solution2 - Solutions to PS 2 Hongyan Zhao September 25,...

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Solutions to PS 2 * Hongyan Zhao September 25, 2009 Question 1 Q3 from PS1 Let’s denote the heights of male college students as random variable Y. We are given Y N (70 , 3 . 5 2 ). Then the sample mean, X N (70 , 3 . 5 2 n ), where n is the sample size. d. Pr ( X < 72) = Pr ( Z < 72 - 70 3 . 5 25 ) = Φ( 20 7 ) = 0 . 9979 e. Pr ( X > 64) = Pr ( Z > 64 - 70 3 . 5 100 ) = 1 - Φ( - 120 7 ) ' 1 f. Pr (67 < X < 72) = Pr ( 67 - 70 3 . 5 144 < Z < 72 - 70 3 . 5 144 ) = Φ( 48 7 ) - Φ( - 72 7 ) ' 1 Q4 from PS1 b. V ar ( ¯ X odd ) = V ar ( 2 n n/ 2 X i =1 X 2 i - 1 ) = 4 n 2 · n 2 · σ 2 X = 2 n σ 2 X As n → ∞ ,V ar ( ¯ X odd ) 0 So, the odds estimator is consistent. c. Let’s look at another estimator–sample mean ¯ X = 1 n n i =1 X i E ( ¯ X ) = E ( X ) ,V ar ( ¯ X ) = σ 2 X n < 2 σ 2 X n It is also unbiased, but the variance is less than the variance of odds estimator. So the odds estimator is less efficient than the sample mean estimator. It is not efficient in the class of unbiased estimators. Q5 from PS1 Let’s denote the math and verbal SAT scores as random variables X and Y respec- tively, the overall SAT score as the random variable W. Then W = X + Y . We are given that X N (500 , 10 , 000) ,Y N (500 , 10 , 000). * If you find any mistake or typo, please email me hyzhao@econ.berkeley.edu 1
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c. E
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This note was uploaded on 02/17/2010 for the course ECON 141 taught by Professor Staff during the Fall '08 term at University of California, Berkeley.

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solution2 - Solutions to PS 2 Hongyan Zhao September 25,...

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