The data file collegetown contains data on 500 single-family houses sold in Baton Rouge, Louisiana, during 2009-2013. The data include sale price in \(\$ 1000\) units, PRICE, and total interior area in hundreds of square feet, \(S Q F T\).

a. Using the quadratic regression model, \(P R I C E=\alpha_{1}+\alpha_{2} S O F T^{2}+e\), test the hypothesis that the marginal effect on expected house price of increasing the size of a 2000 square foot house by 100 square feet is less than or equal to \(\$ 13000\) against the alternative that the marginal effect will be greater than \(\$ 13000\). Use the \(5 \%\) level of significance. Clearly state the test statistic used, the rejection region, and the test \(p\)-value. What do you conclude?

b. Using the quadratic regression model in part (a), test the hypothesis that the marginal effect on expected house price of increasing the size of a 4000 square foot house by 100 square feet is less than or equal to \(\$ 13000\) against the alternative that the marginal effect will be greater than \(\$ 13000\). Use the \(5 \%\) level of significance. Clearly state the test statistic used, the rejection region, and the test \(p\)-value. What do you conclude?

c. Using the quadratic regression model in part (a), estimate the expected price \(E(P R I C E \mid S Q F T)=\) \(\alpha_{1}+\alpha_{2} S Q F T^{2}\) for a house of 2000 square feet. Construct a \(95 \%\) interval estimate of the expected price. Describe your interval estimate to a general audience.

d. Locate houses in the sample with 2000 square feet of living area. Calculate the sample mean (average) of their selling prices. Is the sample average of the selling price for houses with \(S Q F T=20\) compatible with the result in part (c)? Explain.