©2005 by the McGraw-Hill Companies, Inc. All rights reserved. Module G Variables Sampling “USA Today has come out with a new survey-apparently three out of every four people make up 75% of the population.” – David Letterman McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

©2005 by the McGraw-Hill Companies, Inc. All rights reserved. Learning Objectives Define variables sampling and understand when variables sampling is used in the audit examination. Understand the basic process underlying probability proportional to size (PPS) sampling as well as when PPS sampling should be used. Identify the factors affecting the size of a PPS sample and calculate the sample size for a PPS sampling application. McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Learning Objectives (Continued) Evaluate the sample results for a PPS sample by calculating the projected misstatement, incremental allowance for sampling risk, and basic allowance for sampling risk. Understand the basic process underlying classical variables sampling as well as the use of classical variables sampling in an audit examination. Understand the basic process underlying nonstatistical sampling as well as the use of nonstatistical sampling in an audit examination. McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

©2005 by the McGraw-Hill Companies, Inc. All rights reserved. Variables Sampling Used to estimate the amount (or value) of a population Substantive procedures Estimate the account balance or misstatement Compare account balance or misstatement to recorded amount or tolerable error Types of variables sampling approaches Probability proportional to size (PPS) sampling Classical variables sampling McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Probability Proportional to Size (PPS) Sampling Defines the sampling unit as individual dollar in an account balance Auditor will select individual dollars for examination Auditor will verify entire “logical unit” containing the selected dollar Accounts receivable: Customer account Inventory: Inventory item McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Advantages of PPS Sampling Results in smaller sample sizes Includes transactions or components reflecting larger dollar amounts Effective for overstatement errors Generally simpler to use than classical variables sampling McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Disadvantages of PPS Sampling Provides a conservative (higher) estimate of misstatement Not effective for understatement or omission errors Expanding a PPS sample is difficult if the initial conclusion is to reject account balance Requires special consideration for accounts with zero or negative balances McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Major Steps in Variables Sampling Determine the objective Define characteristic of interest Define the population Determine sample size Select the sample Measure sample items Evaluate sample results Planning Performing Evaluating McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Major Steps in Variables Sampling: Planning Determine the objective of sampling Determine whether account balance is fairly stated under GAAP Define characteristic of interest Instance in which audited value of component differs from recorded value Define the population PPS sampling: Dollars comprising account balance Classical variables sampling: Individual components or transactions comprising account balance McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Major Steps in Variables Sampling: Performing Determine sample size Sampling risk (risk of incorrect acceptance) Expected error Tolerable error Population size (recorded account balance) Select sample items Measure sample items McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Effect of Factors on Sample Size in PPS Sampling McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Sampling Risks in Variables Sampling McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

©2005 by the McGraw-Hill Companies, Inc. All rights reserved. PPS Sampling Formulae Sample size Recorded Balance x Reliability Factor Tolerable Error - (Expected Error x Expansion Factor) Sampling Interval Population Size (Recorded Balance) Sample Size Controls exposure to Sampling risk McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Reliability and Expansion Factors Risk of Incorrect Acceptance 1% 5% 10% 15% 20% Reliability Factors 4.61 3.00 2.31 1.90 1.61 (0 Errors) Expansion Factors 1.90 1.60 1.50 1.40 1.30 When incorporated into formula, factors are consistent with inverse relationship between risk of incorrect acceptance and sample size. McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Major Steps in Variables Sampling: Performing Determine sample size Select sample items Select random starting point in population Bypass number of dollars equal to sampling interval Select dollar and identify entire logical unit containing selected dollar Large items may account for more than one selection Measure sample items McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Major Steps in Variables Sampling: Performing Determine sample size Select sample items Measure sample items Perform appropriate substantive procedure Calculate actual misstatement Audited value – Recorded value McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Major Steps in Variables Sampling: Evaluating Evaluate sample results Problem with actual misstatement is that it may result from a nonrepresentative sample Need to “adjust” actual misstatement to control for the risk of incorrect acceptance Calculate an Upper Error Limit McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

©2005 by the McGraw-Hill Companies, Inc. All rights reserved. Upper Error Limit Has a (1 – risk of incorrect acceptance) probability of equaling or exceeding the true amount of misstatement A (risk of incorrect acceptance) probability exists that the true amount of misstatement exceeds the upper error limit McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

©2005 by the McGraw-Hill Companies, Inc. All rights reserved. Upper Error Limit If upper error limit is $50,000 and risk of incorrect acceptance is 5% There is a 5% probability that the true misstatement exceeds $50,000 There is a 95% probability that the true misstatement is less than or equal to $50,000 95% 5% $0 $50,000 McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Components of Upper Error Limit Projected misstatement Incremental allowance for sampling risk Basic allowance for sampling risk McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Projected Misstatement Assumes the entire sampling interval contains the same percentage of misstatement as the item examined by the auditor Tainting % = Amount of Misstatement Recorded Balance of Item Projected = Sampling Interval x Tainting % Misstatement McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Incremental Allowance for Sampling Risk “Adjusts” the projected misstatement to control auditor’s exposure to sampling risk Procedure Rank all projected misstatements less than sampling interval in descending order Determine incremental reliability factor for each misstatement Multiply projected misstatement by incremental reliability factor minus 1.00 McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Basic Allowance for Sampling Risk Provides a statistical measure of the misstatement that may be included in sampling intervals in which a misstatement was not detected Basic Allowance for Sampling Risk = Sampling interval x Reliability factor McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

The Auditor’s Decision Upper error limit  Tolerable error Accept account balance as fairly recorded Upper error limit > Conclude that account balance is not fairly recorded McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Classical Variables Sampling (Sample Size) N x [R(IR) + R(IA)] x SD 2 TE - EE Differences from PPS sampling Standard deviation Risk of incorrect rejection McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Reliability Factors for Classical Variables Sampling Sampling Risk 1% 5% 10% 15% 20% Risk of Incorrect Acceptance 2.33 1.65 1.28 1.04 0.84 Rejection 2.58 1.96 1.65 1.44 1.28 When incorporated into formula, factors are consistent with inverse relationship between sampling risk and sample size. McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Classical Variables Sampling (Evaluating Results) Precision: Closeness of a sample estimate to the true value Precision Interval Estimate Estimate Estimate - Precision + Precision McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Classical Variables Sampling (Evaluating Results) Precision interval has a (1 – risk of incorrect acceptance) probability of including the true balance Decision If recorded account balance falls within the precision interval, accept as fairly recorded If recorded account balance falls outside of the precision interval, conclude that the account is misstated McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

PPS Sampling vs. Classical Variables Sampling PPS is more appropriate when: Overstatement errors are of greater concern The standard deviation is difficult or impractical to estimate No or few misstatements are anticipated The auditor wishes to begin sampling during an interim period McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

PPS Sampling vs. Classical Variables Sampling Classical variables sampling is more appropriate when: The auditor is concerned with both overstatement and understatement errors The standard deviation can be estimated Some levels of misstatement are anticipated The auditor does not begin sampling until after year-end McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Nonstatistical Sampling Does not control the auditor’s exposure to sampling risk Permitted under generally accepted auditing standards Differences Does not consider sampling risk in determining sample size or evaluating sample results May use a nonprobabilistic selection technique (block or haphazard selection) McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

Basic Procedure for Nonstatistical Sampling Select sample Does not explicitly consider sampling risk in determining sample size May use block or haphazard selection methods Measure sample items Evaluate sample results Does not consider sampling risk in projected results to population Compare determined misstatement to tolerable error McGraw-Hill/Irwin ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.