7.8 Setting Up a Test for the Difference of Two Population Means

Part of Inference for Quantitative Data: Means in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Two-Sample t-Test Hypotheses

Parameter: ${\mu_1 - \mu_2}$ = difference in population means

Hypotheses:

${H_0: \mu_1 = \mu_2}$ (equivalently, ${\mu_1 - \mu_2 = 0}$ )
${H_a: \mu_1 \neq \mu_2}$ (two-sided, most common)
${H_a: \mu_1 > \mu_2}$ (right-tailed)
${H_a: \mu_1 < \mu_2}$ (left-tailed)

Conditions (same as two-sample t CI):

Random: both samples random
Independence: 10% condition for each; groups independent
Normal/Large Sample: each group — normal population OR ${n_i \geq 30}$

Test statistic:

t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

Use conservative ${df = \min(n_1-1, n_2-1)}$ by hand; calculator uses Welch df.

Note: Unlike the two-proportion z-test, there is no "pooled" standard error for the two-sample t-test (we don't assume ${\sigma_1 = \sigma_2}$ ).

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