Plus-Four Interval vs Confidence Interval

This is a college-level adjustment and is not part of the AP Statistics course effective Fall 2026, so use the ordinary one-sample z-interval on AP work. You compute $\tilde{p} = \frac{x+2}{n+4}$ (p-tilde, the adjusted proportion) and then form $\tilde{p} \pm z^*\sqrt{\frac{\tilde{p}(1-\tilde{p})}{n+4}}$. For example, 3 successes in 10 trials gives $\tilde{p} = 5/14 = 0.3571$, a standard error of $\sqrt{0.3571(0.6429)/14} = 0.1281$, and a 95% interval of $0.3571 \pm 1.96(0.1281)$, or 0.106 to 0.608. Padding the counts pulls the estimate toward 0.5 and keeps the interval from collapsing to a single point when x is 0 or n.

A confidence interval gives a range for an unknown population parameter instead of a single guess, acknowledging that samples vary. Its width reflects the uncertainty in the estimate: more data and less spread give a narrower interval. For example, a 95% confidence interval of 52% plus or minus 3% for support runs from 49% to 55%. The general form is $\text{estimate} \pm (\text{critical value}) \times (\text{standard error})$, where the standard error measures the typical size of sampling error.