Margin of Error Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Margin of Error.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The margin of error is the maximum expected difference between a sample statistic and the true population parameter, typically expressed as a plus-or-minus value. It equals half the width of a confidence interval and decreases as sample size increases.

When a poll says '52% $\pm$ 3%,' that 3% is the margin of error. It means the true value is probably within 3 percentage points of 52%, so between 49% and 55%.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Margin of Error uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to margin of error but skip the recognition step: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I using sample-to-sample variation to make a population claim with uncertainty stated clearly?

Worked Examples

Example 1

medium

A sample of

n=400

has

\hat{p} = 0.5

. Compute the MOE for a

95\%

CI using

z^* = 1.96

Answer

0.049

First step

= \sqrt{\hat{p}(1-\hat{p})/n} = \sqrt{0.25/400} = 0.025

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Example 2

medium

A poll's

95\%

CI is

(0.42, 0.50)

. What is the margin of error?

Example 3

medium

Two polls report

48\% \pm 3\%

and

51\% \pm 3\%

. Do their

95\%

CIs overlap?

Example 4

hard

90\%

MOE uses

z^* = 1.645

. With SE

= 0.02

, compute the MOE.

Example 5

hard

Two polls each with

n = 1000

report

\hat{p}_1 = 0.50

and

\hat{p}_2 = 0.46

. Approximate the SE of the difference (treat

\hat{p}(1-\hat{p})

0.25

Example 6

challenge

A pollster targets MOE

= 0.02

99\%

confidence (

z^* = 2.576

) with

\hat{p} \approx 0.5

. Roughly what sample size is needed?

Example 7

medium

Find the margin of error for a 95% CI when

\sigma=15

n=225

, and

z^*=1.96

Example 8

medium

A poll has

n=2500

\hat{p}=0.5

z^*=1.96

. Find ME using SE

=\sqrt{\hat{p}(1-\hat{p})/n}

Example 9

medium

We want a 95% CI for a mean with ME no larger than 2. Suppose

\sigma=10

and use

z^*=1.96

. Find the smallest

n

Example 10

hard

A 99% CI for the average household size uses

\sigma=1.6

n=400

, and

z^*=2.576

. Compute ME to four decimals.

Example 11

hard

A 95% margin of error for the difference of two means uses

s_1=4

s_2=3

n_1=n_2=50

, and

t^* \approx 2

. Compute the ME using

\text{SE}=\sqrt{s_1^2/n_1+s_2^2/n_2}

Example 12

medium

A news poll uses

n=1024

and

\hat{p}=0.50

with

z^*=1.96

. Find ME in percentage points to the nearest 0.1.

Example 13

challenge

Design problem: you want a 99% CI for a proportion with ME

\le 0.02

. Use the conservative

\hat{p}=0.5

and

z^*=2.576

. Find the smallest

n

Example 14

hard

A poll of 400 voters found 55% support a policy. Calculate the margin of error for a 95% confidence interval.

Example 15

hard

How does quadrupling the sample size affect the margin of error?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write the formula for the margin of error.

Example 2

easy

With

z^*=2

and SE

=5

, compute the margin of error.

Example 3

easy

A poll says '52%

\pm

3%.' What is the margin of error?

Example 4

easy

The margin of error equals half the ____ of a confidence interval.

Example 5

easy

As sample size increases, the margin of error ____.

Example 6

easy

A poll reports 52% with margin of error 3%. State the confidence interval.

Example 7

easy

Does a larger margin of error indicate a worse poll?

Example 8

easy

Higher confidence levels make the margin of error ____.

Example 9

medium

A sample of

n=100

\sigma=20

, uses

z^*=2

. Compute the 95% margin of error.

Sampling distribution (SE = 2); shaded region

Example 10

medium

A poll of

n=400

with

\hat{p}=0.5

uses SE

=\sqrt{0.25/400}=0.025

and

z^*=2

. Find the margin of error in percentage points.

Proportion sampling distribution (p̂ = 50%, SE = 2.5 pp); CI

Example 11

medium

To halve the margin of error (same confidence, same

\sigma

), how must the sample size change?

Example 12

medium

An election poll gives candidate A 51% with margin of error 4%. Can we confidently say A leads?

Example 13

medium

Doubling the sample size changes the margin by what factor?

Example 14

medium

A 99% margin of error uses

z^*=2.576

. With SE

=2

, compute the margin.

Example 15

medium

A study wants a margin of error of 2 with

\sigma=10

and

z^*=2

. Find the required sample size.

Example 16

medium

Why does a margin of error tell us nothing about non-sampling errors like biased question wording?

Example 17

medium

With

z^*=2

\sigma=15

, and

n=25

, compute the 95% margin of error.

Example 18

challenge

A poll needs a margin of error of at most 3 percentage points at 95% (

z^*=2

). Using the conservative

\hat{p}=0.5

(so SE

=0.5/\sqrt{n}

), find the minimum sample size.

Example 19

challenge

A poll reports 48%

\pm

3% for A and 52%

\pm

3% for B. Can we conclude B leads at 95% confidence? Discuss the overlap.

A: [45%, 51%] and B: [49%, 55%] — overlapping intervals; no winner confirmed

Example 20

challenge

A margin of 4 at 95% comes from

n=100

. A team wants margin 1 at the same confidence and

\sigma

. How many total respondents are needed?

Example 21

easy

A poll has standard error

\text{SE} = 0.02

and uses

z^* = 1.96

. Find the margin of error.

Example 22

easy

95\%

confidence interval has

\hat{p} = 0.48

and MOE

=0.03

. State the interval.

Example 23

easy

A poll reports '47%

\pm

4%.' What is the upper bound of the confidence interval?

Example 24

easy

Which is the standard

z^*

for a

95\%

confidence interval (commonly used)?

Example 25

medium

95\%

MOE is

0.04

n=600

. Approximately what MOE do we get at

n = 2400

(other things equal)?

Example 26

medium

To cut a

95\%

MOE in half, sample size must be multiplied by what factor?

Example 27

medium

99\%

MOE is larger or smaller than a

95\%

MOE (other things equal)?

Example 28

medium

For a sample mean with

\sigma = 10

and

n = 100

, compute the

95\%

MOE using

z^* = 1.96

Example 29

medium

A poll wants MOE

\le 0.03

95\%

confidence with

\hat{p} \approx 0.5

. Roughly what sample size is needed?

Example 30

medium

A claim 'lead beyond the MOE' means the lead exceeds what quantity for two estimates?

Example 31

hard

A poll's

\hat{p} = 0.30

. The MOE is approximately largest when

\hat{p}

equals what value?

Example 32

hard

A poll's MOE is reported as

\pm 3.1\%

95\%

confidence. What is the implied SE (use

z^* = 1.96

Example 33

medium

A salary survey with

n = 256

has

s = \$8000

. Estimate the

95\%

MOE for the mean using

z^* = 1.96

Example 34

medium

95\%

CI for a mean is

(72, 78)

. What is the point estimate and the MOE?

Example 35

easy

A polling firm samples

n = 100

voters. Compare its expected MOE to one with

n = 1000

voters.

Example 36

challenge

A poll reports

\hat{p} = 0.50

with MOE

= 0.04

. A second poll reports

\hat{p} = 0.50

with MOE

= 0.02

. Approximately how many times larger is the second poll's sample size?

Example 37

easy

A 95% CI is

[10,\ 14]

. What is the margin of error?

Example 38

easy

Compute the margin of error if

z^*=1.96

and SE

=2

Example 39

easy

A poll reports 60% with ME of

\pm 4\%

. State the resulting CI.

Example 40

easy

A poll's results are stated as '47%

\pm

5%' at 95% confidence. Write the 95% CI.

Example 41

medium

A 95% CI uses

z^*=1.96

and gives ME

=4

. If we keep the same data but use 90% confidence (

z^*=1.645

), what is the new ME?

Example 42

medium

To shrink ME by a factor of 3 (same

\sigma

, same confidence), how must

n

change?

Example 43

medium

Two polls have the same

\hat{p}=0.55

. Poll A has

n=400

, Poll B has

n=1600

. Whose ME is smaller, and by what factor?

Example 44

medium

A poll's ME is reported as 3 percentage points 'at 95% confidence.' Two candidates are at 49% and 47%. Can we say the leader is genuinely ahead?

Example 45

medium

The margin of error for a poll falls from 4% to 2% across two reports. Which factor cannot explain it: (A) larger

n

, (B) lower confidence level, (C) using a smaller

z^*

, (D) larger population variance?

Example 46

hard

If the population is small and the sample is more than 5% of it, the standard ME formula overstates uncertainty. What correction is applied?

Example 47

hard

A news article says 'the poll has a 3% margin of error' but never states the confidence level. Why is this incomplete?

Example 48

hard

Margin of error captures sampling variability only. Name one other source of error that ME does NOT account for.

Example 49

hard

For a poll with

\hat{p}=0.10

\hat{p}=0.50

at the same

n

and confidence, which has the larger margin of error?

Example 50

medium

A poll cites '52% support, margin of error

\pm 4\%

.' At a 95% confidence level, does this support a claim that more than half the population supports?

Example 51

hard

A 95% CI for a mean is

50 \pm 3

. What is the margin of error, and what would it be if the sample size were quadrupled?

Example 52

hard

If the confidence level is increased from 90% to 99% while the sample size and variability stay the same, what happens to the margin of error?

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Related Concepts

Confidence Interval Standard Error Statistical Significance

Background Knowledge

These ideas may be useful before you work through the harder examples.

confidence intervalstandard error