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2 Margin of Error, Confidence Intervals, and Sample Size

2 Margin of Error, Confidence Intervals, and Sample Size

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Chapter 3

opinion on current topics of interest. As we have said, if properly conducted,

these surveys are amazingly accurate.

When a survey is used to find a proportion based on a sample of only a few

thousand individuals, the obvious question is how close that proportion comes

to the truth for the entire population. The margin of error is a measure of the

accuracy of a sample proportion. It provides an upper limit on the difference

between the sample proportion and the population proportion that holds for at

least 95% of simple random samples of a specific size. In other words, the difference between the sample proportion and the unknown value of the population proportion is less than the margin of error at least 95% of the time, or in

at least 19 of every 20 situations. A conservative estimate of the margin of error

for a sample proportion is calculated as 1> 1n, where n is the sample size. To express results in terms of percentages instead of proportions, simply multiply

everything by 100.


The conservative margin of error for a sample proportion is calculated by

using the formula 1> 1n, where n represents the sample size, the number of

people in the sample. The amount by which the sample proportion differs from

the true population proportion is less than this quantity in at least 95% of all

random samples. Survey results reported in the media are usually expressed as

percentages. The conservative margin of error for a sample percentage is


ϫ 100%


In Chapter 10, we will learn a more complicated and precise formula for the

margin of error that depends on the actual sample proportion. It will never give

an answer larger than the formula given here, so we call this formula a conservative margin of error. For simplicity, many media reports use the conservative

formula. For example, with a sample of 1600 people, we will usually get an estimate that is accurate to within 1> 11600 ϭ 1>40 ϭ 0.025 ϭ 2.5% of the truth.

Confidence Intervals

You might see results such as “Fifty-five percent of respondents support the

President’s economic plan. The margin of error for this survey is plus or minus

2.5 percentage points.” This means that it is almost certain that between 52.5%

and 57.5% of the entire population supports the plan. In other words, add and

subtract the margin of error to the sample value (55% in this example) to create

an interval. If you were to follow this method every time you read the results

of a properly conducted survey, the interval would miss covering the truth

only about 1 in 20 times (5%) and would cover the truth the remaining 95% of

the time.

A confidence interval is an interval of values that estimates an unknown

population value. In the preceding paragraph, the percentage of all U.S. adults

favoring the President’s economic policy is the unknown value we wish to estimate. The confidence interval estimate of this value is 52.5% to 57.5%, calculated as sample percentage Ϯ margin of error.

Sampling: Surveys and How to Ask Questions


The word confidence refers to the fraction or percentage of random samples

for which a confidence interval procedure gives an interval that includes the

unknown value of a population parameter. For the procedure described here,

the confidence is at least 95%, meaning that intervals calculated in this manner

will capture the population proportion for at least 95% of all properly selected

random samples.


95% Confidence Inter val for Population Proportion

For about 95% of properly conducted sample surveys, the interval

sample proportion Ϫ



to sample proportion ϩ



will contain the actual population proportion. This interval is called an approximate “95% confidence interval” for the population proportion. In Chapter 10, a more precise formula will be given.

Another way of writing the approximate 95% confidence interval is

sample proportion Ϯ



To report the results in percentages instead of proportions, multiply everything

by 100%.

Here are three examples of polls, with the margin of error and an approximate 95% confidence interval for each one. In each case, the poll is based on a

nationwide random sample of Americans aged 18 and older.

Example 3.3

The Importance of Religion for Adult Americans For a CNN/USA Today/

Gallup Poll conducted on September 2 to 4, 2002, a random sample of n ϭ 1003

adult Americans was asked, “How important would you say religion is in your

own life: very important, fairly important, or not very important?” (http://www

.pollingreport.com/religion.htm, September 17, 2002). The percentages that

selected each response were as follows:

Very important

Fairly important

Not very important

No opinion





Conservative margin of error:


ϭ .03 or .03 ϫ 100% ϭ 3%


Approximate 95% confidence intervals for the proportion and percentage of

all adult Americans who would say religion is very important are

Proportion: .65 Ϯ .03 or [.65 Ϫ .03 to .65 ϩ .03] or .62 to .68

Percentage: 65% Ϯ 3% or [65% Ϫ 3% to 65% ϩ 3%] or 62% to 68% ■

Example 3.4

Would You Eat Those Modified Tomatoes? An ABCNews.com poll conducted by TNS Intersearch on June 13 to 17, 2001, posed the following question


Chapter 3

Watch a video example at http://

1pass.thomson.com or on your CD.

to a random sample of n ϭ 1024 adult Americans: “Scientists can change the

genes in some food crops and farm animals to make them grow faster or bigger

and be more resistant to bugs, weeds and disease. Do you think this genetically modified food, also known as bio-engineered food, is or is not safe to eat?”

(http://www.pollingreport.com/science.htm, September 17, 2002). The percentages that selected each response were as follows:

Is safe

Is not safe

No opinion




Conservative margin of error:


ϭ .03 or .03 ϫ 100% ϭ 3%


Approximate 95% confidence intervals for the proportion and percentage

of all adult Americans who would say genetically modified food is not safe to

eat are

Proportion: .52 Ϯ .03 or [.52 Ϫ .03 to .52 ϩ .03] or .49 to .55

Percentage: 52% Ϯ 3% or [52% Ϫ 3% to 52% ϩ 3%] or 49% to 55% ■

Example 3.5

Cloning Human Beings A Pew Research Center for the People & the Press

and Pew Forum on Religion & Public Life survey conducted by Princeton Survey Research Associates between February 25 and March 10, 2002, asked a random sample of n ϭ 2002 adult Americans, “Do you favor or oppose scientific

experimentation on the cloning of human beings?” (http://www.pollingreport

.com/science.htm, September 17, 2002). The percentages that selected each response were as follows:



Don’t know/Refused




Conservative margin of error:


ϭ .022 or .022 ϫ 100% ϭ 2.2%


Approximate 95% confidence intervals for the proportion and percentage of

all adult Americans who would say they favor this experimentation are

Proportion: .17 Ϯ .022 or [.17 Ϫ .022 to .17 ϩ .022] or .148 to .192

Percentage: 17% Ϯ 2.2% or [17% Ϫ 2.2% to 17% ϩ 2.2%] or

14.8% to 19.2% ■

Interpreting the Confidence Intervals in Examples 3.3 to 3.5

In each of the three examples of surveys, an approximate 95% confidence interval was found. There is no way to know whether all, some, or none of these

intervals actually covers the population value of interest. For instance, the interval from 62% to 68% may or may not capture the percentage of adult Americans who considered religion to be very important in their lives in September

2002. But in the long run, this procedure will produce intervals that capture the

unknown population values about 95% of the time, as long as it is used with

properly conducted surveys. This long-run performance is usually expressed

after an interval is computed by saying that we are 95% confident that the population value is covered by the interval. We will learn more about how to in-

Sampling: Surveys and How to Ask Questions


terpret a confidence interval and the accompanying confidence level (such as

95%) in Chapter 10.

Choosing a Sample Size for a Survey

Table 3.1 Relationship Between

Sample Size and Margin

of Error for 95% Confidence


Size n

Margin of

Error ‫ ؍‬1/ 1n








.10 (10%)

.05 (5%)

.04 (4%)

.032 (3.2%)

.025 (2.5%)

.02 (2%)

.01 (1%)

When surveys are planned, the choice of a sample size is an important issue.

One commonly used strategy is to use a sample size that provides a desired

margin of error for a 95% confidence interval. Table 3.1 displays the margin of

error for several different sample sizes. The margin of error calculations were

done by using the “conservative” formula 1> 1n. This is commonly done by

polling organizations because, before the sample is observed, there is no way to

know what sample proportion to use in the more exact margin of error formula.

With a table like Table 3.1, researchers can pick a sample size that provides suitable accuracy for any sample proportion, within the constraints of the time and

money available for the survey.

Two important features of Table 3.1 are

1. When the sample size is increased, the margin of error decreases.

2. When a large sample size is made even larger, the improvement in accuracy is relatively small. For example, when the sample size is increased

from 2500 to 10,000, the margin of error decreases only from 2% to 1%. In

general, cutting the margin of error in half requires a fourfold increase in

sample size.

Polling organizations determine a sample size that is accurate enough for

their purposes and is also economical. Many national surveys use a sample size

of about 1000, which, as you can see from Table 3.1, makes the margin of error

roughly 3%. This is a reasonable degree of accuracy for most questions asked in

these surveys.

Some federal government surveys utilize much larger sample sizes, sometimes as large as n ϭ 120,000, to make accurate estimates of quantities such

as the unemployment rate. Also, when researchers want to make accurate

estimates for subgroups within the population, they have to use a very large

overall sample size. For instance, to get information from approximately 1000

African-American women in the 18- to 29-year-old age group, a random sample

of 120,000 Americans might be necessary.

The Effect of Population Size

3.2 Exercises are on page 108.

You might wonder how the number of people in the population affects the accuracy of a survey. The surprising answer is that for most sample surveys, the

number of people in the population has almost no influence on the accuracy of

sample estimates. The margin of error for a sample size of 1000 is about 3%

whether the number of people in the population is 30,000 or 200 million.

The formulas for margin of error in this chapter were derived by assuming

that the number of units in the population is essentially infinite. In practice,

as long as the population is at least ten times as large as the sample, we consider only how sample size affects accuracy and we ignore the specific size of

the population. For small populations, a “finite population correction” is used.

We will not discuss it in this book, but you can consult any book on survey sampling for details.


Chapter 3

t h o u g h t q u e s t i o n 3 . 2 Suppose that a survey of 400 students at your school is conducted to assess student opinion about a new academic honesty policy. Based on Table 3.1, about

what will be the margin of error for the poll? How many students attend your school?

Given this figure, do you think the values in Table 3.1 should be used to estimate the margin of error for a survey of students at your school? Explain.*

3.3 Choosing a Simple Random Sample

The ability of a relatively small sample to accurately reflect a huge population

does not happen haphazardly. It happens only if proper sampling methods are

used. A probability sampling plan is one in which everyone in the population has a specified probability to be selected for the sample. The basic idea is

that everyone in the population must have a specified chance of making it into

the sample.

The most basic probability sampling plan is to use a simple random

sample. Remember that with a simple random sample, every conceivable

group of units of the required size has the same chance of being the selected

sample. In this section, we discuss how to choose a simple random sample. A

variety of other probability sampling methods will be discussed in Section 3.4.

Choosing a simple random sample is somewhat like choosing the winning

numbers in many state lotteries. For instance, in the Pennsylvania Match 6 lottery game, six numbers are randomly selected from the choices 1, 2, . . . , 49.

Every possible set of six numbers is equally likely to be the winning set. There

are actually 13,983,816 different possible sets, which is why the odds of any

specific individual guessing the winning set are so small!

Similarly, the chances of any particular group of units getting selected to

be the random sample from a large population is quite small, but whatever

group is selected is likely to be representative. For instance, in a simple random

sample of 1000 people in your state or country, it is extremely unlikely that you

and your next-door neighbor would both be selected. But it is extremely likely

that someone in the sample will be representative of each of you, having similar opinions to yours.

By the way, if there were 100 million people in a population, the number

of different possible samples of 1000 individuals would be incomprehensively

large. It would take about one and a half pages of this book to show the value.

Approximately, the number of possible different samples of 1000 individuals selected from a population of 100 million is 247 followed by 5430 zeros.

To actually produce a simple random sample, you need only two things.

First, you need a list of the units in the population. For instance, in drawing

the winning lottery numbers, the list of units is the numbers 1, 2, . . . , 49. In

selecting a simple random sample of students from your school, the list of units

is all students in the school (which the registrar can usually produce).

Second, you need a source of random numbers. Computer programs such

as Minitab and Excel or the right calculator can be used to generate random

numbers. Random numbers can also be found in tables called tables of ran-

*H I N T : The sample size is n ϭ 400. Is the number of students at your school more than ten times

the sample size?

Sampling: Surveys and How to Ask Questions


dom digits. If the population isn’t very large, physical methods can be used,

such as in lotteries in which the numbers are written on small, hollow plastic

balls and six of them are physically selected.

Table 3.2 illustrates a portion of a table of random digits. It is organized into

numbered rows to make it easier to find specific sections of the table. There are

only ten rows in Table 3.2, so they are labeled by using the single digits 0 to 9.

A larger table would have to use longer identifying numbers for the rows. The

digits are grouped into columns of five for easier reading. These tables are generated by the equivalent of writing the digits from 0 to 9 on slips of paper, mixing them well, choosing one, and then repeating this process over and over

again with replacement. No single digit, pair of consecutive digits, triplet of digits, and so on is any more or less likely to occur than any others.

Table 3.2 A Table of Random Digits




























































































Using Table 3.2 to Choose a Simple Random Sample

Here are the steps for selecting a simple random sample using Table 3.2.

1. Number the units in the population, using the same number of digits

for each one.

Example: Suppose there are 270 students in a class and the teacher wants

to choose a simple random sample of 10 of them to call on in class. Number the students from 001 to 270.

2. Choose a starting point in Table 3.2. You can close your eyes and point or

use any other method as long as you have not studied the table and then

chosen numbers that give a favorable sample for your purpose.

Example: Start in row 3, column 2 (10484 . . .).

3. From the starting point, read across the row to get numbers with the correct

number of digits to identify a unit. Continue with consecutive rows. For instance, if the units are numbered 001 to 270, read three-digit numbers.

Example: Reading three-digit numbers starting with row 3, column 2 results

in 104, 842, 461, 613, 466, 416, 180, 855, 118, 314, 577, 002, 896, and so on.

4. If a number is in the range of the unit numbers, select that unit number.

Otherwise, continue along the row, choosing more potential unit numbers until you have a sample of the desired size. (But see step 5 for a more

efficient method.) Because each unit can be used only once, if a unit number occurs that has already been selected, simply ignore that number and



Chapter 3

Example: Unit numbers can only be 001 to 270, so most of the numbers

that are chosen are simply ignored. Select units 104, 180, 118, 002, then

keep going until the required 10 students have been selected.

5. Step 4 is very inefficient. To make it more efficient, reassign some of the

higher numbers onto the range of unit numbers in a way that still ensures

that each unit number has an equal chance of selection. For instance,

suppose the units are numbered 001 to 270. If a number between 301

and 570 is chosen, subtract 300 and use it. If a number between 601 and

870 is chosen, subtract 600 and use it. As in step 4, if a unit number occurs more than once, simply ignore subsequent occurrences after the

first one. For instance, in the scenario just given, if unit 017 has already

been selected, then any subsequent occurrence of 017, 317, or 617 would

be discarded.

Example: Now the string generated earlier is more useful. Let’s see what

decision follows each three-digit entry:



Using Step 4,

Choose Unit

























Using Step 5,

Choose Unit


242 (subtracted 600)

161 (subtracted 300)

013 (subtracted 600)

166 (subtracted 300)

116 (subtracted 300)


255 (subtracted 600)


014 (subtracted 300)

Not needed, 10 already selected

Not needed

Using the method in step 4, the first four students who are selected would

be those numbered 104, 180, 118, and 002, and the process would continue until six more were selected. Using the method in step 5, the sample of ten units

would include 104, 242, 161, 013, 166, 116, 180, 255, 118, and 014, and unit 002

would not be needed.


Picking a Random Sample

To create a column of ID numbers, use CalcbMake Patterned Datab

Simple Set of Numbers. In the dialog box, specify a column for storing the

ID numbers, and specify the first and last possible ID number for the


To sample values from a column, use CalcbRandom DatabSample from

Columns. In the dialog box, specify how many items (rows) will be selected from a particular column, and specify a column where the sample

will be stored.

Note: Items can be randomly selected from a column of names or data values, so it may not be necessary to assign ID numbers to the units in the

population in order to select a sample.

Sampling: Surveys and How to Ask Questions

Example 3.6

Watch a video example at http://

1pass.thomson.com or on your CD.


Representing the Heights of British Women In Example 2.17 of Chapter 2,

we examined data on the heights of 199 British women. Suppose you had a list

of these 199 women and wanted to choose ten of them to test-drive a sporty, but

small, automobile model and give their opinions about its comfort. The heights

of the women in the sample should be representative of the range of heights in

the larger group. You would not want your sample of ten to include only short

women or only tall women. Here’s how you could choose a simple random


1. Assign an ID number from 001 to 199 to each woman.

2. Use a table of random digits, a computer, or a calculator to randomly select ten numbers between 001 and 199, and sample the heights of the

women with those ID numbers.

We used the statistical package Minitab to choose a simple random sample of

heights, then used Table 3.2 to choose another one. The samples are listed next

along with their sample means and the list of ten random numbers between 001

and 199 that generated them (with leading zeros dropped). In Chapter 2, the

heights were given in millimeters, but here they have been converted to inches.

Sample 1 (Minitab)

ID numbers of the women selected: 176, 10, 1, 40, 85, 162, 46, 69, 77, 154

Heights: 60.6, 63.4, 62.6, 65.7, 69.3, 68.7, 61.8, 64.6, 60.8, 59.9;

mean ϭ 63.7 inches

Sample 2 (Table 3.2)

ID numbers of the women selected: 41, 93, 167, 33, 157, 131, 110, 180, 185,


Heights: 59.4, 66.5, 63.8, 62.6, 65.0, 60.2, 67.3, 59.8, 67.7, 61.8;

mean ϭ 63.4 inches

3.3 Exercises are on page 109.

Sample 2 was selected by starting with the third set of five digits in the row labeled 5 (04189). Numbers from 001 to 199 can be used directly; for numbers between 201 and 399, subtract 200; for numbers between 401 and 599, subtract 400;

and so on. The only numbers that would need to be discarded in using this

method are 000, 200, 400, 600, and 800. You can test your understanding of the

use of Table 3.2 by starting at 04189 (third set in row 5), selecting ten consecutive sets of three digits, and determining whether you correctly identify the ten

women in sample 2.

As you can see, each sample is different, but each should be representative

of the whole collection of 199 women. Within each sample, the range of heights

is from about 60 inches to about 69 inches. The sample means are both close to

the mean height for the larger group, which was given in Chapter 2 as 1602 mm,

or about 63 inches. ■

3.4 Other Sampling Methods

For large populations, it may not be practical to take a simple random sample

because it may be difficult to get a numbered list of the units. For instance, if a

polling organization wanted to take a simple random sample of all voters or all


Chapter 3

adults in a country or region, the organization would need to get a numbered

list of them, which is simply an impossible task in most cases. Instead, it relies

on more complicated sampling methods, all of which are good substitutes for

simple random sampling in most situations. In fact, they often have advantages

over simple random sampling. For instance, one of these other methods, stratified random sampling, can be used to increase the chance that the sample represents important subgroups within the population.

To visually illustrate the various sampling plans discussed in this chapter,

let’s suppose that a college administration would like to survey a sample of students living in dormitories. The college has undergraduate and graduate dormitories. The undergraduate dormitories have three floors each with 12 rooms per

floor. The graduate dormitories have five floors each with 8 rooms per floor.

Figure 3.1 illustrates a simple random sample of 30 rooms in the dormitories. Any collection of 30 rooms has an equal chance of being the selected

sample. Notice that for the sample illustrated, there are 12 undergraduate

rooms and 18 graduate rooms in the sample.

Undergraduate dormitories

12 rooms per floor

Graduate dormitories

8 rooms per floor

Third floor

Second floor

First floor

Fifth floor

Fourth floor

Third floor

Second floor

First floor

Figure 3.1 ❚ A simple random sample of 30 dorm rooms

Stratified Random Sampling

A stratified random sample is collected by first dividing the population of units

into strata (subgroups of the population) and then taking a simple random

sample from each one. The strata are subgroups that seem important to represent properly in the sample and might differ with regard to values of the response variable(s) measured. For example, public opinion pollsters often take

separate random samples from each region (strata) of the country so that they

can spot regional differences as well as improve the likelihood that all regions

are properly represented in the overall national sample. Or political pollsters

may separately sample from each political party to compare opinions by party

and to be sure that each party is properly represented.

Figure 3.2 illustrates a stratified sample for the college survey. There are two

strata: the undergraduate and graduate dorms. A random sample of 15 rooms is

taken from each of the two strata. Each collection of 15 undergraduate rooms

has an equal chance of being the selected sample for the undergraduate dorms,

Sampling: Surveys and How to Ask Questions

Undergraduate dormitories

12 rooms per floor


Graduate dormitories

8 rooms per floor

Third floor

Second floor

First floor

Fifth floor

Fourth floor

Third floor

Second floor

First floor

Figure 3.2 ❚ A stratified sample of 15 undergraduate and 15 graduate dorm rooms

and each collection of 15 graduate rooms has an equal chance of being the selected sample for the graduate dorms. But the total of 15 rooms to be sampled

within each stratum (undergraduate or graduate) is fixed before the sample is


A principal benefit of stratified sampling is that it can be used to improve the

chance that the selected sample properly represents important subgroups in

the population. It also is used to create more accurate estimates of population

values than we might get from using a simple random sampling method.

So far, we have been focusing on measuring categorical variables, such as

opinions or traits people might have. Surveys are also used to measure quantitative variables, such as age at first intercourse or number of cigarettes smoked per

day. We are often interested in the population average for such measurements.

The accuracy with which we can estimate the average depends on the natural variability among the measurements. The less variable they are, the more

precisely we can assess the population average on the basis of the sample values. For instance, if everyone in a relatively large sample reports that the age at

first intercourse was between 16 years 3 months and 16 years 4 months, then

we can be relatively sure that the average age in the population is close to that.

On the other hand, if the reported ages range from 13 years to 25 years, then we

cannot pinpoint the average age for the population nearly as accurately.

Stratified sampling can help to solve this problem. Suppose we could figure

out how to stratify in such a way that there is little natural variability in the answers within each strata. We could then get an accurate estimate for each stratum and combine these estimates to get a much more precise answer for the

overall group.

Cluster Sampling

In cluster sampling, the population is divided into clusters (subgroups), but

rather than sampling within each cluster, we select a random sample of clusters

and include only members of these selected clusters in the sample. After clusters are selected, it may be that all members of a cluster are included in the


Chapter 3

Undergraduate dormitories

12 rooms per floor

Graduate dormitories

8 rooms per floor

Third floor

Second floor

First floor

Fifth floor

Fourth floor

Third floor

Second floor

First floor

Figure 3.3 ❚ A cluster sample in which five floors (clusters) are randomly selected

sample or that some units are then randomly sampled from each of the selected

clusters. A cluster typically comprises units that are physically close to each

other in some way, such as the students living on one floor of a college dormitory, all individuals listed on a single page of a telephone directory, or all passengers on a particular airplane flight.

Figure 3.3 illustrates a cluster sampling plan for the college survey. Each

floor of each dormitory is a cluster in this particular plan. A random sample of

five floors is selected from the 24 floors of the three undergraduate and three

graduate dorms. Any collection of five floors has an equal chance of being the

selected sample of floors. Once the five floors have been selected, all students

on the five selected floors are surveyed. This plan is efficient logistically because

the data collection team will have to visit only five different dormitory floors to

collect data.

Cluster sampling is often confused with stratified sampling, but it is a radically different concept and can be much easier to accomplish. In most applications of stratified sampling, the population is divided into a few large strata,

such as regions of the country, and a small subset is then randomly sampled

from each of the strata. In most applications of cluster sampling, the population

is divided into small clusters, such as city blocks, a large number of clusters are

randomly sampled, and either everyone or a sample is measured in those clusters selected. In stratified sampling, all subgroups (strata) are represented in the

sample. In cluster sampling, some clusters within the population are included

in the sample, and others are not.

One obvious advantage of cluster sampling is that you need only a list of

clusters instead of a list of all individual units. City blocks are commonly used as

clusters in surveys that require door-to-door interviews. To measure customer

satisfaction, airlines sometimes randomly sample a set of flights, then distribute a survey to everyone on those flights. Each flight is a cluster. It is clearly

much easier for the airline to choose a random sample of flights than it would

be to identify and locate a random sample of individual passengers to whom to

distribute surveys.

If cluster sampling is used, the analysis must proceed differently because

there may be similarities among the members of the clusters that must be taken

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