Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- x̄₁ − x̄₂ is a statistic. It changes from one pair of samples to the next.
- Its sampling distribution describes all the values it could take over many pairs of independent samples.
- It is centred at the true difference μ₁ − μ₂, so it is an unbiased estimator.
- Its spread comes from both samples, so the variances add.
- Larger samples give less spread: multiplying both sample sizes by k divides the standard deviation by √k.
Key relationships
| Item | Formula or rule |
|---|---|
| Mean | μ(x̄₁ − x̄₂) = μ₁ − μ₂ |
| Standard deviation | σ(x̄₁ − x̄₂) = √(σ₁²/n₁ + σ₂²/n₂) |
| Randomization | two independent random samples, or random assignment in an experiment |
| 10% | n₁ ≤ 10% of N₁ and n₂ ≤ 10% of N₂ (sampling without replacement only) |
| Normal | both populations normal (any n₁, n₂) |
| Approximately normal | populations not normal, but n₁ ≥ 30 and n₂ ≥ 30 |
| z-score | z = [(x̄₁ − x̄₂) − (μ₁ − μ₂)] / σ(x̄₁ − x̄₂) |
Assumptions and conventions
- The two samples are independent: different individuals, not linked or paired.
- The population means and standard deviations are given in this topic. When they are unknown (Topic 4.7 onwards), you estimate with s₁ and s₂.
- For an experiment, random assignment meets the randomization condition and the 10% condition is not needed. The shape condition is still needed for a normal model.
- Interpretations name the variable, units, both populations and the order of subtraction.
Mistakes to avoid
- Adding the standard deviations instead of the variances.
- Subtracting the variances because the means are subtracted.
- “n₁ + n₂ ≥ 30” instead of both n₁ ≥ 30 and n₂ ≥ 30.
- Demanding n ≥ 30 when both populations are already normal.
- Using the formula for paired data (the same individuals measured twice).
- Forgetting the order of subtraction when turning words into an inequality.
- “A small probability proves the claim is false.” It only gives reason to doubt it.
Quick self-check
- σ₁ = 10, n₁ = 25, σ₂ = 6, n₂ = 16. Find σ(x̄₁ − x̄₂). (√(4 + 2.25) = √6.25 = 2.5)
- μ₁ = 40 and μ₂ = 46. What is the mean of x̄₁ − x̄₂? (−6)
- Both populations are strongly skewed; n₁ = 20 and n₂ = 50. Can you use a normal model? (No: n₁ < 30)
Next: practice questions.