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Chapter Nine. Audit Sampling: An Application to Substantive Tests of Account Balances. 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:

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## Audit Sampling: An Application to Substantive Tests of Account Balances

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**Chapter Nine**Audit Sampling: An Application to Substantive Tests of Account Balances**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: • Desired confidence level. • Tolerable misstatement. • Expected misstatement. • Population plays a bigger role in some of the sampling techniques used for substantive testing. • Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk.**Book value of inventory account balance**€ 3,000,000 Book value of items sampled € 100,000 Audited value of items sampled 98,000 Total amount of overstatement observed in audit sample € 2,000 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 bestestimate of misstatement of inventory of €60,000.(€3,000,000 × 2%)**Substantive Tests of Details of Account Balances**The results of our audit test depend upon the tolerable misstatement associated with the inventory account. If the tolerable misstatement is €50,000, we cannot conclude that the account is fairly stated because our best estimate of the projected misstatement is greater than the tolerable misstatement.**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.**Monetary-Unit Sampling (MUS)**MUS uses attribute-sampling theory (used primarily to test controls) 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.**Monetary-Unit Sampling (MUS)**Advantages of MUS When the auditor expects no misstatement, MUS usually results in a smaller sample size than classical variables sampling. The calculation of the sample size and evaluation of the sample results are not based on the variation between items in the population. When applied using the probability-proportional-to-size procedure, MUS automatically results in a stratified sample.**Monetary-Unit Sampling (MUS)**Disadvantages of MUS The selection of zero or negative balances generally requires special design consideration. The general approach to MUS assumes that the audited amount of the sample item is not in error by more than 100%. When more than one or two misstatements are detected, the sample results calculations may overstate the allowance for sampling risk.**Steps in MUS Sampling**• Sampling may be used for substantive testing to: • Test the reasonableness of assertions about a financial statement amount (i.e. is the amount fairly stated). This is the most common use of sampling for substantive testing. • Develop an estimate of some amount.**For MUS the population is defined as the monetary value of**an account balance, such as accounts receivable, investment securities or inventory. Steps in MUS Sampling**An individual euro represents the sampling unit.**Steps in MUS Sampling**Steps in MUS Sampling**A misstatement is defined as the difference between monetary amounts in the client’s records and amounts supported by audit evidence.**The auditor selects a sample for MUS by using a systematic**selection approach calledprobability-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 and items or ‘logical units’ greater than the interval will always be selected. Steps in MUS Sampling**Cumulative**Sample Account Balance Euros Item 1001 Ace Emergency Centre € 2,350 € 2,350 € 3,977 1002 Admington Hospital 15,495 17,845 € 3,977 (1) 26,882 1003 Jess Base 945 18,780 € 30,859 1004 Good Hospital Corp. 21,893 40,673 30,859 (2) 1005 Jen Mara Corp. 3,968 44,641 1006 Axa Corp. 32,549 77,190 57,741 (3) 1007 Green River Mfg. 2,246 79,436 1008 Bead Hospital Centres 11,860 91,306 84,623 (4) • • • • • • • • 1213 Andrew Call Medical - 2,472,032 1214 Lilly Health 26,945 2,498,977 2,477,121 (93) 1215 Janyne Ann Corp. 1,023 € 2,500,000 Total Accounts Receivable € 2,500,000 Steps in MUS Sampling 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 €3,977the auditor would select the following items for testing:**After the sample items have been selected, the auditor**conducts the planned audit procedures on the logical units containing the selected euro sampling units. Steps in MUS Sampling**The misstatements detected in the sample must be projected**to the population. Let’s look at the following example: Example Information Book value € 2,500,000 Tolerable misstatement € 125,000 Sample size 93 Desired confidence level 95% Expected amount of misstatement € 25,000 Sampling interval € 26,882 Steps in MUS Sampling**Steps in MUS Sampling**Basic Precision using the Table 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**Tainting**Customer Book Value Audit Value Difference Factor Good Hospital € 21,893 € 18,609 € 3,284 15% Marva Medical Supply 6,705 4,023 2,682 40% Axa Corp. 32,549 30,049 2,500 NA Learn Heart Centres 15,000 - 15,000 100% Steps in MUS Sampling Misstatements DetectedIn the sample of 93 items the following misstatements were found: Because the Axa balance of €32,549is 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. €3,284 ÷ €21,893 = 15%**Steps in MUS Sampling**Compute the Upper Misstatement LimitWe 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. Tainting Sample Projected 95% Upper Upper Misstatement Customer Factor Interval Misstatement Limit 80,646 Basic Precision 1.00 € 26,882 NA 3.0 € Learn Heart Centres 1.00 26,882 26,882 1.7 (4.7 - 3.0) 45,700 Marva Medical 0.40 26,882 10,753 1.5 (6.2 - 4.7) 16,130 5,645 Good Hospital 0.15 26,882 4,032 1.4 (7.6 - 6.2) Add misstatments greater that the sampling interval: 2,500 Axa Corp. NA 26,882 NA 150,621 Upper Misstatement Limit € (0.15 × €26,882 × 1.4 = €5,645)**In our example, the final decision is whether the accounts**receivable balance is materially misstated or not. Steps in MUS Sampling 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.**Steps in MUS Sampling**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: • Increase the sample size. • Perform other substantive procedures. • Request the client adjust the accounts receivable balance. • If the client refuses to adjust the account balance, the auditor would consider issuing a qualified or adverse opinion.**Book**Audit Tainting Customer Value Value Difference Factor Wayne County Medical € 2,000 € 2,200 € (200) -10% 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)**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 transaction. When using non-statistical sampling, the following items must be considered: • Identifying individually significant items. • Determining the sample size. • Selecting sample items. • Calculating the sample results.**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.**Determining the Sample Size and Selecting the Sample**SampleSize Sampling Population book valueTolerable – Expected misstatement = × Confidence factor Auditing standards require that the sample items be selected in such a way that the sample can be expected to represent the population.**If the population total is €200,000, the projected**misstatement would be €20,000 (€200,000 × 10%) 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. This approach is known as ratio projection. Assume the auditor finds €1,500 in misstatements in a sample of €15,000. The misstatement ratio is 10%.**Calculating the Sample Results**A second method is the difference projection. This method projects the averagemisstatement of each item in the sample to all items in the population. Assume misstatements in a sample of 100 items total €300 (for an average misstatement of €3), and the population contains 10,000 items. The projected misstatement would be €30,000 (€3 × 10,000).**Non-Statistical Sampling Example**The auditor’s of Calabro Wireless Service have decided to use non-statistical sampling to examine the accounts receivable balance. Calabro has a total of 11,800(15 + 250 + 11,535) accounts with a balance of €3,717,900. The auditor’s stratify the accounts as follows:**Non-Statistical Sampling Example**• The auditor decides . . . • Based on the results of the tests of controls, the risk of material misstatement is assessed as low. • The tolerable misstatement is €55,000, and the expected misstatement is €15,000. • The desired level of confidence is moderate based on the other audit evidence already gathered. • All customer account balances greater than €25,000 are to be audited.**€3,717,900 – €550,000**€3,167,900€40,000 SampleSize × 1.2 = 95(rounded) = Non-Statistical Sampling Example × Confidence factor SampleSize Sampling population book valueTolerable - Expected misstatement = €55,000 – €15,000**Amount of**Book Value Audit Value Over- Stratum Book Value of Sample of Sample Statement >€25,000 € 550,000 € 550,000 € 549,500 € 500 >€3,000 850,500 425,000 423,000 2,000 <€3,000 2,317,400 92,000 91,750 250 Non-Statistical Sampling Example The auditor sent positive confirmations to each of the 110 (95 + 15) 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**Ratio of Misstatement Projected Stratum Misstatement in Stratum Tested Misstatement >€25,000 € 500 100% € 500 >€3,000 2,000 €2,000 ÷ 425,000 × €850,500 4,002 <€3,000 250 €250 ÷ 92,000 × €2,317,400 6,298 Total projected misstatement € 10,800 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.**Why Did Statistical Sampling Fall Out Of Favour?**• Firms found that some auditors were over relying on statistical sampling techniques to the exclusion of good judgement. • There appears to be poor linkage between the applied audit setting and traditional statistical sampling applications.**Classical Variables 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.**Classical Variables Sampling**A sampling distribution is useful because it allows us to estimate the probability of observing any single sample result. In classical variables sampling, the sample mean is the best estimateof the population mean.**Classical Variables Sampling**• Advantages • When the auditor expects a relatively large number of differences between book and audited values, this method will normally result in smaller sample size than MUS. • The techniques are effective for both overstatements and understatements. • The selection of zero balances generally does not require special sample design considerations.**Classical Variables Sampling**• Disadvantages • Does not work well when little or no misstatement is expected in the population. • To determine sample size, the auditor must estimate the standard deviation of the audit differences. • If few misstatements are detected in the sample data, the true variance tends to be underestimated, and the resulting projection of the misstatements and the related confidence limits are not likely to be reliable.**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.**2**Population size (in sampling units)× CC× SDTolerable misstatement – Estimated misstatement SampleSize = Applying Classical Variables Sampling Determining the Sample Size where CC = Confidence coefficientSD = Estimated standard deviation of audit differences.**Applying Classical Variables Sampling**The Confidence Coefficient (CC) is associated with the desired level of confidence. The desired level of confidence is the complement of the risk that the auditor will mistakenly accept a population as fairly stated when the true population misstatement is greater than tolerable misstatement.**2**SampleSize 5,500 × 1.96 × €31€50,000 – €20,000 = 125 = 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 desired confidence is 95%. Based on work completed last year, the auditor estimates the standard deviation at €31. Let’s calculate sample size.**Meanmisstatementper samplingitem**Total audit differenceSample size = 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 totalled €330.20. Our first calculation is the mean misstatement in an individual account which is calculated as follows: €330.20125 €2.65 =**Projectedpopulationmisstatement**Population size × Mean misstatement per sampling item = (in sampling units) Applying Classical Variables Sampling The mean misstatement must be projected to the population €14,575= 5,500 × €2.65**SampleSize**Mean differenceper sampling item2 × Total squaredaudit differences – Sample size – 1 = €36,018.32 – (125 × 2.652)125 – 1 = €16.83 Applying Classical Variables Sampling The formula for the standard deviation is . . . SD =**SD**Confidencebound Populationsize = × × CC Sample size €16.83 5,500 × 1.96 × = √ 125 Projectedmisstatement Confidencebound Confidenceinterval ± = = €14,575 ± €16,228 Applying Classical Variables Sampling €16,228**Lowerlimit**Projectedmisstatement Upperlimit (€1,653) €14,575 €30,803 Applying Classical Variables Sampling (€50,000) €0 €50,000 Tolerable Misstatement If both limitsare within the bounds of tolerable misstatement, the evidence supports the conclusion that the account is not materially misstated.

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