ConfidenceIntervallen

ConfidenceIntervallen are ranges of values derived from data that are used to estimate a population parameter with a specified level of confidence. In the frequentist framework, the interval is constructed so that, in repeated sampling, a fixed proportion of such intervals would contain the true parameter. The confidence level is typically 90%, 95%, or 99%.

To construct a confidence interval for a mean, one combines a sample statistic with its standard error. If the population variance is known, the interval is x̄ ± zα/2 · σ/√n. If σ is unknown, which is common, the interval uses the t-distribution: x̄ ± tα/2, n-1 · s/√n. For proportions, the interval is p̂ ± zα/2 · √(p̂(1−p̂)/n). For regression coefficients, intervals use estimated standard errors and the t-distribution. Nonparametric or model-free situations use bootstrap or other resampling methods.

Interpretation should be careful: a 95% confidence interval does not guarantee that the true parameter lies in the interval for this data alone. Rather, it reflects the long-run performance of the method. Wider intervals indicate more uncertainty; narrower intervals indicate more precision, all else equal. Confidence intervals are related to hypothesis testing and to Bayesian credible intervals, which have a different interpretation.