chap17_2010

chap17_2010 - Chapter 17 Estimation and Hypothesis Tests:...

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Unformatted text preview: Chapter 17 Estimation and Hypothesis Tests: Two Populations Two Independent Populations 2 2 1 1 2 2 n1 n2 Difference in Means: Variance Known 2 1 2 1 - =- X X 2 2 2 1 2 1 2 1 n n X X + =- Testing Hypothesis on the Difference in Means: Variance Known Alt. Hypothesis P-value Rejection Criterion H1: 0 P(z>z0)+P(z<- z0) z0 > z1-/2 or z0 < z/2 H1: > 0 P(z>z0) z0 > z1- H1: < 0 P(z<-z0) z0 < z Null Hypothesis: H0: 1 - 2 = 0 Test statistic: 2 2 2 1 2 1 2 1 ) ( n n X X Z + -- = Type II Error and Sample Size ) ( ) ( 2 2 2 1 2 1 2 2 2 1 2 1 1 2 2 n n z n n z + - - - + - - =- 2 2 2 2 1 2 1 1 ) ( ) ( ) ( 2 - + + -- z z n Type II Error (OC Curve) Two-sided, =.05 Montgomery, D.C., Runger G.C., Applied Statistics and Probability for Engineers, 5th Ed., 2010, John Wiley Claim: Drying times of 2 different paints are same n1=n2=10, 1= 2 =8, =.05 Null Hypothesis: H0: 1 - 2 = 0 Alt. Hypothesis: H1: 1 > 2 Test statistic: Rejection region: z.95= 1.645 P-value = P(z>2.52)= .0059 Reject H0 .95 .05 C Non-rejection Region Rejection Region 9 1.645 Testing Hypothesis on the Difference of Means with Variance Known Example 112 121 2 1 = = x x 52 . 2 10 8 10 8 ) 112 121 ( ) ( 2 2 2 2 2 1 2 1 2 1 = +- = + -- = n n X X Z Confidence Interval for Difference in Means: Variance Known Confidence Interval One-sided Upper Confidence Bound One-sided Lower Confidence Bound 2 2 2 1 2 1 1 2 1 2 1 2 2 2 1 2 1 2 1 2 2 ) ( ) ( n n z x x n n z x x + +- - + +-- 2 2 2 1 2 1 1 2 1 2 1 ) ( n n z x x + +- -- 2 1 2 2 2 1 2 1 2 1 ) ( - + +- n n z x x Choice of Sample Size If n1 = n2 = n ) ( ) ( 2 2 2 1 2 1 2 + - =- z n Tensile Strength n1=10 n2=12, 1=1 2 =1.5, =.10 Confidence Interval on the Difference of Means with Variance Known Example 5 . 74 6 . 87 2 1 = = x x 2 2 2 1 2 1 1 2 1 2 1 2 2 2 1 2 1 2 1 2 2 ) ( ) ( n n z x x n n z x x + +- - + +-- 12 5 . 1 10 1 ) 645 . 1 ( ) 5 . 74 6 . 87 ( 12 5 . 1 10 1 ) 645 . 1 ( ) 5 . 74 6 . 87 ( 2 2 2 1 2 2 + +- - +- +- 98 . 13 22 . 12 2 1 - Two Independent Populations Variance Unknown 1 2 n1 n2 Case 1: 12=22=2 2 ) , min( ) , max( 2 1 2 1 s s s s Difference in Means: Variance Unknown (but equal) Pooled Estimator of 2 Student t 2 1 2 1 - =- X X 2 ) 1 ( ) 1 ( 2 1 2 2 2 2 1 1- +- +- = n n s n s n s p Testing Hypothesis on the Difference in Means: Variance Unknown (but equal) Null Hypothesis: H0: 1 - 2 = 0 Test statistic: Alt. Hypothesis P-value Rejection Criterion H1: 0 2*P(t>|t0|) t0 > t/2,n1+n2-2 or t0 < -t/2, n1+n2-2 H1: > 0 P(t>t0) t0 > t, n1+n2-2 H1: < 0 P(t<-t0) t0 <- t, n1+n2-2 2 1 2 1 1 1 ) ( n n s X X T p + -- =...
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chap17_2010 - Chapter 17 Estimation and Hypothesis Tests:...

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