What you'll learn
Inference about population means using t-distributions (~10–18%).
Why t
When the population SD σ is unknown (almost always), use the t-distribution with n−1 degrees of freedom — wider than z to account for extra uncertainty.
One-sample t
- Interval: x̄ ± t*·(s/√n).
- Test of H₀: μ = μ₀ using t = (x̄ − μ₀)/(s/√n).
- Conditions: random; 10% (independence); normal/large sample (n ≥ 30 by CLT, or roughly symmetric for small n).
Paired data
For before/after or matched pairs, analyze the differences with a one-sample t procedure.
Two-sample t
Compare μ₁ − μ₂ using a two-sample t (don't pool unless told). Conditions for both groups.
Interpretation
Same logic as proportions: interpret intervals and p-values in context; compare p-value to α.
Exam tips
- Recognize paired vs two independent samples (a common decision).
- Use t, not z, when σ is unknown.
Common mistakes
- Using z instead of t.
- Treating paired data as two independent samples.