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:
- 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.
- Coverage depends on assumptions (e.g., normality via CLT for small \(n\); less critical for \(n \geq 30\)).
- In practice, we only compute one interval, but its reliability is tied to the theoretical behavior of many intervals. ## Lab Prep Connection
Before starting Lab 6, run the simulation below and then repeat it with different sample sizes and confidence levels. This will prepare you to interpret coverage and interval width in the lab’s simulation exercises.