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Unformatted text preview: Statistics 403 Problem Set 9 Due in lab on Friday, November 19th 1. Suppose we are planning to collect paired data X i ,Y i with the goal of estimating EX- EY . We want to have 80% power to detect a raw effect size EX- EY = 1, and based on previous experience we are confident that SD( X ) 2 and SD( Y ) 3. However we do not know much about the correlation between X and Y . (a) What sample size is required if cor( X,Y ) = 0 . 3? (b) If we perform the study using the sample size from part (a), and the correlation coefficient is actually 0 . 5, then what power do we actually have? Solution: The expected test statistic is ET = n EX- EY SD( X- Y ) = n 1 4 + 9- 12 r = q n/ (13- 12 r ) , where r = cor( X,Y ). Note that when r increases, ET increases, so the power increases. To get 80% power we solve . 8 = P ( T > 2) = P ( T- ET > 2- ET ) = P Z > 2- q n/ (13- 12 r ) . So we set 2- q n/ (13- 12 r ) =- . 84 (since- . 84 is the 20 th = (100- 80) th percentile of the standard normal distribution), and solve for n to get n = 2 . 84 2 (13- 12 r ) . Substitute r = 0 . 3 to get n = 76. To get the power when r = 0 . 5, we calculate P Z > 2- q 76 / (13- 12 . 5) , and we get that the power is approximately 0 . 9....
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- Winter '12