Wooldridge IE AISE SSM ch02 - CHAPTER 2 SOLUTIONS TO...

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This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 3 CHAPTER 2 SOLUTIONS TO PROBLEMS 2.2 (i) Let y i = GPA i , x i = ACT i , and n = 8. Then x = 25.875, y = 3.2125, 1 n i = ( x i x )( y i y ) = 5.8125, and 1 n i = ( x i x ) 2 = 56.875. From equation (2.9), we obtain the slope as 1 ˆ β = 5.8125/56.875 .1022, rounded to four places after the decimal. From (2.17), 0 ˆ = y 1 ˆ x 3.2125 – (.1022)25.875 .5681. So we can write n GPA = .5681 + .1022 ACT n = 8. The intercept does not have a useful interpretation because ACT is not close to zero for the population of interest. If ACT is 5 points higher, n GPA increases by .1022(5) = .511. (ii) The fitted values and residuals — rounded to four decimal places — are given along with the observation number i and GPA in the following table: i GPA n GPA ˆ u 1 2.8 2.7143 .0857 2 3.4 3.0209 .3791 3 3.0 3.2253 –.2253 4 3.5 3.3275 .1725 5 3.6 3.5319 .0681 6 3.0 3.1231 –.1231 7 2.7 3.1231 –.4231 8 3.7 3.6341 .0659 You can verify that the residuals, as reported in the table, sum to .0002, which is pretty close to zero given the inherent rounding error. (iii) When ACT = 20, n GPA = .5681 + .1022(20) 2.61.
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This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 4 (iv) The sum of squared residuals, 2 1 ˆ n i i u = , is about .4347 (rounded to four decimal places), and the total sum of squares, 1 n i = ( y i y ) 2 , is about 1.0288. So the R -squared from the regression is R 2 = 1 – SSR/SST 1 – (.4347/1.0288) .577. Therefore, about 57.7% of the variation in GPA is explained by ACT in this small sample of students. 2.3 (i) Income, age, and family background (such as number of siblings) are just a few possibilities. It seems that each of these could be correlated with years of education. (Income and education are probably positively correlated; age and education may be negatively correlated because women in more recent cohorts have, on average, more education; and number of siblings and education are probably negatively correlated.) (ii) Not if the factors we listed in part (i) are correlated with educ . Because we would like to hold these factors fixed, they are part of the error term. But if u is correlated with educ then E( u|educ ) 0, and so SLR.4 fails.
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Wooldridge IE AISE SSM ch02 - CHAPTER 2 SOLUTIONS TO...

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