Using data from 2013 on 64 black females, the estimated log-linear regression between WAGE (earnings per hour, in \$) and years of education, EDUC is \(\widehat{\ln (W A G E)}=1.58+0.09 E D U C\). The reported \(t\)-statistic for the slope coefficient is 3.95. 

a. Test at the \(5 \%\) level of significance, the null hypothesis that the return to an additional year of education is less than or equal to \(8 \%\) against the alternative that the rate of return to education is more than \(8 \%\). 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.

b. Testing the null hypothesis that the return to education is \(8 \%\), against the alternative that it is not \(8 \%\), we obtain the \(p\)-value 0.684. What is the \(p\)-value for the test in part (a)? In a sketch, show for the test in part (a) the \(p\)-value and the \(5 \%\) critical value from the \(t\)-distribution.

c. Construct a \(90 \%\) interval estimate for the return to an additional year of education and state its interpretation.