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 $49 and the estimated standard deviation is about $8.
(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 not normal.
The sampling distribution of x is approximately normalwith mean ?x = 49 and standard error?x = $8.
The sampling distribution of x is approximately normalwith mean ?x = 49 and standard error?x = $0.13.
The sampling distribution of x is approximately normalwith mean ?x = 49 and standard error?x = $1.03.
Is it necessary to make any assumption about the xdistribution? Explain your answer.
It is not necessary to make any assumption about the xdistribution because ? is large.
It is necessary to assume that x has an approximatelynormal distribution.
It is not necessary to make any assumption about the xdistribution because n is large.
It is necessary to assume that x has a largedistribution.
(b) What is the probability that x is between $47 and $51?(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 $47 and $51? (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 sample size is smaller for the x distribution thanit is for the x distribution.
The standard deviation is smaller for the x distributionthan it is for the xdistribution.
The x distribution is approximately normal while thex distribution is not normal.
The mean is larger for the x distribution than it is forthe x distribution.
The standard deviation is larger for the x distributionthan it is for the x 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 limit theorem tells us that the standard deviation ofthe sample mean is much smaller than the population standarddeviation. Thus, the average customer is more predictable than theindividual customer.
