Interval estimation for proportions

To estimate proportions, percentiles or rates we assume we sample from a binomial distribution with size of $n$ with probability $p$ of the event

Examples

$X \sim B i n (n, p)$ , find $95 % C I$ for $p$

\begin{aligned} X = \sum Y_{i}, Y_{1}, \dots, Y_{n} \sim B e r n o u l l i (p) \equiv B i n (1, p) \\ E (X) = n p, V a r (X) = n p (1 - p), n iis the sample size \\ For a large sample size (n \geq 25), we apply CLT \\ \bar{y} = \frac{x}{n} = \sum \frac{y_{i}}{n} = \hat{p}, V a r (\hat{y}) = \frac{V a r (y)}{n} = \frac{p (1 - p)}{n} \\ Z = \frac{\hat{y} - p}{\sqrt{V a r (\hat{y})}} = \frac{\hat{p} - p}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \sim N (0, 1) \\ P (p_{l} \leq p \leq p_{u}) = 1 - α \\ P (\frac{\hat{p} - p_{u}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \leq \frac{\hat{p} - p}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \leq \frac{\hat{p} - p_{l}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}}) = 1 - α \\ P (- Z_{1 - α / 2} \leq \frac{\hat{p} - p}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \leq \frac{\hat{p} - p_{l}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}}) = 1 - α \\ C I for p is (\hat{p} -) \end{aligned}

\begin{aligned} X = \sum Y_{i}, Y_{1}, \dots, Y_{n} \sim B e r n o u l l i (p) \equiv B i n (1, p) \\ E (X) = n p, V a r (X) = n p (1 - p), n is the sample size \\ For a large sample size (n \geq 25), we apply the Central Limit Theorem (CLT) \\ \bar{Y} = \frac{X}{n} = \sum \frac{Y_{i}}{n} = \hat{p}, V a r (\bar{Y}) = \frac{V a r (Y)}{n} = \frac{p (1 - p)}{n} \\ Z = \frac{\hat{p} - p}{\sqrt{V a r (\hat{p})}} = \frac{\hat{p} - p}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \sim N (0, 1) \\ P (p_{l} \leq p \leq p_{u}) = 1 - α \\ P (\frac{\hat{p} - p_{u}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \leq \frac{\hat{p} - p}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \leq \frac{\hat{p} - p_{l}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}}) = 1 - α \\ P (- Z_{1 - α / 2} \leq \frac{\hat{p} - p}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}} \leq Z_{1 - α / 2}) = 1 - α \\ C I for p is (\hat{p} - Z_{1 - α / 2} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}, \hat{p} + Z_{1 - α / 2} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}) \end{aligned}

\begin{aligned} P (p_{l} \leq p \leq p_{u}) = P (\frac{\hat{p} - p_{u}}{\sqrt{\frac{p (1 - p)}{n N}}} \leq \frac{\hat{p} - p}{\sqrt{\frac{p (1 - p)}{n N}}} \leq \frac{\hat{p} - p_{l}}{\sqrt{\frac{p (1 - p)}{n N}}}) \\ C I for p : (\hat{p} - Z_{1 - \frac{α}{2}} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n N}}, \hat{p} + Z_{1 - \frac{α}{2}} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n N}}) \end{aligned}