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ps04 - Statistics 403 Problem Set 4 Due in lab on Friday...

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Statistics 403 Problem Set 4 Due in lab on Friday, October 8th 1. Suppose we are interested in comparing the support for a certain political candidate in Michigan to the support for the candidate in Ohio. We conduct independent surveys in both states, and obtain responses X 1 , . . . , X m in Michigan and Y 1 , . . . , Y n in Ohio. Let μ m be the population level of support in Michigan (i.e. EX i = μ m ) and let μ o be the population level of support in Ohio ( EY i = μ o ). Our goal is to estimate μ m - μ o . (a) Construct a 95% confidence interval for μ m - μ o centered at ¯ X - ¯ Y . Solution: In order to get 95% coverage probability, we need the interval to cover two standard deviations on either side of the estimate. The standard deviation of the estimate is SD( ¯ X - ¯ Y ) = q σ 2 x /m + σ 2 y /n. Thus the interval is ¯ X - ¯ Y ± 2 q σ 2 x /m + σ 2 y /n. (b) Suppose the true levels of support are μ m = 0 . 6 and μ o = 0 . 54. When the sample sizes in the two states are equal ( m = n ), what sample size is required so that the confidence interval is 0.04 units wide? Hint: since the X i and Y i are binary, you can determine their variances using their expected values. Solution: Since the X i and Y i are binary, we need to have a general expression for the variance of a binary random variable. Suppose P ( X = 1) = p and P ( X = 0) = 1 - p , so the expected value is p · 1 + (1 - p ) · 0 = p, and therefore the variance is p (1 - p ) 2 + (1 - p )(0 - p ) 2 = p (1 - p ) .
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