Overview — Confidence Intervals and Interval Estimation

Topic 3 built point estimators; Topic 4 built the Bayesian alternative to them. Neither, on its own, tells a risk committee how much to trust the number it is looking at. This topic closes that gap by converting a point estimate into an interval with a guaranteed coverage probability — and it does so in a deliberate order, from the most exact construction available down to the most general. It opens with pivotal quantities, the single reusable trick (find a statistic whose distribution does not depend on the unknown parameter, then invert it) that produces exact, finite-sample-valid intervals whenever a clean pivot exists, such as the

t

and

\chi^2

pivots already built in Topic 1. It then relaxes that requirement: the Wald and likelihood-ratio intervals extend the same idea to essentially any regular parametric model, using the maximum likelihood estimator's asymptotic normality as an approximate pivot instead of an exact one. A dedicated topic then handles the two-sample case — comparing two means, two variances, or two paired series — before the topic closes with the bootstrap, which drops the requirement of a tractable pivot entirely and estimates the sampling distribution directly by simulation.

Running underneath all four lessons is a single unifying idea, made explicit in the first lesson and reused throughout: every confidence interval is secretly the acceptance region of a family of hypothesis tests, worn inside out. This duality is what lets the topic move fluidly between "here is an interval for $\theta$ " and "here is a test of $H_0:\theta=\theta_0$ " without ever doing the work twice, and it is exactly the bridge the next topic, Neyman-Pearson theory, needs to convert its optimal tests into optimal intervals.

By the end of this topic you should be able to:

define a pivotal quantity, verify a proposed statistic is pivotal, and construct an exact confidence interval by pivoting a monotone CDF;
derive and apply the exact $t$ -interval for a normal mean and the exact $\chi^2$ -interval for a normal variance;
state and apply the duality between confidence intervals and hypothesis test acceptance regions;
construct the Wald, delta-method, and profile-likelihood confidence intervals from the MLE's asymptotic normality, and identify when the Wald interval undercovers;
construct the pooled, Welch, and paired two-sample $t$ -procedures, and the two-sample $F$ -test for variance ratios, choosing correctly between them;
run the nonparametric bootstrap and construct the normal, pivotal, percentile, and BCa confidence intervals it produces.

In this lesson

Contents
The big picture

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