Confidence Intervals for a Proportion
A confidence interval for a proportion turns a single sample percentage into a plausible range for the true population proportion, plus a statement of how confident you are that the range captured it.
Definition
A one-sample proportion confidence interval estimates an unknown population proportion p from a sample proportion p̂. Instead of reporting one number, you report a range that, at a stated confidence level, is likely to contain the true proportion. The confidence level (say 95%) describes the long-run success rate of the method: if you repeated the sampling and rebuilt the interval many times, about 95% of those intervals would capture the true p.
The formula & procedure
Interval = point estimate ± margin of error, i.e. p̂ ± z* · √( p̂(1 − p̂) / n ). Here z* is the critical value (1.96 for 95% confidence, 1.645 for 90%). Check three conditions first: the data come from a random sample; the sample is at most 10% of the population (so observations are effectively independent); and the counts are large enough, n·p̂ ≥ 10 and n(1 − p̂) ≥ 10.
Worked example
A campus coffee bar surveys 200 randomly chosen visitors and finds 46 order oat milk, so p̂ = 46/200 = 0.23. Conditions check out: random (stated), 10% (200 is well under 10% of all visitors), and large counts (200·0.23 = 46 ≥ 10 and 200·0.77 = 154 ≥ 10). For 95% confidence, z* = 1.96. The standard error is √(0.23·0.77/200) = √0.00089 ≈ 0.0298, so the margin of error is 1.96·0.0298 ≈ 0.058. The interval is 0.23 ± 0.058 = (0.172, 0.288). We are 95% confident that between about 17% and 29% of all visitors order oat milk.
The classic mistake
Reading the confidence level as a probability about one specific interval — saying "there's a 95% chance the true proportion is between 0.172 and 0.288." Once the interval is computed it either contains p or it doesn't; the 95% describes the method across many samples, not this one result. A second trap is confusing the confidence level with the percentage of the population.
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