Confidence Interval Examples in Statistics

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 Statistics.

Concept Recap

A confidence interval is a range of values, calculated from sample data, constructed so that the procedure captures the true population parameter a specified percentage of the time (e.g., 95%). It quantifies the uncertainty inherent in using a sample to estimate a population value.

Instead of saying 'the average is 50,' you say 'I'm 95% confident the average is between 47 and 53.' The interval acknowledges uncertainty from sampling.

Read the full concept explanation →

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: Confidence Interval uses a sample result and a variation model to make a careful population statement.

Common stuck point: Students often know a procedure related to confidence interval 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

From

n=64

samples,

\bar{x}=120

\sigma=16

known. Find the 99% CI for

\mu

Answer

[114.85, 125.15]

First step

SE=16/8=2

;

z^*=2.576

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Example 2

medium

How large should

n

be so that a 95% z-CI for

\mu

(with

\sigma=12

) has margin

\le 2

Example 3

hard

\bar{x}=68

s=12

n=9

. Construct a 95% t-CI for

\mu

. Use

t^*=2.306

(df

=8

Example 4

challenge

We want a 99% CI for

\mu

with margin

\le 1

unit. From a pilot,

\sigma\approx 8

. Find the required sample size.

Example 5

medium

A sample of

n=100

has mean

\bar{x}=80

with

\sigma=10

. Build a 95% z-interval using

z^*=1.96

Sampling distribution of x̄ (n = 100, σ = 10, SE = 1); 95% CI = [78.04, 81.96]

Example 6

medium

A poll of

n=900

voters has

\hat{p}=0.60

. Using SE

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

and

z^*=1.96

, find the 95% CI to the nearest 0.01.

Example 7

hard

We want a 95% CI for a proportion with ME

\le 0.03

, using worst-case

\hat{p}=0.5

and

z^*=1.96

. Find the smallest

n

Example 8

hard

Two independent samples give

\bar{x}_A = 70

\bar{x}_B=66

, with

\text{SE}(\bar{x}_A - \bar{x}_B)=1.5

. Build a 95% CI for

\mu_A - \mu_B

using

z^*=1.96

Example 9

medium

A study reports a 95% CI of

[2.0,\ 4.0]

kg for average weight loss. Translate this into estimate-and-ME form.

Example 10

challenge

A 95% CI for

\mu

uses

\bar{x}=50

s=12

n=36

, and

t^* \approx 2.030

(with 35 df). Build the interval and round to 0.01.

Example 11

hard

A sample of 100 students has a mean test score of

\overset{ˉ}{x} = 72

with population standard deviation

σ = 10

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

Example 12

hard

A 95% confidence interval for the mean weight of apples is (150g, 170g). Interpret this interval.

Practice Problems

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

Example 1

easy

Write the general form of a confidence interval.

Example 2

easy

A sample mean is 50 with margin of error 3. State the 95% confidence interval.

Example 3

easy

Does a 95% confidence interval mean 95% of the data lies inside it?

Example 4

easy

Which is wider: a 90% confidence interval or a 99% confidence interval from the same data?

Example 5

easy

As sample size grows, what happens to the width of a confidence interval (other things equal)?

Example 6

easy

A 95% CI for a mean is [12, 18]. What is the point estimate (center)?

Example 7

easy

A 95% CI for a mean is [12, 18]. What is the margin of error?

Example 8

easy

Fill in: a confidence interval quantifies the ____ in using a sample to estimate a population value.

Example 9

medium

A sample of

n=64

has mean 100, with

\sigma=16

. Build a 95% CI using

z^*=2

Sampling distribution of x̄: shaded region is the 95% CI [96, 104]

Example 10

medium

A 95% CI for a difference in means is [-1, 5]. Does it provide evidence of a real difference at the 5% level?

Example 11

medium

Going from 95% (

z^*=2

) to 99.7% (

z^*=3

) confidence, by what factor does the margin of error grow?

Increasing z* from 2 (95%) to 3 (99.7%) stretches the margin of error by a factor of 1.5

Example 12

medium

A poll reports 54% approval with a 95% CI of [50%, 58%]. Interpret what '95% confident' means here.

Example 13

medium

To halve the width of a confidence interval (same confidence, same

\sigma

), how must

n

change?

Example 14

medium

A 90% CI uses

z^*=1.645

. With SE

=4

and estimate 30, give the 90% CI.

Example 15

medium

A 95% CI for a mean is entirely above 0, say [2, 8]. What does this imply about the parameter?

Example 16

medium

Two 95% CIs for a mean: study A gives [48, 52], study B gives [40, 60]. Which study had the smaller standard error?

Example 17

medium

A 95% CI for a mean from

n=36

\sigma=18

\bar{x}=40

uses

z^*=2

. Find the interval.

Sampling distribution of x̄: SE = 18/√36 = 3; 95% CI

Example 18

challenge

A sample of

n=400

has

\hat{p}=0.5

. Build a 95% CI for the proportion using

z^*=2

and SE

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

Example 19

challenge

A 95% CI for a mean from

n=100

\sigma=20

is desired with total width at most 4. Is

n=100

enough? If not, find the needed

n

Example 20

challenge

Explain why a single computed 95% CI like [47, 53] should NOT be described as 'the parameter has a 95% probability of lying in [47, 53]'.

Example 21

easy

A sample of

n=100

gives

\bar{x}=20

with known

\sigma=5

. Compute the 95% CI for

\mu

Sampling distribution of x̄ (n = 100, σ = 5, SE = 0.5); 95% CI

Example 22

easy

A 95% CI for

\mu

[10, 16]

. What is the point estimate?

Example 23

easy

n=400

sampled voters give

\hat{p}=0.52

. Compute the 95% CI for

p

Example 24

easy

\bar{x}=80

\sigma=10

known,

n=25

. Compute the 90% CI for

\mu

Example 25

medium

\bar{x}=15.2

s=2.4

n=16

. Compute the 95% t-CI for

\mu

. Use

t^*=2.131

Example 26

medium

A 95% CI for

\mu

[48,\ 52]

. Interpret correctly.

Example 27

medium

n=900

\hat{p}=0.30

. Find the 90% CI for

p

Example 28

medium

n=25

\bar{x}=72

s=10

. Find the 95% t-CI for

\mu

. Use

t^*=2.064

Example 29

medium

Two polls each report 95% CIs

[42,48]

and

[45,51]

for

p

in two populations. Does this show the populations differ?

Example 30

medium

n=200

\hat{p}=0.25

. Compute the 95% CI for

p

Example 31

medium

A 95% CI for

\mu

[40,46]

. Construct an approximate 95% lower bound for

\mu

Example 32

medium

\bar{x}=200

s=24

n=36

. Find the 95% t-CI for

\mu

. Use

t^*=2.030

Example 33

medium

Two-sample 95% CI for

\mu_1-\mu_2

[-1.5,\ 4.5]

. Can we reject

H_0\!:\mu_1=\mu_2

\alpha=0.05

Example 34

hard

Plan: estimate a proportion within

\pm 0.03

at 95% confidence with no prior info. What is the conservative required

n

Example 35

hard

A 95% CI for

p

[0.40,\ 0.50]

. What is the approximate sample size if the CI uses

z^*=1.96

and

\hat{p}=0.45

Example 36

hard

Two-sample data:

\bar{x}_1-\bar{x}_2=4.0

, pooled

SE=1.5

. Approximate 95% CI for

\mu_1-\mu_2

using

z^*=1.96

Example 37

hard

A 95% CI for

\mu_d

(paired differences) is

[-0.2,\ 1.4]

. Conclude about a paired test of

H_0\!:\mu_d=0

Example 38

easy

A 90% confidence interval for a mean is

[42,\ 50]

. What is the point estimate?

Example 39

easy

A 95% CI for a proportion is

[0.42,\ 0.58]

. What is the margin of error?

Example 40

easy

A 95% CI for the mean test score is

[72,\ 78]

. State the interval width.

Example 41

easy

A sample proportion is

\hat{p}=0.40

with margin of error

0.05

. State the 95% confidence interval.

Example 42

medium

A 99% CI for a difference in means is

[2,\ 8]

. Does the data give evidence that the means differ?

Example 43

medium

Going from

n=100

n=400

(same data spread, same confidence), by what factor does the margin of error shrink?

Example 44

medium

A 95% CI for the average commute time is

[28,\ 34]

minutes. Which interpretation is correct: (A) 95% of commuters are in

[28,34]

; (B) the procedure produces an interval containing the true mean 95% of the time?

Example 45

medium

A 95% CI for

\mu

[10,\ 18]

. Without new data, what is the 95% CI for

\mu/2

Example 46

medium

A 95% CI for a treatment effect is

[-0.5,\ 2.5]

. Does the data show statistically significant evidence of an effect at

\alpha=0.05

Example 47

medium

Why does using a t-distribution instead of a z give a wider CI for the mean when

\sigma

is unknown?

Example 48

hard

A 95% CI for

\mu

has half-width 4. To cut the half-width to 1 at the same confidence (same

\sigma

), what new sample size is required if the current

n=50

Example 49

hard

A 95% CI for the ratio

\mu_A/\mu_B

[0.9,\ 1.4]

. Does the data support that the two means differ?

Example 50

hard

Researcher A builds a 95% CI:

[10,\ 20]

. Researcher B uses the same data but builds an 80% CI. Will B's interval contain the value 15?

Example 51

hard

A claim says 'the true mean is in

[42,\ 50]

with probability 95%.' Under the frequentist interpretation, what is wrong with the wording?

Example 52

hard

A 95% CI for one mean is

[5,\ 15]

and an independent 95% CI for another mean is

[10,\ 20]

. Can you conclude the means differ at

\alpha=0.05

just because the intervals overlap?

Example 53

hard

A sample of 64 has

\overset{ˉ}{x} = 50

and

σ = 8

. Find the 99% confidence interval (

z^{*} = 2.576

Example 54

hard

A 90% confidence interval for a population mean is given as 68 to 74. What are the sample mean and the margin of error?

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Related Concepts

Standard Error Sampling Distribution Margin of Error Hypothesis Testing

Background Knowledge

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

standard errorsampling distribution