Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A test asks whether sample data give convincing evidence against a claimed value of a parameter.
- For a quantitative variable, σ is unknown, so use s and a t-distribution with df = n − 1.
- Matched pairs: two values on the same individual (before/after, both treatments) or natural pairs. Analyse one sample of differences.
- This topic is State and Plan. Topic 4.5 does Do and Conclude.
Key relationships
| Situation | Procedure | Parameter | H₀ | Hₐ options |
|---|---|---|---|---|
| One sample, claimed mean μ₀ | One-sample t-test for a population mean | μ = true mean [variable, units] of all [population] | μ = μ₀ | μ < μ₀, μ > μ₀, μ ≠ μ₀ |
| Paired data | One-sample t-test for a population mean difference | μd = true mean of (A − B) for [population] | μd = 0 | μd < 0, μd > 0, μd ≠ 0 |
| Two separate groups | Two-sample t-test (Topic 4.9) | — | — | — |
Conditions
| Condition | Check |
|---|---|
| Random | Random sample, or random assignment (random order of treatments for pairs) |
| 10% | n ≤ 10% of N when sampling without replacement; not needed for an experiment with no sampling |
| Sample data | Population approximately normal, or n ≥ 30, or (n < 30) no strong skew and no outliers in the sample. For pairs: apply to the differences |
Mistakes to avoid
- Hypotheses with x̄ instead of μ.
- Two-sample analysis of paired data. Take the differences.
- No order of subtraction for μd, or an Hₐ that does not match it.
- “np₀ ≥ 10” for a mean. That is a proportion check.
- “n < 30, so we cannot test.” Check the data first.
- Direction chosen from the sample. Use the question’s wording.
Quick self-check
- A fictional brand says its dishwashers use “at most 9.0 litres of water per cycle” on average, and a tester suspects more. Write the hypotheses. (H₀: μ = 9.0, Hₐ: μ > 9.0, where μ is the true mean water use, in litres per cycle, of all the brand’s dishwashers.)
- Each of 25 workers is timed doing a task with an old tool and a new tool. Which procedure? (One-sample t-test for a population mean difference, on the 25 differences.)
- A random sample of 18 values has an outlier, and nothing says the population is normal. Is the sample data condition met? (No: with n < 30 the data must be free from outliers.)
Next: practice questions.