Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- p̂ = X / n, where X is the number of successes in the sample. It varies from sample to sample.
- The sampling distribution of p̂ is the distribution of p̂ over all random samples of size n.
- Its mean is p, so p̂ is unbiased (Topic 3.1).
- Its standard deviation shrinks as n grows; multiply n by 4 to halve it.
- With large enough expected counts, its shape is approximately normal, so you can use z-scores.
Key relationships
| Quantity | Formula or rule | Needs |
|---|---|---|
| Mean | μp̂ = p | Random sample |
| Standard deviation | σp̂ = √(p(1 − p) / n) | Independence: random, and n ≤ 10% of N when sampling without replacement |
| Shape | approximately normal | np ≥ 10 and n(1 − p) ≥ 10 |
| z-score | z = (p̂ − p) / σp̂ | Normal shape |
| Sample size for σp̂ ≤ d | n ≥ p(1 − p) / d², rounded up | — |
Interpretation templates
- Mean: “In repeated random samples of [n] [individuals] from [population], the sample proportion who [success] averages [p].”
- Standard deviation: “In random samples of [n] [individuals], the sample proportion who [success] typically differs from the population proportion [p] by about [σp̂].”
- Probability: “If [p] of [population] [success], about [probability] of random samples of [n] would give a sample proportion [event].”
Mistakes to avoid
- Using p̂ in the formula or the large-counts check when p is given.
- Checking n ≥ 30 instead of the expected counts np and n(1 − p).
- Forgetting the 10% condition, or checking it the wrong way round (it is n ≤ 0.10N).
- Thinking doubling n halves σp̂ (it divides it by √2).
- Rounding a required sample size down.
- Writing “it varies less” without saying which distribution.
- Concluding a claim is proved false from a small probability.
Quick self-check
- p = 0.7 and n = 84. Find σp̂ and check large counts. (σp̂ = √(0.21 / 84) = √0.0025 = 0.05; np = 58.8 and n(1 − p) = 25.2, both ≥ 10.)
- A random sample of 100 is taken from a club of 600 members. Is the 10% condition met? (No: 10% of 600 is 60, and 100 > 60.)
- σp̂ = 0.04 when n = 150. What is σp̂ when n = 600 (same p)? (0.02: four times the sample size halves σp̂.)
Next: practice questions.