By ToolzHive · Updated
A frequentist 95% confidence procedure has 95% long-run coverage under its assumptions. It is not a 95% probability statement about a fixed parameter after observing the data.
Reference table
| Situation | Common approach | Check |
|---|---|---|
| Mean, unknown population SD | t: x̄ ± t* s/√n | Independence and distribution assumptions |
| Mean, known population SD | z: x̄ ± z* σ/√n | Population SD must actually be known |
| Binomial proportion | Wilson or appropriate exact method | Sample size and proximity to boundaries |
| Independent mean difference | Welch t when appropriate | Do not assume equal variances automatically |
| Paired difference | Within-pair differences | Retain pairing |
| RR or OR | Often log-scale interval, then exponentiate | Sparse and zero cells need care |
Worked example
For n = 25, x̄ = 100 and s = 10, SE = 2. A two-sided 95% t critical value of about 2.064 with 24 degrees of freedom gives 100 ± 4.128, approximately [95.87, 104.13].
Interval width
More observations generally narrow intervals; more variation or a higher confidence level widens them. An interval does not correct selection bias or confounding.
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- Confidence Interval Calculator — Confidence intervals for means, proportions, differences, odds ratios and relative risks.