The theory of labor supply indicates that more labor services will be offered at higher wages. Suppose that HRSWK is the usual number of hours worked per week by a randomly selected person and WAGE is their hourly wage. Our regression model is specified as \(H R S W K=\beta_{1}+\beta_{2} W A G E+e\). Using a sample of 9799 individuals from 2013, we obtain the estimated regression \(\widehat{H R S W K}=41.58+0.011\) WAGE. The estimated variances and covariance of the least squares estimators are as follows:

a. Test the null hypothesis that the relationship has slope that is less than, or equal to, zero at the 5\% level of significance. State the null and alternative hypotheses in terms of the model parameters. Using the results, do we confirm or refute the theory of labor supply?

b. Use Statistical Table 1 of normal probabilities to calculate an approximate \(p\)-value for the test in (a). Draw a sketch representing the \(p\)-value.

c. Under assumptions SR1-SR6 of the simple regression model, the expected number of hours worked per week is \(E(H R S W K \mid W A G E)=\beta_{1}+\beta_{2} W A G E\). Construct a \(95 \%\) interval estimate for the expected number of hours worked per week for a person earning \(\$ 20 / \mathrm{h}\).

d. In the sample, there are 203 individuals with hourly wage \(\$ 20\). The average number of hours worked for these people is 41.68. Is this result compatible with the interval estimate in (c)? Explain your reasoning.

e. Test the null hypothesis that the expected hours worked for a person earning \(\$ 20\) per hour is 41.68, against the alternative that it is not, at the \(1 \%\) level of significance.

Transcribed Image Text:

INTERCEPT INTERCEPT WAGE 0.02324 -0.00067 WAGE -0.00067 0.00003