Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- Use a two-sample t-interval for μ₁ − μ₂ when a quantitative response is measured in two independent groups.
- If the values come in linked pairs (same subject twice, matched pairs), the data are paired: use a one-sample t-interval on the differences instead.
- The parameter names the order of subtraction, the response variable (with units) and both populations or treatments.
- The interval is point estimate ± t × SE*, with t from a t-distribution because σ₁ and σ₂ are unknown.
Key relationships
| Quantity | Formula or rule |
|---|---|
| Point estimate | x̄₁ − x̄₂ |
| Standard error | SE = √(s₁²/n₁ + s₂²/n₂) |
| Degrees of freedom | from technology; always between the smaller of n₁ − 1 and n₂ − 1, and n₁ + n₂ − 2 |
| Conservative df | smaller of n₁ − 1 and n₂ − 1 (slightly wider interval) |
| Margin of error | t* × SE |
| Interval | (x̄₁ − x̄₂) ± t* √(s₁²/n₁ + s₂²/n₂) |
| Reverse the order of subtraction | both limits change sign and swap places |
Conditions
- Randomization: two independent random samples, or a randomized experiment.
- 10%: n₁ ≤ 10% N₁ and n₂ ≤ 10% N₂, when sampling without replacement. Not needed for a randomized experiment.
- Sample data: both n ≥ 30, or both populations approximately normal. If either n < 30, both samples must be free from strong skewness and outliers (look at a dot plot or use the 1.5 × IQR rule).
Mistakes to avoid
- Adding standard deviations (s₁/√n₁ + s₂/√n₂) instead of variance terms.
- Using z* instead of t*.
- Using df = n₁ + n₂ − 2 as if it were exact. It is only the upper bound.
- Choosing “pooled” on the calculator.
- Treating paired data as independent.
- Checking a condition for one group only.
- A parameter with no order of subtraction, or one that describes the samples instead of the populations.
Quick self-check
- n₁ = 20, s₁ = 3; n₂ = 20, s₂ = 4. Find the SE. (√(9/20 + 16/20) = √1.25 ≈ 1.118)
- A 95% interval for μ₁ − μ₂ is (2.4, 9.0). Find the point estimate and margin of error. (Point estimate 5.7, margin of error 3.3)
- n₁ = 12 and n₂ = 18. Between which values must the technology df lie? (11 and 28)
- Ten patients have blood pressure measured before and after a drug. Two-sample t-interval? (No: the data are paired. Use a one-sample t-interval on the 10 differences.)
Next: practice questions.