Plus-four interval

The plus-four interval is the ordinary one-proportion z-interval computed as though the sample had held two more successes and two more failures than it did. Add 2 to the success count $x$ and 4 to the sample size, giving $\tilde{p} = \frac{x+2}{n+4}$ (p-tilde), then form $\tilde{p} \pm z^*\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n+4}}$. Both the center and the standard error use the padded counts. It is a college-level adjustment, not part of the AP Statistics course effective Fall 2026, so use the plain z-interval on AP work.

What it fixes is undercoverage. The plain interval puts $\hat{p}$ (p-hat) where the unknown $p$ belongs inside the standard error, and that swap costs it some of its stated confidence level. Sum the binomial probabilities of every count whose interval captures $p$ and you get the exact long-run coverage: at $n = 50$ and $p = 0.60$ a nominal 95% interval covers 94.1% of the time, and at $n = 20$ and $p = 0.90$ it covers 87.6%. The plus-four version covers 95.7% in that second case. The site's coverage simulator computes those same numbers.

With 3 successes in 10 trials, $\tilde{p} = 5/14 = 0.3571$, the standard error is $\sqrt{0.3571(0.6429)/14} = 0.1281$, and the 95% interval is $0.3571 \pm 1.96(0.1281)$, or 0.106 to 0.608. The plain interval on the same data runs 0.016 to 0.584.

"It is just a wider interval, padded to be safe" is the wrong picture. On those counts the plus-four interval is narrower, 0.502 against 0.568. The adjustment pulls the center toward 0.5 and divides by $n+4$, so it widens the interval at extreme counts, where 1 success in 10 gives 0.430 against 0.372, and narrows it in the middle. Coverage improves because the center moved, not because the interval was inflated.

The case it rescues outright is $x = 0$ or $x = n$. There $\hat{p}(1-\hat{p})$ is zero, so the plain interval has zero width and cannot contain anything at all. Once the sample carries at least 10 successes and 10 failures the two intervals nearly agree, which is why the AP course checks that condition instead.