07_31ans - STAT 410 Examples for 07/31/2009 Summer 2009 H :...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STAT 410 Examples for 07/31/2009 Summer 2009 H : = vs. H 1 : = 1 . Likelihood Ratio: ( 29 ( 29 ( 29 ,..., , ; ,..., , ; ,..., , 2 1 1 2 1 2 1 L L n n n x x x x x x x x x = . Neyman-Pearson Theorem : C = { ( x 1 , x 2 , , x n ) : ( 29 k x x x n ,..., , 2 1 }. ( Reject H if ( 29 k x x x n ,..., , 2 1 ) is the best (most powerful) rejection region. 4. Let X 1 , X 2 , , X n be a random sample of size n from the distribution with probability density function ( 29 ( 29 ( 29 2 X X ln 1 ; x x x f x f - = = , x > 1, > 1. Find the form of the uniformly most powerful rejection region for testing H : = 2 vs. H 1 : > 2. Let > 2. ( 29 ( 29 ( 29 ( 29 = = - = = n i i i n i i i n n n x x x x x x x x x x x x x 1 2 1 2 2 1 1 2 1 2 1 ln 1 ln ,..., , ; ,..., , ; ,..., , H L H L . ( 29 ( 29 n n i i n x x x x 2 2 1 2 1 1 ,..., , - - = = . ( 29 k x x x n ,..., , 2 1 c x n i i = 1 ( since > 2 ). 4. Let X have an Exponential distribution with mean 1 / . That is, f X ( x ) = x e - , x > 0. Consider the test H : = 5 vs. H 1 : = 2 . a) Use the likelihood ratio to find the best rejection region. ( 29 ( 29 ( 29 x x x e e e x x x 3 2 5 1 5 . 2 2 5 ; ; L L --- = = = . ( 29 k x < x > c . Reject H if X > c . b) If the rejection region is Reject H if X > 0.50, find the significance level of the test. = P ( Reject H | H is true ) = P ( X > 0.50 | = 5 ) = - 5 . 5 5 dx x e = e 2.5 0.082 . c) If the rejection region is Reject H if X > 0.50, find the power of the test at = 2. Power = P ( Reject H | H is NOT true ) = P ( X > 0.50 | = 2 ) = - 5 . 2 2 dx x e = e 1 0.3679 . d) Find the rejection region with the significance level = 0.05....
View Full 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 / 11

07_31ans - STAT 410 Examples for 07/31/2009 Summer 2009 H :...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online