Focus Problem: Impulse Buying Let x represent the dollar

Chapter , Problem 16

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Focus Problem: Impulse Buying Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a Denver Post article, the mean of the x distribution is about $20 and the estimated standard deviation is about $7.

(a) Consider a random sample of n = 100 customers, 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 of \(\overline{x}\), the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the \(\overline{x}\) distribution? Is it necessary to make any assumption about the x distribution? Explain.

(b) What is the probability that \(\overline{x}\) is between $18 and $22?

(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $18 and $22?

(d) Interpretation: In part (b), we used \(\overline{x}\), the average amount spent, computed for 100 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen? In this example, \(\overline{x}\) is a much more predictable or reliable statistic than x. Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?