3. You work as a manager for the restaurant chain, Tex-Mex. You would like to estimate the percentage of customers who use a two-for one coupon from the Entertainment Book when paying for their dinner. You randomly select 80 customers and 16 of them used a coupon.

(a) What is the point estimate of the proportion of customers who use a coupon?

(b) Construct a 99% confidence interval for the percentage of customers who use a coupon.

(c) You decide to take a larger sample to reduce the error on the estimate of the percentage of customers who use a coupon. How many more customers would need to be sampled to reduce the margin of error to 4% at the 99% confidence level?

(d) You would also like to estimate the percentage of customers who order an appetizer among those who use a coupon. You check the bills for the 16 customers who used a coupon and found that 12 out of the 16 ordered an appetizer. How many additional customers would need to be sampled if you want the estimate to be accurate to within 3%, 18 times out of 20?

(e) You examine the bills for the 16 customers who used an Entertainment coupon and obtain a mean expenditure of $75 and standard deviation of $15.30. You want to know if they typically spend more for dinner (before discount) than customers who don't use a coupon. Construct a 95% confidence interval for the true mean amount spent (before discount) for the customers who use a coupon. Assume the amounts spent are normally distributed.

(f) You think that the margin of error on the estimate of the mean restaurant bill is too large. Among the customers who used a coupon, how many restaurant bills need to be sampled to reduce the margin of error to $3 at the 95% confidence level? Assume the sample standard deviation of $15.30 is a fairly close estimate of the population standard deviation.