Sampling and Centralizing Tendencies

Abstract

This note addresses sampling concepts introduced in a formulaic manner in managerial statistics textbooks like Anderson et. al. [ASW02] or Levin et. al. [LR97] that are used in Quantitative Methods III course. Students with quantitative backgrounds like engineers are unsatisfied with this approach and ask for further information that cannot be covered in a short duration course looking at practical implications of data analysis. At times, when proofs are provided in these textbooks, some of the reasoning or the assumptions used or both are suspect. An example of this from a popular textbook is noted in section [2.1]. This note answers the following oft repeated questions specifically and is accessible to students with very rudimentary familiarity with concepts of probability theory, expectations etc: • Why finiteness correction factor is used in simple random sampling without replacement schemes applied to small populations? A term, (^ET) appears in the efficiency measures associated with sample mean and sample proportion statistics. In addition, a term (^p) that appears in the sample variance statistic's definition, is hardly ever discussed in most textbooks.• Why estimator of population variance has a n — 1 in its denominator instead of n? • What is the mathematical basis for central limit theorem? Section [1] recaps basic concepts involving expectations of random variables, conditional expectations, covariances, variances, moments and moment generating functions. Results derived in section [1] are used subsequently. Section [2] addresses simple random sampling without replacement. Sampling statistics, unbiased and efficient estimators, estimators of mean, variance, proportion are introduced. This section also explains the finiteness correction factor that arises in simple random sampling. Extensions to sampling with replacement are described in section [3]. Section [4] quantifies the "centralizing" effect of sample mean using Chebyshev inequality and the section is used as a motivation for cental limit theorem, while section [5] proves the central limit theorem under an assumption.