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Chapter 8: Confidence Intervals

Overview

Confidence intervals quantify the uncertainty in estimating population parameters by providing a range of plausible values together with a specified level of confidence. This chapter develops the theory and construction of confidence intervals for means, proportions, and variances across one-sample, two-sample, and paired-sample designs. It also addresses the practical question of how large a sample is needed to achieve a desired margin of error.

Chapter Structure

8.1 Foundations

The foundational theory behind confidence intervals, including what they are, how to interpret them correctly, and the most common misconceptions:

  • Confidence Level and Coverage -- Defines confidence intervals formally, explains the role of the confidence level (90%, 95%, 99%), and derives the general form as point estimate plus or minus margin of error.
  • Interpretation and Common Pitfalls -- Clarifies the repeated-sampling interpretation of confidence intervals and debunks the common misconception that a 95% CI means there is a 95% probability that the parameter lies in the interval.

8.2 One-Sample Intervals

Confidence intervals for a single population parameter estimated from one sample:

  • CI for the Mean (mu) -- Constructs z-intervals (known variance) and t-intervals (unknown variance) for the population mean, including validity conditions and the role of the Central Limit Theorem.
  • CI for a Proportion (p) -- Derives the Wald z-interval for a population proportion, discusses the Wilson and Clopper-Pearson alternatives, and states the normal approximation conditions.
  • CI for the Variance (sigma squared) -- Uses the chi-square distribution to build a confidence interval for the population variance, requiring the normality assumption.

8.3 Two-Sample Intervals

Confidence intervals for comparing parameters from two independent populations:

  • CI for the Difference of Means (mu1 minus mu2) -- Covers the z-interval (known variances), pooled t-interval (equal variances), and Welch t-interval (unequal variances) using the Welch-Satterthwaite degrees of freedom.
  • CI for the Difference of Proportions (p1 minus p2) -- Presents the Wald interval for the difference of two proportions and compares it with the Newcombe and Clopper-Pearson alternatives.
  • CI for the Variance Ratio (sigma1 squared over sigma2 squared) -- Constructs an F-distribution-based confidence interval for the ratio of two population variances.

8.4 Paired-Sample Intervals

Confidence intervals for data where observations are naturally paired:

  • CI for the Mean of Differences (mu_D) -- Applies the one-sample t-interval to the paired differences, estimating the mean difference between two related measurements.
  • When to Use Paired vs Independent Designs -- Provides guidance on choosing between paired and independent designs based on expected within-pair correlation and study logistics.
  • Paired Interval for Proportions (McNemar) -- Addresses paired binary data using McNemar's approach, focusing on discordant pairs to construct a confidence interval for the difference in paired proportions.

8.5 Sample Size Determination

Formulas for determining the minimum sample size required to achieve a target precision:

  • Sample Size for Desired Margin of Error -- Derives sample size formulas for estimating a mean and a proportion with a specified margin of error and confidence level.
  • Sample Size for Comparing Two Groups -- Extends sample size calculations to two-sample designs for comparing means and proportions, incorporating effect size and desired power.

8.6 Code

Complete Python implementations:

  • confidence_intervals.py -- Comprehensive demonstrations of confidence interval construction across multiple settings.
  • sample_size.py -- Sample size calculation utilities for one-sample and two-sample designs.
  • ci_mean_sim.py -- Monte Carlo simulation verifying coverage of the mean confidence interval.
  • ci_prop_sim.py -- Coverage simulation for the proportion confidence interval.
  • ci_var_sim.py -- Coverage simulation for the variance confidence interval using the chi-square distribution.
  • ci_paired_sim.py -- Coverage simulation for the paired-sample mean difference interval.
  • ci_diff_means_sim.py -- Coverage simulation for the two-sample difference of means interval.
  • ci_mean_calc.py -- One-sample mean CI computation with step-by-step output.
  • ci_prop_calc.py -- One-sample proportion CI computation with step-by-step output.

8.7 Exercises

Practice problems covering CI interpretation, construction of one-sample and two-sample intervals, sample size determination, and the relationship between confidence level and interval width.

Prerequisites

This chapter builds on:

  • Chapter 5 (Sampling Distributions) -- The normal, t, chi-square, and F sampling distributions that serve as the basis for constructing pivotal quantities.
  • Chapter 6 (Statistical Estimation) -- Maximum likelihood and method of moments estimators, including properties such as bias and consistency.
  • Chapter 7 (Estimation of mu and sigma squared) -- The sample mean and sample variance as estimators, Bessel's correction, and their sampling distributions.

Key Takeaways

  1. A confidence interval provides a range of plausible values for an unknown parameter; the confidence level refers to the long-run coverage rate under repeated sampling, not the probability that any single interval contains the parameter.
  2. The choice between z-intervals and t-intervals depends on whether the population variance is known and the sample size; t-intervals are more appropriate when sigma is unknown and the sample is small.
  3. Two-sample intervals allow comparison of means, proportions, and variances across independent groups, with Welch's approach providing robustness when variances are unequal.
  4. Paired-sample intervals exploit the within-pair correlation to reduce variability and produce narrower intervals than independent-sample designs when pairing is meaningful.
  5. Sample size formulas translate a desired margin of error and confidence level into the minimum number of observations required, enabling researchers to plan studies with sufficient precision.