Confidence Intervals for a Proportion

A confidence interval for a proportion is simply a range of plausible values for a percentage you can't measure directly, like the share of all voters who support something, built from a poll. The formula has the same shape as the one for means: $\hat p$ (the percentage in your sample) plus or minus $z^*$ (how many standard errors, 1.96 for 95%) times $\sqrt{\frac{\hat p(1 - \hat p)}{n}}$ (the standard error of a proportion, from the CLT unit). The sample-size formula is that margin of error solved for $n$. Same idea you've already seen, new costume.

Why $\hat p$ in the standard error? The true standard error is $\sqrt{\frac{p(1 - p)}{n}}$, but we don't know $p$; that's the whole point of the interval. So we plug in our best estimate, $\hat p$. (A hypothesis test is different: it assumes a value $p_0$ and uses that.)

Conditions: a random sample with at least 10 successes and 10 failures, so the normal approximation holds.

Planning a survey. Before you've collected data, you don't have $\hat p$ yet. But $\hat p(1 - \hat p)$ is largest at $\hat p = 0.5$, so using 0.5 gives the worst case and guarantees the margin of error. For $\pm3\%$ at 95% confidence that's about 1,068 people, which is why national polls survey around a thousand people.

Diminishing returns. Quadrupling the sample only halves the margin of error, because $n$ sits under a square root.

A common mistake. It's tempting to think the margin of error covers every way a poll can go wrong. It only covers random sampling luck. If the poll only reached people who answer unknown numbers, or the question was worded to push an answer, the true value can be far outside the interval, and no sample size fixes that.

Election polls, approval ratings, defect rates and vaccine efficacy estimates are all proportion intervals.