Let x represent the dollar amount spent on supermarket impulse
buying in a 10-minute (unplanned) shopping interval. Based on a
certain article, the mean of the x distribution is about $25 and
the estimated standard deviation is about $8.
1. Consider a random sample of n = 70 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?
(a) The sampling distribution of x is not normal.
(b) The sampling distribution of x is approximately normal
with mean μx = 25 and standard error σx = $0.96.
(c) The sampling distribution of x is approximately normal
with mean μx = 25 and standard error σx = $0.11.
(d) The sampling distribution of x is approximately normal
with mean μx = 25 and standard error σx = $8.
2. Is it necessary to make any assumption about the x
distribution? Explain your answer.
(a) It is not necessary to make any assumption about the x
distribution because n is large.
(b) It is necessary to assume that x has an approximately
normal distribution.
(c) It is necessary to assume that x has a large
distribution.
(d) It is not necessary to make any assumption about the x
distribution because μ is large.
3. What is the probability that x is between $23 and $27?
(Round your answer to four decimal places.)
4. Let us assume that x has a distribution that is
approximately normal. What is the probability that x is between $23
and $27? (Round your answer to four decimal places.)
5. In part (2), we used x, the average amount spent, computed
for 70 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?
(a) The sample size is smaller for the x distribution than it
is for the x distribution.
(b) The standard deviation is smaller for the x distribution
than it is for the x distribution.
(c) The x distribution is approximately normal while the x
distribution is not normal.
(d) The mean is larger for the x distribution than it is for
the x distribution.
(e) The standard deviation is larger for the x distribution
than it is for the x distribution.
6. 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?
(a) 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.
(b) 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.