STATISTICAL INFERENCE FOR MEANS AND PROPORTIONS

PARAMETER	$C$% CONFIDENCE INTERVAL	TEST STATISTIC	DISTRIBUTION OF TEST
	ENDPOINTS	$H_0$: Parameter = value	STATISTIC WHEN $H_{0}$ TRUE
$\mu$ ($\sigma$ known)	$\overline{x} \pm z^{*} \displaystyle{\frac{\sigma}{\sqrt{n}}}$	$z = \displaystyle{\frac{\overline{x} - \mu_{0}}{\sigma / \sqrt{n}}}$	$N(0,1)$
$\mu$ ($\sigma$ unknown)	$\overline{x} \pm t^{*} \displaystyle{\frac{s}{\sqrt{n}}}$	$t = \displaystyle{\frac{\overline{x} - \mu_{0}}{s / \sqrt{n}}}$	$t$ with $n-1$ df
$p$	$\tilde{p} \pm z^{*}\displaystyle{ \sqrt{\frac{\tilde{p} (1-\tilde{p})}{n+4}}}$	$z = \displaystyle{\frac{\hat{p}-p_{0}}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}}}$	approx $N(0,1)$
	Wilson: Add two successes and two failures; get $\tilde{p}$.
$\mu_{1}-\mu_{2}$ ( $\sigma_{1}, \sigma_{2}$ known)	$(\overline{x}_{1} - \overline{x}_{2}) \pm z^{*} \displaystyle{\sqrt{\frac{\sigma_{1}^{2}}{n_1} + \frac{\sigma_{2}^{2}}{n_{2}}}}$	$z = \displaystyle{\frac{\overline{x}_{1} - \overline{x}_{2} - (\mu_1-\mu_2)} {\sqrt{\frac{\sigma_{1}^{2}}{n_1} + \frac{\sigma_{2}^{2}}{n_{2}}}}}$	$N(0,1)$
			approx $t$ with df =
$\mu_{1}-\mu_{2}$	$(\overline{x}_{1} - \overline{x}_{2}) \pm t^{*} \displaystyle{\sqrt{\frac{s_{1}^{2}}{n_1} + \frac{s_{2}^{2}}{n_{2}}}}$	$t = \displaystyle{\frac{\overline{x}_{1} - \overline{x}_{2} - (\mu_1-\mu_2)} {\sqrt{\frac{s_{1}^{2}}{n_1} + \frac{s_{2}^{2}}{n_{2}}}}}$	$${\frac{\left ( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2} {... ..._1^2}{n_1} \right )^2 + \frac{1}{n_2 - 1} \left(\frac{s_2^2}{n_2} \right )^2}}$$
( $\sigma_{1}, \sigma_{2}$ unknown)
$\mu_{1}-\mu_{2}$	$(\overline{x}_{1} - \overline{x}_{2}) \pm t^{*} \displaystyle{\sqrt{s_{p}^2\left ( \frac{1}{n_1} + \frac{1}{n_{2}}\right )}}$	$t = \displaystyle{\frac{\overline{x}_{1} - \overline{x}_{2} - (\mu_1-\mu_2)} {\sqrt{s_{p}^2\left ( \frac{1}{n_1} + \frac{1}{n_{2}}\right )}}}$	$t$ distribution with
( $\sigma_{1} = \sigma_{2}$ unknown)	$${s_p^2 = ((n_1 - 1)s_1^2 + (n_2 - 1)s_2^2)/(n_1 + n_2 - 2)}$$		df = $n_{1} + n_{2} - 2$
$p_{1}-p_{2}$	$(\tilde{p}_{1}-\tilde{p}_{2}) \pm z^{*} \displaystyle{\sqrt{\frac{\tilde{p}_{1}(1-\tilde{p}_{1})}{n_{1}+2}+ \frac{\tilde{p}_{2}(1-\tilde{p}_{2})}{n_{2}+2}}}$	$z = \displaystyle{\frac{\hat{p}_{1}-\hat{p}_{2} - (0)} {\sqrt{\hat{p}(1-\hat{p}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}}}$	approx $N(0,1)$
	Wilson: Add one success and one failure in each sample.	where $\hat{p} = \displaystyle{\frac{x_{1}+x_{2}}{n_{1}+n_{2}}}$	if $H_0: p_1 = p_2$.