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 n 1 n 2 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 H 1 : P(z&gt;z )+P(z&lt;-z ) z &gt; z 1- /2 or z &lt; z /2 H 1 : &gt; P(z&gt;z ) z &gt; z 1- H 1 : &lt; P(z&lt;-z ) z &lt; z Null Hypothesis: H : 1- 2 = 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, 5 th Ed., 2010, John Wiley Claim: Drying times of 2 different paints are same n 1 =n 2 =10, 1 = 2 =8, =.05 Null Hypothesis: H : 1 - 2 = 0 Alt. Hypothesis: H 1 : 1 &gt; 2 Test statistic: Rejection region: z .95 = 1.645 P-value = P(z&gt;2.52)= .0059 Reject H 112 121 2 1 = = x x .95 .05 C Non-rejection Region Rejection Region 9 1.645 Testing Hypothesis on the Difference of Means with Variance Known Example 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 ) ( ) ( 2 2 2 1 2 1 2 + - =- z n If n 1 = n 2 = n Tensile Strength n 1 =10 n 2 =12, 1 =1 2 =1.5, =.10 5 . 74 6 . 87 2 1 = = x x Confidence Interval on the Difference of Means with Variance Known Example 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 n 1 n 2 Case 1: 1 2 = 2 2 = 2 2 ) , min( ) , max( 2 1 2 1 s s s s Difference in Means: Variance Unknown (but equal) 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 Pooled Estimator of 2 Student t Testing Hypothesis on the Difference in Means: Variance Unknown (but equal) Null Hypothesis: H : 1- 2 = Test statistic: 2 1 2 1 1 1 ) ( n n s X X T p + -- = Alt. Hypothesis P-value Rejection Criterion H 1 : 2*P(t&gt;|t |) t &gt; t /2,n1+n2-2 or t &lt; -t /2, n1+n2-2 H 1 :...
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## This note was uploaded on 12/03/2011 for the course EIN 5226 taught by Professor Staff during the Fall '11 term at FIU.

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

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