(a) Consider a random sample of
n = 50 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 x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?
The sampling distribution of x is approximately normal with mean μx = 40 and standard error σx = $0.18.The sampling distribution of x is approximately normal with mean μx = 40 and standard error σx = $1.27. The sampling distribution of x is approximately normal with mean μx = 40 and standard error σx = $9.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 thex distribution becausen is large.It is necessary to assume thatx has a large distribution. It is not necessary to make any assumption about thex distribution because μ is large.It is necessary to assume thatx has an approximately normal distribution.
(b) What is the probability that x is between $38 and $42? (Round your answer to four decimal places.)
(c) Let us assume that
x has a distribution that is approximately normal. What is the probability that
x is between $38 and $42? (Round your answer to four decimal places.)
(d) In part (b), we used x, the
average amount spent, computed for 50 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?
The x distribution is approximately normal while thex distribution is not normal.The standard deviation is larger for the x distribution than it is for thex distribution. The mean is larger for the x distribution than it is for thex distribution.The standard deviation is smaller for the x distribution than it is for thex distribution.The sample size is smaller for the x distribution than it is for thex distribution.
In this example, 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?
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.