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What is a confidence interval?
A confidence interval (CI) is a range of values, computed from sample data, that — at a given confidence level (e.g. 95%) — contains the true value of the population parameter.
The correct interpretation is frequentist: if we repeated the sampling many times and computed the CI each time, approximately 95% of those intervals would contain the true parameter. It does not mean the parameter is "inside the interval with 95% probability."
General structure of a CI
Most classic intervals follow the form:
\( \text{CI} = \hat{\theta} \pm \text{critical value} \times \text{standard error} \)
where \(\hat{\theta}\) is the point estimate of the parameter, the critical value depends on the confidence level and the distribution used, and the standard error measures the variability of the estimate.
Confidence level and width
- A higher confidence level (closer to 1) makes the interval wider.
- A larger sample size makes the interval narrower (more precise).
- More variability in the data makes the interval wider.
Relationship with hypothesis testing
A 95% CI and a two-sided test with α = 0.05 are complementary: if the hypothesized value \(\mu_0\) falls outside the CI, the test would reject H₀. Both approaches give compatible information from different angles.
How to interpret a confidence interval
A confidence interval combines a point estimate with its uncertainty. If we repeated the procedure many times, approximately the stated percentage of intervals would contain the true population value. The width depends on the sample size, the variability and the confidence level.
Mean
Use it for continuous variables such as time, weight, spend or average score.
Proportion
Use it for percentages such as conversion, satisfaction or prevalence.
Differences
Use it to compare two groups and quantify the effect size.
Worked example: mean with unknown σ
With n = 25, mean = 72 and sample standard deviation = 10, the 95% interval for the mean uses Student's t with 24 degrees of freedom. The CI for one mean tool returns the range compatible with that data.