Using data from 2013 on 64 black females, the estimated linear regression between WAGE (earnings per hour, in \$) and years of education, EDUC is \(\widehat{W A G E}=-8.45+1.99 E D U C\).

a. The standard error of the estimated slope coefficient is 0.52. Construct and interpret a \(95 \%\) interval estimate for the effect of an additional year of education on a black female's expected hourly wage rate.

b. The standard error of the estimated intercept is 7.39. Test the null hypothesis that the intercept \(\beta_{1}=0\) against the alternative that the true intercept is not zero, using the \(\alpha=0.10\) level of significance. In your answer, show (i) the formal null and alternative hypotheses, (ii) the test statistic and its distribution under the null hypothesis, (iii) the rejection region (in a figure), (iv) the calculated value of the test statistic, and (v) state your conclusion, with its economic interpretation.

c. Estimate the expected wage for a black female with 16 years of education, \(E(W A G E \mid E D U C=16)\).

d. The estimated covariance between the intercept and slope is -3.75 . Construct a \(95 \%\) interval estimate for the expected wage for a black female with 16 years of education.

e. It is conjectured that a black female with 16 years of education will have an expected wage of more than \(\$ 23\) per hour. Use this as the "alternative hypothesis" in a test of the conjecture at the \(10 \%\) level of significance. Does the evidence support the conjecture or not?