A 𝛽-content tolerance interval (TI) is a statistical interval which contains at least some fraction (proportion) of the population with a given confidence level. When we are interested in the precision of a quality characteristic, a TI for the sample variance is useful. In this paper, we consider an exact two-sided 𝛽-content TI for the sample variance from a normal distribution with a specified ratio of the tail probabilities. The proposed tolerance interval allows the practitioner more control over how the probabilities in the tails are distributed, which may be useful in certain applications. A comparison with an existing two-sided 𝛽-content TI shows that the proposed TI is better on the basis of expected coverage and standard deviation of the coverage. In addition, the proposed TI is shown to require fewer subgroups to achieve a specific accuracy level. Moreover, a phase II control chart with guaranteed performance is obtained from the proposed TI. Finally, a real and a simulated data are used for illustration.
Multivariate statistical process control (MSPC) charts are particularly useful when there is a need to simultaneously monitor several quality characteristics of a process. Most of the control charts in MSPC assume that the quality characteristics follow some parametric multivariate distribution, such as the normal. This assumption is almost impossible to be justified in practice. Distribution-free MSPC charts are attractive, as they can overcome this hurdle by guaranteeing a stable (or in-control (IC)) performance of the control chart without the assumption of a parametric multivariate process distribution. Utilizing an existing distribution-free multivariate tolerance interval, we propose a Phase II Shewhart-type distribution-free MSPC chart for individual observations, with control limits based on Phase I order statistics. In addition to being easy to interpret, the proposed chart preserves the original scale of measurements and can easily identify out-of-control variables after a signal. The exact in-control performance based on the conditional and unconditional perspectives is presented and examined along with the control limits determination. The out-of-control performance of the chart is studied by simulation for data from a number of multivariate distributions. Illustrative examples are provided for chart implementation, using both real and simulated data, along with a summary and conclusions.
Black women are struggling to properly control their hypertension. Black women continue to be disproportionately affected by hypertension due to historically systemic and structural barriers. The Social Determinants of Health related to race, gender, and access are major contributing factors. The Black woman’s plight is explained through the concepts of Intersectionality Theory and the Superwoman Schema. With interventions varying on controlling hypertension, research agrees that adherence to preventative and follow-up care is essential to improving blood pressure results. During COVID-19, there was a resurgence in the use of telehealth to improve health outcomes without the need of in-person visits. Through the evaluation of literature and content analysis, this author utilized this information to encourage the use of telehealth as a workable intervention to provide Black women with additional resources to improve efforts to properly control their hypertension.