ams572_notes_10

# Ams572_notes_10 - AMS 572 Lecture Notes Oct 15 2010 Review Inference on two population means 2 1 Two normal pops 12 2 are known exact Z 2 Two large

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AMS 572 Lecture Notes Oct . 1 5 , 2010. Review: Inference on two population means 1. Two normal pops, 2 1 σ & 2 2 are known. exact Z 2. Two large samples → approximate Z 3. Two normal pops, 2 1 & 2 2 are unknown but 2 2 2 2 1 = = Pooled variance t (exact) P.Q. df p t n n S Y X T ~ 1 1 ) ( 2 1 2 1 + - - - = μ Exact 2 . . 2 1 - + = n n f d (SAS) 4. Two normal pops, 2 1 & 2 2 are unknown, 2 2 2 1 approximate t P.Q. df t n S n S Y X T ~ ) ( 2 2 2 1 2 1 2 1 + - - - = More accurate d.f. Satterthwaite method (SAS) Quick & dirty ) 1 , 1 min( . . 2 1 - - = n n f d in-class exam 5. other situations nonparametric method Ma nn -Whitney U -test = Wilcoxon Rank- S um Test (SAS) 6. Modern nonparametric method Bootstrap Resampling method 7. Transformation to Normal distribution Box-Cox transformation Inference on two population variances * Both pop’s are normal, two independent samples Sample 1 : ) , ( ~ , , , 2 1 1 . . . 2 1 1 N X X X d i i n 2 1 2 1 2 1 1 1 ~ ) 1 ( - - n S n χ 1

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Sample 2 : ) , ( ~ , , , 2 2 2 . . . 2 1 2 σ μ N Y Y Y d i i n 2 1 2 2 2 2 2 2 ~ ) 1 ( - - n S n χ 1. point estimator : 2 1 2 1 ˆ S = (parameter of interest : 2 2 2 1 ) 2 2 2 2 ˆ S = Def. F-distribution Let 2 1 1 ~ k W , 2 2 2 ~ k W , 2 1 , W W are independent. Then, 2 1 , 2 2 1 1 ~ k k F k W k W F = ) ( , , , 2 1 lower k k F F P α = ) 1 1 ( , , , 2 1 lower k k F F P = 1 2 , ~ 1 k k F F upper k k lower k k F F , , , , , , 1 2 2 1 1 = 2
1 , 1 2 2 2 1 2 2 2 1 2 2 2 2 2 2 1 2 1 2 1 1 2 1 ~ ) 1 ( ) 1 ( ) 1 ( ) 1 ( - - = - - - - = n n F S S n S n n S n F σ 2. )% 1 ( 100 α - CI for 2 2 2 1 ) ( 1 , 2 2 2 2 1 2 2 2 1 , 2 upper lower F S S F P = - ) 1 1 ( , 2 2 2 2 1 2 2 2 1 , 2 upper lower F S S F P = ) ( , 2 2 2 2 1 2 2 2 1 , 2 2 2 2 1 lower upper F S S F S S P = 3. Test 1 : 2 2 2 1 0 = H 1 : 2 2 2 1 a H Test Statistic 1 , 1 2 2 2 1 0 2 1 0 ~ - - = n n H F S S F At the significance level , we reject 0 H if 0 F is too large or too small. 1 0 c F , 2 0 c F * conventional boundries / thresholds upper n n F c , 2 , 1 , 1 1 2 1 - - = lower n n F c , 2 , 1 , 1 2 2 1 - - = 3

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SAS program for test on 2 pop means 1. paired samples sample 1 10 23 16 18 … 33 sample 2 15 28 21 29 … 58 data paired; input IQ1 IQ2;
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## This note was uploaded on 01/31/2011 for the course AMS 572 taught by Professor Weizhu during the Fall '10 term at SUNY Stony Brook.

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Ams572_notes_10 - AMS 572 Lecture Notes Oct 15 2010 Review Inference on two population means 2 1 Two normal pops 12 2 are known exact Z 2 Two large

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