## Plus Four Confidence Interval for Proportion

Let $X$ be the observed number of individuals possessing certain attributes (number of successes) in a random sample of size $n$ from a large population with population proportion $p$. The estimator of the population proportion of success based on plus four rule is $\hat{p}=\frac{X+2}{n+4}$.

## Formula

$100(1-\alpha)$% plus four confidence interval for population proportion is

### $\hat{p} - E \leq p \leq \hat{p} + E$

where,

• $\hat{p}=\dfrac{X+2}{n+4}$ is the estimate of population proportion based on plus four rule,
• $E = Z_{\alpha/2} \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n+4}}$ is the margin of error,
• $1-\alpha$ is the confidence coefficient,
• $Z_{\alpha/2}$ is the critical value of $Z$.