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 quantifies the precision of an estimate—it says 'here's my best guess, plus or minus the uncertainty.'

Common stuck point: A 95\% CI does NOT mean there's a 95\% probability the true parameter is in this specific interval. It means 95\% of similarly constructed intervals would contain the true parameter.

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.

Solution

  1. 1
    Standard error: SE = \frac{s}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2
  2. 2
    Critical value for 95% CI: z^* = 1.96
  3. 3
    Margin of error: E = z^* \times SE = 1.96 \times 2 = 3.92
  4. 4
    95% CI: \bar{x} \pm E = 85 \pm 3.92 = (81.08, 88.92)

Answer

95% CI: (81.08, 88.92). We are 95% confident the population mean is in this interval.
A 95% confidence interval means: if we repeated this sampling procedure many times, 95% of the resulting intervals would contain the true population mean. The interval does NOT say there's a 95% chance the mean is in this specific interval — the mean is fixed, the interval is random.

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.

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.
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Background Knowledge

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

sampling distributioncentral limit theoremz score