Confidence Intervals

A confidence interval (CI) is an interval estimate for a population parameter constructed so that, under repeated random sampling, a specified proportion (1-α) of intervals will contain the true parameter. It solves the problem that a point estimate x̄ alone gives no measure of precision. Principle: x̄ varies by Standard Error SE = σ/√n (or s/√n when σ unknown). By CLT, for random independent samples, x̄ ≈ Normal(μ, SE²) if population is normal or n large enough. The CI is point estimate ± critical value × SE. For mean, σ known: x̄ ± z_{α/2}·σ/√n. σ unknown: x̄ ± t_{α/2, n-1}·s/√n, where t accounts for extra uncertainty in s, with df = n-1. As n grows, t → z. For proportion p̂: p̂ ± z_{α/2}·√[p̂(1-p̂)/n]. Width depends on: confidence level (1-α) via z or t, variability σ/s, and sample size n via 1/√n. Higher confidence → larger critical value → wider interval. Larger n → narrower interval. Assumptions: random sample, independent observations, n < 10% of population or with correction, approximate normality of x̄ (check n≥30 or population normal, no severe skew/outliers for t). For proportion, need n·p̂ ≥5 and n·(1-p̂) ≥5. Interpretation: A 95% CI does not mean 95% probability that this specific interval contains μ, nor that 95% of data lie in it. It means the procedure has 95% long-run coverage. CI is not a prediction interval for individual observations and is invalid if assumptions fail or bias exists.