Algorithm Providing Ordered Integer Sequences for Sampling with Replacement Confidence Intervals

Sampling with replacement occurs when drawing without removing individuals from finite populations. It is a common distribution technique used in physics, biology, and medicine. It is used in state analysis of qubits, the physics of particle interactions, studies of genetic variation and variability, and analyzing the treatment effects from clinical trial analyses. When applied, sample statistics should be accompanied by confidence intervals. The major difficulty in expressing the confidence intervals in sampling with replacement consists of discreetness regarding the probability distribution. As a result, no mathematical formula can handle an optimum solution. Using a simple algorithm is proposed in order to obtain confidence intervals for sampling with replacement variables (x from m trials with replacement) and their proportion (x/m). A question-based discussion is presented. Traditional confidence intervals often require large sample sizes. Confidence intervals, constructed in a deterministic way provided by the proposed algorithm for sampling with replacement, allow constructing intervals without constraints.