Stratified Sampling Under a Tight Budget

Abstract

This note addresses stratified 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. The authors argue that the sequential process for determining sample sizes suggested by authors like Anderson et al. [ASW02, pages 866-867] is suboptimal. In addition identical cost assumptions of the Neyman allocations are untenable in many busi¬ness scenarios. The rationale for using these inaccuracies according to some of the authors is "simplicity". Since most of these textbooks do not delve into proofs of the formulae presented and the formulae for the "complex" scenario are almost the same, this line of reasoning does not hold water. In short, authors argue that stratified sampling is a very important tool and it should be taught with appropriate formulae. The reason¬ing behind the formulae will be beyond the reach of a typical MBA class and requires understanding constrained optimization. But the intuition can be conveyed to a class without going into proofs. Section [1] provides a motivation of when stratified sampling should be used with an example. The appropriate definitions are provided in section [2] while developing a framework for stratified sampling. Section [3] compares simple random sampling with stratified sam¬pling and conditions when stratification is a better option. A nonlin¬ear programming model for stratified sampling motivated by Ben-Israel [BI88] is described in section [4]. The optimization model will be used to prove that the sequential decision process is suboptimal. In section [4] it is argued that budgetary constraint will always be binding or "tight" at an optimal solution. In section [5] two stage adaptive stratification strategies are discussed when the variances are unknown. Traditional sequential approach to sample size determina¬tion recommended by many managerial statistics books is explained in section [6].