04_07ans - STAT 410 Examples for Spring 2008 1 Let X 1 X 2...

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Unformatted text preview: STAT 410 Examples for 04/07/2008 Spring 2008 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 : 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. Test Statistic: ( ) 2 2 2 2 2 39 15 1 s & s ¡ ⋅ = ⋅ =- n . Reject H if 2 2 α > χ χ ( n – 1 ) = 2 0.05 ¡ ( 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 | H is not true ) = P ( S 2 > 2535 | σ = 66.7 ) = P ( ( ) 2 2 & 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 . ¡ ( ) r 2 975 . ¡ ( ) r 2 95 . ¡ ( ) r 2 90 . ¡ ( ) r 2 10 . ¡ ( ) r 2 05 . ¡ ( ) r 2 025 . ¡ ( ) r 2 01 . ¡ 15 5.229 6.262 7.261 8.547 22.31 25.00 27.49 30.58 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, where 0 < θ . The hypothesis H : 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. 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 & & ¡ ¢ £ ¤ ¥ x , 0 < x < θ ....
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04_07ans - STAT 410 Examples for Spring 2008 1 Let X 1 X 2...

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