MODERN SOLUTIONS TO CLASSICAL PROBLEMS: INNOVATIONS IN CONFIDENCE INTERVAL METHODS

Statistical inference frequently involves estimating confidence intervals for binomial parameters, especially the proportion ppp. Among the most commonly used techniques is the Wald interval, which utilizes the sample proportion p^\hat{p}p^, the sample size nnn, and the standard normal quantile zαz_\alphazα. Despite its simplicity, the Wald interval is known for its poor performance with small sample sizes and when ppp is near 0 or 1, often leading to inaccurate coverage probabilities.To overcome these limitations, a range of alternative methods has been proposed. The Clopper-Pearson "exact" interval ensures a minimum coverage probability of 1−α1 - \alpha1−α for all values of ppp, though it tends to be conservative. The Score interval, introduced by Wilson and refined by subsequent researchers like Guan, offers improved accuracy and stability. Bayesian approaches, including those based on non-informative priors such as the Jeffreys prior, also provide flexible and effective solutions for constructing intervals. Additional techniques like the Arcsin and Logit transformations further expand the set of tools available for inference on binomial proportions.This article reviews these key methods, comparing their theoretical properties, practical strengths and weaknesses, and applicability to real-world statistical problems. Emphasis is placed on understanding when and why certain methods are preferable, depending on factors like sample size, target coverage level, and the range of the proportion being estimated. Through this comparative analysis, the study highlights the intricacies involved in constructing reliable confidence intervals for binomial data and related linear functions.