Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Do: calculate t and the p-value. Conclude: compare the p-value with α and answer the question in context.
- The t-statistic counts how many standard errors x̄₁ − x̄₂ is from the null value 0.
- The p-value assumes H₀ (equal population means) is true.
- Random samples allow generalising to the populations; random assignment allows cause and effect.
Key relationships
| Item | Rule |
|---|---|
| Standard error | SE = √(s₁²/n₁ + s₂²/n₂) |
| Test statistic | t = ((x̄₁ − x̄₂) − 0) / SE |
| Degrees of freedom | From technology; between min(n₁ − 1, n₂ − 1) and n₁ + n₂ − 2 |
| Conservative df | min(n₁ − 1, n₂ − 1): gives a slightly larger p-value |
| Hₐ: μ₁ − μ₂ > 0 | p-value = P(t ≥ observed t) |
| Hₐ: μ₁ − μ₂ < 0 | p-value = P(t ≤ observed t) |
| Hₐ: μ₁ − μ₂ ≠ 0 | p-value = 2 × P(t ≥ |observed t|) |
| Decision | p-value ≤ α: reject H₀; p-value > α: fail to reject H₀ |
Templates
- p-value: “Assuming the true mean [response] of [group 1] and [group 2] are equal, there is a [p] probability of getting a difference in sample means of [observed] or more extreme [in the direction of Hₐ] by chance alone.”
- Conclusion: “Because [p] ≤ (>) [α], we reject (fail to reject) H₀. There is (not) convincing evidence that the mean [response] of [population 1] is [greater than / less than / different from] that of [population 2].”
Mistakes to avoid
- p-value = probability H₀ is true. It assumes H₀ is true.
- “Accept H₀” or “the means are equal.”
- “Proves.” Use “convincing evidence”.
- Not doubling for a two-sided test.
- df = n₁ + n₂ − 2 or pooling.
- Comparing t, not the p-value, with α.
- Causal claims from random samples.
Quick self-check
- x̄₁ − x̄₂ = 3.0 and SE = 1.2. What is t? (t = 3.0 ÷ 1.2 = 2.5)
- Samples of 15 and 22. What are the conservative df and the largest possible df? (14 and 35)
- p-value = 0.074, α = 0.05, Hₐ: μ₁ ≠ μ₂. Decision? (0.074 > 0.05, fail to reject H₀: not convincing evidence that the population means differ.)
Next: practice questions.