Sampling, estimation and hypothesis testing
Sampling, estimation and hypothesis testing
This domain tests how sample data support statements about a population: sampling methods, standard errors, confidence intervals and significance tests for means and proportions.
Core rules
- A simple random sample gives every subset of size n an equal chance of selection; a sampling frame lists the population. Bias arises from poor selection, non-response or faulty measurement.
- For sample mean x̄ from a population with mean μ and standard deviation σ, E(x̄) = μ and standard error = σ/√n. The central limit theorem makes x̄ approximately normal for large n.
- Confidence interval for μ: x̄ ± z(σ/√n), or x̄ ± t(s/√n) when σ is unknown and the sample is small from a normal population.
- Hypothesis test: state H0 and H1, choose significance level α, calculate the test statistic, compare with critical values or use a p-value, then conclude in context.
- Type I error: rejecting a true H0. Type II error: failing to reject a false H0.
Worked example
A sample of 100 gives x̄ = 52; assume σ = 10. Test H0: μ = 50 against H1: μ > 50 at 5%. Standard error = 10/√100 = 1, so z = (52 − 50)/1 = 2. The 5% one-tail critical value is 1.645; since 2 > 1.645, reject H0. There is evidence that μ > 50.
Common traps
Confusing standard deviation with standard error; using a two-tail critical value for a one-tail test; saying a p-value is the probability that H0 is true; interpreting a confidence interval as a probability statement about μ; and ignoring assumptions such as independence.