The owners of a motel discovered that a defective product was used during construction. It took 7 months to correct the defects during which approximately 14 rooms in the 100 -unit motel were taken out of service for 1 month at a time. The data are in the file motel.

a. Plot MOTEL_PCT and COMP_PCT versus TIME on the same graph. What can you say about the occupancy rates over time? Do they tend to move together? Which seems to have the higher occupancy rates? Estimate the regression model MOTEL_PCT \(=\beta_{1}+\beta_{2} C O M P \_P C T+e\). Construct a 95\% interval estimate for the parameter \(\beta_{2}\). Have we estimated the association between MOTEL_PCT and COMP_PCT relatively precisely, or not? Explain your reasoning.

b. Construct a \(90 \%\) interval estimate of the expected occupancy rate of the motel in question, \(M O T E L_{-} P C T\), given that \(C O M P \_P C T=70\).

c. In the linear regression model MOTEL_PCT \(=\beta_{1}+\beta_{2} C O M P_{-} P C T+e\), test the null hypothesis \(H_{0}: \beta_{2} \leq 0\) against the alternative hypothesis \(H_{0}: \beta_{2}>0\) at the \(\alpha=0.01\) level of significance. Discuss your conclusion. Clearly define the test statistic used and the rejection region.

d. In the linear regression model MOTEL_PCT \(=\beta_{1}+\beta_{2} C O M P \_P C T+e\), test the null hypothesis \(H_{0}: \beta_{2}=1\) against the alternative hypothesis \(H_{0}: \beta_{2} eq 1\) at the \(\alpha=0.01\) level of significance. If the null hypothesis were true, what would that imply about the motel's occupancy rate versus their competitor's occupancy rate? Discuss your conclusion. Clearly define the test statistic used and the rejection region.

e. Calculate the least squares residuals from the regression of \(M O T E L_{-} P C T\) on \(C O M P \_P C T\) and plot them against TIME. Are there any unusual features to the plot? What is the predominant sign of the residuals during time periods 17-23 (July, 2004 to January, 2005)?