Starting in 1998, a series of credit card usage studies have been performed by Sallie Mae, a major provider of educational loans and savings programs. In a recent pilot study, a random pool of 122 loan applicants attending four-year colleges had their credit card data pulled for analysis. The sample mean credit card balance was $3173. The standard deviation was $3216.
(a) Compute a 95% confidence interval for the mean credit card balance among all undergraduate loan applicants (studying at four-year colleges).
(b) Provide the ‘correct' interpretation of your interval.
(c) Sally Mae would like to repeat this survey but with a sample size large enough to achieve a margin of error of $100. How many credit card records should they analyze in order to achieve this margin of error with 95% confidence? You should use the results of the pilot survey in part (a) in answering this question.
(d) Researchers at Sally Mae were interested not only in the actual balances in the pilot survey described above. They were also interested in using the 122 records to estimate the percentage of all credit card balances that exceeded $10,000. In fact, 23 of the 122 balances in the pilot survey exceeded $10,000. Use this result to find an 80% confidence interval for the percentage of all credit card balances that exceeded $10,000.
(e) To the extent possible, check the validity of the conditions associated with your confidence interval in part (d).
(f) Explain carefully how you would use your confidence interval in part (d) to perform a two-sided test of whether a quarter of all balances exceed $10,000. What decision is appropriate? What level of significance is implied in this test?