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# Lec23 - IOE/Stat IOE/Stat 265 Fall 2009 Lecture#23...

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IOE/Stat 265, Fall 2009 Lecture #23: Hypothesis Testing for Two Variances and Two Proportions Case 5: Tests for 2 Proportions (Section 9-4) Case 6: Tests for 2 Variances (S ti 9 5) 1 (Section 9-5) Case 5: Tests of Two Proportions Examples: Test if defect rate from 1st Shift is the same as the defect rate on 2nd Shift. Test if one model has a higher repair rate than another model. Key Assumptions Require large # samples from each group to use the normal approximation. 2 Proportion Test Null Hypothesis: H : p p 0 or p p Null Hypothesis: H 0 : 1 - 2 = 0 or 1 = 2 Test Statistic: 1 2 ˆ ˆ (1 )(1 / 1 / ) ˆ ˆ o p p z p p n n = + 1 2 1 2 ˆ x x p + = Alt Hypothesis Reject Region 1 2 n n + 1 1 2 1 1 2 : 0 : 0 o o H p p z z H p p z z α α > < ≤ − 3 1 1 2 /2 /2 : 0 or o o H p p z z z z α α ≤ − Example 1: Molding Machines T diff t i j ti ldi hi d Two different injection molding machines are compared. (assume α = 0.05) Machine 1: 15 out of 300 are defective (p 1 = 0.050) Machine 2: 7 out of 290 are defective (p 2 = 0.024) Note: Compute Z 0 Z α /2 (Z 0 > Z 0.025 = 1.96 or Z 0 < - Z 0.025 = -1.96 ) Is it reasonable to conclude that both machines produce the same fraction defective? 4

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Step by Step Approach Step-by-Step Approach 1) Identify the parameter of interest Identify the parameter of interest. 2) State the Null Hypothesis H 0 . 3) Specify Alt. Hypothesis H 1 . 4) Choose a significance level α . 5) State Test Statistic. 6) State the Rejection Region. 7) Substitute sample statistics and compute. 7 8) State statistical conclusion in problem context. Proportion Test Using Minitab STAT » Basic Statistics » 2 Proportions Note: Minitab provides p-value If l R j t H If p-value α → Reject H 0 , If p-value > α → Fail to Reject H 0 T t d CI f T P ti Test and CI for Two Proportions Sample X N Sample p 1 15 300 0.050000 Are the proportions different?
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