We have data on 2323 randomly selected households consisting of three persons in 2013. Let ENTERT denote the monthly entertainment expenditure (\$) per person per month and let INCOME \((\$ 100)\) be monthly household income. Consider the simple linear regression model ENTERT \({ }_{i}=\) \(\beta_{1}+\beta_{2}\) INCOME \(_{i}+e_{i}, i=1, \ldots, 2323\). Assume that assumptions SR1-SR6 hold. The least squares estimated equation is \(\widehat{E N T E R T}_{i}=9.820+0.503 I N C O M E_{i}\). The standard error of the slope coefficient estimator is \(\operatorname{se}\left(b_{2}\right)=0.029\), the standard error of the intercept estimator is \(\operatorname{se}\left(b_{1}\right)=2.419\), and the estimated covariance between the least squares estimators \(b_{1}\) and \(b_{2}\) is -0.062 .

a. Construct a \(90 \%\) confidence interval estimate for \(\beta_{2}\) and interpret it for a group of CEOs from the entertainment industry.

b. The CEO of AMC Entertainment Mr. Lopez asks you to estimate the average monthly entertainment expenditure per person for a household with monthly income (for the three-person household) of \(\$ 7500\). What is your estimate?

c. AMC Entertainment's staff economist asks you for the estimated variance of the estimator \(b_{1}+75 b_{2}\). What is your estimate?

d. AMC Entertainment is planning to build a luxury theater in a neighborhood with average monthly income, for three-person households, of \(\$ 7500\). Their staff of economists has determined that in order for the theater to be profitable the average household will have to spend more than \(\$ 45\) per person per month on entertainment. Mr. Lopez asks you to provide conclusive statistical evidence, beyond reasonable doubt, that the proposed theater will be profitable. Carefully set up the null and alternative hypotheses, give the test statistic, and test rejection region using \(\alpha=0.01\). Using the information from the previous parts of the question, carry out the test and provide your result to the AMC Entertainment CEO.

e. The income elasticity of entertainment expenditures at the point of the means is \(\varepsilon=\) \(\beta_{2}(\overline{I N C O M E} / \overline{E N T E R T})\). The sample means of these variables are \(\overline{E N T E R T}=45.93\) and \(\overline{I N C O M E}=71.84\). Test the null hypothesis that the elasticity is 0.85 against the alternative that it is not 0.85, using the \(\alpha=0.05\) level of significance.

f. Using Statistical Table 1, compute the approximate two-tail \(p\)-value for the \(t\)-statistic in part (e). Using the \(p\)-value rule, do you reject the null hypothesis \(\varepsilon=\beta_{2}(\overline{I N C O M E} / \overline{E N T E R T})=0.85\), versus the alternative \(\varepsilon eq 0.85\), at the \(10 \%\) level of significance? Explain.