7.9 Carrying Out 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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Complete Two-Sample t-Test

Problem: Do students in rural and urban schools differ in mean math scores?

Rural: ${n_1 = 40}$ , ${\bar{x}_1 = 71.2}$ , ${s_1 = 12.4}$ Urban: ${n_2 = 35}$ , ${\bar{x}_2 = 76.8}$ , ${s_2 = 10.1}$

H: ${H_0: \mu_R = \mu_U}$ vs. ${H_a: \mu_R \neq \mu_U}$

A: Random (assume) ✓; 10% condition ✓; ${n_1 = 40 \geq 30}$ ✓, ${n_2 = 35 \geq 30}$ ✓

N: Two-sample t-test

t = \frac{71.2 - 76.8}{\sqrt{12.4^2/40 + 10.1^2/35}} = \frac{-5.6}{\sqrt{3.844 + 2.915}} = \frac{-5.6}{\sqrt{6.759}} = \frac{-5.6}{2.600} \approx -2.15

{df = \min(39, 34) = 34}

O: ${p = 2 \cdot P(t_{34} \leq -2.15) \approx 2(0.019) = 0.038}$

M/S: Since ${0.038 < 0.05}$ , reject ${H_0}$ . We have convincing evidence that rural and urban students differ in mean math scores.

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