The probability of obtaining a sample result as extreme or more extreme than observed, assuming the null hypothesis is true.

A statement that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error.

The hypothesis that sample observations are influenced by some non-random cause.

A hypothesis test where the alternative hypothesis includes values greater than the null hypothesis value.

A hypothesis test where the alternative hypothesis includes values less than the null hypothesis value.

A hypothesis test where the alternative hypothesis includes values that are different from the null hypothesis value (both greater and less).

Statistical significance indicates that the observed result is unlikely to have occurred by random chance alone, assuming the null hypothesis is true. A small p-value suggests statistical significance.

The significance level (alpha) is the probability of rejecting the null hypothesis when it is actually true (Type I error). It sets the threshold for determining statistical significance; typically, alpha is set to 0.05.

A Type I error occurs when the null hypothesis is rejected when it is actually true (false positive). The probability of making a Type I error is equal to the significance level (alpha).

A Type II error occurs when the null hypothesis is not rejected when it is actually false (false negative).

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ where $\hat{p}$ is the sample proportion, $p_0$ is the hypothesized population proportion, and n is the sample size.