June 7, 2019

CHAPTER 9

Determining Sample Size and the Sample Plan

LEARNING OBJECTIVES

To become familiar with the basic concepts involved in sampling

To learn how to calculate the minimum sample size needed to achieve a predetermined level

of accuracy

To understand the difference between “probability” and “nonprobability” sampling plans

To become acquainted with the specifics of four probability and four nonprobability

sampling plans

CHAPTER OUTLINE

Basic Concepts in Samples and Sampling

Determining Size of a Sample

The Accuracy of a Sample

Formula to Determine Sample Accuracy

How to Calculate Sample Size When Estimating a Percentage

Variability: p times q

Level of Confidence: z

Desired Accuracy: e

Explaining the Logic of Our Sample Size Formula

How to Calculate Sample Size When Estimating a Mean

The Effects of Incidence Rate and Nonresponse on Sample Size

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

1

Using the XL Data Analyst to Calculate Sample Size

How to Select a Representative Sample

Probability Sampling Methods

Simple Random Sampling

Using the XL Data Analyst to Generate Random Numbers

Systematic Sampling

Cluster Sampling

Stratified Sampling

Nonprobability Sampling Methods

Convenience Samples

Judgment Samples

Referral Samples

Quota Samples

Online Sampling Techniques

KEY TERMS

Accuracy of a sample

Area sampling

Census

Cluster sampling

Completed interviews

Completion rate

Proportionate sample size

Proportionate stratified sample

Quota sample

Random digit dialing

Random numbers technique

Random online intercept sampling

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

2

Confidence interval formula for sample

size

Convenience sample

Demographic incidence

Desired accuracy

Disproportionate sample size

Geographic incidence

Incidence

Invitation online sampling

Judgment sample

Level of confidence

Nonprobability sampling method

One-step area sample

Online panel sampling

Population

Probability sample methods

Product incidence

Random sample

Random starting point

Referral sample

Sample

Sample frame

Sample frame error

Sampling error

Simple random sampling

Skip interval

Stratified sampling

Surrogate measure

Systematic sampling

Table of random numbers

Two-step area sample

Variability

Weighted average

Working phone rate

TEACHING SUGGESTIONS

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

3

1. National opinion polls tend to be around 1,000–1,200 in sample size, and this can be easily

verified. Have students do background research on opinion polls (perhaps there is a political

campaign underway) and bring their findings to class on the sample size and reported errors.

The findings illustrate that these companies are using the sample size for a percentage

formula with p and q equal to 50 percent and with a 95 percent level of confidence.

2. Figure 10.1 illustrates visually how large sample sizes fail to add to the accuracy of a survey.

Because the graph is difficult to read with precision, consider using the following table to

illustrate to students how little additional accuracy is gained with increases in the sample

size, especially with large samples.

Sample Size 95% Accuracy Accuracy Increase

100 9.8% -

250 6.2% 3.6%

500 4.4% 1.8%

750 3.6% 0.8%

1,000 3.1% 0.5%

1,250 2.8% 0.3%

1,500 2.5% 0.3%

1,750 2.3% 0.2%

2,000 2.2% 0.1%

3. The confidence interval approach to sample size involves several statistical concepts.

Although they are described in the chapter, it is necessary to review them thoroughly in class.

Students may memorize the sample size determination formula, but it is important that they

comprehend the factors in the formulas, namely, (1) the amount of variability believed to be

in the population, (2) the desired accuracy, and (3) the level of confidence required in

estimates of the population values.

4. Students do not easily grasp the concept of a sampling distribution. If you have a class of,

say, 30 students, you can pair them up and simulate 15 different samples. Have each pair give

its average age (other variables might be: hours carried this term, hours worked per week,

minutes to commute to campus), and plot all of the sample means in a distribution. If you

select a variable where students are quite similar (e.g., age), the sampling distribution will be

quite compact, but if you use a variable where students are dissimilar (e.g. distance of home

town away from campus), the sampling distribution will be much less compact.

5. Use the 95 percent level of confidence in sample size determination examples. The 1.96 z

value is about 2, and students have an easier time following the squaring computations than

they do with the 2.58 z value for the 99 percent level of confidence.

6. The sample size determination function in the XL Data Analyst provides sensitivity analysis

tables where the estimated “p” and the allowable error are varied slightly. This feature is

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

4

provided so users do not need to run the analysis several times. It shows how sensitive

sample size is to variations in p of ±5% and ±10% and variations in the allowable error of

±.5% and ±1%. An extension of this feature is to add a cost factor (such as x dollars per

respondent) and to show how the sensitivity estimates affect the cost of the sample.

7. The equiprobable aspects of a random sample can be demonstrated in a number of different

ways. Here are two examples:

Some Internet sites are devoted to lotteries and gambling. Although we do not have a

specific one in mind, you might consider assigning a student or a team of students to

search out one or more that lists the lottery numbers selected over time, perhaps for your

state if you have a state lottery. Have students calculate the percent of times each number

was drawn and comment on whether or not they have found the random selection process

used to embody equal probability in all cases.

Put 3 x 5 index cards with students’ names in a hat or a box and have a series of actual

blind draw samples. Replace the drawn names to the population pool after each sample.

Maintain a record of how often each student’s name is selected.

8. The greater efficiency of systematic sampling over simple random sampling can be

demonstrated with a class exercise. Identify two groups of students and give each a copy of

the same page from the telephone book. Tell the first group to select 10 household names

using a table of random numbers or random numbers generated via a spreadsheet program

such as Microsoft Excel (simple random sampling), and tell the second group to select 10

names using systematic sampling. The second group should finish before the first group.

9. A class exercise that illustrates basic differences between various sample methods is to use a

data set with 10 or fewer variables listed in labeled columns and containing 200 or more

respondents (an Excel or other spreadsheet printout will work). Break the class into several

teams and give specific instructions for each team to draw 30 respondents and calculate the

percentage distribution or mean of the variables. Each team should be assigned a different

sampling method (convenience, simple random sampling, etc.). Have students time their

teams or raise their hands when they have completed the work. Student teams using

nonprobability sampling (except quota) will finish first, while teams using the more tedious

probability methods such as stratified simple random sampling will finish last. However,

nonprobability sampling method teams will have findings different from the true percentage

distributions and means, although probability sampling method teams will have findings

close to the actuals. (Note: Instructors will have to provide a table or list of random numbers

to teams needing them. Such tables can be generated by the random numbers feature of the

XL Data Analyst.)

10. When learning about the skip interval used in systematic sampling, students sometimes ask

how to determine the population size when a directory or phone book is used. In the absence

of a precise number, the size is usually estimated by multiplying the approximate number of

names on each page by the number of pages. Some adjustments may need to be made for

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

5

multiple listings such as children’s phone, discontinued phones, or areas of “white space” in

the directory.

11. Some students may have difficulty understanding the weighted average calculations in

stratified sampling. It may be necessary to illustrate how the mean changes with different

stratum configurations. Here are some comparisons than can be used to demonstrate the

effects of three different weighted average situations.

Stratum Mean Estimated Population Mean

A B 50%/50% 40%/60%

10%/90%

5 8 6.5 6.8

7.7

12. Students should come to realize that the success of quota sampling is greatly dependent on a

prior knowledge of the population’s characteristics. One way to facilitate this understanding

is to ask students what quota characteristics should be used in the following two cases.

Case One. Kellogg’s wants to know the reactions of parents to a new children’s cereal called,

“Cheery-Os.”

Case Two. Proctor & Gamble wants the reactions of potential buyers of its new hair rinse

called, “Gentle Care.”

With case one, the quota characteristics would be: (1) parents (percent female versus male),

(2) marital status (percent married versus separated), and (3) age of youngest child (percent

4, 5, 6, etc.). With case two, however, the target market is not identified well other than it is

implicitly made up of women.

13. Instructors can use the random numbers function of the XL Data Analyst to illustrate the

nature of random numbers. Use it to generate a large number of random numbers, then

use Excel average and standard deviation functions to compute these values of each row and

each column.

In theory, the average of any random number table row or any column should be

approximately equal to the average of any other row or column. The standard deviations

should be approximately equal as well.

14. When students work with a familiar population, they are better able to apply sample methods.

Ask how the full-time students in your university would be sampled using each sample

method described in the chapter. For example, where would they station interviewers for a

convenience sample? How would they set up clusters or strata using student characteristics?

15. Here is a table the summarizes the differences between probability sampling methods and

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

6

nonprobability sampling methods and may be useful as a means of helping students

understand differences between these two types of sampling methods.

Probability Sampling Nonprobability Sampling

Known chance of selection Unknown chance of selection

Takes more time Takes less time

Higher cost Lower cost

Can compute sample error Cannot compute sample error

Copyright © 2012 Pearson Education, Inc. publishing as Prentice Hall

7