Mathfolis

Confidence Interval for the Difference of Two Means

Unit 7 · Inference for Quantitative Data: Means

What AP Stats asks here

A two-sample t-CI estimates μ1−μ2\mu_1 - \mu_2 from independent samples using the unpooled SE s12/n1+s22/n2\sqrt{s_1^2/n_1 + s_2^2/n_2}. The conservative df is min⁡(n1,n2)−1\min(n_1, n_2) - 1; calculators default to the Welch-Satterthwaite df. Two recurring traps: (1) applying two-sample procedures to paired data and (2) claiming 'no difference' from overlap of two one-sample CIs.

Two-sample t-CI

Estimate ± ME
(xˉ1−xˉ2)±t∗s12n1+s22n2(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}
Conservative df
min⁡(n1,n2)−1\min(n_1, n_2) - 1

Conditions

Independent random samples
from each population\text{from each population}
Normality (each)
population normal OR ni≥30 OR symmetric sample\text{population normal OR } n_i \ge 30 \text{ OR symmetric sample}
10% (each)
ni<0.10Nin_i < 0.10 N_i
AP Tip: The smaller sample dominates the SE. To shrink the difference CI, grow the smaller group first.
Caution: Overlap of two one-sample CIs does not prove the means are equal. Use the two-sample CI as the decision tool.
Type 1

Construct the two-sample t-CI

Use the unpooled SE — never assume σ1=σ2\sigma_1 = \sigma_2. Use min⁡(n1,n2)−1\min(n_1, n_2) - 1 for the conservative df if the calculator's Welch value is not available.

Example 1
Group A: nA=32n_A = 32, xˉA=78.4\bar{x}_A = 78.4, sA=9.2s_A = 9.2. Group B: nB=30n_B = 30, xˉB=73.1\bar{x}_B = 73.1, sB=10.5s_B = 10.5. Using conservative df = 29 and t∗≈2.045t^* \approx 2.045, the 95% CI for μA−μB\mu_A - \mu_B is: (A) (−5.14,5.14)(-5.14, 5.14) (B) (0.16,10.44)(0.16, 10.44) (C) (−10.44,0.16)(-10.44, 0.16) (D) (5.30,7.81)(5.30, 7.81)

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Type 2

Decision via whether 0 is in the CI

If the CI excludes 0, there is evidence of a difference. The sign of the bounds tells the direction.

Example 2
A 95% CI for μ1−μ2\mu_1 - \mu_2 is (−15,−2)(-15, -2). Which conclusion is correct? (A) No evidence of a difference because the interval is short. (B) Evidence that μ1<μ2\mu_1 < \mu_2 — the entire interval is negative, so 0 is not plausible at the 95% level. (C) Evidence that μ1>μ2\mu_1 > \mu_2 because −15<−2-15 < -2. (D) The two means are equal because the interval contains negative values.

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Type 3

Independent vs paired

Random assignment to two separate groups ⇒ independent. Twins or before/after on the same subject ⇒ paired.

Example 3
Which scenario uses a two-sample (independent) t-CI? (A) Each of 40 patients takes both drug and placebo in randomized order. (B) 40 patients are randomly assigned to drug or placebo; blood pressure is measured at the end of the study. (C) Twin pairs are randomly split into drug and placebo. (D) The same group of students is tested in September and again in May.

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Confidence Interval for the Difference of Two Means | AP Statistics — Mathfolis