Normalintervalls

Normalintervalls, in statistics, are interval estimates for a population parameter derived under the assumption that the sampling distribution of the estimator is normal or approximately normal. They provide a range within which the parameter is believed to lie, given a chosen confidence level.

Common constructions: For a population mean μ with known population standard deviation σ, a 100(1-α)% Normalintervall is x̄ ± z_{1-α/2} (σ/√n). When σ is unknown, replace σ with s and use the t distribution: x̄ ± t_{n-1,1-α/2} (s/√n). For large n, t approximates z. For a population proportion p, the normal interval for p is p̂ ± z_{1-α/2} sqrt(p̂(1-p̂)/n).

Interpretation: If the data-generating process satisfies the assumptions, the long-run proportion of such intervals that contain the true parameter equals the confidence level (1-α). It does not imply that a specific interval has a 0.95 probability of containing the parameter.

Limitations: They depend on normality or large-sample justification; for small samples or highly skewed data, intervals may be inaccurate. They rely on accurate standard error estimation, and discrete data or boundary values can distort coverage. Alternatives include bootstrap or Bayesian credible intervals.

See also: Confidence interval, Normal distribution, Central limit theorem.