Chapter 8 Notes(1)

8/10/2019 Chapter 8 Notes(1)

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BADM220 Chapter 8

Chapter 8: Sampling Methods and the Central Limit Theorem

Introduction

In this chapter, we begin our study of sampling and how samples are used to infer something

about a population.

Sampling Methods

I. Reasons to sample

A. Contacting the whole population would be time consuming.

B. The cost of studying all the items in a population may be prohibitive.

C. It may be physically impossible to check all the items in the population.

D. Some tests might destroy the items in the population.

E. Sample results are usually adequate.

II. Sampling Methods

A. Simple Random Sample A sample selected so that each item or person in the

population has the same chance of being included.

1. Example: Self-Review 8-1, p. 251

The following class roster lists the students enrolling in an introductory course in

business statistics. Three students are to be randomly selected and asked

various questions regarding course content and method of instruction.

(a) The numbers 00 through 45 are handwritten on slips of paper and

placed in a bowl. The three numbers selected are 31, 7, and 25. Which

students would be included in the sample?


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(b) Now use the table of random digits, Appendix B.4, to select your

own sample. (Hint: Read the first paragraph on page 250 for an

explanation of how to use the table of random digits.)

(c) What would you do if you encountered the number 59 in the table

of random digits?

B. Systematic Random Sample A random starting point is selected, and then every kth

member of the population is selected. kis calculated as the population size divided

by the sample size. A random number between 1 and kis chosen as the starting point

(and included in the sample), then every kthmember is selected.

1. Example: Self-Review 8-2, page 254

Refer to Self-Review 8-1 and the class roster on page 251. Suppose a systematic

random sample will select every ninth student enrolled in the class. Initially, the

fourth student on the list was selected at random. That student is numbered

03. Remembering that the random numbers start with 00, which students will

be chosen to be members of the sample?

C. Stratified Random Sample A population is divided into subgroups, called strata, and

a sample is randomly selected from each stratum. For example, the students on the

class roster of the Self-Review problem could be divided into subgroups based on

class rank. Then, a random sample is chosen from each group.

1. Example: How can the students on the class roster in Self-Review 8-1 be

divided into strata?

(Solution: The students could be divided into groups based on gender: male and

female. Another possibility is by class standing: freshman, sophomore, junior,

and senior. )

D. Cluster Sample A population is divided into clusters using naturally occurring


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geographic or other boundaries. Then, clusters are randomly selected and a sample is

collected by randomly selecting from each cluster. For example, the students on the

class roster could be divided by their county of residence.

Sampling Error

I. Sampling Error

A. Since a sample is only a part or portion of the population, it is unlikely that the

sample mean would be exactly equal to the population mean. Likewise, it is unlikely

that the sample standard deviation would be exactly equal to the population

standard deviation. We expect a difference between the sample statistic and the

population parameter.

B. Sampling Error - The difference between a sample statistic and its corresponding

population parameter.

C. For the mean, the sampling error is the difference between the sample mean and

the population mean, .

Sampling Distribution of the Sample Mean

I. Sampling Distribution of the Sample Mean - A probability distribution of all possible sample

means of a given sample size.

A. Example: Self-Review 8-3, p. 260 (Remember to check your answers in Appendix E.)

The lengths of service of all of the executives employed by Standard Chemicals are:

Name Years

Mr. Snow 20

Ms. Tolson 22

Mr. Kraft 26

Ms. Irwin 24

Mr. Jones 28

(a) Using the combination formula, how many samples of size 2 are possible?


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(b) List all possible samples of 2 executives from the population and compute their means.

Sample Number Executives in Sample Years of Service Sample Mean

1

2

3

45

6

7

8

9

10

(c) Organize the means into a sampling distribution.

Sample Mean Number (frequency) Probability

(d) Compare the population mean and the mean of the sample means.

(e) Compare the dispersion in the population with that in the distribution of the sample mean.

(Hint: Calculate the range as the measure of dispersion for both the population data and

the sample data.)

(f) A chart portraying the population values follows. Is the distribution of population values

normally distributed (bell-shaped)?

(g) Is the distribution of the sample mean computed in part (c) starting to show some tendency

toward being bell-shaped?


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BADM220 Chapter 8

The Central Limit Theorem

I. Central Limit Theorem - If all samples of a particular size are selected from any population, the

sampling distribution of the sample mean is approximately a normal distribution. This

approximation improves with larger samples. Most statisticians consider a sample of 30 or

more to be large enough for the central limit theorem to be employed.

A. Mean of the Sampling Distribution:

B. Standard Deviation of the Sampling Distribution of the Sample Mean/Standard Error

of the Mean:

Using the Sampling Distribution of the Sample Mean

I. Formula for finding the z value:

(different version of the formula from book)

II. Complete the four-step process for finding probability of a normal distribution:

1) translate the question into probability notation

2) convert to z score

3) rewrite the probability in terms of z

4) sketch the area under the curve and calculate the area using Table B.3, page 732.

III. Example: p. 272, #16

A normal population has a mean of 75 and a standard deviation of 5. You select a

sample of 40. Compute the probability the sample mean is:

(a) Less than 74.

(b) Between 74 and 76.

(c) Between 76 and 77.


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(d) Greater than 77.

Solutions to above: (Note: I did not provide the sketches of the areas under the curve, but would

strongly recommend that you make the sketches.)

(a) P(< 74); find

; P(z < -1.26) = .5 - .3962 = .1038

(b) P(74 76); find

; P(-1.26 z 1.26) = .3962 + .3962 = .7924

(c) P(76 77); find

; P(1.26 z 2.53) = .4943 - .3962 = .0981

(d) P(> 77) = P(z > 2.53) = .5 - .4943 = .0057

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Chapter 8 Notes(1)