Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-1 Chapter Nine Audit Sampling: An Application to Substantive Tests of Account Balances Chapter Nine Audit Sampling: An Application to Substantive Tests of Account Balances

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-2 Substantive Tests of Details of Account Balances The statistical concepts we discussed in the last chapter apply to this chapter as well. Three important determinants of sample size are The statistical concepts we discussed in the last chapter apply to this chapter as well. Three important determinants of sample size are 1.Desired confidence level. 2.Tolerable misstatement (error). 3.Estimated misstatement (error). Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk. Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk. The statistical concepts we discussed in the last chapter apply to this chapter as well. Three important determinants of sample size are The statistical concepts we discussed in the last chapter apply to this chapter as well. Three important determinants of sample size are 1.Desired confidence level. 2.Tolerable misstatement (error). 3.Estimated misstatement (error). Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk. Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-3 Substantive Tests of Details of Account Balances Consider the following information about the inventory account balance of an audit client: The ratio of misstatement in the sample is 2% (€2,000 ÷ €100,000) Applying the ratio to the entire population produces a best estimate of misstatement of inventory of €60,000. (€3,000,000 × 2%) Book value of inventory account balance3,000,000€ Book value of items sampled100,000€ Audited value of items sampled98,000 Total amount of overstatement observed in audit sample2,000€

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-4 Substantive Tests of Details of Account Balances The results of our audit test depend upon the tolerable error associated with the inventory account. If the tolerable error is €50,000, we cannot conclude that the account is fairly stated because our best estimate of the projected error is greater than the tolerable error. The results of our audit test depend upon the tolerable error associated with the inventory account. If the tolerable error is €50,000, we cannot conclude that the account is fairly stated because our best estimate of the projected error is greater than the tolerable error.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-5 Monetary-Unit Sampling (MUS) MUS uses attribute-sampling theory to express a conclusion in monetary amounts (e.g. in euros or other currency) rather than as a rate of occurrence. It is commonly used by auditors to test accounts such as accounts receivable, loans receivable, investment securities and inventory.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-6 Monetary-Unit Sampling (MUS) MUS uses attribute-sampling theory to estimate the percentage of monetary units in a population that might be misstated and then multiplies this percentage by an estimate of how much the euros are misstated.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-7 Monetary-Unit Sampling (MUS) Advantages of MUS 1.When the auditor expects no misstatement, MUS usually results in a smaller sample size than classical variables sampling. 2.The calculation of the sample size and evaluation of the sample results are not based on the variation between items in the population. 3.When applied using the probability-proportional-to-size procedure, MUS automatically results in a stratified sample. 1.When the auditor expects no misstatement, MUS usually results in a smaller sample size than classical variables sampling. 2.The calculation of the sample size and evaluation of the sample results are not based on the variation between items in the population. 3.When applied using the probability-proportional-to-size procedure, MUS automatically results in a stratified sample.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-8 Monetary-Unit Sampling (MUS) Disadvantages of MUS 1.The selection of zero or negative balances generally requires special design consideration. 2.The general approach to MUS assumes that the audited amount of the sample item is not in error by more than 100%. 3.When more than one or two misstatements are detected, the sample results calculations may overstate the allowance for sampling risk. 1.The selection of zero or negative balances generally requires special design consideration. 2.The general approach to MUS assumes that the audited amount of the sample item is not in error by more than 100%. 3.When more than one or two misstatements are detected, the sample results calculations may overstate the allowance for sampling risk.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-9 Steps in MUS Sampling Application

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-10 Steps in MUS Sampling Application Sampling may be used for substantive testing to: 1.Test the reasonableness of assertions about a financial statement amount. 2.Develop an estimate of some amount. Sampling may be used for substantive testing to: 1.Test the reasonableness of assertions about a financial statement amount. 2.Develop an estimate of some amount.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-11 Steps in MUS Sampling Application For MUS the population is defined as the monetary value of an account balance, such as accounts receivable, investment securities or inventory.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-12 Steps in MUS Sampling Application An individual euro represents the sampling unit.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-13 Steps in MUS Sampling Application A misstatement is defined as the difference between monetary amounts in the client’s records and amounts supported by audit evidence.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-14 Steps in MUS Sampling Application

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-15 Steps in MUS Sampling Application The auditor selects a sample for MUS by using a systematic selection approach called probability- proportionate-to-size selection. The sampling interval can be determined by dividing the book value of the population by the sample size. Each individual euro in the population has an equal chance of being selected.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-16 Steps in MUS Sampling Application Assume a client’s book value of accounts receivable is €2,500,000, and the auditor determined a sample size of 93. The sampling interval will be €26,882 (€2,500,000 ÷ 93). The random number selected is €3,977 the auditor would select the following items for testing: 3,977€ 26,882 30,859€

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-17 Steps in MUS Sampling Application After the sample items have been selected, the auditor conducts the planned audit procedures on the logical units containing the selected euro sampling units.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-18 Steps in MUS Sampling Application The misstatements detected in the sample must be projected to the population. Book value2,500,000€ Tolerable misstatement125,000€ Sample size93 Desired confidence level5% Expected amount of misstatement25,000€ Sampling interval26,882€ Example Information

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-19 Steps in MUS Sampling Application Basic Precision Basic Precision If no misstatements are found in the sample, the best estimate of the population misstatement would be zero euros. €26,882 × 3.0 = €80,646 upper misstatement limit

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-20 Steps in MUS Sampling Application Misstatements Detected Misstatements Detected In the sample of 93 items the following misstatements were found: €3,284 ÷ €21,893 = 15% Because the Axa balance of €32,549 is greater than the interval of €26,882, no sampling risk is added. Since all the euros in the large accounts are audited, there is no sampling risk associated with large accounts. CustomerBook ValueAudit ValueDifference Tainting Factory Good Hospital21,893€ 18,609€ 3,284€ 15% Marva Medical Supply6,705 4,023 2,682 40% Axa Corp.32,549 30,049 2,500 NA Learn Heart Centers15, %

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-21 Steps in MUS Sampling Application Compute the Upper Misstatement Limit Compute the Upper Misstatement Limit We compute the upper misstatement limit by calculating basic precision and ranking the detected misstatements based on the size of the tainting factor from the largest to the smallest. (0.15 × €26,882 × 1.4 = €5,645) Customer Tainting Factor Sample Interval Projected Misstatement 95% Upper Limit Upper Misstatement Basic Precision ,882€ NA3.080,646€ Learn Heart Centers ,882 (26,882) 1.7 ( )45,700 Marva Medical ,882 (10,753) 1.5 ( )16,130 Good Hospital ,882 (4,032) 1.4 ( )5,645 Add misstatments greater that the sampling interval: Axa Corp.NA26,882 NA2,500 Upper Misstatement Limit150,621€

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-22 Steps in MUS Sampling Application We compare the tolerable misstatement to the upper misstatement limit. If the upper misstatement limit is less than or equal to the tolerable misstatement, we conclude that the balance is not materially misstated. In our example, the final decision is whether the accounts receivable balance is materially misstated or not.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-23 Steps in MUS Sampling Application In our example the upper misstatement limit of €150,621 is greater than the tolerable misstatement of €125,000, so the auditor concludes that the accounts receivable balance is materially misstated. When faced with this situation, the auditor may: 1.Increase the sample size. 2.Perform other substantive procedures. 3.Request the client adjust the accounts receivable balance. 4.If the client refuses to adjust the account balance, the auditor would consider issuing a qualified or adverse opinion. When faced with this situation, the auditor may: 1.Increase the sample size. 2.Perform other substantive procedures. 3.Request the client adjust the accounts receivable balance. 4.If the client refuses to adjust the account balance, the auditor would consider issuing a qualified or adverse opinion.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-24 Risk When Evaluating Account Balances

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-25 Why is Sample Size Not Used in Evaluating MUS Results? Most MUS evaluation approaches use the misstatement factors and increments associated with a sample size of 100, regardless of the actual sample size used by the auditor.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-26 Effect of Understatement Misstatements MUS is not particularly effective at detecting understatements. An understated account is less likely to be selected than an overstated account. The most likely error will be reduced by €2,688 (– 0.10 × €26,882) Customer Book Value Audit ValueDifference Tainting Factor Wayne County Medical2,000€ 2,200€ (200)€ -10%

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-27 Non-statistical Sampling for Tests of Account Balances The sampling unit for non-statistical sampling is normally a customer account, an individual transaction, or a line item on a transactions. When using non-statistical sampling, the following items must be considered: o Identifying individually significant items. o Determining the sample size. o Selecting sample items. o Calculating the sample results. o Identifying individually significant items. o Determining the sample size. o Selecting sample items. o Calculating the sample results.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-28 Identifying Individually Significant Items The items to be tested individually are items that may contain potential misstatements that individually exceed the tolerable misstatement. These items are tested 100% because the auditor is not willing to accept any sampling risk.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-29 Determining the Sample Size Sample Size = Population book value Tolerable misstatement × Assurance factor

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-30 Selecting Sample Items Auditing standards require that the sample items be selected in such a way that the sample can be expected to represent the population.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-31 Calculating the Sample Results One way of projecting the sampling results to the population is to apply the misstatement ratio in the sample to the population. If the population total is €200,000, the projected misstatement would be €20,000 (€200,000 × 10%) Assume the auditor finds €1,500 in misstatements in a sample of €15,000. The misstatement ratio is 10%.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-32 Calculating the Sample Results A second method is the difference estimation. This method projects the average misstatement of each item in the sample to all items in the population. The projected misstatement would be €30,000, (€300 ÷ 100 = €3 × 10,000). Assume misstatements in a sample of 100 items total €300, and the population contains 10,000 items.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-33 Non-statistical Sampling Example The auditor’s of Calabro Paging Service have decided to use non-statistical sampling to examine the accounts receivable balance. Calabro has 11,800 accounts with a balance of €3,717,900. The auditor’s stratify the accounts as follows:

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-34 Non-statistical Sampling Example The auditor’s decide... o There is a low assessment for inherent and control risk. o The tolerable misstatement is €40,000, and the expected misstatement is €15,000. o There is a moderate risk that other auditing procedures will fail to detect material misstatements. o All customer account balances greater than €25,000 are to be audited. The auditor’s decide... o There is a low assessment for inherent and control risk. o The tolerable misstatement is €40,000, and the expected misstatement is €15,000. o There is a moderate risk that other auditing procedures will fail to detect material misstatements. o All customer account balances greater than €25,000 are to be audited.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-35 Non-statistical Sampling Example Sample Size = Population book value Tolerable misstatement × Assurance factor Sample Size = €3,167,900 €40, × 1.2 = 95 rounded €3,717,900 – €550,000

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-36 Non-statistical Sampling Example The auditor sent positive confirmations to each of the 110 ( ) accounts selected. Either the confirmations were returned or alternative procedures were successfully used. Four customers indicated that their accounts were overstated and the auditors determined that the misstatements were the result of unintentional error by client personnel. Here are the results of the audit testing: Amount of Book ValueAudit ValueOver- StratumBook Valueof Sample Statement >€25,000550,000€ € 549,500€ 500€ >€3,000850, , ,000 2,000 <€3,0002,317,400 92,000 91,

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-37 Non-statistical Sampling Example As a result of the audit procedures, the following projected misstatement was prepared: The total projected misstatement of €10,800 is less than the expected misstatement of €15,000, so the auditors may conclude that there is a low risk that the true misstatement exceeds the tolerable misstatement. Amount ofProjected StratumMisstatement >€25,000500€ € >€3,0002,000 4,002 <€3, ,298 Total projected misstatement10,800€ €250 ÷ 92,000 × €2,317,400 Ratio of Misstatement in Stratum Tested 100% €2,000 ÷ 425,000 × €850,500

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-38 Why Did Statistical Sampling Fall Out Of Favor? 1.Firms found that some auditors were over relying on statistical sampling techniques to the exclusion of good judgment. 2.There appears to be poor linkage between the applied audit setting and traditional statistical sampling applications.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-39 Classical Variable Sampling Classical variables sampling uses normal distribution theory to evaluate the characteristics of a population based on sample data. Auditors most commonly use classical variables sampling to estimate the size of misstatement. Sampling distributions are formed by plotting the projected misstatements yielded by an infinite number of audit samples of the same size taken from the same underlying population.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-40 Classical Variables Sampling A sampling distribution is useful because it allows us to estimate the probability of observing any single sample result.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-41 Classical Variables Sampling In classical variables sampling, the sample mean is the best estimate of the population mean.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-42 Classical Variables Sampling Advantages 1.When the auditor expects a large number of differences between book and audited values, this method will result in smaller sample size than MUS. 2.The techniques are effective for both overstatements and understatements. 3.The selection of zero balances generally does not require special sample design considerations. Advantages 1.When the auditor expects a large number of differences between book and audited values, this method will result in smaller sample size than MUS. 2.The techniques are effective for both overstatements and understatements. 3.The selection of zero balances generally does not require special sample design considerations.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-43 Classical Variables Sampling Disadvantages 1.To determine sample size, the auditor must estimate the standard deviation of the audited value or differences. 2.If few misstatements are detected in the sample data, the true variance tends to be underestimated, and the resulting projection of the misstatements to the population is likely not to be reliable. Disadvantages 1.To determine sample size, the auditor must estimate the standard deviation of the audited value or differences. 2.If few misstatements are detected in the sample data, the true variance tends to be underestimated, and the resulting projection of the misstatements to the population is likely not to be reliable.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-44 Applying Classical Variables Sampling Defining the Sampling Unit The sampling unit can be a customer account, an individual transaction, or a line item. In auditing accounts receivable, the auditor can define the sampling unit to be a customer’s account balance or an individual sales invoice included in the account balance.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-45 Applying Classical Variables Sampling Determining the Sample Size where Z IA = One-tailed Z value for the specified level of the risk of incorrect acceptance. SD = Estimated standard deviation. Sample Size = Population size × Z IA × SD Tolerable misstatement – Estimated misstatement 2

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-46 Applying Classical Variables Sampling The risk of incorrect acceptance is the risk that the auditor will mistakenly accept a population as fairly stated when the true population misstatement is greater than tolerable misstatement.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-47 Applying Classical Variables Sampling The year-end balance for accounts receivable contains 5,500 accounts with a book value of €5,500,000. The tolerable misstatement for accounts receivable is set at €50,000. The expected misstatement has been judged to be €20,000. The risk of incorrect acceptance is 2.5%. Based on work completed last year, the auditor estimates the standard deviation at €31. Let’s calculate sample size. Sample Size 5,500 × 1.96 × €31 €50,000 – €20,000 2 = 125 = 125

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-48 Applying Classical Variables Sampling Calculating the Sample Results The sample selection usually relies on random- selection techniques. Upon completion, 30 of the customer accounts selected contained misstatements that totaled € Our first calculation is the mean misstatement in an individual account which is calculated as follows: Mean misstatement per sampling item = Total audit difference Sample size = € €2.65 = €2.65

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-49 Applying Classical Variables Sampling The mean misstatement must be projected to the population. €14,575 = 5,500 × €2.65 = €14,575 Population size × Mean misstatement per sampling item Projected population misstatement = (in sampling units)

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-50 Applying Classical Variables Sampling Point estimate of accounts receivable balance... Accounts receivable point estimate Book value – Projected population misstatement = = €5,485,425 €5,500,000 – €14,575 = €5,485,425 The sum of the audited differences squared is equal to €36, We will use this value to calculate the standard deviation.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-51 Applying Classical Variables Sampling The formula for the standard deviation is... Total audit differences squared – Mean difference per sampling item 2 Sample Size × Sample size – 1 SD = = €36, – (125 × ) 124 €16.83 = €16.83

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-52 Applying Classical Variables Sampling Confidence bound Population size Z IA SD Sample size × ×= = 5,500 × 1.96 × √ €16,228 = €16,228 Confidence interval Population point estimate Confidence bound ± = = €5,485,425 ± €16,228

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-53 Applying Classical Variables Sampling Lower bound €5,469,197 Point estimate €5,485,425 Upper bound €5,501,652 Book value €5,500,000 Confidence interval If the precision interval includes the book value, the evidence supports the conclusion that the account is not materially misstated.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-54 Applying Classical Variables Sampling Lower bound €5,469,197 Point estimate €5,485,425 Upper bound €5,501,652 Book value €5,508,000 Confidence interval (1)(2) (3) (4) When the evidence indicates that the account may be materially misstated the auditor might consider (1) increasing sample size, (2) performing additional substantive procedures, (3) adjusting the account, or (4) issue a qualified or adverse opinion.

Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 9-55 End of Chapter 9