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 $11 and the estimated standard deviation is about $9.

(a) Consider a random sample of n = 120 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 μ_x = 11 and standard error σ_x = $0.08.

- The sampling distribution of x is approximately normal with mean μ_x = 11 and standard error σ_x = $9.

- The sampling distribution of x is approximately normal with mean μ_x = 11 and standard error σ_x = $0.82.

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 n is large.

- It is not necessary to make any assumption about the x distribution because μ 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 $9 and $13? (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 $9 and $13? (Round your answer to four decimal places.)


(d) In part (b), we used x, the average amount spent, computed for 120 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 the x distribution is not normal.

- The sample size is smaller 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.The mean is larger 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

a)

(i) The sampling distribution of x is approximately normal with mean μ_x = 11 and standard error σ_x = $0.82.

    standard error σ_x = σ/n= 9/120 = $0.82.

(ii)

It is necessary to assume that x has an approximately normal distribution.Since N > 30.

NOTE: Usually we switch to parametric distribution if we find N 30.

b)

The provided lower limit of the distribution is a=9, and the upper limit is b=1.Pr⁡(9≤X≤13)

Therefore, :

Pr⁡(9≤X≤13)=13−9/13-9 =1

c) Assume,X~Normal( 11,0.6724) ; =0.82; = 11

Pr⁡(9≤X≤13) = 0.9853

d) (i) The x distribution is approximately normal while the x distribution is not normal.

        Reason: In part c we considered x~normal but in part b x ~uniform.

    (ii) 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.

NOTE:

The standard deviation of the sample means (known as the standard error of the mean) will be smaller than the population standard deviation and will be equal to the standard deviation of the population divided by the square root of the sample size.