Hypothesis Testing Cheat Sheet

Parameter	Statistic/Estimate	Conditions	(Theoretical) Sampling Distribution	Confidence Interval	Hypothesis	Test Statistic
p	$\hat{p}$	1. independent observations 2. np>= 10 and n(1-p)>=10	$\hat{p} \sim N(\mu_{\hat{p}} = p, \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}})$	$\hat{p} \pm z^{\ast} \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$	$H_{0} : p = p_{0}$ $H_{A} : p \ne p_{0}$	$z = \frac{\hat{p} - p_{0}}{\sqrt{\frac{p_{o} (1 - p_{o})}{n}}}$
$p_1 - p_2$	$\hat{p_1} - \hat{p_2}$	1. Independent Observations and groups 2. $n_{1}\hat{p_1} \ge 5, n_{1}(1 - \hat{p_1}) \ge 5$ $n_{2}\hat{p_2} \ge 5, n_{2}(1 - \hat{p_2}) \ge 5$	$\hat{p_1} - \hat{p_2} \sim N(\mu_{\hat{p_1} - \hat{p_2}} = p_1 - p_2, \sigma_{\hat{p_1} - \hat{p_2}} = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}})$	$\hat{p_1} - \hat{p_2} \pm z^{\ast} \sqrt{\frac{\hat{p_1}(1 - \hat{p_1})}{n_1} + \frac{\hat{p_2}(1 - \hat{p_2})}{n_2}}$	$H_0 : p_1 - p_2 = 0$ $H_A : p_1 - p_2 \ne 0$	$z = \frac{\hat{p_1} - \hat{p_2}}{\sqrt{\frac{\hat{p_1}(1 - \hat{p_1})}{n_1} + \frac{\hat{p_2}(1 - \hat{p_2})}{n_2}}}$
$\mu$	$\bar{x}$	1. Independent observations 2. $\sigma$ is known (yes or no) 3. Population is normal OR $n \ge 30$	If $\sigma$ is known : $\bar{X} \sim N(\mu_{\bar{x}} = \mu, \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ If $\sigma$ is not known: $\bar{X} \sim t_{df}$ , where df = n-1	If $\sigma$ is known: $\bar{X_1} \pm Z^{\ast} * \frac{\sigma}{\sqrt{n}}$ If $\sigma$ is not known: $\bar{X} \pm t^{\ast} * \frac{s}{\sqrt{n}}$ where s is the sample standard deviation	$H_{0} : \mu = \mu_{0}$ $H_{A} : \mu \ne \mu_{0}$	If $\sigma$ is known: $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$ If $\sigma$ is not known: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$
$\mu_1 - \mu_2$	$\bar{x}_1 - \bar{x}_2$	1. Independent observations and groups 2. $\sigma_1$ and $\sigma_2$ known? (yes or no) 3. Both populations are normal OR $n_1 \ge 30$ and $n_2 \ge 30$	If $\sigma_1$ and $sigma_2$ are known: $\bar{X_1} - \bar{X_2} \sim N(\mu_{\bar{x_1} - \bar{x_2}} = \mu_1 - \mu_2, \sigma_{\bar{x_1} - \bar{x_2}} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}})$ If $\sigma_1$ and $\sigma_2$ are unknown: $\bar{X_1} - \bar{X_2} \sim t_{df}$ where $df = min(n_1, n_2) - 1$	If $\sigma_1$ and $\sigma_2$ are known: $\bar{x_1} - \bar{x_2} \pm z^{\ast} * \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$ If $\sigma_1$ and $\sigma_2$ are unknown: $\bar{x_1} - \bar{x_2} \pm t^{\ast} * \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	$H_0 : \mu_1 - \mu_2 = \mu_0$ $H_A : \mu_1 - \mu_2 \ne \mu_0$	If $\sigma_1$ and $\sigma_2$ are known: $z = \frac{\bar{x_1} - \bar{x_2} - \mu_0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$ If $\sigma_1$ and $\sigma_2$ are unknown: $t = \frac{\bar{x_1} - \bar{x_2} - \mu_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
$\mu_d$	$\bar{x}_d$	1. Independent Observations and Dependent groups 2. Population distribution of differences is approximately normal	$\bar{X_d} \sim t_{df}$ where df = n-1, where n is the number of pairs	$\bar{x_d} \pm t^{\ast} * \frac{s_d}{sqrt{n}}$	$H_{0} : \mu_d = 0$ $H_{A} : \mu_d \ne 0$	$t = \frac{\bar{x}_d - 0}{\frac{sd}{\sqrt{n}}}$