📊 Inference for Two Sample Means (CI and Test)

Comparing the means of two populations—such as treatment vs. control, or two manufacturing processes—is one of the most common inference problems. We use two-sample t-procedures to estimate and test the difference $\mu_1 - \mu_2$.

Setting Up the Problem

Let $\mu_1$ and $\mu_2$ be the true means of two independent populations. We draw independent samples of sizes $n_1$ and $n_2$, obtaining sample means $\bar{x}_1$ and $\bar{x}_2$, and sample standard deviations $s_1$ and $s_2$.

We focus on the difference $\bar{x}_1 - \bar{x}_2$.

Sampling Distribution of $\bar{x}_1 - \bar{x}_2$

When conditions are met:

• Mean: $\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$
• Standard error: $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
• Distribution: Approximately $t$ with complicated degrees of freedom (calculator handles this)

Two-Sample t-Interval for $\mu_1 - \mu_2$

Confidence Interval: $(\bar{x}_1 - \bar{x}_2) \pm t^* \cdot SE$

where $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ and $t^*$ is the critical $t$-value based on the degrees of freedom.

Degrees of Freedom (Welch's adjustment): $df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}$

(Most calculators compute this automatically; always use the more conservative estimate if doing by hand.)

Two-Sample t-Test for $\mu_1 = \mu_2$

Null Hypothesis: $H_0: \mu_1 = \mu_2$ (or equivalently, $\mu_1 - \mu_2 = 0$)

Test Statistic: $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE}$

Compare to the $t$-distribution with the appropriate df to find the p-value.

Alternative Hypotheses:

• Two-tailed: $H_a: \mu_1 \neq \mu_2$
• One-tailed: $H_a: \mu_1 > \mu_2$ or $H_a: \mu_1 < \mu_2$

Conditions for Two-Sample t-Procedures

All methods require:

Robustness of t-Tests

t-procedures are robust to departures from Normality, especially for large samples and balanced designs ($n_1 \approx n_2$). However:

• With very small samples ($n < 15$) and skewed data, be cautious.
• With very large samples ($n > 100$), slight departures from Normality are not a concern.

Worked Example 1: Two-Sample t-Interval

Two high schools compare math proficiency test scores:

• School A: $n_1 = 25$, $\bar{x}_1 = 78$, $s_1 = 8$
• School B: $n_2 = 30$, $\bar{x}_2 = 75$, $s_2 = 10$

Construct a 95% CI for $\mu_A - \mu_B$.

Standard error: $SE = \sqrt{\frac{8^2}{25} + \frac{10^2}{30}} = \sqrt{\frac{64}{25} + \frac{100}{30}} = \sqrt{2.56 + 3.333} = \sqrt{5.893} \approx 2.427$

Degrees of freedom (conservative estimate: use $n - 1 = 24$ from smaller sample): Using $df = 24$ and 95% confidence, $t^* \approx 2.064$.

CI: $(78 - 75) \pm 2.064(2.427) = 3 \pm 5.008 = (-2.008, 8.008)$

We are 95% confident that School A's mean exceeds School B's by between −2.0 and +8.0 points.

Worked Example 2: Two-Sample t-Test

Test $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 \neq \mu_2$ at $\alpha = 0.05$ using the data above.

Test statistic: $t = \frac{3 - 0}{2.427} \approx 1.236$

p-value (two-tailed with $df \approx 24$): $p\text{-value} = 2 \times P(T > 1.236) \approx 2(0.114) \approx 0.228$

Since $p\text{-value} = 0.228 > 0.05$, we fail to reject $H_0$. Insufficient evidence that the school means differ.

Common Pitfalls

⚠️ Confusing SE with $s$: $SE = \sqrt{s_1^2/n_1 + s_2^2/n_2}$ is the standard error of $\bar{x}_1 - \bar{x}_2$. Do not use the sample standard deviations directly as if they were the population SDs.

⚠️ Forgetting Independence: Two-sample t-tests require independent samples. If the samples are paired (same individuals measured twice), use a paired t-test instead.

⚠️ Misinterpreting CI: A 95% CI does not mean there is a 95% probability that $\mu_1 - \mu_2$ is in the interval for this specific data. Rather, the method has 95% long-run success rate.

Calculator Tip

💡 TI-84 / TI-Nspire: Use 2-SampTInt for confidence intervals and 2-SampTTest for hypothesis tests. Enter the summary statistics ($\bar{x}_1, s_1, n_1, \bar{x}_2, s_2, n_2$) and the alternative hypothesis. The calculator computes df and the interval or test automatically (with or without assuming equal variances).