Let x represent the dollar amount spent on supermarket impulse buying in a $10-$minute $($unplanned$)$ shopping interval. Based on a newspaper article, the mean of the x distribution is about $$48$ and the estimated standard deviation is about $$6.$

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$($a$)$

Consider a random sample of n $=40$ 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 not normal.

The sampling distribution of x is approximately normal with mean $$x $=48$ and standard error $$x $=$ $$6.$

The sampling distribution of x is approximately normal with mean $$x $=48$ and standard error $$x $=$ $$0\mathrm{.}15.$

The sampling distribution of x is approximately normal with mean $$x $=48$ and standard error $$x $=$ $$0\mathrm{.}95.$

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

It is necessary to assume that x has an approximately normal distribution.

It is not necessary to make any assumption about the x distribution 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 x distribution because $$ is large.

$($b$)$

What is the probability that x is between $$46$ and $$50?($Round your answer to four decimal places.$)$

$($c$)$

Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $$46$ and $$50?($Round your answer to four decimal places.$)$

$($d$)$

In part $($b$),$ we used x$,$ the average amount spent, computed for $40$ 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 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.

The mean is larger for the x distribution than it is for the x distribution.

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.

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.