Technical Notes On The AICPA Audit Guide Audit Sampling .

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Technical Noteson the AICPA Audit GuideAudit SamplingMarch 1, 2012Trevor R. StewartDeloitte (Retired), and Rutgers UniversityMember of the 2008 Audit Sampling Guide Task Force

2012 Edition, Version 1.0While this document is believed to contain correct information, neither the AICPA nor the author makes anywarranty, express or implied, or assumes any legal responsibility for its accuracy, completeness, or usefulness. Reference herein to any specific product or publication does not necessarily constitute or imply its endorsement, recommendation, or favoring by the AICPA or the author. The views and opinions of the authorexpressed herein do not necessarily state or reflect those of the AICPA.Microsoft and Microsoft Office Excel are registered trademarks of Microsoft Corporation.Copyright 2008, 2012 byAmerican Institute of Certified Public Accountants, Inc.New York, NY 10036-8775All rights reserved. For information about the procedure for requesting permission to make copies of any partof this work, please visit www.copyright.com or call (978) 750-8400.

PREFACEThis paper contains technical notes on the 2012 edition of the AICPA Audit Guide Audit Sampling. It updatesthe technical notes on the 2008 edition of the guide. Because there are no changes in the guide’s statisticaltables these notes are substantially unchanged from 2008. References to the guide have been updated wherenecessary, and there are a number of minor expositional and other improvements.Trevor R. StewartNew York, NYAugust 2012trsny@verizon.netiii

PREFACE TO THE 2008 EDITIONThis paper contains technical notes on the 2008 edition of the AICPA Audit Guide Audit Sampling. I havewritten the paper to document the key statistical tables in the guide for the benefit of statistical specialists,educators, students, and others. It will help firms extend the tables to cover their specific policies and guidance, individual practitioners tailor their sampling techniques to specific audit circumstances, and developerswrite software to augment or replace tables. While I have provided some theoretical background, I have assumed that the reader is familiar with the basics of audit sampling and have focused on the application of theory to the tables.In the interest of clarity and with practical computation in mind, I have explained matters in terms offunctions that are available in Microsoft Office Excel (2007 and 2003 versions), software that is widely usedby auditors. These functions can be readily translated into their equivalent in other software, and the use ofExcel for the purposes of this paper is not an endorsement of that product or its suitability for statistical calculations.Section 1, “Definitions and Conventions,” defines the terms and conventions used in the paper. Section2, “Theory and Algorithms,” provides enough theory to anchor the discussion in established statistical termsand explains specific formulas and algorithms. Section 3, “Statistical Functions in Excel,” shows how the statistical functions required for the tables may be implemented in Excel. Section 4, “Computation of Tableswith Examples,” shows how each key table can be computed, referring back to the preceding material.I wish to acknowledge the help I have received from fellow task force members, especially AbeAkresh and Bill Felix.2008 Audit Sampling Guide Task ForceLynford E. Graham, ChairmanAbraham D. AkreshMark D. MayberryJohn P. BrollyDouglas F. PrawittMichael F. CampanaTrevor R. StewartMark S. ChapinJohn R. TroyerWilliam L. Felix, Jr.Phil D. WedemeyerKenneth C. GarrettHarold I. ZeidmanAICPA StaffWilliam S. BoydTechnical Manager, Accounting & Auditing PublicationsI would also like to acknowledge the assistance obtained from several others including Lucas Hoogduin ofKPMG LLP and Paul van Batenburg of Deloitte. Further, I thank Donald Roberts of the University of Illinoisat Urbana-Champaign for his review of this document.Trevor R. StewartNew York, NYJune 2008v

CONTENTS1Definitions and Conventions .12Theory and Algorithms .32.12.22.32.42.53Statistical Functions in Excel .193.13.24The Hypergeometric Probability Distribution . 32.1.1Attributes Sample Sizes Using the Hypergeometric Distribution . 52.1.2Practical Limitations of the Hypergeometric Distribution . 5The Binomial Probability Distribution . 52.2.1Attributes Sample Sizes Using the Binomial Distribution . 62.2.2The Beta Distribution and its Relationship to the Binomial . 72.2.3Calculating the Upper Error Rate Limit . 8The Poisson Probability Distribution . 82.3.1The Gamma Distribution and its Relationship to the Poisson. 102.3.2Evaluating MUS Samples . 112.3.3MUS Sample Sizes . 12Conservatism of Binomial- and Poisson-Based Sample Design and Evaluation. 13Precision and Tolerable Misstatement in Classical Variables Sampling (Table D.1 of the Guide) . 15Built-in Excel Functions . 19Special-Purpose Functions in VBA . 203.2.1Binomial Sample Sizes. 203.2.2Poisson Sample Sizes . 213.2.3MUS Sample Design Factors . 22Computation of Tables with Examples 4.154.164.174.18Table 3.1: Effect on Sample Size of Different Levels of Risk of Overreliance and Tolerable DeviationRate . 25Table 3.2: Effect of Tolerable Rate on Sample Size . 25Table 3.3: Relative Effect of the Expected Population Deviation Rate on Sample Size . 25Table 3.4: Limited Effect of Population Size on Sample Size . 26Table 4.2: Table Relating RMM, Analytical Procedures (AP) Risk, and Test of Details (TD) Risk . 26Table 4.5: Illustrative Sample Sizes . 27Table 4.6: Confidence (Reliability) Factors. 27Table 6.1: Confidence (Reliability) Factors. 27Table 6.2: 5% of Incorrect Acceptance . 27Table A.1: Statistical Sample Sizes for Tests of Controls—5% Risk of Overreliance . 28Table A.2: Statistical Sample Sizes for Tests of Controls—10% Risk of Overreliance . 28Table A.3: Statistical Sampling Results Evaluation Table for Tests of Controls—Upper Limits at 5%Risk of Overreliance . 28Table A.4: Statistical Sampling Results Evaluation Table for Tests of Controls—Upper Limits at 10%Risk of Overreliance . 28Table C.1: Monetary Unit Sample Size Determination Tables. 28Table C.2: Confidence Factors for Monetary Unit Sample Size Design . 29Table C.3: Monetary Unit Sampling—Confidence Factors for Sample Evaluation . 29Table D.1: Ratio of Desired Allowance for Sampling Risk to Tolerable Misstatement . 29Tables Not Described . 30References .31vii

TABLE OF FIGURESFigure 1, The Binomial Distribution and Related Beta Distribution . 7Figure 2, The Poisson Distribution and Related Gamma Distribution . 11Figure 3, Conceptual Relationship Between Cumulative Distributions as Functions of n . 15Figure 4, Controlling for Both α and β Risks in an Overstatement Test . 17Figure 5, BinomSample VBA Function . 21Figure 6, PoissonSample VBA Function . 22Figure 7, MUSFactor VBA Function . 23viii

Audit Sampling: Technical Notes1 DEFINITIONS AND CONVENTIONSSymbols and terms used in this paper are set forth in the following table.TermDefinitionErrorTDeviation or misstatement.Tolerable error.Risk of incorrect rejection: the risk of concluding that the total error or the error rateexceeds what is tolerable when it does not.Risk of incorrect acceptance: the risk of concluding that the total error or error rate istolerable when it is not.Monetary total of the population.Number of items in the population.Number of errors in the population; also LT for the tolerable number of errors.Sample size.Population error rate; also, pE, pT, and pU for expected, tolerable, and upper limitrates, respectively.Number of errors in sample; also, kE for the expected number of errors; may alsodenote sum of the error taints in monetary unit sampling applications.Mean number of errors in samples of size n; also, rT and rU for tolerable and upperlimit mean number of errors, respectively.Precision of a classical variables sampling estimate.Standard deviation of an estimator for sample of size n. In the statistical literature, σnis often called the standard error of the estimator, but in auditing such use of theterm error can be confusing.Standard normal deviate such that the probability is 1 ε that z zε.Monetary unit sampling.The variable x rounded up to the nearest δ decimal places. When δ 0, x is roundedup to the nearest integer.The constant e (Euler’s e 2.7182 ) raised to the power of x. For x 0, it is theinverse (the antilogarithm) of ln(x), thus exp(ln(x)) x.The natural logarithm (base e) of x. This is the inverse of exp(x); thus ln(exp(x)) x.Microsoft Visual Basic for Applications, as implemented in Excel 2007 and 2003.αβMNLnpkrAσnzεMUSRoundUp(x, δ)exp(x)ln(x)VBAThe following mathematical conventions are used for probability distributions. For discrete probabilitydistributions, namely the hypergeometric, binomial, and Poisson distributions, the probability mass function isthe probability of obtaining exactly k errors. These are denoted as Hyp(k, ), Bin(k, ), and Poi(k, ). Thecumulative distribution function is the probability of obtaining k or fewer errors and is the sum of the massfunction from zero to k. Cumulative distribution functions are denoted by prefixing the letter C to the symbolfor the mass function; thus CHyp(k, ), CBin(k, ), and CPoi(k, ).For continuous probability distributions, the probability density function at point x is the ordinate of thefunction at that point. The density function defines the probability curve (for example, the familiar bell-1

Audit Sampling: Technical Notesshaped curve for normal distributions). The cumulative distribution function is the probability that total erroror error rate, depending on the context, does not exceed x and is represented by the area under the curve to theleft of x. Mathematically, it is the integral of the density function to the left of x. The beta and gamma densityfunctions are denoted by b(x, ) and g(x, ), respectively, while their cumulative probability distributions aredenoted by B(x, ) and G(x, ). The cumulative

This paper contains technical notes on the 2012 edition of the AICPA Audit Guide Audit Sampling. It updates the technical notes on the 2008 edition of the guide. Because there are no changes in the guide’s statistical tables these notes are substantially unchanged from 2008. References to the guide have been updated whereFile Size: 342KBPage Count: 39