Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- σ is rarely known, so you estimate the standard deviation of x̄ with the standard error s/√n.
- Standardising with s gives a t statistic, which follows a t-distribution with df = n − 1.
- t-distributions: symmetric, bell-shaped, centred at 0, with heavier tails than the standard normal. As df grows they approach the standard normal.
- Heavier tails mean t* > z*, so intervals are wider, especially for small n.
- Matched pairs (two linked measurements per individual): subtract in a stated order and treat the differences as one sample.
Key relationships
| Quantity | Formula or rule |
|---|---|
| Point estimate | x̄ (or x̄d) |
| Standard error | SE = s/√n |
| Degrees of freedom | df = n − 1 (n = number of differences for pairs) |
| Critical value | t* for the central C% of t(df); invT(1 − (1 − C)/2, df) |
| Margin of error | ME = t* × s/√n |
| Interval | x̄ ± t* × s/√n |
Conditions
| Condition | Check |
|---|---|
| Randomization | Random sample or randomized experiment |
| 10% | n ≤ 10% of N, when sampling without replacement |
| Sample data | Population approximately normal, or n ≥ 30, or (n < 30) data free from strong skewness and outliers. For pairs: check the differences |
Mistakes to avoid
- Using z* when σ is unknown.
- df = n instead of n − 1.
- Two-sample methods for paired data. Pairs → differences → one sample.
- No order of subtraction in a μd definition.
- “n ≥ 30” written for a small sample. Look at the data and describe it.
- Checking each column of paired data instead of the differences.
- Parameter without context: name the mean, the variable, the units and the population.
Quick self-check
- What is t* for a 90% interval from a sample of 20? (df = 19, t ≈ 1.729)*
- A sample of 36 has s = 12 cm. What is the standard error? (12/√36 = 2 cm)
- Fifteen students each take a test before and after a course. What are the degrees of freedom for an interval for μd? (15 differences, so df = 14)
Next: practice questions.