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mtermw04_soln - EC120B Winter 2004 Midterm Solution Prof...

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1 EC120B Midterm Solution Prof. Berman Winter 2004 (20) 1. A. Given a random sample of 40 observations of variables Y and X from some population, provide a formula to estimate β 1 , the slope of the population linear regression line. Estimate for β 1 : β ˆ 1 = S xy /S xx = Σ (X i - X )(Y i - Y )/ Σ (X i - X ) 2 B. Write a formula to estimate a 95% confidence interval for β 1 . 95% confidence interval for β 1 : β ˆ 1 ± 1.96 * se( β ˆ 1 ) C. Now assume that 140 of your classmates sample from that population and estimate a confidence interval as in part A. Let z be the number of confidence intervals (between 0 and 140) that do not contain β 1 . Estimate the expected value of z, E(z)? Using the CLT to approximate, prob(interval does not contain β 1 ) 0.05 for each classmate. E(z) 140(0.05)=7 (10 Bonus) D. What’s the variance of z, V(z)? The variance of a binary variable is p(1-p), where p is the probability of a “1.” The variance of the sum of n independent binary variables is just n times the variance of each. V(z) np(1-p)=140(0.05)(1-0.05)=6.65
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2 (40) 2. For each statement below, indicate if it is true or false and write a sentence justifying your answer. A. If X 1 , ... X N are a random sample from a population with E(X) =μ : then Σ (X i - μ)=0 True [ ] / False [X] Explain: Σ (X i - X )=0 But X μ in a finite sample So Σ (X i - μ) 0 B. In random sampling from a population with expectation E(Y) = μ : , Σ (Y i - Y ) 2 is greater than Σ (Y i - μ ) 2 since the mean of Y is an imprecise estimate of the population mean : .
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