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

This preview shows pages 1–4. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 8

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online