Letxrepresent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of thexdistribution is about$27and the estimated standard deviation is about$9.

(a) Consider a random sample ofn=100customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution ofx, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of thexdistribution?

Is it necessary to make any assumption about thexdistribution? Explain your answer.

(b) What is the probability thatxis between$25and$29? (Round your answer to four decimal places.)

(c) Let us assume thatxhas a distribution that is approximately normal. What is the probability thatxis between$25and$29? (Round your answer to four decimal places.)

(d) In part (b), we usedx, theaverageamount spent, computed for100customers. In part (c), we usedx, the amount spent by onlyonecustomer. The answers to parts (b) and (c) are very different. Why would this happen?

In this example,xis a much more predictable or reliable statistic thanx. Consider that almost all marketing strategies and sales pitches are designed for theaveragecustomer andnot the individualcustomer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?