Sampling and Estimation

• A population is the whole group; a sample is the part observed. Samples must be random to avoid bias.
• X̄ has E(X̄) = μ and Var(X̄) = σ²/n.
• If X is normal, X̄ is exactly normal; for large n, X̄ is approximately normal for any X (Central Limit Theorem).
• Unbiased estimates: x̄ for μ, and s² = (Σx² − (Σx)²/n)/(n − 1) for σ².
• Confidence interval for μ: x̄ ± z σ/√n. For a proportion (large sample): p̂ ± z√(p̂(1 − p̂)/n).

Randomness and the CLT

A random sample gives every member of the population an equal chance of selection, independently. Choosing people outside one shop, at one time, or from volunteers is unsatisfactory because some members are more likely to be chosen — the sample may not be representative. Random numbers (from a calculator or table) can be used: number the population and select the members whose numbers come up.

The mean of a random sample of size n varies from sample to sample. Its distribution has mean μ and variance σ²/n. If X ~ N(μ, σ²) then X̄ ~ N(μ, σ²/n) exactly; by the Central Limit Theorem, for large n (say n ≥ 30) X̄ is approximately normal whatever the distribution of X.

x̄ and s²

An estimator is unbiased if, although individual estimates vary, on average it gives the true value. The sample mean x̄ is an unbiased estimate of μ. Dividing by n underestimates the variance on average, so the unbiased estimate of σ² divides by n − 1.

Estimate ± z × standard error

A 95% confidence interval for μ is an interval constructed so that, in repeated sampling, 95% of such intervals would contain μ. Use it when the population is normal with known variance, or when the sample is large (then the CLT applies and s may replace σ). For a population proportion from a large sample, the standard error is √(p̂(1 − p̂)/n).