Let x represent the dollar amount spent on supermarket impulsebuying in a 10-minute (unplanned) shopping interval. Based on acertain article, the mean of the x distribution is about $19 andthe estimated standard deviation is about $7.
(a) Consider a random sample of n = 40 customers, each of whomhas 10 minutes of unplanned shopping time in a supermarket. Fromthe central limit theorem, what can you say about the probabilitydistribution of x, the average amount spent by these customers dueto impulse buying? What are the mean and standard deviation of thex distribution?
The sampling distribution of x is approximately normal with meanμx = 19 and standard error σx = $1.11.
The sampling distribution of x is approximately normal with meanμx = 19 and standard error σx = $7.
The sampling distribution of x is approximately normal with meanμx = 19 and standard error σx = $0.18.
The sampling distribution of x is not normal.
Is it necessary to make any assumption about the x distribution?Explain your answer.
It is not necessary to make any assumption about the xdistribution because n is large.
It is necessary to assume that x has a large distribution.
It is not necessary to make any assumption about the xdistribution because μ is large.
It is necessary to assume that x has an approximately normaldistribution.
(b) What is the probability that x is between $17 and $21?(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 is between $17and $21? (Round your answer to four decimal places.)
(d) In part (b), we used x, the average amount spent, computedfor 40 customers. In part (c), we used x, the amount spent by onlyone customer. The answers to parts (b) and (c) are very different.Why would this happen?
The mean is larger for the x distribution than it is for the xdistribution.
The standard deviation is smaller for the x distribution than itis for the x distribution.
The x distribution is approximately normal while the xdistribution is not normal.
The sample size is smaller for the x distribution than it is forthe x distribution.
The standard deviation is larger for the x distribution than itis for the x distribution.
In this example, x is a much more predictable or reliablestatistic than x. Consider that almost all marketing strategies andsales pitches are designed for the average customer and not theindividual customer. How does the central limit theorem tell usthat the average customer is much more predictable than theindividual 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 deviationof the sample mean is much smaller than the population standarddeviation. Thus, the average customer is more predictable than theindividual customer.
