Sampling
Warm-up
Statistical investigation
Evidence
Sampling: each sample produces a mean. Increasing sample size usually makes these means less variable.
Live sampling plot
Make your choice, then check your reasoning.
The maths
Sampling mean: E(x̄) = μ. Standard error SE(x̄) = σ/√n.
Central Limit Theorem: for independent identically distributed observations with finite variance, the standardised sample mean approaches a standard normal distribution as n increases.
Unbiased sample variance: s² = [Σx² − (Σx)²/n] / (n − 1).
Z-score: z = (x̄ − μ₀)/(σ/√n).
Right-tail test: p = P(Z ≥ z). Left-tail test: p = P(Z ≤ z).
Two-tailed test: p = 2 min[P(Z ≤ z), P(Z ≥ z)].
Reject H₀ if p ≤ α; otherwise do not reject H₀.
A sample variance uses n − 1 because estimating its mean removes one degree of freedom.