Two-Sample t-Interval and t-Test
Construct a two-sample t-interval and perform a two-sample t-test for μ₁ − μ₂. Use the conservative df = min(n₁−1, n₂−1) or the calculator's Welch approximation. Distinguish paired from two-sample procedures.
A lesson of AP Statistics, in Inference for Quantitative Data: Means, about 10 minutes.
Two-Sample t Procedures
By the end of this lesson, you will be able to construct and interpret a two-sample t confidence interval and significance test for comparing two population means.
When to Use Two-Sample t Procedures
Use two-sample t procedures when you want to compare the means of two independent groups.
Parameter: $\mu_1 - \mu_2$ (the difference in population means)
Distinguish from paired procedures: Two-sample t is for independent groups; paired t is for matched pairs. The key question: can you identify a natural pairing between specific individuals in Group 1 and Group 2? If not, it's two-sample.
Conditions
1. Random: Both samples must be randomly selected (or randomly assigned in an experiment).
2. Normal: Both populations approximately normal, or both $n_1 \geq 30$ and $n_2 \geq 30$ (CLT). If either n < 30, check for skewness/outliers in that sample.
3. Independent:
- The two groups must be independent of each other
- Within each group: 10% condition applies ($n_1 < 10%$ of population 1; $n_2 < 10%$ of population 2)
The Two-Sample t Test
Hypotheses:
- $H_0: \mu_1 = \mu_2$ (or equivalently, $\mu_1 - \mu_2 = 0$)
- $H_a: \mu_1 \neq \mu_2$, $\mu_1 > \mu_2$, or $\mu_1 < \mu_2$
Test statistic:
$$t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$
Degrees of freedom: Use technology. Conservative approximation: df = min($n_1 - 1$, $n_2 - 1$).
The Two-Sample t Confidence Interval
$$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$
Interpretation: "We are C% confident that the true mean [variable] for [Group 1] is [direction] the true mean for [Group 2] by between [L] and [U] [units]."
If the interval contains 0: plausible that $\mu_1 = \mu_2$ (no significant difference).
If the interval is entirely positive: evidence that $\mu_1 > \mu_2$.
Worked Example
A school tests two teaching methods. 30 students used Method A ($\bar{x}_1 = 82$, $s_1 = 10$), and 35 students used Method B ($\bar{x}_2 = 76$, $s_2 = 12$). Is there evidence that Method A produces higher scores?