Confidence Intervals

Confidence intervals estimate unknown population parameters (e.g., a mean \(\mu\) or proportion) using sample data. While a point estimator like the sample mean \(\bar{X}\) provides a single-value estimate, it lacks a measure of reliability. An interval estimator (confidence interval) addresses this by producing a range of values with an associated confidence level (e.g., 95%). This confidence level reflects the long-run success rate of the method: if we repeatedly sampled the population and constructed intervals, 95% of those intervals would contain the true parameter. This “repeated sampling” interpretation emphasizes that the confidence level describes the method’s reliability, not the probability that a single interval contains the parameter.

To illustrate, let’s simulate 95% confidence intervals for a population mean \(\mu = 50\) using R. We’ll generate 50 samples (each with \(n=30\) observations) from a normal distribution, compute their confidence intervals, and visualize results. Plotting these intervals shows most (≈95%) capture \(\mu\), while a few miss it – mirroring the confidence level’s meaning. The proportion of intervals containing \(\mu\) will typically hover near 0.95, with deviations due to random sampling.

Key insights:

1. A 95% CI does not mean “there’s a 95% chance \(\mu\) lies in this interval” – \(\mu\) is fixed, and the interval either contains it or not.
   
2. Coverage depends on assumptions (e.g., normality via CLT for small \(n\); less critical for \(n \geq 30\)).
   
3. In practice, you only compute one interval, but its reliability is tied to the theoretical behavior of many intervals.