Mathfolis

Significance Test for the Difference of Two Means

Unit 7 · Inference for Quantitative Data: Means

What AP Stats asks here

A two-sample t-test asks whether two population means differ. Use the unpooled SE s12/n1+s22/n2\sqrt{s_1^2/n_1 + s_2^2/n_2}, the conservative df min⁡(n1,n2)−1\min(n_1, n_2) - 1 on free response (calculators give Welch). AP problems also test whether you can distinguish two-sample from paired and how sample size interacts with the same observed gap to flip a p-value.

Test statistic

Unpooled t
t=xˉ1−xˉ2s12/n1+s22/n2t = \dfrac{\bar{x}_1 - \bar{x}_2}{\sqrt{s_1^2/n_1 + s_2^2/n_2}}
Conservative df
min⁡(n1,n2)−1\min(n_1, n_2) - 1

Conditions

Independent samples
required\text{required}
Normality
ni≥30 OR each sample roughly symmetricn_i \ge 30 \text{ OR each sample roughly symmetric}
10%
ni<0.10Nin_i < 0.10 N_i
AP Tip: Statistical vs practical significance: a huge n can make a tiny effect look 'significant'. Always report the CI for the difference alongside the test to communicate the magnitude.
Caution: Two-sample t on paired data inflates the SE and destroys power. Check the structure before choosing the procedure.
Type 1

Set up and compute t

Translate the wording, compute t with the unpooled SE, and use the conservative df.

Example 1
Two versions of an online task: V1 (n1=40n_1 = 40, xˉ1=14.2\bar{x}_1 = 14.2, s1=3.6s_1 = 3.6) and V2 (n2=35n_2 = 35, xˉ2=12.5\bar{x}_2 = 12.5, s2=3.1s_2 = 3.1). For a two-sided test at α=0.05\alpha = 0.05 with conservative df = 34: (A) t≈2.20t \approx 2.20, p-value ≈ 0.035, reject — evidence that mean times differ. (B) t≈2.20t \approx 2.20, p-value ≈ 0.020, reject. (C) t≈1.10t \approx 1.10, p-value ≈ 0.28, fail to reject. (D) z≈2.20z \approx 2.20, p-value ≈ 0.014, reject.

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Type 2

Independent vs paired test

Two-sample t requires independent samples. Same subjects measured twice ⇒ paired.

Example 2
Which scenario uses a two-sample t-test? (A) 30 students take a placement test in September, then take it again in May. (B) 30 randomly selected freshmen take a test; 30 different randomly selected seniors take the same test. (C) Identical-twin pairs are split: one twin uses Method A, the other Method B. (D) Each of 25 people drinks two beverages and rates each.

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Type 3

Sample size vs effect size

The same observed difference can produce very different p-values depending on n. Statistical significance ≠ practical significance.

Example 3
Two studies report the same point estimate xˉ1−xˉ2=5\bar{x}_1 - \bar{x}_2 = 5 and same SDs s1=s2=10s_1 = s_2 = 10. Study A has n=25n = 25; Study B has n=400n = 400. For a two-sided test at α=0.05\alpha = 0.05: (A) Study A rejects; Study B fails to reject — small samples are more powerful. (B) Both reject at the same p-value. (C) Both fail to reject because the gap is only 5. (D) Study A fails to reject (p ≈ 0.09); Study B rejects overwhelmingly (p < 0.001). Larger samples detect smaller effects, but practical significance is a separate question.

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