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 $\hat{p}=\frac{X+2}{n+4}$.


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

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


  • $\hat{p}=\dfrac{X+2}{n+4}$ 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 $Z$.

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