Confidence Intervals

Confidence Intervals

Overview

A confidence interval is a range computed from sample data such that, before observing data, . After observing data, we compute specific bounds and say we are confident the interval contains . The general structure is point estimate ± critical value × standard error. For means with known , use the z-interval. When is unknown (almost always), use the t-interval which has heavier tails reflecting the additional uncertainty from estimating . For proportions, the Wilson score interval is preferred over the basic Wald interval, especially when is near 0 or 1. Bootstrap CIs make no distributional assumptions and work for any statistic.

Key Concepts

Common Critical Values

Factors Affecting CI Width

Quick Example

Key Takeaways

Deep Dive

For detailed explanations, worked examples, and Python implementations, explore the dedicated statistics lessons:

Confidence Intervals for Means

• Confidence Intervals for the Mean — Z-interval and t-interval with full derivations, examples, and Python code

Confidence Intervals for Proportions

• Confidence Intervals for Proportions — Wilson score interval, Agresti-Coull, and when each is appropriate

Margin of Error

• Margin of Error — Factors affecting ME, the rate, and practical guidelines for study planning

Sample Size Determination

• Sample Size Determination — Formulas for means and proportions, power considerations, and conservative estimates

Bootstrap Methods

• Bootstrap Confidence Intervals — Percentile, BCa, and pivotal methods for distribution-free inference

Related Topics

• Central Limit Theorem — Foundation for why CIs work: the sampling distribution of the mean is approximately Normal
• Point Estimation — MLE and the point estimates that CIs are built around
• Hypothesis Testing — CIs and hypothesis tests are duals: same information, different format
• t-Distribution — The heavier-tailed distribution used when σ is unknown