ZIMSEC A Level · Statistics · Form 6
Sampling methods
Simple random, systematic, stratified, quota, cluster and convenience sampling. Good samples are representative and free from bias.
Distribution of the sample mean
If X has mean μ and variance σ², the sample mean X̄ of n observations has mean μ and variance σ²/n. The standard error is σ/√n.
The central limit theorem
For large n (often n ≥ 30), X̄ is approximately normal whatever the distribution of X.
Unbiased estimates
x̄ estimates μ; s² = Σ(x − x̄)²/(n − 1) is an unbiased estimate of σ².
Confidence intervals
A 95% confidence interval for μ (σ known or large sample): x̄ ± 1.96 σ/√n. For 99%, use 2.576.
Worked example: A sample of 64 bags has mean mass 52 kg. σ = 8 kg. Find a 95% confidence interval for μ.
Standard error = 8/√64 = 1
52 ± 1.96 × 1 → (50.04, 53.96) kg
Interpretation
If many samples were taken, about 95% of such intervals would contain the true mean.
Check yourself
- State the central limit theorem.
- σ = 12 and n = 36. Find the standard error of the mean.
- Find a 99% confidence interval for μ if x̄ = 100, σ = 10, n = 25.
Show answers
- For large samples, the sample mean is approximately normally distributed, whatever the parent distribution.
- 12/6 = 2
- 100 ± 2.576 × 2 → (94.85, 105.15)
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