right-tailed. n = 20. a) Suppose we decided to use the rejection region “Reject H 0 if X 15.” Find the significance level α associated with this Rejection Region. α = P ( Type I error ) = P ( Reject H 0 | H 0 true ) = P ( X 15 | p = 0.60 ) = 1 – CDF ( 14 | p = 0.60 ) = 1 – 0.874 = 0.126 . b) Find the “best” rejection region with the significance level α closest to 0.05. Rejection Region for a Right – tailed test: Find b such that P ( X b – 1 ) 1 – α . Then the Rejection Region is “Reject H 0 if X b .” Decision rule: Reject H 0 if X b . Want P ( Type I error ) = 0.05. P
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This note was uploaded on 11/15/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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10_19_11ans - STAT 409 Fall 2011 Examples for 10/19/2011...

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