Abstract
The Bernoulli parameter, denoted as p, is the probability of success in a trial or the probability that a nonconforming result is found when a unit is sampled. In our world, the Bernoulli parameter is widely applied in various fields, for example, simulation input modeling, statistical inference, and process control. Therefore, understanding the nature and the application of the Bernoulli parameter is an important topic in both the engineering and scientific fields. In this study, four issues related to the Bernoulli parameter are presented: 1. In the first part, we investigate the rule of thumb for the standard Wald confidence interval of a Bernoulli parameter, p. The analytical results are derived to compute the coverage probability of p and to explain the sample sizes for which the oscillation phenomenon occurs. Moreover, we correctly interpret the rule of thumb and show that satisfying a necessary condition of the rule of thumb “n >μ+ 10σ and 0<μ−10σ” can guarantee a good coverage probability. 2. In the second part, we expand on the work of Braun (1999) to consider the effects of the initial reference sample size, m, in Phase I and the on-line sampling size, n, in Phase II on the statistical performance of a conventional p chart. The conditional and marginal distributions of run length are provided, and the numerical results from in-control and out-of-control conditions with known and unknown p0 are computed for evaluation and comparison. In addition, we find that the oscillation of ARL and the right skewness of the distribution of P and P0 cause the minimal required sampling size to exceed those suggested in current textbooks. 3. In the third part, we improve upon previously published results by depicting 80 univariate probability distributions in one user-friendly ten-by-eightmatrix-format display. The figure is logically organized, and thereby allows users to easily locate a particular distribution. These 80 distributions and associated relationships provide rapid access to information that must otherwise be found through a time-consuming search of numerous sources. 4. In the last part, a discrete-event simulation approach was used tomodel the patient flow of Emergency Department’s (ED), including several estimated Bernoulli parameters, to investigate the effect of inpatient boarding on the ED efficiency in terms of the National Emergency Department Crowding Scale (NEDOCS) score and the rate of patients who leave without being seen (LWBS). The decision variable in this model was the boarderreleased-ratio, defined as the ratio of admitted patients whose boarding time is zero, to all admitted patients. Our analysis shows that the Overcrowded+ (a NEDOCS score over 100) ratio decreased from 88.4% to 50.4%, and the rate of LWBS patients decreased from 10.8% to 8.4% when the boarder-released-ratio changed from 0% to 100%. These results show that inpatient boarding significantly impacts both the NEDOCS score and the rate of LWBS patient, and this analysis provides a quantification of the impact of boarding on emergency department patient crowding.