Chapter 3

Review of Statistics

Solutions to Empirical Exercises

1. (a) Average Hourly Earnings, Nominal $’s Mean AHE1992 AHE2004 AHE2004 − AHE1992 (b) Average Hourly Earnings, Real $2004 Mean AHE1992 AHE2004 AHE2004 − AHE1992 15.66 16.77 Difference 1.11 SE(Mean) 0.086 0.098 SE(Difference) 0.130 95% Confidence Interval 15.49−15.82 16.58−16.96 95% Confidence Interval 0.85−1.37 11.63 16.77 Difference 5.14 SE(Mean) 0.064 0.098 SE(Difference) 0.117 95% Confidence Interval 11.50−11.75 16.58−16.96 95% Confidence Interval 4.91−5.37

(c) The results from part (b) adjust for changes in purchasing power. These results should be used. (d) Average Hourly Earnings in 2004 Mean High School College College−High School 13.81 20.31 Difference 6.50 SE(Mean) 0.102 0.158 SE(Difference) 0.188 95% Confidence Interval 13.61−14.01 20.00−20.62 95% Confidence Interval 6.13−6.87

Solutions to Empirical Exercises in Chapter 3

109

(e)

Average Hourly Earnings in 1992 (in $2004)

Mean High School College College−High School (f) 13.48 19.07 Difference 5.59

SE(Mean) 0.091 0.148 SE(Difference) 0.173

95% Confidence Interval 13.30−13.65 18.78−19.36 95% Confidence Interval 5.25−5.93

Average Hourly Earnings in 2004 Mean AHEHS,2004 − AHEHS,1992 AHECol,2004 − AHECol,1992 Col–HS Gap (1992) Col–HS Gap (2004) Gap2004 − Gap1992 0.33 1.24 SE(Mean) 0.137 0.217 95% Confidence Interval 0.06−0.60 0.82−1.66

5.59 6.50 Difference 0.91

0.173 0.188 SE(Difference) 0.256

5.25−5.93 6.13−6.87 95% Confidence Interval 0.41−1.41

Wages of high school graduates increased by an estimated 0.33 dollars per hour (with a 95% confidence interval of 0.06 − 0.60); Wages of college graduates increased by an estimated 1.24 dollars per hour (with a 95% confidence interval of 0.82 − 1.66). The College − High School gap increased by an estimated 0.91 dollars per hour. (g) Gender Gap in Earnings for High School Graduates Year 1992 2004

Ym

14.57 14.88

sm 6.55 7.16

nm 2770 2772

Yw 11.86 11.92

sw 5.21 5.39

nw 1870 1574

Ym − Yw SE( Ym − Yw ) 2.71 2.96 0.173 0.192

95% CI 2.37−3.05 2.59−3.34

There is a large and statistically significant gender gap in earnings for high school graduates. In 2004 the estimated gap was $2.96 per hour; in 1992 the estimated gap was $2.71 per hour (in $2004). The increase in the gender gap is somewhat smaller for high school graduates than it is for college graduates.

Chapter 4

Linear Regression with One Regressor

Solutions to Empirical Exercises

1. (a) · = 3.32 + 0.45 × Age AHE Earnings increase, on average, by 0.45 dollars per hour when workers age by 1 year. (b) Bob’s predicted earnings = 3.32 + 0.45 × 26 = $11.70 Alexis’s predicted earnings = 3.32 + 0.45 × 30 = $13.70 (c) The R2 is 0.02.This mean that age explains a small fraction of the variability in earnings across individuals. (a)

2.

5

Course Evaluation

4

3

2 -2 -1 0 Beauty Index 1 2

There appears to be a weak positive relationship between course evaluation and the beauty index. · (b) Course _ Eval = 4.00 + 0.133 × Beauty. The variable Beauty has a mean that is equal to 0; the estimated intercept is the mean of the dependent variable (Course_Eval) minus the estimated slope (0.133) times the mean of the regressor (Beauty). Thus, the estimated intercept is equal to the mean of Course_Eval. (c) The standard deviation of Beauty is 0.789. Thus Professor Watson’s predicted course evaluations = 4.00 + 0.133 × 0 × 0.789 = 4.00 Professor Stock’s predicted course evaluations = 4.00 + 0.133 × 1 × 0.789 = 4.105

Solutions to Empirical Exercises in Chapter 4

111

(d) The standard deviation of course evaluations is 0.55 and the standard deviation of beauty is 0.789. A one standard deviation increase in beauty is expected to increase course evaluation by 0.133 × 0.789 = 0.105, or 1/5 of a standard deviation of course evaluations. The effect is small. (e) The regression R2 is 0.036, so...