Sampling and estimation
Sampling
- A sample is a small group from the whole population; good sampling needs randomness.
- The sample mean $\bar{X}$ is itself a random variable:
$$E(\bar{X}) = \mu, \qquad \text{Var}(\bar{X}) = \frac{\sigma^2}{n}$$
- By the Central Limit Theorem, for a large sample $\bar{X}$ is approximately normal, whatever the population's shape.
Practice
A population has σ = 8. For a sample of n = 64, what is Var(X̄) = σ²/n?
Var(X̄) = σ²/n = 64/64 = 1.
Practice
By the Central Limit Theorem, the sample mean is approximately normal for a large sample, whatever the population shape.
The CLT makes X̄ approximately normal as the sample size grows.
Confidence intervals
- A confidence interval gives a range that probably contains the true mean.
- For a normal population with known $\sigma$ (or a large sample), a 95% interval is:
$$\bar{x} \pm 1.96\,\frac{\sigma}{\sqrt{n}}$$
Practice
A sample of n = 64 has σ = 8. What is the margin 1.96 × σ/√n for a 95% interval? (2 dp)
1.96 × 8/√64 = 1.96 × 8/8 = 1.96.
You've got it
Key idea
- $E(\bar{X}) = \mu$, $\text{Var}(\bar{X}) = \dfrac{\sigma^2}{n}$; CLT makes $\bar X$ ≈ normal for large $n$
- a 95% confidence interval: $\bar{x} \pm 1.96\dfrac{\sigma}{\sqrt{n}}$
- good sampling needs randomness