Example 2: Confidence Interval for sample proportion

Quantities such as the fraction of dry days etc. can be estimated by using the sample proportion. The sample proportion of a binary event is given by the sample mean $\overline{X}$ of a Bernoulli distributed variable $X\sim Be(p)$. The sampling distribution of the number of cases $X=1$ is given by $n\overline{X}\sim Bin(n,p)$. For large sample size ($n\geq 30$), the binomial distribution $Bin(n,p)$ approximates the normal distribution $N(np,np(1-p))$, and hence the sampling distribution becomes $\overline{X}\sim N(p,p(1-p)/n)$. Therefore, for large enough samples, the proportion is estimated by $\hat{p}=\overline{X}$ with a standard error of $s_{\hat{p}}=\sqrt{\hat{p}(1-\hat{p})/n}$. Note the standard errors shrink when $\hat{p}$ gets close to either zero or one. For small samples, the normal approximation can not be used and the C.I.'s are asymmetric due to the skewed nature of the binomial distribution.