$\mu > \mu_0$ $Z > Z_\alpha$ $\mu \neq \mu_0$ $|Z| > Z_{\alpha/2}$ $\sigma$ is unknown, $n \le 30$ $\mu < \mu_0$ $T = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}$ $T < -t_{\alpha, \nu=n-1}$ $\mu > \mu_0$ $T > t_{\alpha, \nu=n-1}$ $\mu \neq \mu_0$ $|T| > t_{\alpha/2, \nu=n-1}$ $\sigma$ is unknown, $n > 30$ $\mu < \mu_0$ $Z = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}$ $Z < -Z_\alpha$ $\mu > \mu_0$ $Z > Z_\alpha$ $\mu \neq \mu_0$ $|Z| > Z_{\alpha/2}$

Remarks:

Formulas hold strictly for random samples from a normal distribution.
The third test (unknown $\sigma$, $n > 30$) provides a good approximate test when the distribution is not normal, provided $n \ge 30$.

Hypothesis Testing for Proportion (Single Population)

$H_0: p = p_0$

If the unknown proportion is not expected to be too close to 0 or 1 and $n$ is large, a large sample approximation is given by:

Test Statistic	Alternative Hypothesis	Region of Rejection
$Z = \frac{Y - np_0}{\sqrt{np_0(1 - p_0)}}$	$p < p_0$	$Z < -Z_\alpha$
	$p > p_0$	$Z > Z_\alpha$
	$p \neq p_0$	$\|Z\| > Z_{\alpha/2}$

where $Y$ is the number of "successes" in the sample.

Hypothesis Tests for Difference of Means (Two Populations)

For Independent Sampling, $H_0: \mu_X - \mu_Y = d_0$

Let $(X_1, ..., X_{n_1})$ be a random sample with mean $\mu_X$ and variance $\sigma_X^2$.

Let $(Y_1, ..., Y_{n_2})$ be a random sample with mean $\mu_Y$ and variance $\sigma_Y^2$.

Let $\bar{X}$ and $\bar{Y}$ denote the sample mean and $S_X^2$ and $S_Y^2$ denote the sample variance of the two random samples, respectively.

Case 1: $\sigma_X^2$ and $\sigma_Y^2$ are known

Alternative Hypothesis	Test Statistic	Region of Rejection
$\mu_X - \mu_Y < d_0$	$Z = \frac{(\bar{X} - \bar{Y}) - d_0}{\sqrt{\frac{\sigma_X^2}{n_1} + \frac{\sigma_Y^2}{n_2}}}$	$Z < -Z_\alpha$
$\mu_X - \mu_Y > d_0$		$Z > Z_\alpha$
$\mu_X - \mu_Y \neq d_0$		$\|Z\| > Z_{\alpha/2}$

Case 2: $\sigma_X^2$ and $\sigma_Y^2$ are unknown but $\sigma_X^2 = \sigma_Y^2$

Alternative Hypothesis	Test Statistic	Region of Rejection
$\mu_X - \mu_Y < d_0$	$T = \frac{(\bar{X} - \bar{Y}) - d_0}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$ $S_p = \sqrt{\frac{(n_1-1)S_X^2 + (