Point Estimation of a Population Proportion

Parameter: Population proportion p
Data: X = Number having the characteristic in a random sample of size n
Estimator: $\displaystyle\hat{p}=\frac{X}{n}$

$S.E.(\hat{p})=\sqrt{\frac{pq}{n}}$ and estimated $S.E.(\hat{p})=\sqrt{\frac{\hat{p}\hat{q}}{n}}$.

For large n, an approximate $100(1-\alpha)\%$ error margin is $z_{\alpha/2}\sqrt{\hat{p}\hat{q}/n}$

例
有一抽樣調查中心, 做某一問卷調查, 在 9000 個名單中選出 250 位寄發問卷. 在這些問卷中贊成的人數為 70 位, 則 n=250, X=70, 所以母體比例的估計為,

$\begin{array}{rcl} \hat{p}&=&\frac{70}{250}=.28 \\ \mbox{Estimated } S.E.(\hat... ...}{250}}=.028 \\ 95.4\%\mbox{ error margin}&=&2\times .028=.056 \\ \end{array}$

因此, 估計母體比例 $\hat{p}=.28$, 95.4% 的誤差範圍約在 $\pm .056$ 之間.