About the sample size

When planning statistical studies, it is often necessary to determine the required sample size. The rules for finding the required sample size are quite widely discussed in the works of specialists in various fields of science and practice — engineering and technology, economics and management, medicine, sociology, etc. However, usually only individual specific statements of the problem are considered, and certain recipes for calculating the required quantity are given without justification. sample size, and often erroneous. The article is devoted to the strict formulation of methods for calculating the required sample size, developed to date, as well as the justification of the corresponding algorithms from the standpoint of modern applied mathematical statistics. Four approaches are considered. In the first of them, the sample size is determined by the available resources (material, financial, personnel, etc.) that can be used to obtain sample elements (measurements, observations, analyses, experiments, tests, surveys). In the second, problems of estimating the theoretical characteristics of a random variable are considered. The required sample size is determined by the specified estimation accuracy, by which we mean the half-width of the confidence interval. The third approach is based on the theory of statistical hypothesis testing. We are talking about choosing between the null and alternative hypotheses, based on the level of significance and power of the criterion. One-sided alternatives are discussed in detail. The fourth is developed within the framework of interval data statistics. It developed a general approach based on the principle of equalizing statistical and metrological errors. Based on the known accuracy of the initial measurements, we determine the rational sample size, which is necessary from the point of view under consideration. As basic examples, the rules for calculating the required sample size when studying probability values and mathematical expectations are considered. Limit theorems of probability theory and mathematical statistics are used as the basis of the mathematical apparatus. In further research, it is advisable to study the rate of convergence in the obtained asymptotic expressions and consider finite sample sizes.

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