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# part12 - Public Health 6450 Fall 2011 Andrew Mugglin and...

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Public Health 6450 – Fall 2011 Andrew Mugglin and Lynn Eberly Division of Biostatistics School of Public Health University of Minnesota [email protected] Part 12

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Review Two-Sample Statistical Inference Where are we going? Previously: One-sample hypothesis testing and CIs for H 0 : μ = μ 0 z -tests for known σ , t -tests for unknown σ Errors in and power of signiﬁcance tests Linking one-sample tests with CIs Hypothesis tests for matched pairs Current topics: Two-sample hypothesis testing and CIs for H 0 : μ 1 - μ 2 = μ 0 z -tests for known σ 1 and σ 2 , t -tests for unknown σ 1 and σ 2 (Chapter 7.2) Special case when σ 2 1 = σ 2 2 (Chapter 7.2) Linking two-sample tests with CIs (Chapter 7.2) Mugglin and Eberly PubH 6450 Fall 2011 Part 12 2 / 42
Review Two-Sample Statistical Inference Two-sample problems The goal of inference here is to compare a variable (some measured characteristic) across two groups. We assume: Each group is obtained as a random sample from a distinct population. The responses in each group are independent of those in the other group. The ﬁrst group’s values are drawn from a N ( μ 1 , σ 1 ), with sample mean ¯ X 1 and sample size n 1 . The second group’s values are drawn from a N ( μ 2 , σ 2 ), with sample mean ¯ X 2 and sample size n 2 . Mugglin and Eberly PubH 6450 Fall 2011 Part 12 3 / 42

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Review Two-Sample Statistical Inference Distribution of ¯ X 1 - ¯ X 2 Using the assumptions on the previous slide, we can use Rules for Means and Rules for Variances (M,M&C Chapter 4.4) to show that: ¯ X 1 - ¯ X 2 N μ 1 - μ 2 , s σ 2 1 n 1 + σ 2 2 n 2 , so we can standardize just as we have before: z = ( ¯ X 1 - ¯ X 2 ) - ( μ 1 - μ 2 ) q σ 2 1 n 1 + σ 2 2 n 2 N (0 , 1) by subtracting the expectation and dividing by the standard error. Mugglin and Eberly PubH 6450 Fall 2011 Part 12 4 / 42
Review Two-Sample Statistical Inference Distribution of ¯ X 1 - ¯ X 2 (cont’d) E [ ¯ X 1 - ¯ X 2 ] = E [ ¯ X 1 ] - E [ ¯ X 2 ] = μ 1 - μ 2 Var [ ¯ X 1 - ¯ X 2 ] = ± (1) ¯ X 1 + ( - 1) ¯ X 2 ² = (1) 2 [ ¯ X 1 ] + ( - 1) 2 [ ¯ X 2 ] = [ ¯ X 1 ] + [ ¯ X 2 ] = σ 2 1 / n 1 + σ 2 2 / n 2 Mugglin and Eberly PubH 6450 Fall 2011 Part 12 5 / 42

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Review Two-Sample Statistical Inference Steps in hypothesis testing Now that we know how to standardize appropriately for the two-sample problem, hypothesis testing is done exactly as we did for the one-sample problem. 1. State the null hypothesis and the alternative hypothesis. 2. Calculate the value of the test statistic. 3. Find the p-value for this test statistic. 4. State a conclusion that connects the statistics back to the science. Mugglin and Eberly PubH 6450 Fall 2011 Part 12 6 / 42
Review Two-Sample Statistical Inference Step 1: One-sided vs. two-sided hypotheses Deﬁnition For H 0 : μ 1 - μ 2 = μ 0 , H a is one of the following: one-sided : H a : μ 1 - μ 2 > μ 0 one-sided : H a : μ 1 - μ 2 < μ 0 two-sided : H a : μ 1 - μ 2 6 = μ 0 We choose the alternative that is most appropriate for the scientiﬁc question of interest.

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part12 - Public Health 6450 Fall 2011 Andrew Mugglin and...

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