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=64n=64 samples, xˉ=120\bar{x}=120, σ=16\sigma=16 known. Find the 99% CI for μ\mu.

Answer

[114.85, 125.15][114.85,\ 125.15]

First step

1
SE=16/8=2SE=16/8=2; z∗=2.576z^*=2.576.

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Example 2

medium
How large should nn be so that a 95% z-CI for μ\mu (with σ=12\sigma=12) has margin ≤2\le 2?

Example 3

hard
xˉ=68\bar{x}=68, s=12s=12, n=9n=9. Construct a 95% t-CI for μ\mu. Use t∗=2.306t^*=2.306 (df =8=8).

Example 4

challenge
We want a 99% CI for μ\mu with margin ≤1\le 1 unit. From a pilot, σ≈8\sigma\approx 8. Find the required sample size.

Example 5

medium
A sample of n=100n=100 has mean xˉ=80\bar{x}=80 with σ=10\sigma=10. Build a 95% z-interval using z∗=1.96z^*=1.96.

Example 6

medium
A poll of n=900n=900 voters has p^=0.60\hat{p}=0.60. Using SE =p^(1−p^)/n= \sqrt{\hat{p}(1-\hat{p})/n} and z∗=1.96z^*=1.96, find the 95% CI to the nearest 0.01.

Example 7

hard
We want a 95% CI for a proportion with ME ≤0.03\le 0.03, using worst-case p^=0.5\hat{p}=0.5 and z∗=1.96z^*=1.96. Find the smallest nn.

Example 8

hard
Two independent samples give xˉA=70\bar{x}_A = 70, xˉB=66\bar{x}_B=66, with SE(xˉA−xˉB)=1.5\text{SE}(\bar{x}_A - \bar{x}_B)=1.5. Build a 95% CI for μA−μB\mu_A - \mu_B using z∗=1.96z^*=1.96.

Example 9

medium
A study reports a 95% CI of [2.0, 4.0][2.0,\ 4.0] kg for average weight loss. Translate this into estimate-and-ME form.

Example 10

challenge
A 95% CI for μ\mu uses xˉ=50\bar{x}=50, s=12s=12, n=36n=36, and t∗≈2.030t^* \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 xˉ=72\bar{x} = 72 with population standard deviation σ=10\sigma = 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=64n=64 has mean 100, with σ=16\sigma=16. Build a 95% CI using z∗=2z^*=2.

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∗=2z^*=2) to 99.7% (z∗=3z^*=3) confidence, by what factor does the margin of error grow?

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 σ\sigma), how must nn change?

Example 14

medium
A 90% CI uses z∗=1.645z^*=1.645. With SE =4=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=36n=36, σ=18\sigma=18, xˉ=40\bar{x}=40 uses z∗=2z^*=2. Find the interval.

Example 18

challenge
A sample of n=400n=400 has p^=0.5\hat{p}=0.5. Build a 95% CI for the proportion using z∗=2z^*=2 and SE =p^(1−p^)/n=\sqrt{\hat{p}(1-\hat{p})/n}.

Example 19

challenge
A 95% CI for a mean from n=100n=100, σ=20\sigma=20 is desired with total width at most 4. Is n=100n=100 enough? If not, find the needed nn.

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=100n=100 gives xˉ=20\bar{x}=20 with known σ=5\sigma=5. Compute the 95% CI for μ\mu.

Example 22

easy
A 95% CI for μ\mu is [10,16][10, 16]. What is the point estimate?

Example 23

easy
n=400n=400 sampled voters give p^=0.52\hat{p}=0.52. Compute the 95% CI for pp.

Example 24

easy
xˉ=80\bar{x}=80, σ=10\sigma=10 known, n=25n=25. Compute the 90% CI for μ\mu.

Example 25

medium
xˉ=15.2\bar{x}=15.2, s=2.4s=2.4, n=16n=16. Compute the 95% t-CI for μ\mu. Use t∗=2.131t^*=2.131.

Example 26

medium
A 95% CI for μ\mu is [48, 52][48,\ 52]. Interpret correctly.

Example 27

medium
n=900n=900, p^=0.30\hat{p}=0.30. Find the 90% CI for pp.

Example 28

medium
n=25n=25, xˉ=72\bar{x}=72, s=10s=10. Find the 95% t-CI for μ\mu. Use t∗=2.064t^*=2.064.

Example 29

medium
Two polls each report 95% CIs [42,48][42,48] and [45,51][45,51] for pp in two populations. Does this show the populations differ?

Example 30

medium
n=200n=200, p^=0.25\hat{p}=0.25. Compute the 95% CI for pp.

Example 31

medium
A 95% CI for μ\mu is [40,46][40,46]. Construct an approximate 95% lower bound for μ\mu.

Example 32

medium
xˉ=200\bar{x}=200, s=24s=24, n=36n=36. Find the 95% t-CI for μ\mu. Use t∗=2.030t^*=2.030.

Example 33

medium
Two-sample 95% CI for μ1−μ2\mu_1-\mu_2: [−1.5, 4.5][-1.5,\ 4.5]. Can we reject H0 ⁣:μ1=μ2H_0\!:\mu_1=\mu_2 at α=0.05\alpha=0.05?

Example 34

hard
Plan: estimate a proportion within ±0.03\pm 0.03 at 95% confidence with no prior info. What is the conservative required nn?

Example 35

hard
A 95% CI for pp is [0.40, 0.50][0.40,\ 0.50]. What is the approximate sample size if the CI uses z∗=1.96z^*=1.96 and p^=0.45\hat{p}=0.45?

Example 36

hard
Two-sample data: xˉ1−xˉ2=4.0\bar{x}_1-\bar{x}_2=4.0, pooled SE=1.5SE=1.5. Approximate 95% CI for μ1−μ2\mu_1-\mu_2 using z∗=1.96z^*=1.96.

Example 37

hard
A 95% CI for μd\mu_d (paired differences) is [−0.2, 1.4][-0.2,\ 1.4]. Conclude about a paired test of H0 ⁣:μd=0H_0\!:\mu_d=0.

Example 38

easy
A 90% confidence interval for a mean is [42, 50][42,\ 50]. What is the point estimate?

Example 39

easy
A 95% CI for a proportion is [0.42, 0.58][0.42,\ 0.58]. What is the margin of error?

Example 40

easy
A 95% CI for the mean test score is [72, 78][72,\ 78]. State the interval width.

Example 41

easy
A sample proportion is p^=0.40\hat{p}=0.40 with margin of error 0.050.05. State the 95% confidence interval.

Example 42

medium
A 99% CI for a difference in means is [2, 8][2,\ 8]. Does the data give evidence that the means differ?

Example 43

medium
Going from n=100n=100 to n=400n=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][28,\ 34] minutes. Which interpretation is correct: (A) 95% of commuters are in [28,34][28,34]; (B) the procedure produces an interval containing the true mean 95% of the time?

Example 45

medium
A 95% CI for μ\mu is [10, 18][10,\ 18]. Without new data, what is the 95% CI for μ/2\mu/2?

Example 46

medium
A 95% CI for a treatment effect is [−0.5, 2.5][-0.5,\ 2.5]. Does the data show statistically significant evidence of an effect at α=0.05\alpha=0.05?

Example 47

medium
Why does using a t-distribution instead of a z give a wider CI for the mean when σ\sigma is unknown?

Example 48

hard
A 95% CI for μ\mu has half-width 4. To cut the half-width to 1 at the same confidence (same σ\sigma), what new sample size is required if the current n=50n=50?

Example 49

hard
A 95% CI for the ratio μA/μB\mu_A/\mu_B is [0.9, 1.4][0.9,\ 1.4]. Does the data support that the two means differ?

Example 50

hard
Researcher A builds a 95% CI: [10, 20][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][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][5,\ 15] and an independent 95% CI for another mean is [10, 20][10,\ 20]. Can you conclude the means differ at α=0.05\alpha=0.05 just because the intervals overlap?

Example 53

hard
A sample of 64 has xˉ=50\bar{x} = 50 and σ=8\sigma = 8. Find the 99% confidence interval (z∗=2.576z^* = 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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Background Knowledge

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

standard errorsampling distribution