9
Audit CHAPTER Sampling
Review Questions 9-1 Nonstatistical sampling is an audit sampling technique in which the risk of sampling error is estimated by the auditors using professional judgment rather than by the laws of probability. Statistical sampling involves the quantification of the risk of sampling error through the use of mathematics and laws of probability. Sampling risk is the possibility that the auditors will make an erroneous decision based on a sample result. To control sampling risk the auditors increase the size of their samples. Nonsampling risk is the risk of erroneous conclusions by the auditors based on any factor other then sampling. For example, the auditors may perform inappropriate tests, or they may not recognize errors in the sample items examined. Nonsampling risk may be controlled by adequate planning and supervision of engagements, and the establishment of effective quality control policies and procedures. The physical representation of the actual population is the recorded value that represents the population. For example, if the auditors use a computer printout of recorded accounts payable from which to sample, they must attempt to determine that it properly includes all accounts payable. All three of the methods of selecting items for examination (random number table selection, systematic selection, random number generator selection) will produce a random sample if properly applied. However, when using systematic sampling on a population that is not in random order it may be necessary to stratify the population into segments, or to use a relatively large number of starting points to produce a random sample. "Systematic selection" in auditing means drawing every nth item from the population of items to be sampled. For example, the auditors might draw every tenth check from a file of paid checks. To help insure a random sample, the auditors should ascertain that the population is in random sequence, and is not, for example, classified by size of check. If the population is not in random order, the auditors may be able to stratify it into segments, each of which is in random order. They may also guard against a nonrandom sample by using several random starting points. Systematic selection is easily applied to unnumbered documents merely by counting off the sampling interval between documents to be selected.
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Sampling without replacement means that once an item is drawn for inclusion in a sample, it is not replaced into the population prior to drawing the remaining items. Therefore, one item cannot be included more than once in a given sample. Sampling with replacement means that selected items are replaced into the population prior to drawing the next item. Under this method, it is possible for one item in the population to be drawn several times, thus representing several sample items in a given sample. Sampling without replacement results in a smaller sample because the population is treated as finite. If a particular account is drawn twice when sampling with replacement, the item is included twice in the sample. Note that when sampling without replacement the item is only included once. The three major factors that determine the sample size for an attributes sampling plan are (1) the risks of assessing control risk too low, (2) the tolerable deviation rate, and (3) the expected population deviation rate. In addition, the size of the population has a small effect on the sample size. In estimating the expected population deviation rate the auditors often use (1) the sample results from prior years, (2) experience with similar tests for other clients, or (3) results of a small pilot sample. Combining several types of attributes in a definition of a deviation is acceptable, provided that the attributes are of similar audit significance. Note, however, that if various attributes of differing importance are combined, the significance of a specified deviation rate becomes obscured. When an auditor believes that the expected rate exceeds the tolerable deviation rate, no test of controls will generally be performed. This is because control risk will normally be assessed at the maximum level since the control is believed to be ineffective since the expected deviation rate exceeds the deviation rate exceeding the tolerable deviation rate. A dual-purpose test is one which tests and internal control procedure and substantiates the dollar amount of an account balance. When performing dual-purpose test the sample size is determined by calculating the sample sizes for each of the purposes and then using the larger of the two. Yes. If the sample was drawn on a random basis, the sample results should represent the true population characteristic within a specified allowance for sampling risk and with a specified level of sampling risk. Expected deviation rate is a term associated with attributes sampling. In sampling for attributes, the required sample size will vary directly with the expected deviation rate. The higher the expected deviation rate, the larger the sample required to give an estimate of the true deviation rate with a specified sampling risk and allowance for sampling risk. Some students may point out that the higher the "true" deviation rate the smaller the sample size needed in discovery sampling. While this observation is correct, the "expected" deviation rate in a discovery sampling plan is always zero. Attributes sampling would estimate the percentage of extensions that are in error, and variables sampling would estimate total dollar amount of misstatements in the schedule. In general, attributes
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sampling estimates the deviation rate (occurrence rate) of a characteristic, and the variables sampling estimates the dollar value of a characteristic. 9-18 In using a sequential sampling plan, the auditors select a small initial sample, and based on the results of examining it they decide (1) to assess control risk at the planned level, (2) assess control risk at a level higher than the planned level, or (3) to increase their sample size to get more information about operating effectiveness of the procedure. If they decide to increase their sample size, they make a similar decision after examining the additional items. The process usually continues for no more than four stages. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) True True True True True True True False (The larger the standard deviation, the larger the required sample size.) True False (Unrestricted random sampling normally results in a larger sample size than stratified sampling.)
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In performing substantive tests of the book value of a population, the two aspects of sampling risk are: (a) The risk of incorrect rejection--that is, the possibility that the sample will indicate the existence of material misstatement when, in fact, material misstatement does not exist. The risk of incorrect acceptance--that is, the possibility that the sample results will cause the auditor to accept the population as free of material misstatement when, in fact, material misstatement does exist.
(b)
The risk of incorrect acceptance is of more significance to auditors than the risk of incorrect rejection. If auditors incorrectly reject a population, they will perform more audit procedures to determine the nature and extent of the assumed material misstatement. These additional procedures will, presumably, lead the auditors to the correct conclusion. Thus, the risk of incorrect rejection causes inefficiency in the audit process, but does not cause the auditors to overlook material misstatements in the financial statements. The risk of incorrect acceptance, on the other hand, can lead directly to material misstatement being present in the audited statements. The condition can result in losses to those relying upon the financial statements, and possible legal liability for the auditors. 9-21 The auditors may obtain an estimate of the dollar amount of the standard deviation by procedures such as by (1) using the results obtained in previous years, (2) calculating the standard deviation of the population's book values, or (3) taking a small pilot sample. The planned allowance for sampling risk is first used as an input into the formula for determining the appropriate sample size. It is then used for evaluating sample results when the sample's standard deviation is equal to the estimated standard deviation. Note, in cases in which the estimated standard deviation differs from that obtained in the sample, an adjusted allowance for sampling risk may be calculated.
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An adjusted allowance for sampling risk will be calculated when the estimated standard deviation differs from that obtained in the sample. This method keeps the risk of incorrect acceptance at it's planned level. Students may point out that it can be calculated in all circumstances for evaluation purposes in that it provides the planned allowance for sampling risk in situations in which the estimated and sample's (audited value) standard deviations are equal. The mean-per-unit sampling method's estimated total audited value of the population is $20 x 10,000 accounts, or $200,000. The use of ratio or difference estimation techniques requires that (1) each population item has a book value, (2) an audited value may be ascertained for each sample item, and (3) differences between audited and book values (misstatements) are relatively frequent. When the client's book value falls outside the interval, the auditor may decide to (1) increase the sample size of the test, (2) perform other audit tests of the account, or (3) work with the client's personnel to locate other misstated items in the account. When evaluating results using the mean-per-unit method the auditors do not directly use the difference between the audited values and the book values of the individual items in their sample. The estimated total audited value is compared to the book value of the entire population. It is not correct to say that when using nonstatistical variables sampling it is impossible to project the misstatements identified to the entire population. In fact, auditors are required to project the misstatements identified in the sample to the population regardless of whether statistical or nonstatistical sampling is used. When using nonstatistical sampling, the projected misstatement must be much lower than the tolerable misstatement in order for the auditors to conclude that an account balance is acceptable. Because statistical sampling is not being used, the auditors using nonstatistical sampling will apply judgment in determining whether the projected misstatement and the tolerable misstatement differ by a adequate amount to led to a conclusion that the account balance is acceptable.
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Questions Requiring Analysis 9-30 (a) Relative to nonstatistical sampling, statistical techniques may provide the auditors with the following advantages: (1) (2) Designing efficient samples (that may avoid "overauditing"). Measuring the sufficiency of the evidence obtained (this potentially could be helpful in a court proceeding or to help justify work to a client who was critical of the extent of testing). Objectively evaluating sample results.
(3)
(b)
Decisions requiring professional judgment in performing statistical tests of controls include: (1) Identifying the controls to be tested. This involves consideration of the types of misstatements that might occur, identifying the controls that should prevent these misstatements, and deciding whether consideration of those controls to reduce the auditors' assessment of control risk would sufficiently reduce the required amount of substantive testing to justify the related tests of controls. Defining a "deviation." If the test results are to be meaningful, all exceptions included in the definition of "deviation" must be similar in their potential audit significance. Determining the maximum tolerable deviation rate. This involves judgment because deviations do not necessarily correspond directly with misstatements in the financial statements. A single deviation may result in material misstatement, while numerous deviations may result in no misstatement. Establishing an appropriate risk of assessing control risk too low. Defining the population to be tested. The auditor may want to consider stratifying the population by time period if, for example, changes in internal control may have occurred or new personnel may have been hired. Evaluating the results. In addition to drawing statistical conclusions, the CPAs should evaluate the qualitative characteristics of the exceptions found in the sample. Also, if the sample results do not support reliance on the control, the CPA must determine the nature and extent of appropriate substantive tests.
(2) (3)
(4) (5)
(6)
(c)
When the CPAs have applied attributes sampling techniques and discovered that the deviation rate exceeds their stipulated maximum tolerable deviation rate, the following courses of action are available to them: (1) (2) (3) An investigation might be made to determine the reasons for the unexpectedly high deviation rate and to ascertain its potential effect upon the financial statements. The CPAs might extend the size of their original sample to provide a more precise estimate of the population deviation rate. In light of the higher than anticipated deviation rate, the auditors must increase their assessment of control risk in the related area. This may necessitate expanding the planned substantive test procedures for assertions that are potentially affected by the control weakness.
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(a)
The auditor's justification for accepting the uncertainties that are inherent in the sampling process is based upon the premises that (1) the cost of examining all of the financial data would usually outweigh the benefit of the added reliability of a complete (100%) examination, and (2) the time required to examine all of the financial data would usually preclude issuance of a timely auditor's report. Sampling risk arises from the possibility that, when a test is restricted to a sample, the auditor's conclusions may be different from the conclusions that might be reached if the test were applied in the same way to all items in the account balance or class of transactions. That is, a particular sample may contain proportionately more or less monetary misstatements or deviations than exist in the balance or class as a whole. Nonsampling risk includes all the aspects of audit risk that are not due to sampling. An auditor may apply a procedure to all transactions or balances and still fail to detect a material misstatement or a material internal control weakness. Nonsampling risk includes the possibility of selecting audit procedures that are not appropriate to achieve the specific objective, or failing
(b)
to recognize errors in documents examined, which would render the procedure ineffective even if all items were examined. The auditor should apply professional judgment in assessing sampling risk. In performing substantive tests of details the auditor is concerned with two aspects of sampling risk: The risk of incorrect acceptance is the risk that the sample supports the conclusion that the recorded account balance is not materially misstated when it is materially misstated. The risk of incorrect rejection is the risk that the sample supports the conclusion that the recorded account balance is materially misstated when it is not materially misstated. In performing tests of controls the auditor is also concerned with two aspects of risk: The risk of assessing control risk too low is the risk that the assessed level of control risk based on the sample is less than the true operating effectiveness of the internal control. The risk of assessing control risk too high is the risk that the assessed level of control risk based on the sample is greater than the true operating effectiveness of the internal control. The risk of incorrect acceptance and the risk of assessing control risk too low relate to the effectiveness of an audit in detecting an existing material misstatement. The risk of incorrect rejection and the risk of assessing control risk too high relate to the efficiency of the audit. 9-32 (a) (b) Attributes sampling. The appropriate table for determining sample size is illustrated in Figure 9-4. Using the row for a 2 percent expected deviation rate and the column for a tolerable deviation rate of 8 percent, we find a required sample size of 77 items, with 2 allowable deviations. No. Since 4 deviations is in excess of the allowable deviations (2) in Figure 9-4, the CPAs know that the upper limit is in excess of 8 percent. Figure 9-5 may be used to determine that when the sample contains 4 deviations the upper limit is 11.8 percent (using the table values for a sample of 75 items).
(c)
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Discovery sampling can provide a high degree of confidence of locating an example of an exception when the exception has an occurrence rate in the population of as low as one half of 1 percent. If the number of forgeries is very small, however, discovery sampling could give little assurance of locating a forged check. For example, if only one forged check exists, the probability of it being included in a sample is exactly equal to the percentage of the population examined. Discovery sampling cannot rule out the possibility of one or a few forgeries with much assurance. (a) The acceptable level of risk of incorrect rejection should be determined in light of the cost of making that type of error--that is, having to perform additional audit procedures in the event that the sample results do not allow acceptance of the population. The cost of these follow-up procedures varies with the nature of the audit work. Confirming receivables, for example, is both costly and time consuming. Therefore, auditors might limit the risk of incorrect rejection to lower levels in testing receivables than, say, in estimating sales revenue based on invoices readily available within the client's system.
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(b)
The maximum tolerable misstatement in the population is based upon the concept of materiality. It is the amount of misstatement in the account that the auditors believe would result in a material misstatement in the financial statements. In determining the acceptable level of risk of incorrectly accepting a population, Stafford should consider the assessed level of control risk and the evidence obtained from analytical procedures, as well as the risk he is willing to accept of material misstatement in the particular account.
(c)
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Case 1 (1) The required sample size from population 1 is larger than from population 2. An increase in population size, holding other factors constant, requires an increase in sample size.
Case 2 (4) The required sample size from population 1 relative to population 2 cannot be determined from this information. The greater variability of population 1 indicates the need for a larger sample, but the larger tolerable misstatement permits a smaller sample. Since the magnitude of these opposing factors is not known, the overall effect upon sample size cannot be determined.
Case 3 (1) A larger sample size is required for population 1 than for population 2. The larger size of population 1 and the smaller tolerable misstatement both indicate the need for a larger sample.
Case 4 (4) The required sample size from population 1 relative to population 2 cannot be determined from this information. The smaller size and variability of population 1 indicates the need for a smaller sample, but the lower planned risk of incorrect acceptance requires a larger sample. Since the magnitude of the factors is unknown, the overall effect cannot be determined.
Case 5 (3) A smaller sample size is required from population 1 relative to population 2. The smaller population size, larger tolerable misstatement and higher risk of incorrect acceptance all indicate the need for a smaller sample.
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There are several major problems with this sampling plan and the method used to evaluate the sample results. (1) The method of selection is likely to lead to a sample that is not representative of the population. Items on the same page of an inventory listing may be of similar type and cost. The cost of the items may have been determined by the same employee. The auditor should have considered taking a stratified sample of the inventory items. A stratified sample is more efficient and, in this case, probably more effective because it is unlikely that a $50,000 sample would be adequate to estimate the misstatement in a $5,000,000 inventory account. The variance in inventory account populations is usually very large.
(2)
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The auditor failed to project the misstatement in the sample to the population. The projected misstatement is equal to $500,000 ($5,000 x [$5,000,000/$50,000]), which is significantly greater than the $100,000 amount that is considered to be a material misstatement in the account. The auditor failed to consider the sampling risk that the true misstatement in the inventory might be significantly more than the projected misstatement. The auditor failed to propose an adjustment to the inventory for the $5,000 in known misstatements. Even though the amount is not material, the auditor should suggest that the adjustment be made.
(4)
(5)
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The risk of incorrect acceptance is controlled by tightening or widening the allowance for sampling risk (ASR). The relationship between the planned allowance for sampling risk, sampling risk, and the maximum tolerable misstatement is expressed below:
Planned ASR =
Tolerable misstateme nt 1 + (Incorrect acceptance coefficien t / Incorrect rejection coefficien t)
The tolerable misstatement was given as $500,000. The risk coefficients are taken from Figure 9-8; the coefficient for incorrect acceptance is 1.28 and the coefficient for incorrect rejection is 1.15. Thus, the required planned allowance for sampling risk may be determined as follows:
Planned ASR =
$500,000 = $236,630 1 + (1.28 / 1.15)
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(a)
(b)
Ratio and difference estimation techniques are appropriate when all of the following conditions are met: (1) every item in the population has a known book value, (2) an audited or correct value may be ascertained for every item included in the sample, and (3) differences between audited and book values are relatively frequent. The relationship of the differences in audited and book values to book values determines whether ratio or difference estimation will be more effective. Ratio estimation is more effective when the differences are nearly proportional to book values. Thus, the ratios of audited to book values will be relatively constant. This occurs more frequently than might be apparent because of the tendency for transactions (and errors) to be nearly proportional to account balances. When misstatements not are nearly proportional to account balances, the variability of the ratios of audited to book values increases. This may cause difference estimation to become the more efficient technique.
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The required sample size is calculated as:
Sample size
Population value x Reliability factor Tolerable misstatement $330,000 x 3.00 $30,000
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The required sample size is calculated as:
Sample size =
Population size x Incorrect rejection coefficien t x Estimate d standard deviation Planned allowance for sampling risk
2
Sample size =
5,000 x 2.00 x $30 $25,000
2 =
$300,000 $25,000
2 = 144 accounts
The problem states that the population consists of 5,000 invoices. Figure 9-8 shows the incorrect rejection coefficient corresponding to 4.6 percent as 2.00. The planned standard deviation and allowance for sampling risk are provided as $30, and $25,000, respectively. Multiple Choice Questions 9-41 (a) (2) Sampling risk is the risk of making a wrong decision based on a sample result. The other choices are elements of nonsampling risk. The risk of incorrect rejection and assessing control risk too high relate to the efficiency of the audit since the related errors result in additional, unneeded, audit procedures. The risks of incorrect acceptance and assessing control risk too low relate to audit effectiveness. Block sampling is least desirable. It consists of all items during a selected time period, numerical sequence, or alphabetical sequence. Due to the relatively large number of blocks needed to form a reasonable audit conclusion, block sampling cannot generally be relied upon to efficiently produce a representative sample.
(b)
(1)
(c)
(2)
(d)
(3)
A sample that is representative of the audit population must be obtained. The sample is representative in the sense that it allows the auditor to project sample results to the overall population. The sample may, but need not be random, stratified, or large. Discovery sampling is a modified case of attributes sampling. Its purpose is to detect at least one deviation, with a predetermined risk of assessing control risk too low, if the deviation rate in the population is greater than the specified tolerable deviation rate. It is effectively used when auditors are looking for critical deviations that are not expected to be frequent in number. A larger variance in the dollar value of accounts (as measured by the standard deviation) results in a larger sample size because the standard deviation is in the numerator of the equation to calculate sample size--increasing the numerator increases sample size. The other factors all result in a smaller sample size. Attribute sampling is used for test of controls since it is a plan that enables auditors to estimate the rate of deviation in a population. Tolerable misstatement is directly related to materiality. That is, as the measure of materiality increases, the tolerable misstatement allocated to the various accounts is increased. Tolerable misstatement is materiality at the account balance level. When the expected deviation rate exceeds the tolerable deviation rate it is unlikely that tests of controls will be performed. This is because in such a situation testing is only likely to reveal to the auditors that the system is not operating effectively as they expected. Mean-per-unit estimation is best because no "book value" is available for the trucks. The difference, ratio, and probability-proportional-to-size methods all require the comparison of a book value to an audited value. The estimated total audited value of the population using the mean-per-unit method is calculated as $203 (mean audited value in sample) x 1000 accounts = $203,000. When calculating the estimated audited value of the population using the difference method the average misstatement is first calculated as $203 - $200 = $3. The projected misstatement is then calculated by multiplying the average misstatement times the number of accounts in the population, or $3 x 1000 = $3000. Because the average misstatement is an understatement, the projected misstatement of $3000 is added to the population book value of $198,000 to calculate the estimated audited value of $201,000.
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Problems 9-42 SOLUTION: Judgment in sampling (Estimated time: 30 minutes) (a) Areas requiring the auditors to make judgment decisions when statistical sampling techniques are employed (only four required): (1) Defining the population, characteristics to be tested, and deviations. Unless a relationship is defined between the occurrence rate of deviations in the population and either the
(2)
(3)
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(5) (6)
propriety of the client's financial statements or the strength of internal control, little useful information is gained by estimating the occurrence rate. Determining the appropriate statistical selection techniques for drawing a random sample. The auditors must recognize the advantages and disadvantages of stratified selection, unstratified selection, and systematic selection, and determine which technique is appropriate for selecting an economical random sample. Establishing the required maximum tolerable deviation rate and the risk of assessing control risk too low for the procedure. This requires judgment decisions regarding materiality, time, cost, and the planned assessed level of control risk. Interpreting sample results. This requires a decision as to whether the results support the auditors' planned assessed level of control risk, or whether additional sampling is necessary to reach a conclusion. Following up on the discovery of critical errors or unacceptable deviation rates. Determining the circumstances under which statistical sampling is appropriate, and those in which other techniques should be used in lieu of, or to supplement, the statistical sampling techniques.
This is an open-ended question. The student may identify numerous other areas in which the auditors must make judgment decisions. (b) If the CPAs' sample shows an unacceptable deviation rate, they may take the following actions: (1) (2) (3) They may enlarge their sample to increase the precision of their estimate. They may isolate the type of deviation and expand their examination as it relates to the transactions that give rise to that type of deviation. The auditors' usual response to an unacceptably high deviation rate is to increase their assessed level of control risk. Accordingly, the auditors would increase the intensity of their substantive tests.
(c)
Techniques for selecting a unstratified random sample of accounts payable vouchers include the following: Random Sample A random sample is a sample of a given size drawn from a population in a manner such that every possible sample of that size is equally likely to be drawn. Items may be selected randomly by: (1) Table of Random Numbers. Use one of a number of published tables. Using four columns in the table, select the first 80 numbers which fall within the range of 1 to 3,200. The starting point in the table should be selected randomly and the path to be followed through the table should be set in advance and followed consistently. Random Number Generator. Using generalized audit software, generate a list of 80 random numbers.
(2)
Systematic Sample. A systematic sample is drawn by selecting every nth item beginning with one or more random starts. (1) One random starting point. Select every 40th voucher after selecting the initial voucher (from 1 to 40) randomly.
(2)
Several random starting points. For example, use two random starting points and select 40 of the 80 vouchers from each of the two sequences. Select every 80th voucher (3,200/40) after each of the two random starting points between 1 and 80 for each of the two sequences.
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SOLUTION: Sampling for Attributes (Estimated time: 30 minutes) (a) (1) Since the results of tests of controls typically play a significant role in determining the nature, timing, and extent of other audit procedures, the auditors usually specify a low level of risk of assessing control risk too low. It is usually set at 5 or 10 percent. In determining the tolerable deviation rate, an auditor should consider the planned assessed level of control risk and the extent of assurance desired from the evidential matter included in the sample. In determining the expected population deviation rate, an auditor should consider the results of prior years' tests, the overall control environment, or utilize a pilot sample. There is a decrease in sample size if the acceptable level of the risk of assessing control risk too low is increased. There is a decrease in sample size if the tolerable deviation rate is increased. There is an increase in sample size if the expected population deviation rate is increased.
(2)
(3)
(b)
(1)
(2) (3) (c)
Using a statistical sampling approach, Figure 9-5 reveals that 7 deviations in a sample of size 100 results in an achieved upper deviation rate of 12.8%, well in excess of the tolerable deviation rate (8%). The sample results should thus be interpreted as not supporting the planned assessed level of control risk. Using a nonstatistical sampling approach, the 7% estimated population deviation rate identified in the sample (7 deviations/100 sample items) approaches the tolerable deviation rate of 8%. Therefore, using a nonstatistical approach, the sample would result also be interpreted as not supporting the planned assessed level of control risk.
(d)
Statistical sampling allows the auditors to quantify sampling risk. As described in part (c), only when statistical sampling is used do the auditors know that the achieved upper deviation rate is 12.8%.
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SOLUTION: Abba Company (Estimated time: 20 minutes) (a) The additional factors affecting the size of the sample include the risk of assessing control risk too low, the tolerable deviation rate, and, to a lesser extent, the population size. Sources of the estimate of the population deviation rate include (1) pilot sample results, (2) a review of prior working papers, and (3) professional judgment based on discussions with management, etc. Table 9-5 indicates that when 4 deviations are identified (4% of 100) for a sample of 100 with a 5% risk of assessing control risk too low an achieved upper deviation rate of 9% is obtained.
(b)
(c)
(d) +Among the actions possible when computed sample results exceed the established tolerable maximums include increasing the assessed level of control risk or expanding the sample size.
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SOLUTION: Ingo Corporation (Estimated time: 20 minutes) (a) As indicated in the table, a tolerable deviation rate of 4% would indicate an assessment of control risk at the low level. From Figure 9.4 we get a sample size of 117 with one allowable deviation. Using a sample size of 100 (the largest table sample size that does not exceed the actual size) and no deviations, Figure 9.5 indicates that the achieved upper deviation rate is 3%. The assessed level of control risk would be low. Using a sample size of 100 (the largest table sample size that does not exceed the actual size) and 2 deviations, Figure 9.5 indicates that the achieved upper deviation rate is 6.2%. The assessed level of control risk would still be low as it is less than 7%. Using a sample size of 100 (the largest table sample size that does not exceed the actual size) and 8 deviations, Figure 9.5 indicates that the achieved upper deviation rate is 14.0%. The assessed level of control risk would be slightly below the maximum because the upper deviation rate falls between 11 and 20%.
(b) (c)
(d)
(e)
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SOLUTION: Scott Duffney, CPA (Estimated time: 20 minutes) (1) The mean-per-unit method estimates the total value of the population by (1) using the sample audited mean as an estimate of the true population mean, and (2) extending this estimated population mean by the number of items in the population as follows: $990 x 3,000 accounts = $2,970,000 estimated audited value The projected misstatement is equal to the client's book value minus the estimated audited value: $3,000,000 - $2,970,000 = $30,000 projected misstatement (overstatement) (2) The projected misstatement using the ratio method is calculated as follows:
Sample net misstatement x Population book value Book value of sample
Sample net misstatement = 100 x ($990 998) = -$800 Book value of the sample = 100 x $998 = $99,800
Projected misstatement =
$800 x $3,000,000 $99,800
= $24,048 overstatement The estimated audited value would be calculated as follows: Estimated audited value = Book value projected overstatement = $3,000,000 - 24,048 = $2,975,952 (3) The projected misstatement using difference estimation would be calculated as follows: Projected misstatement =
Sample net misstatement x Population items Sample items
=
$800 x 3,000 100
= $24,000 overstatement The estimated audited value would be calculated as follows: Estimated misstatement = Book value projected misstatement = $3,000,000 24,000 = $2,976,000
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SOLUTION: Mean-per-Unit Estimation (Estimated time: 25 minutes) (a) To calculate the required sample size the auditors must first calculate the planned allowance for sampling risk (ASR):
Planned ASR =
Tolerable misstateme nt 1 + (Incorrect acceptance coefficien t/Incorrec t rejection coefficien t)
Planned ASR =
$60,000 1 + (1.28 / 1.96)
= $36,296
(The tolerable misstatement is given, and the incorrect acceptance and rejection coefficients are from Figure 9-8.) The sample size may next be determined as:
Sample size =
Pop. size x Incorrect rej. coef. x Est. std. dev.
Planned allowance for sampling risk 10,000 x 1.96 x $25 $36,296 2 = 183 accounts
2
Sample size =
(b) (1)
Using mean-per-unit sampling, the point estimate of the accounts receivable audited balance is calculated by multiplying the average audited value in the sample ($146) times the number of accounts in the population (10,000), or: $146 x 10,000 accounts = $1,460,000.
(2)
Since the client's book value is given as $1,500,000, the projected misstatement is $40,000 ($1,500,000 - $1,460,000). The formula for the adjusted allowance (ASR) for sampling risk is:
(3)
Tol. mis. -
Pop. value x Inc. accept. coef. x Est. std. dev. Sample size 10,000 x 1.28 x $28 183
$60,000
(4)
$33,506
Because the interval calculated by using the point estimate of the account's audited value and adjusted ASR ($1,460,000 + $33,506) does not include the client's book value ($1,500,000) the auditor is unable to accept the population as not including misstatement in excess of the tolerable amount. SOLUTION: Jexel (Estimated time: 20 minutes) (a) To calculate the required sample size the auditors must first calculate the planned allowance for sampling risk (ASR):
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Planned ASR =
Tolerable misstatement 1 (Incorrect accept. coef./Incorrect rej. coef)
=
$182,000 1 (1.64 / 2.00)
= $100,000 (The tolerable misstatement is given, and the incorrect acceptance and rejection coefficients are from Figure 9-8.) The sample size may next be determined as:
Sample size =
Population size x Incorrect rejection coefficien t x Estimated std . dev . Planned allowance for sampling risk
2
2
10,000 x 2.00 x $25 Sample size = $100,000
= 25
(b) To calculate the acceptance interval, the point estimate and the adjusted allowance for sampling risk (ASR) must be calculated as illustrated below:
Point estimate = $122 x 10,000 = $1,220,000
Adjusted ASR = Tolerable misstatement -
N x Incorrect accept . coef x Est . std . dev . Sample size
= $182,000 -
10,000 x 1.64 x $25 25
= $100,000 Because the book value ($1,250,000) falls within the acceptance interval of the point estimate $1,220,000 + the adjusted allowance for sampling risk ($100,000), the balance may be accepted as not being materially misstated. The projected misstatement is calculated as follows: Projected misstatement = Book value Point estimate = $1,250,000 - $1,220,000 = $30,000 overstatement (c) To calculate the acceptance interval, the adjusted allowance for sampling risk (ASR) and the point estimate must first be calculated:
Adjusted ASR = Tolerable misstatement -
N x Incorrect accept . coef x Est . std . dev . Sample size
= $182,000 -
10,000 x 1.64 x $20 25
= $116,400 The point estimate is still $1,220,000. Because the book value ($1,250,000) falls within the acceptance interval of the point estimate $1,220,000 + the adjusted allowance for sampling risk ($116,400), the balance may be accepted as not being materially misstated. The projected misstatement is still $30,000.
9-49
SOLUTION: Potomac Mills (Estimated time: 30 minutes) (a) (1) Mean-per-unit estimates the total value of a population by (1) using the sample mean as an estimate of the true population mean, and (2) extending this estimated population mean by the number of items in the population. The computations are as follows: (1) Estimated population mean = $582,000 / 200 (lots) = $2,910 (per lot) (2) Estimated total audited value = $2,910 (per lot) x 2,000 (lots) = $5,820,000 (2) Ratio estimation estimates the total population value by (1) using the ratio of the sample audited values to book values as an estimate of the ratio of population audited value to book value, and (2) applying this estimated ratio to the population book value. The computations are as follow: (1) Estimated ratio of audited to book values = $582,000 / $600,000 = 97% (2) Estimated total audited value = 97% x $5,900,000 = $5,723,000 (3) Difference estimation estimates the total population value by (1) using the average difference between the audited and book values of sample items as an estimate of the average difference for all population items, (2) extending the estimated average difference by the number of items in the population, and (3) using the resulting estimate of the total difference between audited and book values to compute the estimated total value. The computations are as follow: (1) Estimated average difference in audit and book values = ($582,000 - $600,000)/200 (lots) = -$90 (per lot) (2) Estimated total difference = -$90 (per lot) x 2,000 (lots) = -$180,000
(3)
Estimated total audited value = $5,900,000 - $180,000 = $5,720,000
(b)
The sample contains an element of sampling error with respect to the average dollar value of production lots. The mean book value of the population is $2,950 ($5,900,000 / 2,000 lots),
while the mean book value in the sample is $3,000 ($600,000 / 200 lots). Mean-per-unit estimation uses the mean value of the sample as the basis for estimating total value. Thus, if the sample contains a disproportionate number of higher (or lower) priced items, this sampling error will affect the estimate of the total population value. The estimate of total value developed in ratio estimation is based upon the ratio of audited values to book values, rather than upon mean dollar value. If this ratio has no tendency to vary with the dollar value of the lot, the estimate of total value is not affected by the mean value of items in the sample. However, sampling error may still be present if the sample lots are not representative of the population with respect to the ratio of audited values to book values. 9-50 SOLUTION: Evaluating Sample Results (Estimated time: 30 minutes) (a) The auditors would project the misstatement found in the sample to the population using either the ratio or difference approach. The ratio approach would result in a projected misstatement of $65,500. This may be computed by first calculating the ratio of the audited to book value as 1.0132 ($23,100/$22,800 [since there is a net understatement of $300, the audited value is $23,180]) and estimating the audited value of the population as: 1.0132 x $5,000,000 = $5,066,000 (rounded) The projected misstatement is thus $66,000 under the ratio method. (Note: The student who rounded to 1.0, will have a book value of $5,050,000 and a projected misstatement of $50,000.) The difference approach results in an average difference of $1.50 ($300 net difference divided by 200 items). Multiplying by the 100,000 invoices indicates a projected misstatement of $62,400 ($1.50 x 41,600). The auditors would then compare the projected misstatement to the tolerable misstatement (which is not provided) to determine whether to accept or reject the population. (b) As indicated in part (a) the average difference of $1.50 is projected to a total misstatement of $62,400. However, without information on the standard deviation of the differences we are unable to determine sampling risk. If the sample of 200 invoices was selected as part of a valid sampling attributes plan, the auditors could use the 5 deviations observed in the sample to estimate the deviation rate in the population. Figure 9-5 reveals that for a sample of 200 items containing 5 deviations the achieved upper deviation rate is 5.2 percent, at a 5 percent risk of assessing control risk too low. Since the achieved upper deviation rate exceeds the stipulated maximum deviation rate of 5 percent, the sample results have not fulfilled the auditors' original requirements. Absent other circumstances, the auditors will increase the assessed level of control risk based on the results of the sample.
(c)
9-51
Landi Corporation (Estimated Time: 20 minutes) a. The required sample size is calculated as:
Sample size
Pop. value x Reliability factor Tolerable misstatement $2,500,000 x 2.30 $57,500
100
b.
(1) Ratio Estimation Method:
Projected misstateme nt =
Sample net misstateme nt Book value of sample
x Book value of population
Projected misstatement
$800 x $2,500,000 $117,647.06 $17,000
(2) Difference Estimation Method: The average difference is $8 ($800/100 items); the projected misstatement is equal to $8 x 15,000 accounts, or $120,000. c. Because the projected misstatement far exceeds the tolerable misstatement of $57,500, the auditors would reject the population. This would normally result in additional procedures aimed at identifying the other misstated accounts.
9-52
SOLUTION: Kaplan Corporation (Estimated time: 25 minutes) a. The required sample size is calculated as:
Sample Size =
Population book value x Reliabilit y factor Tolerable misstateme nt
Sample Size =
$1,000,000 x 1.6 $50,000
= 32
b. The book value of the three accounts with misstatements is $1,512 ($100 + 512 + 900) while the audited value is $1,210 ($90 + 600 + 520). Since the remainder of the sample, 29 items, has an average book and audited value of $501, the total is $14,529 ($510 x 29). Accordingly, the sample total audited and book value is as follows:
3 accounts with misstatements Other 29 accounts Total
Book Value $ 1,512 14,529 $ 16,041
Audited Value $ 1,210 14,529 $ 15,739
Accordingly, the misstatement in the sample is equal to $302 ($16,041 - $15,739)
The projected misstatement may be calculated as:
Projected misstateme nt =
Sample net misstateme nt Book value of sample
x Book value of population
$302 x $1,000,000 $18,826 $16,041
The population would be accepted because $18,826 does not approach $50,000.