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On Estimating the Size of a Statistical Audit

It is important at this point to determine if assessments about organizational practices will be made internally or by outside experts. The advantages to doing it internally are that direct costs are likely to be lower and the process may become an engaging organizational exercise that builds communications capacity in and of itself. The advantages to using outside experts are their objectivity, time and availability, the knowledge they bring from other organizations for comparison purposes, and the credibility that may accompany their credentials and expertise | On Estimating the Size of a Statistical Audit Ronald L. Rivest Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge MA 02139 rivest@mit.edu November 14 2006 Abstract We develop a remarkably simple and easily-calculated estimate for the sample size necessary for determining whether a given set of n objects contains b or more bad objects n 1 - exp -3 b 1 This is for sampling without replacement and a confidence level of 95 . The basis for this estimate is the following procedure a estimate the sample size t needed if sampling were to be done with replacement b estimate the expected number u of distinct elements seen in such a sample and finally c draw a sample of size u without replacement. This formula is also remarkably accurate experiments show that for n 5000 this formula gives results that are never too small with some exceptions when b 1 but are never too large by more than 4 additively . The latest version of this paper can always be found at http theory.csail.mit. edu rivest Rivest-OnEstimatingTheSizeOfAStatisticalAudit.pdf 1 1 Introduction Given a universe of n objects how large a sample should be tested to determine with high confidence whether a given number b of them or more are bad We first present a simple approximate rule of thumb the Rule of Three for estimating how big such a statistical sample should be when using sampling with replacement. This Rule of Three is simple and known although perhaps not particularly well-known. Jovanovic and Levy 5 discuss the Rule of Three its derivation and its application to clinical studies. See also van Belle 12 . We then consider the question of how many distinct elements such a sample really contains. We finally provide an Improved Rule of Three for use with sampling without replacement it corrects for the bias in the Rule of Three due to sampling with replacement rather than sampling without replacement by only sampling now without replacement the expected .