Constructing a Confidence Interval for a Population Proportion

A one-proportion z-interval is $\hat{p} \pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, with $z^{*}$ equal to 1.645, 1.96, or 2.576 for 90%, 95%, and 99% confidence. Because $p$ is unknown, the standard error uses $\hat{p}$, and the large-counts condition is checked as $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$, alongside random selection and the 10% condition, each written with the study's own numbers. For 248 supporters out of 400, $\hat{p} = 0.62$, $SE \approx 0.0243$, and the 95% interval is about $(0.572, 0.668)$. Raising the confidence level widens the interval; quadrupling $n$ halves its width; the population's size does not enter.

The reporting is where this topic is lost. An interval estimates a parameter, so a sentence about what fraction of individuals fall inside it, or about a 95% probability that $p$ lies in it, or about where future sample proportions will land, has changed the subject. The confidence level belongs to the method across repeated samples, not to the single interval in hand, which has already caught $p$ or missed it. The mechanical errors sit alongside: $p_0$ or a bare $p$ used in an interval's standard error where $\hat{p}$ belongs, and conditions named rather than checked against the observed counts of 248 and 152.