ioe265f11-Lec20 - Lec20 - Two Sample Hypothesis Tests...

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Lec20 - Two Sample Hypothesis Tests IOE 265 F11 1 1 Two-Sample Hypothesis Tests 2 Topics I. Two Sample Mean Tests II. Test of Two Proportions III. F-Test (2 Variances)
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Lec20 - Two Sample Hypothesis Tests IOE 265 F11 2 3 I. Two Sample Mean Tests Case USE Test Statistic Formula I known ’s , and normal populations z o II unknown ’s , and normal populations (unequal variances) t o III t o Tests of Two Population Means n m y x z 2 2 2 1 0 n s m s y x t 2 2 2 1 0 unknown ’s , and normal populations (equal variances) n m s y x t p 1 1 2 0 IV t o Paired t-test n s d t d 0 4 Case I: Two-Sample Z Test Z Tests and Confidence Intervals for a Difference between the means of two different population distributions: 1 - 2 Assumptions: X i ’s are a random sample from a population with mean 1 and variance 1 2 Y i ’s are a random sample from a population with mean 2 and variance 2 2 The X i ’s and Y i ’s are independent from one another The variances of both populations are known
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Lec20 - Two Sample Hypothesis Tests IOE 265 F11 3 5 2 Sample Z test Since both populations are normal, then X-bar and Y-bar are also normal and we can get the following standard normal variable:        n m Y X Y X Z Y X 2 2 2 1 2 1 2 2 1 samples Y of Number samples X of Number n m 6 Z test If we set the difference between the population means as: Null Hypothesis: Test statistic: Alternative Hypothesis: 2 1 0 0 2 1 0 : H n m y x z 2 2 2 1 0 2 2 0 2 1 0 2 1 0 2 1 or : : : z z z z H z z H z z H a a a
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Lec20 - Two Sample Hypothesis Tests IOE 265 F11 4 7 P-Values The P-value (or observed significance level) is the smallest level of significance (probability) at which H 0 would be rejected when a specified test procedure is used on a given data set. Basic rules: The bigger the P-value, the higher the chance of a type I error for the calculated test statistic (e.g. z 0 or t 0 ). Therefore, we should not reject H 0 The smaller the P-value, the lower the chance of a type I error. Therefore, we should reject H 0 level) (at reject 0 H value P level) (at reject not do 0 H value P 8 What is the P-value then? It is customary to call the data significant when H 0 is rejected and not significant when we fail to reject H 0 The p-value is then the tail area captured by the computed value of the test statistic (e.g. z 0 or t 0 )
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Lec20 - Two Sample Hypothesis Tests IOE 265 F11 5 9 P-Value Areas (Z test) z 0 Upper-Tailed Lower-Tailed z 0 Two-Tailed -z 0 z 0   0 1 z value P   0 z value P   0 1 2 z value P 10 Z test - Example The drying times for two formulations of paint are tested: standard vs. express. It is known that the population standard deviation
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ioe265f11-Lec20 - Lec20 - Two Sample Hypothesis Tests...

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