Question

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 $18 and the estimated standard deviation is about $7.

(a) 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?

The sampling distribution of *x* is approximately normal
with mean *μ*_{x} = 18 and standard error
*σ*_{x} = $0.84.The sampling distribution
of *x* is not normal. The sampling
distribution of *x* is approximately normal with mean
*μ*_{x} = 18 and standard error
*σ*_{x} = $7.The sampling distribution of
*x* is approximately normal with mean
*μ*_{x} = 18 and standard error
*σ*_{x} = $0.10.

Is it necessary to make any assumption about the *x*
distribution? Explain your answer.

It is not necessary to make any assumption about the *x*
distribution because *μ* is large.It is not necessary to
make any assumption about the *x* distribution because
*n* is large. It is necessary to
assume that *x* has an approximately normal distribution.It
is necessary to assume that *x* has a large
distribution.

(b) What is the probability that *x* is between $16 and $20?
(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 $16 and $20? (Round your answer to four decimal
places.)

(d) In part (b), 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?

The sample size is smaller for the *x* distribution than
it is for the *x* distribution.The *x* distribution
is approximately normal while the *x* distribution is not
normal. The mean is larger for the
*x* distribution than it is for the *x*
distribution.The standard deviation is larger for the *x*
distribution than it is for the *x* distribution.The
standard deviation is smaller for the *x* distribution than
it is for the *x* 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 small sample sizes have small standard deviations on average. Thus, the average customer is 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.

Answer #1

**The standard deviation is smaller for the
distribution than it is for the x distribution.**

**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.**

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