One Proportion, Large Samples

This hypothesis testing is useful when it involves a single sample to inference the population proportion.

Suppose $p$ denotes the proportion in the population. The test statistic to examine is

z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}

where $p_0$ is the claimed population proportion, $\hat{p}$ is the sample proportion and $n$ is the sample size. And $z$ should follow $\mathcal{N}(0,1)$ under $H_0$ hypothesis.

$H_0$	$H_1$	$p$ -value	Decision Rule, if reject $H_0$
$p=p_0$ or $p\le p_0$	$p\gt p_0$	-	$z\gt z_\alpha$
$p=p_0$ or $p\ge p_0$	$p\lt p_0$	-	$z\lt -z_{\alpha}$
$p=p_0$	$p\ne p_0$	-	$\vert z\vert \gt z_{\alpha/2}$

Confidence Interval

\hat{p} \plusmn z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}