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Unformatted text preview: Problem Set 3 ECON 231W Spring 2010 Suggested Solutions 1. From the definition of covariance: ˆ σ XY = 1 n n X i =1 ( X i ¯ X )( Y i ¯ Y ) This is identical to: ˆ σ XY = 1 n n X i =1 ( X i ¯ X ) Y i ( X i ¯ X ) ¯ Y (1) and ˆ σ XY = 1 n n X i =1 ( Y i ¯ Y ) X i ( Y i ¯ Y ) ¯ X (2) Equation (1) can be rewritten as: ˆ σ XY = 1 n n X i =1 ( X i ¯ X ) Y i 1 n n X i =1 ( X i ¯ X ) ¯ Y = 1 n n X i =1 ( X i ¯ X ) Y i ¯ Y 1 n n X i =1 ( X i ¯ X ) = 1 n n X i =1 ( X i ¯ X ) Y i ¯ Y · = 1 n n X i =1 ( X i ¯ X ) Y i Analogously for equation (2): ˆ σ XY = 1 n n X i =1 ( Y i ¯ Y ) X i 1 n n X i =1 ( Y i ¯ Y ) ¯ X = 1 n n X i =1 ( Y i ¯ Y ) X i ¯ X 1 n n X i =1 ( Y i ¯ Y ) = 1 n n X i =1 ( Y i ¯ Y ) X i ¯ X · = 1 n n X i =1 ( Y i ¯ Y ) X i 2. a. For the people with at least one parent who went to college, a confidence interval for μ yrsed c is given by 1 yrsed c ± 1 . 96 · SE ( yrsed c ) We know that yrsed c = 14 . 8 and SE ( yrsed c ) = s c √ n c = 1 . 74 √ 954 = . 056 . Therefore the confidence interval is yrsed c ± 1 . 96 · SE ( yrsed c ) = 14 . 8 ± 1 . 96 · . 056 = (14 . 69 , 14 . 91) For the people for which neither parent went to college, a confidence interval for μ yrsed nc is given by yrsed nc ± 1 . 96 · SE ( yrsed nc ) In this case, yrsed nc = 13 . 5 and SE ( yrsed nc ) = s nc √ n nc = 1 . 72 √ 2842 = . 032 . Therefore the confidence interval is yrsed nc ± 1 . 96 · SE ( yrsed nc ) = 13 . 5 ± 1 . 96 · . 032 = (13 . 44 , 13 . 56) b. The null and alternative hypotheses are written: H : μ c μ nc = 0 H A : μ c μ nc 6 = 0 To test this hypothesis we can either construct a confidence interval (using z α/ 2 = 2 . 58) or simply calculate the pvalue...
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 Spring '10
 JoshuaKinsler
 Null hypothesis, Yi, Yi Xi, xi xi

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