## 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 $Z$.