Let x represent the dollar amount spent on supermarketimpulse buying in a 10-minute (unplanned) shopping interval. Basedon a certain article, the mean of the x distribution isabout $39 and the estimated standard deviation is about $7.
(a) Consider a random sample of n = 60 customers, eachof whom has 10 minutes of unplanned shopping time in a supermarket.From the central limit theorem, what can you say about theprobability distribution of x, the average amount spent bythese customers due to impulse buying? What are the mean andstandard deviation of the x distribution?
The sampling distribution of x is approximately normalwith mean ?x = 39 and standard error?x = $0.90.The sampling distributionof x is approximately normal with mean?x = 39 and standard error?x = $7. Thesampling distribution of x is approximately normal withmean ?x = 39 and standard error?x = $0.12.The sampling distributionof x is not normal.
Is it necessary to make any assumption about the xdistribution? Explain your answer.
It is necessary to assume that x has a largedistribution.It is not necessary to make any assumption about thex distribution because n islarge. It is not necessary to make anyassumption about the x distribution because ? islarge.It is necessary to assume that x has anapproximately normal distribution.
(b) What is the probability that x is between $37 and $41?(Round your answer to four decimal places.)
(c) Let us assume that x has a distribution that isapproximately normal. What is the probability that x isbetween $37 and $41? (Round your answer to four decimalplaces.)
(d) In part (b), we used x, the average amountspent, computed for 60 customers. In part (c), we used x,the amount spent by only one customer. The answers toparts (b) and (c) are very different. Why would this happen?
The standard deviation is larger for the x distributionthan it is for the x distribution.The mean is larger forthe x distribution than it is for the xdistribution. The x distribution isapproximately normal while the x distribution is notnormal.The standard deviation is smaller for the xdistribution than it is for the x distribution.The samplesize is smaller for the x distribution than it is for thex distribution.
In this example, x is a much more predictable or reliablestatistic than x. Consider that almost all marketingstrategies and sales pitches are designed for the averagecustomer and not the individual customer. How does thecentral limit theorem tell us that the average customer is muchmore predictable than the individual customer?
The central limit theorem tells us that small sample sizes havesmall standard deviations on average. Thus, the average customer ismore predictable than the individual customer.The central limittheorem tells us that the standard deviation of the sample mean ismuch smaller than the population standard deviation. Thus, theaverage customer is more predictable than the individualcustomer.
