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Unformatted text preview: AMS 572 Lecture Notes 8 Oct. 8 th , 2010 Ch8 Inference on two population means (and two population variances). 1. The samples are paired paired sample ttest 2. The samples are independent independent sample ttest a) 2 2 1 2 = pooledvariance ttest b) 2 2 1 2 unspooledvariance ttest (Note: to check if 2 2 1 2 = , use Ftest) 2a.Inference on 2 population means, when both populations are normal. We have 2 independent samples, the population variances are unknown but equal ( 2 2 2 1 2 = = ) pooledvariance ttest. Data: 1 2 1 1 1 , , ( , ) iid n X X N K : 2 2 1 2 2 , , ~ ( , ) iid n Y Y N K Goal: Compare 1 and 2 1) Point estimator: 2 2 1 2 1 2 1 2 1 2 ~ ( , ) X Y N n n  = + 2 1 2 1 2 1 1 ( ,( ) ) N n n  + 2) Pivotal quantity: 1 2 1 2 ( ) ( ) ~ (0,1) 1 1 X Y Z N n n  = + not the PQ, since we dont know 2 Q 1 2 2 1 1 1 2 ( 1) ~ n n S  , 2 2 2 2 2 1 2 ( 1) ~ n n S  , and they are independent ( 2 1 S & 2 2 S are independent because these two samples are independent to each other) 1 2 2 2 2 1 1 2 2 2 2 2 ( 1) ( 1) ~ n n n S n S W + = + Definition: 2 2 2 1 2 k W Z Z Z = + + + L , when . . . ~ (0,1) i i d i Z N , then 2 ~ k W . Definition: tdistribution: ~ (0,1) Z N , 2 ~ k W , and Z & W are independent, then Z T W k = ~ k t 1 2 1 2 1 2 1 2 2 2 2 1 1 2 2 2 2 1 2 1 2 1 2 ( ) ( ) 1 1 ( ) ( ) ~ 1 1 ( 1) ( 1) 2 2 n n p X Y n n X Y Z T t W n S n S S n n n n n n + + = = = + + + + . where 2 2 2 1 1 2 2 1 2 ( 1) ( 1) 2 p n S n S S n n + = + is the pooled variance. This is the PQ of the inference on the parameter of interest 1 2 ( )  3) Confidence Interval for 1 2 ( )  1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2, 2, 2 2 1 2 2, 2, 2 2 1 2 1 2 2, 2, 1 2 1 2 2 2 1 2 2, 1 2 2 1 ( ) ( ) ( ) 1 ( ) 1 1 1 1 1 1 1 ( ( ) ( ) ) 1 1 1 ( n n n n n n n n p p p n n n n p n n n n P t T t X Y P t t S n n P t S X Y t S n n n n P X Y t S X Y t n n + + + + + + + + =  = + = +  + = +   + 2, 1 2 2 1 1 ) p S n n  + This is the 100(1 )%  C.I for 1 2 ( )  4) Test: Test statistic: T = 1 2 2 1 2 ( ) ~ 1 1 H n n p X Y c t S n n + + a) 1 2 1 2 : : a H c H c  =  (The most common situation is c =0 1 2 : H = ) At the significance level , we reject H in favor of a H iff 1 2 2, n n T t + If (  ) P value...
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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.
 Fall '10
 WeiZhu

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