Using 2011 data on 141 U.S. public research universities, we examine the relationship between cost per student and full-time university enrollment. Let \(A C A=\) real academic cost per student (thousands of dollars), and let FTESTU = full-time student enrollment (thousands of students). The least squares fitted relation is \(\widehat{A C A}=14.656+0.266 F T E S T U\).

a. For the regression, the \(95 \%\) interval estimate for the intercept is \([10.602,18.710]\). Calculate the standard error of the estimated intercept.

b. From the regression output, the standard error for the slope coefficient is 0.081. Test the null hypothesis that the true slope, \(\beta_{2}\), is 0.25 (or less) against the alternative that the true slope is greater than 0.25 using the \(10 \%\) level of significance. Show all steps of this hypothesis test, including the null and alternative hypotheses, and state your conclusion.

c. On the regression output, the automatically provided \(p\)-value for the estimated slope is 0.001 . What is the meaning of this value? Use a sketch to illustrate your answer.

d. A member of the board of supervisors states that ACA should fall if we admit more students. Using the estimated equation and the information in parts (a)-(c), test the null hypothesis that the slope parameter \(\beta_{2}\) is zero, or positive, against the alternative hypothesis that it is negative. Use the \(5 \%\) level of significance. Show all steps of this hypothesis test, including the null and alternative hypotheses, and state your conclusion. Is there any statistical support for the board member's conjecture?

e. In 2011, Louisiana State University (LSU) had a full-time student enrollment of 27,950. Based on the estimated equation, the least squares estimate of \(E(A C A \mid F T E S T U=27,950)\) is 22.079, with standard error 0.964. The actual value of \(A C A\) for LSU that year was 21.403. Would you say that this value is surprising or not surprising? Explain.