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Variables Sampling 689 I have edited a portion of Module G from your textbook so that it more closely follows my lecture. I need to acknowledge that this is not my original work and much of it is taken word for word from the 2nd edition of Auditing & Assurance Services by Louwers, Ramsay, Sinason and Strawser. Tad Miller Classical Variables Sampling LEARNING OBJECTIVE Understand the basic process underlying classical variables sampling in an audit examination. When performing substantive procedures, one approach is classical variables sampling. Classical variables sampling methods use normal distribution theory and the Central Limit Theorem to provide a range estimate of the account balance. The auditor uses the sample estimates to determine whether the account balance is fairly stated. The Central Limit Theorem indicates larger sample sizes provide a sampling distribution that more closely reflects a normal distribution. Therefore, larger sample sizes will yield a lower level of sampling risk. In this section, we briefly illustrate mean-per-unit classical variables sampling. We illustrate the manual calculations necessary to determine sample size and evaluate sample results. However, if clients maintain records in electronic format, auditors typically use computer software to perform these tasks. Classical Variables Sampling: Planning In the planning stages of classical variables sampling, the auditor determines the objective of sampling, defines the attribute of interest, and defines the population. We will utilize the basic information from Rice, Inc.'s accounts receivable introduced in the previous section. Recall that Rice's accounts receivable are comprised of 1,505 individual customer accounts which are recorded at $416,000. Also recall that the auditor is interested in evaluating the existence or occurrence assertion and the valuation and allocation assertion. The following assessments or judgments have been made prior to selecting individual customer accounts for confirmation: Overall exposure to audit risk = 5 percent (goal or objective) Inherent risk for the assertions = 100 percent Control risk related to the assertions = 50 percent β risk (incorrect acceptance) = 10 percent (based on above risks). α risk (incorrect rejection) = 15 percent (based on above risks). Tolerable error = $6,000 AR = IR * CR * DR DR = AR / ( IR * CR) DR = 1.00 / ( 1.00 *0.50 ) AR = 5% IR = 100% CR = 50% β = 10% α = 15% DR = 0.10 Assume that Rice's five largest customers have balances totaling $120,000 and that each of these balances is larger than the $6,000 tolerable error. Also, assume that no other customer had a balance larger than $2,000. Auditors should always examine each item that exceeds the tolerable 1 error. In this case, the auditor will audit each of the accounts for the five largest customers. The auditor would then sample from among the remaining 1,500 accounts which have a balance of $296,000. This is a form of stratification. When using classical variables sampling, we can reduce variability and ensure the selection of individually significant items by subdividing the population into different (more homogenous) groups based on account size. Stratification can reduce the necessary sample size. In the above scenario, the auditor divides the population into two strata: the five largest dollar accounts with a balance of $120,000 and the remaining 1,500 accounts with a balance of $296,000. Assuming we find no errors in the five largest account balances, then our objective is to gather evidence that the remaining accounts are not materially overstated. Although we cannot prove the recorded balance is correct using with statistics, we can provide persuasive evidence that the account balance is not materially overstated. If tolerable error is $6,000, then we want persuasive evidence that the balance in the account exceeds $410,000 ($416,000 - $6,000) or that the balance of the 1,500 remaining accounts exceeds $290,000. If we are 90% confident that the true balance of these accounts exceeds $290,000, then we would be 90% confident that the true balance of the account exceeds $410,000 and the account is not materially overstated. Classical Variables Sampling: Determining Sample Size The formula for calculating sample size using mean-per-unit estimation is: book value critical value -3 -2 -1.44 hypothetical mean 0 0 292,824 296,000 195.216 197.333 2 1.28 290,000 292,824 193.333 195.216 TE = [α/2+ β] * σ/√n N n N α β σ TE EE 1 -1 OR average 3 average n= risk of incorrect acceptance β risk = 10% one-tail risk of incorrect rejection α risk = 15% two-tail => 7.5% in each ta N*[α/2+ β] * σ) 2 TE = Sample size = Population size (number of accounts) = Reliability factor for the risk of incorrect rejection (alpha Type I) α = 15% = Reliability factor for the risk of incorrect acceptance (beta Type II) β = 10% = Standard deviation from page 4 S = 31 = Tolerable error $6,000 = Expected error The remainder of this section focuses on the two new factors that are considered in classical variables sampling: the risk of incorrect rejection and standard deviation. 2 Risk of Incorrect Rejection Classical variables sampling explicitly considers both β risk (incorrect acceptance) and α risk (incorrect rejection) to determine the sample size. α risk is the probability that the auditor concludes that the account balance is materially misstated when, in fact, it is fairly stated. As with any sampling risk, achieving lower levels of α risk would increase the necessary sample size. That is, α risk has an inverse relationship with sample size. Prior to proposing an adjustment to the financial statements based on our sample results, we would ordinarily expand the sample to include additional customers. This greatly reduces the likelihood that we will actually reject a balance that is correct. So, the issue of interest related to α risk (incorrect rejection) is efficiency; what does it cost the auditor to expand the sample? EXHIBIT G.11 Reliability Factors for Different Levels of Sampling Risk in Classical Variables Sampling no. of level of risk level No. of std dev std dev α 2-tail β 1-tail of risk α 2-tail β 1-tail 2.58 .01 .005 0.01 2.58 2.33 2.33 .02 .01 .012 2.25 1.96 .05 .025 0.05 1.96 1.65 1.65 .10 .05 0.10 1.65 1.28 1.44 .15 .075 0.15 1.44 1.04 1.28 .20 .10 0.20 1.28 0.84 1.04 .30 .15 0.84 .40 .20 0.56 .288 In contrast, if the cost sample is relatively high, the auditor would be concerned about these costs and would ordinarily choose a lower level of α risk (incorrect rejection). This lower risk would, in turn, result in an increased sample size. How are sampling risks incorporated in the determination of sample size? Exhibit G .11 provides a listing of reliability factors for various levels of α/2 risk (incorrect rejection) and β risk (incorrect acceptance). These factors represent the proportions of observations that fall within a certain number of standard deviations in a normally distributed population. For Rice, Inc. the auditor assesses α risk at 15 percent. β risk was previously established at 10 percent. Standard Deviation The standard deviation represents the variability of the population being examined; it is calculated by summing the squared differences between each item in the population and the population mean. As the population is more variable (i.e., the items comprising the population differ more widely 3 with respect to dollar amount), the standard deviation increases. When the standard deviation of dollar amounts is larger, it is more difficult for the auditor to select a representative sample. To do so, he or she needs to increase the necessary sample size. Thus, the standard deviation has a direct relationship with sample size. That is, as the standard deviation is higher, the sample size Increases. How can the auditor determine the standard deviation? In mean-per-unit estimation, the auditor is interested in knowing the standard deviation of audited values of customer accounts. The auditor can either rely on experience from prior years' audits or use a small pilot sample in the current year. Assume that the sample standard deviation for Rice, Inc.'s accounts receivable is $31. Calculating Sample Size At this point, the sample size can be determined as follows: n = (N * [α/2 + β] * σ) 2 TE (1,500 * [1.44 + 1.28] * $31) 2 $6,000 = 444.37 or 445 accounts N (the number of accounts) can be readily determined from the client's records; in this case, Rice's accounts receivable were comprised of 1,505 customer accounts, of which we have already audited each of the five individual accounts that exceeded tolerable error. The factors for α risk and β risk correspond to 15 percent α risk and 10 percent β risk are drawn from Exhibit G.11. Based on prior years' audits, the standard deviation was estimated as $31. Tolerable error was determined based on the recorded account balance and overall financial statement materiality and was established at $6,000. If the auditor had not decided to examine the five individually significant customer account balances the variability of the population would have been significantly larger. Review Checkpoints G.21 How does classical variables sampling provide the auditor with evidence as to the fairness of an account balance or class of transactions? G.23 What is the standard deviation? How does it affect the necessary sample size? Classical Variables Sampling: Selecting the Sample One of the basic tenants of statistical sampling is that each sampling unit has an equal probability of selection. The sample of Rice, Inc.'s customer accounts could be selected in either of the following ways: 1. Identify 445 random numbers and select the corresponding customer accounts for confirmation (unrestricted random selection). 2. Randomly select a starting point (or a number of starting points) in the population and select every nth customer account thereafter for confirmation (systematic random selection). Classical variables sampling defines the sampling unit as the customer account balance. As a result, the auditor will select 445 of Rice's remaining 1,500 customer account balances for examination. Without stratification, classical variables sampling does not provide the auditor with assurance that the largest dollar account balances are selected for examination. 4 Classical Variables Sampling: Measuring Sample Items Once the sample size has been determined and the sample has been selected, the auditor measures the sample items. In the audit of accounts receivable, this will require the auditor to determine the audited value of the customers' accounts receivable. This will be done using accounts receivable confirmation procedures as well as additional procedures necessary to follow up on any discrepancies revealed by the confirmation procedures. This is the stage where the auditor may be exposed to nonsampling risk if his or her substantive procedures fail to detect misstatements. Assume that the auditor's examination of the 445 customer accounts receivable revealed a total audited value of $87,220. Therefore, the mean audited value per unit is $196 ($87,220 / 445 = $196). In addition, the standard deviation of the audited values is $25, which is different than we used for planning our tests. Classical Variables Sampling: Evaluating Sample Results using critical value approach If the standard deviation of the sample was the same as that used to plan the tests, then we would compare the sample mean ($196) to the critical value ($195.22). Because the sample mean exceeds the critical value the auditor could assume that there is less than a 10% probability that the sample mean of $196 would come from a population with a true mean of $193.33. However, the sample standard deviation was $25. Therefore, the auditor must evaluate the sample results using this new information. The auditor could calculate a new critical value using the standard deviation from the sample. critical value = μ + β * σ /√n Or critical value = (BV – TE) +N* β * σ /√n 193.333 + 1.28*25/√445 290,000 + 1,500*1.28*25/√445 $ 194.85 $ 292,275.42 n = Sample size N = Population size (number of accounts) β = Reliability factor for the risk of incorrect acceptance (beta risk Type II error) σ = Standard deviation (for items selected for examination) Because the sample mean of $196 exceeds the critical value, there is less than a 10% probability that a sample mean greater than $194.85 would come from a population with a mean of $193.333. Therefore there is less than a 10% probability that the true mean is less than $193.33. There is less than a 10% probability that the recorded book value is materially overstated. Evaluating Sample Results using achieved precision approach Another approach would be to use the sample information to calculate the achieved precision. With this approach, the auditor compares the sample mean ($196) to the hypothetical mean ($193.333) and calculates the probability of obtaining such results. We would divide the $2.667 difference between our sample mean and the hypothetical mean by the standard deviation ($25/√445), or $2.667/$1.185, to determine the sample mean is 2.25 standard deviations greater than the hypothetical mean (or tolerable mean). From table G1.11, we can see that there is a 1.2% probability of a sample mean being 2.25 standard deviations greater. The achieved precision would be greater than 98.8%. We can be more than 98.8% confident that the recorded book value is not materially overstated. 5 Repeat the exercise assuming a sample mean of $194 Evaluating Sample Results using critical value approach If the standard deviation of the sample was $31, the same as that used to plan the tests, a comparison of the sample mean ($194) to the critical value ($195.22) would not allow the auditor to state that the account balance is not materially overstated. However, the sample standard deviation was $25. Therefore, when the auditor evaluates the sample results using this new information, (s)he would calculate a new critical value using the standard deviation from the sample. critical value = μ + β * σ /√n Or critical value = (BV – TE) +N* β * σ /√n 193.333 + 1.28*25/√445 290,000 + 1,500*1.28*25/√445 $194.85 $292,275.42 n = Sample size N = Population size (number of accounts) β = Reliability factor for the risk of incorrect acceptance (beta risk Type II error) σ = Standard deviation (for items selected for examination) Because the sample mean of $194 is less than the revised critical value, there is still greater than a 10% probability that a sample mean greater than $194 could come from a population with a mean of $193.333. Therefore, there is more than a 10% probability that the recorded book value is materially overstated. Achieved precision approach Another approach would be to use the sample information to calculate the achieved precision. With this approach, the auditor would compare the sample mean ($194) to the hypothetical mean ($193.333) and calculate the probability of obtaining such results. We would divide the $0.667 difference between our sample mean and the hypothetical mean by the standard deviation ($25/√445), or $0.667/$1.185 to determine that the sample mean is 0.56 standard deviations greater than the hypothetical mean (or tolerable mean). From table G1.11, we can see that there is greater than a 29% probability of a sample mean being 0.56 standard deviations greater. The achieved precision would be 71%. There is an unacceptable probability that the recorded book value might be materially overstated. 6