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chapter7_s - CHAPTER 7 SOLUTIONS TO PROBLEMS 7.1(i The...

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CHAPTER 7 SOLUTIONS TO PROBLEMS 7.1 (i) The coefficient on male is 87.75, so a man is estimated to sleep almost one and one-half hours more per week than a comparable woman. Further, t male = 87.75/34.33 2.56, which is close to the 1% critical value against a two-sided alternative (about 2.58). Thus, the evidence for a gender differential is fairly strong. (ii) The t statistic on totwrk is .163/.018 9.06, which is very statistically significant. The coefficient implies that one more hour of work (60 minutes) is associated with .163(60) 9.8 minutes less sleep. (iii) To obtain 2 r R , the R -squared from the restricted regression, we need to estimate the model without age and age 2 . When age and age 2 are both in the model, age has no effect only if the parameters on both terms are zero. 7.3 (i) The t statistic on hsize 2 is over four in absolute value, so there is very strong evidence that it belongs in the equation. We obtain this by finding the turnaround point; this is the value of hsize that maximizes (other things fixed): 19.3/(2 ˆ sat 2.19) 4.41. Because hsize is measured in hundreds, the optimal size of graduating class is about 441. (ii) This is given by the coefficient on female (since black = 0): nonblack females have SAT scores about 45 points lower than nonblack males. The t statistic is about –10.51, so the difference is very statistically significant. (The very large sample size certainly contributes to the statistical significance.) (iii) Because female = 0, the coefficient on black implies that a black male has an estimated SAT score almost 170 points less than a comparable nonblack male. The t statistic is over 13 in absolute value, so we easily reject the hypothesis that there is no ceteris paribus difference. (iv) We plug in black = 1, female = 1 for black females and black = 0 and female = 1 for nonblack females. The difference is therefore –169.81 + 62.31 = 107.50. Because the estimate depends on two coefficients, we cannot construct a t statistic from the information given. The easiest approach is to define dummy variables for three of the four race/gender categories and choose nonblack females as the base group. We can then obtain the t statistic we want as the coefficient on the black female dummy variable. 7.5 (i) Following the hint, = n colGPA 0 ˆ β + 0 ˆ δ (1 – noPC ) + 1 ˆ hsGPA + 2 ˆ ACT = ( 0 ˆ + 0 ˆ ) 0 ˆ noPC + 1 ˆ hsGPA + 2 ˆ ACT . For the specific estimates in equation (7.6), 0 ˆ = 1.26 and 0 ˆ = .157, so the new intercept is 1.26 + .157 = 1.417. The coefficient on noPC is –.157. (ii) Nothing happens to the R -squared. Using noPC in place of PC is simply a different way of including the same information on PC ownership. 34
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(iii) It makes no sense to include both dummy variables in the regression: we cannot hold noPC fixed while changing PC . We have only two groups based on PC ownership so, in addition to the overall intercept, we need only to include one dummy variable. If we try to include both along with an intercept we have perfect multicollinearity (the dummy variable trap).
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