Confidence Intervals at Harrigan

Harrigan University is a liberal arts university in the Midwest that attempts to attract the highest quality students, especially from its region of the country. It has gathered data on 178 applicants who were accepted by Harrigan (a random sample from all acceptable applicants over the past several years). The data are in the file C08_01.xlsx. The variables are as follows:
Accepted: whether the applicant accepts Harrigan's offer to enroll
MainRival: whether the applicant enrolls at Harrigan's main rival university
HSClubs: number of high school clubs applicant served as an officer
HSSports: number of varsity letters applicant earned
HSGPA: applicant's high school GPA
HSPctile: applicant's percentile (in terms of GPA) in his or her graduating class
HSSize: number of students in applicant's graduating class
SAT: applicant's combined SAT score
Combined Score: a combined score for the applicant used by Harrigan to rank applicants

The derivation of the combined score is a closely kept secret by Harrigan, but it is basically a weighted average of the various components of high school performance and SAT. Harrigan is concerned that it is not getting enough of the best students, and worse yet, that many of these best students are going to Harrigan's main rival. Solve the following problems and then, based on your analysis, comment on whether Harrigan appears to have a legitimate concern.

1. Find a 95% confidence interval for the proportion of all acceptable applicants who accept Harrigan's invitation to enroll. Do the same for all acceptable applicants with a combined score less than 330, with a combined score between 330 and 375, and then with a combined score greater than 375. (Note that 330 and 375 are approximately the first and third quartiles of the Combined Score variable.)

2. Find a 95% confidence interval for the proportion of all acceptable students with a combined score less than the median (356) who choose Harrigan's rival over Harrigan. Do the same for those with a combined score greater than the median.

3. Find 95% confidence intervals for the mean combined score, the mean high school GPA, and the mean SAT score of all acceptable students who accept Harrigan's invitation to enroll. Do the same for all acceptable students who choose to enroll elsewhere. Then find 95% confidence intervals for the differences between these means, where each difference is a mean for students enrolling at Harrigan minus the similar mean for students enrolling elsewhere.

4. Harrigan is interested (as are most schools) in getting students who are involved in extracurricular activities (clubs and sports). Does it appear to be doing so? Find a 95% confidence interval for the proportion of all students who decide to enroll at Harrigan who have been officers of at least two clubs. Find a similar confidence interval for those who have earned at least four varsity letters in sports.

5. The combined score Harrigan calculates for each student gives some advantage to students who rank highly in a large high school relative to those who rank highly in a small high school. Therefore, Harrigan wonders whether it is relatively more successful in attracting students from large high schools than from small high schools. Develop one or more confidence intervals for relevant parameters to shed some light on this issue.