S - Chapter 8.2 Margin of Error

Lesson 8.2
What Affects the Margin of Error?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

8

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

After this lesson you should be able to:

Interpret a confidence interval.

Describe how the confidence level and
sample size affect the margin of error.

Explain how practical issues like nonresponse,
undercoverage, and response bias can affect the
interpretation of a confidence interval.

Learning Targets

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In this activity, you will use the Confidence Intervals applet to learn what it means to say
that we are “95% confident” that our confidence interval captures the parameter value.

1. Go to https://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html and launch the applet. Use the default settings:

Confidence Level 95% and Sample Size n = 20.

2. Click “SAMPLE” to choose an SRS and display the resulting confidence interval.

The 20 values in the sample are marked with yellow dots. The confidence interval is displayed as a horizontal line segment with a dot in the middle. What do you think the black dot represents?

3. Did the interval “hit” the target? That is, did it capture the population mean µ

represented by the green vertical line?

Click “SAMPLE” a total of 10 times. How many of the intervals captured the
population mean µ? Note: So far, you have used the applet to select 10 SRSs,
each of size n = 20. Be sure you understand the difference between sample size and
the number of samples selected.

Activity: Investigating confidence level with the

Confidence Intervals applet

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

4. Reset the applet. Click “SAMPLE 25” twice to

choose 50 SRSs and display the confidence
intervals based on those samples. How many
intervals captured the parameter µ?

Keep clicking “SAMPLE 25” and observe the
value of “Percent hit.” What do you notice?

5. Repeat Step 4 using a 90% confidence level.

6. Repeat Step 4 using an 80% confidence level.

7. Summarize what you have learned about the

relationship between confidence level and
capture rate (percent hit) after taking many
samples.

We will investigate the effect of changing the sample size later.

Activity: Investigating confidence level with the

Confidence Intervals applet

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In the preceding lesson, we claimed to be “95% confident
that the interval from 0.10 to 0.18 captures the proportion
of all U.S. adults who would say that health care costs are
the biggest financial problem facing their family today.”

The claim of “95% confidence” means that we are using a method
that successfully captures the parameter in 95% of all possible
samples.

That is, if we were to select many random samples and construct a 95% confidence interval using each sample, about 95% of those intervals would capture p, the proportion of all U.S. adults who would answer health care costs.

What Affects the Margin of Error?

p

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Make sure not to interpret the confidence interval when you are asked to interpret
a confidence level—and vice versa.

Confidence levels describe the process used to make an interval, not any one
particular interval.

What Affects the Margin of Error?

How to Interpret a Confidence Level

To interpret the confidence level C, say,

“If we were to select many random samples from a population and
construct a C% confidence interval using each sample, about C%
of the intervals would capture the [parameter in context].”

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

The National Center for Health Statistics asked a random sample of 3626 U.S.
adults about a variety of topics. According to the study, the 95% confidence
interval for the true proportion of adults who eat fast food on a given day is
0.350 to 0.382. Interpret the confidence level.

Example: Do you eat fast food?

​“If we were to select many random samples from a population and
construct a C% confidence interval using each sample, about C%
of the intervals would capture the [parameter in context].”

​How to Interpret a Confidence Level

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

The National Center for Health Statistics asked a random sample of 3626 U.S.
adults about a variety of topics. According to the study, the 95% confidence
interval for the true proportion of adults who eat fast food on a given day is 0.350
to 0.382. Interpret the confidence level.

SOLUTION:

If the National Center for Health Statistics selects many random
samples of U.S. adults and constructs a 95% confidence interval using
each sample, about 95% of these intervals would capture the true
proportion of U.S. adults who eat fast food on a given day.

Example: Do you eat fast food?

​Confidence levels describe the process used to make an interval, not any one particular interval.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

The confidence level does not tell
us the probability that a particular
confidence interval captures the
population parameter.

Once a particular confidence interval is
calculated, its endpoints are fixed.

A particular confidence interval either
includes the parameter (probability = 1)
or doesn’t include the parameter (probability = 0).

What Affects the Margin of Error?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In general, we prefer narrow (short) confidence intervals. That is, we prefer
confidence intervals with a small margin of error.

Knowing that tomorrow’s high temperature
will be between −40°F and 200°F won’t help
you pick out your clothes in the morning!

But knowing that tomorrow’s high temperature
will be between 80 F° and 82 F° would be quite useful.

To reduce the margin of error, we can change two factors:

sample size and confidence level.

What Affects the Margin of Error?

?

n

C%

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In this activity, you will use the applet to explore the relationship between the
confidence level, the sample size, and the margin of error.

Part 1: Adjusting the Confidence Level

1. Go back to the same applet and launch

the Confidence Intervals applet. Use the
default settings: Confidence Level 95%
and Sample Size n = 20.
Click “SAMPLE 25” to select 25 SRSs
and make 25 confidence intervals.

2. Change the confidence level to 99%.

What happens to the width of the
confidence intervals?

Activity: Exploring margin of error with the

Confidence Intervals applet

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Part 1: Adjusting the Confidence Level
3. Now change the confidence level to

90%. What happens to the width of the
confidence intervals?

4. Finally, change the confidence level to

80%. What happens to the width of the
confidence intervals?

5. Summarize what you learned about the

relationship between the confidence
level and the margin of error for a fixed
sample size.

Activity: Exploring margin of error with the

Confidence Intervals applet

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Part 2: Adjusting the Sample Size
6. Reset the applet to the default settings: Confidence Level 95% and Sample

Size n = 20. Click “SAMPLE 25” to select 25 SRSs and make 25 confidence
intervals.

7. Using the slider, increase the sample size to n =100 and click “SAMPLE 25.”

What happens to the width of the confidence intervals?

8. Using the slider, increase the sample size to n = 200 and click “SAMPLE 25.”

What happens to the width of the confidence intervals?

9. Summarize what you learned about the relationship between the sample size

and the margin of error for a fixed confidence level.

Activity: Exploring margin of error with the

Confidence Intervals applet

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

As the activity illustrates, the price we pay for

greater confidence is a wider interval.

If we’re satisfied with 80% confidence,
then our interval of plausible values for
the parameter will be much narrower
than if we insist on

What Affects the Margin of Error?

90%, 95%, or 99% confidence.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Intervals constructed at an 80% confidence level will capture the true value of the
parameter much less often than intervals that use a 99% confidence level.

What Affects the Margin of Error?

80% confidence
intervals for a
population
proportion
using n = 20

99% confidence
intervals for a
population
proportion
using n = 20

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

We can get a more precise estimate of a parameter by increasing the sample size.
Larger samples typically yield narrower confidence intervals.
However, larger samples cost more time and money to obtain.

What Affects the Margin of Error?

95% confidence
intervals for a
population
proportion
using n = 20

95% confidence
intervals for a
population
proportion
using n = 50

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

What Affects the Margin of Error?

Decreasing the margin of error

In general, we prefer an estimate with a small margin of error. The
margin of error gets smaller when:
§ The confidence level decreases. To obtain a smaller margin of

error from the same data, you must be willing to accept a
smaller capture rate.

§ The sample size n increases. In general, increasing the sample

size n reduces the margin of error for any fixed confidence level.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In the preceding example, you read about a random
sample of 3626 U.S. adults who were surveyed by the
National Center for Health Statistics. According to the
study, the 95% confidence interval for the true
proportion of adults who eat fast food on a given day
is 0.350 to 0.382.

(a) Explain what would happen to the width of

the interval if the confidence level were increased
to 99%.

(b) Explain what would happen to the width of the

interval if the sample size were increased to 5000.

Example: Do you want fries with that?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In the preceding example, you read about a random sample of 3626 U.S. adults who were surveyed by the National Center for Health Statistics.
According to the study, the 95% confidence interval for the true proportion
of adults who eat fast food on a given day is 0.350 to 0.382.

(a) Explain what would happen to the width of the interval if the

confidence level were increased to 99%.

SOLUTION:

The confidence interval will be wider because increasing the confidence
level increases the margin of error.

Example: Do you want fries with that?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

In the preceding example, you read about a random sample of 3626 U.S.
adults who were surveyed by the National Center for Health Statistics.
According to the study, the 95% confidence interval for the true proportion
of adults who eat fast food on a given day is 0.350 to 0.382.

(b) Explain what would happen to the width of the interval if the sample

size were increased to 5000.

SOLUTION:

The confidence interval would be narrower because increasing the
sample size typically decreases the margin of error.

Example: Do you want fries with that?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

When we calculate a confidence interval,
we include the margin of error because we
expect the value of the point estimate to
vary somewhat from the parameter.

The margin of error accounts for only the
variability due to random sampling.

What Affects the Margin of Error?

Simulated sampling distribution of the sample
mean year ̅𝑥 in 50 samples of size n = 5 from
a population of pennies.

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

How might practical difficulties, such as nonresponse in a sample survey, affect the estimates for a parameter?
These problems can result in estimates that are much farther from the
parameter than the margin of error would suggest.

Remember this unpleasant fact when reading the
results of an opinion poll or other sample survey.

The margin of error does not account for any
sources of bias in the data collection process.

What Affects the Margin of Error?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

As part of a project about response bias, Ellery surveyed a random sample
of 25 students from her school. One of the questions in the survey required
students to state their GPA aloud. Based on the responses, Ellery said she
was 90% confident that the interval from 3.14 to 3.52 captures the mean
GPA for all students at her school.

(a) Why did Ellery include a margin of error with her estimate?

(b) Describe one potential source of bias in Ellery’s study that is not

accounted for by the margin of error.

Example: What’s your GPA?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

As part of a project about response bias, Ellery surveyed a random sample of 25
students from her school. One of the questions in the survey required students to
state their GPA aloud. Based on the responses, Ellery said she was 90% confident
that the interval from 3.14 to 3.52 captures the mean GPA for all students at her
school.

(a) Why did Ellery include a margin of error with her estimate?

SOLUTION:

To account for sampling variability—the fact that different
samples of the same size from the same population will give
different estimates.

Example: What’s your GPA?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

As part of a project about response bias, Ellery surveyed a random sample of 25
students from her school. One of the questions in the survey required students to
state their GPA aloud. Based on the responses, Ellery said she was 90% confident
that the interval from 3.14 to 3.52 captures the mean GPA for all students at her
school.

(b) Describe one potential source of bias in Ellery’s study that is not accounted for

by the margin of error.

SOLUTION:

The margin of error doesn’t account for the fact that many students
might lie about their GPAs when having to respond without
anonymity.

The mean GPA for all students might be even less than 3.14!

Example: What’s your GPA?

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

At the end of 2019, Morning Consult surveyed a
random sample of U.S. adults about New Year’s
Resolutions. Among the 655 people in the sample
who made a resolution, 402 said they had kept
their resolution. The 95% confidence interval for
the proportion of all resolution-making U.S. adults
who kept their resolution is 0.576 to 0.651.

1. Interpret the confidence interval.

2. Name two actions Morning Consult could take to reduce the margin of

error. What drawbacks do these actions have?

3. Describe one potential source of bias in this survey. Does the stated

margin of error account for this possible bias? Explain your answer.

LESSON APP 8.2

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

What did you learn?

Interpret a confidence interval.

Describe how the confidence level and
sample size affect the margin of error.

Explain how practical issues like nonresponse,
undercoverage, and response bias can affect the
interpretation of a confidence interval.

Learning Targets

Starnes/Tabor/Wilcox, Statistics and Probability with Applications, 4e

© 2021 BFW Publishers

Section 8.2: 3, 8, 13, 14, 19

Homework