Confidence Interval Examples in Math

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

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

Concept Recap

A range of values, computed from sample data, that is likely to contain the true population parameter with a specified level of confidence.

You can't know the exact average height of all Americans, but after measuring 200 people you can say: 'I'm $95\%$ confident the true average is between 167 cm and 173 cm.' It's like casting a net—wider nets catch the true value more often, but narrower nets are more useful. A $95\%$ confidence level means that if you repeated this process 100 times, about 95 of those nets would contain the true value.

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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: A confidence interval is a range from sample data that likely contains the true population value, at a stated confidence level.

Common stuck point: The procedure for confidence interval is the easy part; the trap is saying '95% chance the true value is in THIS interval'. Asking "Am I building a range from a sample that likely contains the true population value at a stated confidence?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I building a range from a sample that likely contains the true population value at a stated confidence?

Worked Examples

Example 1

medium

A sample of

n=64

has

\bar{x}=85

and

s=16

. Construct a 95% confidence interval for the population mean.

Answer

95% CI:

(81.08, 88.92)

. We are 95% confident the population mean is in this interval.

First step

Standard error:

SE = \frac{s}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2

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Example 2

hard

Compare 90% and 99% confidence intervals for

\bar{x}=100

s=15

n=36

. Calculate both and explain the trade-off between confidence and precision.

Example 3

hard

A factory needs a

95\%

CI for mean bolt length with

E \le 0.5\text{mm}

. Known

\sigma = 4\text{mm}

z^*=1.96

. Smallest

n

Example 4

challenge

95\%

CI for a mean is

(80, 90)

with

n=64

. Estimate

\sigma

assuming

z^* = 1.96

Practice Problems

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

Example 1

easy

A 95% CI for a population mean is

(42, 58)

. Find the sample mean

\bar{x}

and the margin of error

E

Example 2

hard

To achieve a margin of error of

E=3

with 95% confidence, given

\sigma=20

, find the required sample size

n

Example 3

easy

A sample gives point estimate

\bar{x} = 50

with margin of error

4

. Write the

95\%

confidence interval.

Example 4

easy

95\%

CI for a mean is

(46, 54)

. What is the point estimate (the center)?

Example 5

easy

95\%

CI is

(46, 54)

. What is the margin of error?

Example 6

easy

True or false: increasing the confidence level from

90\%

99\%

(same data) makes the interval wider.

Example 7

easy

True or false: increasing the sample size

n

(same confidence) narrows the interval.

Example 8

easy

Which critical value

z^*

corresponds to a

95\%

confidence interval?

Example 9

easy

Should you use a z-interval or a t-interval when

\sigma

is unknown and

n

is small?

Example 10

easy

90\%

CI for a proportion is

(0.40, 0.50)

. What is the point estimate

\hat{p}

Example 11

medium

A sample has

\bar{x} = 100

, known

\sigma = 15

n = 36

. Construct the

95\%

confidence interval (

z^* = 1.96

Example 12

medium

A sample has

\bar{x} = 200

\sigma = 40

n = 64

. Find the

90\%

CI (

z^* = 1.645

Example 13

medium

95\%

CI for the mean is

(95.1, 104.9)

. A colleague claims 'there is a

95\%

probability the true mean lies in this interval.' What is the correct interpretation?

Example 14

medium

95\%

CI for the difference in means is

(-2, 6)

. Does this suggest a significant difference at

\alpha = 0.05

Example 15

medium

To halve the width of a

95\%

CI for a mean (same

\sigma

, same confidence), by what factor must

n

increase?

Example 16

medium

A poll of

n = 400

finds

\hat{p} = 0.50

. Find the

95\%

CI (

z^* = 1.96

) using SE

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

Example 17

medium

99\%

CI (

z^* = 2.576

) uses

\bar{x} = 60

\sigma = 10

n = 25

. Find the interval.

Example 18

challenge

Two

95\%

CIs are computed from the same data: one for the mean of single observations and one for the sample mean. Why is the CI for the sample mean narrower, and by what factor for

n=25

Example 19

challenge

95\%

CI for

\mu

(48, 52)

. A two-sided test of

H_0: \mu = 50

\alpha = 0.05

would reach what conclusion, and why?

Example 20

challenge

A researcher wants a

95\%

CI for a mean with margin of error at most

1

, where

\sigma = 8

. What is the smallest sample size

n

(

z^* = 1.96

Example 21

medium

A sample has

\bar{x} = 80

, known

\sigma = 12

n = 144

. Build the

95\%

CI (

z^* = 1.96

Example 22

medium

95\%

CI for a proportion is

(0.30, 0.42)

. State the point estimate and the margin of error.

Example 23

easy

95\%

CI for

\mu

(20, 30)

. State the point estimate and the margin of error.

Example 24

easy

True or false: a CI is a statement about the population parameter, not about future samples.

Example 25

easy

95\%

CI for

\mu

(8, 12)

. What is the width of the interval?

Example 26

easy

True or false: holding everything else constant, a smaller sample size yields a wider CI.

Example 27

easy

A poll reports a margin of error of

\pm 3\%

95\%

confidence. The estimate is

\hat{p}=0.47

. Write the CI.

Example 28

medium

Compute a

95\%

CI for

\mu

given

\bar{x}=50

\sigma=10

n=100

z^*=1.96

Example 29

medium

Compute a

90\%

CI for

\mu

\bar{x}=72

\sigma=18

n=81

z^*=1.645

Example 30

medium

For a proportion with

\hat{p}=0.6

n=100

, build a

95\%

CI (

z^*=1.96

Example 31

medium

95\%

CI is

(10, 14)

. What is the

E

, and what was

\bar{x}

Example 32

medium

True or false: 'The probability that the true mean is in this

95\%

CI is

0.95

.' Justify briefly.

Example 33

medium

Given

\bar{x}=120

\sigma=20

n=400

. Build a

99\%

CI (

z^*=2.576

Example 34

medium

When should you use a

t

-interval instead of a

z

-interval?

Example 35

medium

95\%

CI for the difference of means is

(1.2, 4.8)

. Is the difference significantly different from zero?

Example 36

hard

95\%

CI for a proportion:

\hat{p}=0.32

n=400

z^*=1.96

. Find the CI.

Example 37

hard

Find the smallest

n

for a

95\%

CI for a proportion with

E \le 0.03

, assuming worst case

\hat{p}=0.5

z^*=1.96

Example 38

hard

100

different

95\%

CIs are constructed from independent samples, about how many are expected to miss the true mean?

Example 39

hard

Two

95\%

CIs for treatment means overlap. Does overlap automatically imply no significant difference?

Example 40

hard

98\%

CI for

\mu

with

\bar{x}=50

\sigma=20

n=100

z^*=2.326

. Find the CI.

Example 41

challenge

Researcher reports CI

(2.3, 5.7)

95\%

for the mean difference. Estimate the

t

statistic value, assuming a

t

-CI with

E =

half-width.

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Related Concepts

Sampling Distribution Central Limit Theorem Z-Score Margin of Error Hypothesis Testing

Background Knowledge

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

sampling distributioncentral limit theoremz score