Modernizing How Students Compute Interval Estimates in Mathematical Statistics Courses

The calculation of confidence intervals is a fundamental aspect of statistical inference that undergraduate students are expected to master when taking calculus-based mathematical statistics courses. In this work, we describe an interval estimation framework that we present to statistics majors at Carnegie Mellon University, one that supersedes the traditionally taught pivotal method. When utilizing this framework, which we dub “cdf inversion,” students determine the cumulative distribution functions of interval-estimating statistics through any of a number of approaches, including analytic derivations, numerical simulations, and the bootstrap; importantly, these statistics need not be pivotal quantities. Given the cdf, students then derive interval bounds analytically (when possible) or by coding a numerical root-finding algorithm. We find that utilizing cdf inversion offers numerous advantages: for instance, by removing the need for pivotal quantities, we greatly expand the space of interval-estimation problems that mathematical statistics students can solve, while at the same time making solutions easier to formulate. In addition, because it is possible for students to fully implement cdf inversion via code, we feel that elements of the cdf-inversion framework can, with proper scaffolding, be incorporated into introductory statistics and data science class settings.