A travel agency is studying the behavior of potential customers who are about to book a vacation package directly on the agency's website. The website visitor is considered a potential customer if they select their destination, enter their personal information, and just need to commit to a credit card payment to complete their booking. There are 1,800 potential customers in the data that were collected within a 24-hour period, and the behavior of these customers seems representative of all potential customers. The average purchase in this group is $520 with a sample standard deviation of $210. Those who did not complete their booking are considered to have made a $0 purchase. Out of these 1,800 potential customers, 400 did not complete a purchase and 950 paid more than $500 for their purchase. (a) Construct a 95% confidence interval for the average purchase of potential customers. (b) The management is willing to assume that \sigma , the standard deviation of the purchase amount of a potential customer, is at least $150 and at most $240. With this assumption, what is the smallest sample size that would allow estimating the average purchase amount of potential customers within 5% of \sigma at 95% confidence?