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$23and the estimated standard deviation is about$9.

(a) Consider a random sample ofn=80customers, 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?

The sampling distribution ofxis approximately normal with mean_x= 23 and standard error_x= $1.01.

The sampling distribution ofxis not normal.

The sampling distribution ofxis approximately normal with mean_x= 23 and standard error_x= $0.11.

The sampling distribution ofxis approximately normal with mean_x= 23 and standard error_x= $9.

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

It is not necessary to make any assumption about thexdistribution becauseis large.

It is necessary to assume thatxhas an approximately normal distribution.

It is necessary to assume thatxhas a large distribution.

It is not necessary to make any assumption about thexdistribution becausenis large.

(b) What is the probability thatxis between$21and$25? (Round your answer to four decimal places.)

(c) Let us assume thatxhas a distribution that is approximately normal. What is the probability thatxis between$21and$25? (Round your answer to four decimal places.)

(d) In part (b), we usedx, theaverageamount spent, computed for80customers. In part (c), we usedx, the amount spent by onlyonecustomer. The answers to parts (b) and (c) are very different. Why would this happen?

The standard deviation is larger for thexdistribution than it is for thexdistribution.

Thexdistribution is approximately normal while thexdistribution is not normal.

The sample size is smaller for thexdistribution than it is for thexdistribution.

The standard deviation is smaller for thexdistribution than it is for thexdistribution.

The mean is larger for thexdistribution than it is for thexdistribution.

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?

The central limit theorem tells us that the standard deviation of the sample mean is much smaller than the population standard deviation. Thus, the average customer is more predictable than the individual customer.

The central limit theorem tells us that small sample sizes have small standard deviations on average. Thus, the average customer is more predictable than the individual customer.