et 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 $10 and the estimated standard deviation is about $7. (a) Consider a random sample of n = 80 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 = 10 and standard error ?x = $0.09. The sampling distribution of x is approximately normal with mean ?x = 10 and standard error ?x = $7.The sampling distribution of x is approximately normal with mean ?x = 10 and standard error ?x = $0.78. 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 ? 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 a large distribution. (b) What is the probability that x is between $8 and $12? (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 $8 and $12? (Round your answer to four decimal places.)

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

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 $10 and the estimated standard deviation is about $7.

(a)

Consider a random sample of n = 80 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 = 10 and standard error ?_x = $0.09. The sampling distribution of x is approximately normal with mean ?_x = 10 and standard error ?_x = $7.The sampling distribution of x is approximately normal with mean ?_x = 10 and standard error ?_x = $0.78.

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 ? 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 a large distribution.

(b)

What is the probability that x is between $8 and $12? (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 $8 and $12? (Round your answer to four decimal places.)

(d)

In part (b), we used x, the average amount spent, computed for 80 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 smaller 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 standard deviation is larger 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.

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.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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