Chapter 8, Problem 62PS

Scarlett and Heather, the owners of an upscale restaurant in Dayton, Ohio, want to study the dining characteristics of their customer. They decide to focus on two variables: the amount of money spent by customers and whether customer order dessert. The results from a sample of 60 customers are as follows:

Amount spent $\bar{X}=\$38.54,S=\$\mathrm{7.26.}$

Eighteen customers purchased dessert.

a. Construct a $95\%$ confidence interval estimate for the population mean amount spent per customer in the restaurant.

b. Construct a $90\%$ confidence interval estimate for the population proportion of customers who purchase dessert.

Jeanine, the owner of a competing restaurant, wants to conduct a similar survey in her restaurant. Jeanine does not have access to the information that Scarlett and Heather have obtained from the survey they conducted. Answer the following questions:

a. What sample size is needed to have $95\%$ confidence of estimating the population mean amount spent her restaurant to within $\pm \$1.50,$ assuming that the standard deviation is estimated to be $\$8?$

b. How many customer need to be selected to have $90\%$ confidence of estimating the population proportion of customer who purchase dessert to within $\pm 0.004?$

c. Based on your answer to (c) and (d), how large a sample should Jeanine take?