Two-Sample Tests (t-test, z-test) Cheat Sheet

Key Facts

• Use a two-sample t-test for independent means when population standard deviations are unknown, with test statistic $t = \frac{\bar{x}_1 - \bar{x}_2 - \Delta_0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$.
• Use a two-sample z-test for independent means when population standard deviations are known, with test statistic $z = \frac{\bar{x}_1 - \bar{x}_2 - \Delta_0}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$.
• For two proportions, use $z = \frac{\hat{p}_1 - \hat{p}_2 - 0}{\sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$, where $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$ for a hypothesis test with $H_0: p_1 = p_2$.
• A confidence interval for the difference of two means is $(\bar{x}_1 - \bar{x}_2) \pm t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ when using sample standard deviations.
• A confidence interval for the difference of two proportions is $(\hat{p}_1 - \hat{p}_2) \pm z^*\sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$.
• The null hypothesis usually states no difference, such as $H_0: \mu_1 - \mu_2 = 0$ or $H_0: p_1 - p_2 = 0$.
• Reject $H_0$ when the p-value is less than the significance level $\alpha$, such as $p < 0.05$.
• Two-sample tests require independent samples, random sampling or random assignment, and approximately normal sampling distributions.

Vocabulary

Two-sample test

A statistical test used to compare a parameter, such as a mean or proportion, between two independent groups.

Null hypothesis

The claim being tested that usually says there is no difference, such as $H_0: \mu_1 - \mu_2 = 0$.

Alternative hypothesis

The claim that represents a difference or direction of change, such as $H_a: \mu_1 - \mu_2 \ne 0$.

Standard error

The estimated standard deviation of a sampling distribution, such as $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ for two sample means.

P-value

The probability of getting a test statistic at least as extreme as the observed one, assuming the null hypothesis is true.

Significance level

The cutoff probability $\alpha$ used to decide whether evidence is strong enough to reject the null hypothesis.