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 $45 and the estimated standard deviation is about $8.

(a) Consider a random sample of n = 60 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 = 45 and standard error σ_x = $0.13.The sampling distribution of x is approximately normal with mean μ_x = 45 and standard error σ_x = $1.03.     The sampling distribution of x is not normal.The sampling distribution of x is approximately normal with mean μ_x = 45 and standard error σ_x = $8.

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

(b) What is the probability that x is between $43 and $47? (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 $43 and $47? (Round your answer to four decimal places.)


(d) In part (b), we used x, the average amount spent, computed for 60 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 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.

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.     

The personnel office at a large electronics firm regularly schedules job interviews and maintains records of the interviews. From the past records, they have found that the length of a first interview is normally distributed, with mean μ = 38minutes and standard deviation σ = 8 minutes. (Round your answers to four decimal places.)

(a) What is the probability that a first interview will last 40 minutes or longer?


(b) Twelve first interviews are usually scheduled per day. What is the probability that the average length of time for the twelve interviews will be 40 minutes or longer?

Answer 1

(a) The sampling distribution of (sample mean) is Normal with mean and standard error,

Note : By central limit theorem, the sampling distribution of sample mean is normal with mean = population mean and standard error = , if sample size is large , regardless of distribution of population .

It is not necessary to make any assumption of x distribution (population distribution) as n (sample size) is large .

(b) follows Normal with mean =45 and standard error =1.03

then

To find

= 0.9476 (from z table)

(c )

X follows Normal with mean =45 and standard deviation =8

then

To find

= 0.1974 (from z table)

(d) The standard deviation is smaller for (sample mean)distribution , than it is for x (population)distribution .

The central limit theorem tells us that standard deviation of sample mean is much smaller than population standard deviation , thus average customer is more predictable than individual customer.