Classical variables sampling considers each record as a sampling unit. Therefore each record has an equal chance of being selected for the sample, unlike MUS, which favors higher dollar value records. MUS treat each dollar as a sampling unit. While MUS is best used in situations where there are few expected errors, CVS is a better fit for dealing with many exceptions.

If the purpose is to look for material misstatements in an account balance or class of transactions, CVS is a good choice. However, CVS typically requires a larger sample size than other sampling methods. It is mainly used to perform substantive audit tests.

Three common types of CVS are mean-per-unit estimation, ratio estimation, and difference estimation.

- Mean per unit uses the statistical concept of mean or average discussed earlier in the Measures of Center section. Using the average value (mean) of items in your sample, you can estimate the population value.

If you had a data set of 1,000 records and determined that your sample size should be 20 out of the population of 1,000, you would total the 20 individual values and divide by the sample size to obtain the mean. Assume that the total amount for the 20 records is $5,000, so the mean is $250 ($5,000/20). With 1,000 records in your population, then the mean estimate is $250,000 ($250 x 1,000). Mean estimates can be used in conjunction with confidence level, sample risk, and error rate to project to the entire population.

- Ratio estimation uses the method of applying the sample ratio to the entire population. Assuming that the 20 audited samples result in total errors of $500 out of the total sample amount of $5,000, your misstatement ratio is 10 percent ($500/$5,000). If the total for the data set is $225,000, then the projected misstatement would be $22,500 ($225,000 x 10%).

- Similar to the ratio estimation, difference estimation uses a ratio but also incorporates the records in the population or data set. Suppose your data set has 1,000 records and your sample is 20 of those records. Your audit procedures determine $500 worth of errors. The estimated misstatement would be $25,000 (($500/20) x 1,000)).

IDEA calculates results for these three methods and more. To perform CVS, you need to determine a confidence level and the desired precision. Desired precision is the difference between the tolerable error and expected error. Small precision amounts allow for less margin of error that requires larger samples, while large precision amounts offer more leeway and require smaller samples. Determining the precision requires knowledge of the business, historical data, and past experience. Precision needs to be determined by management along with the auditor.

For statistical sampling to be considered valid, the sample size must be determined by statistical calculations and the samples taken randomly. The results of the review of the samples must then be evaluated statistically.

In this example, we believe that there may be issues with how sales are voided. From our sales database of 89,979 records, we extract 1,828 records of void sales. The total value of voided transactions is $52,104.18. We will perform CVS on the voided records.

Management believes that there are weaknesses in the POS system and that it may contain errors anywhere from 1 in 5 (20 percent) to as many as 1 in 3 cases (33 percent).

By selecting the Preparation option in the Variables sample area of IDEA (Figure 4.18), the Classical Variables: Prepare-Stratify window appears as displayed

FIGURE 4.18 Classical Variables Sampling Feature of IDEA

in Figure 4.19. Normally, there is a considerable variance between the highest and the lowest values in data sets. Stratifying the data in bands or ranges produces better samples. However, this data set is from a fast-food restaurant where each record or transaction is of low value, so there would not be a need to stratify in this case. The number of strata is entered as 1 and the automatic sampling of high-value items box is unchecked.

FIGURE 4.19 Preparing for Classical Variables Sampling

By selecting the Next button, the Classical Variables: Stratification Result screen displays, as in Figure 4.20. The mean or average is 28.50 and the standard deviation is 32.41. The histogram is right-skewed, indicating that there are outliers in the higher value ranges.

A confidence level of 95 percent, desired precision of $6,773.54, and an expected proportion of errors in the population of 20 percent are entered to compute the sample size needed.

Due to the high confidence level of 95 percent, IDEA calculates a required sample size of 256, as shown in Figure 4.21.

The next step is to randomly extract256 records to test. IDEA completes this process for you, including generating a random number seed as in Figure 4.22 .

IDEA creates the AUDIT_AMT field where the auditor can input the correct amount if the audited amount from the sample is found to be incorrect. In this case, the audit determined that 21 of 256 samples were found to be incorrectly entered into the POS system.

In the Classical Variables: Evaluate box, shown in Figure 4.23, are displays for entry of the audit results from the samples extracted as outlined in Figure 4.22.

The graph in Figure 4.24 displays six types of statistical evaluations.

1. Mean per unit

2. Difference

FIGURE 4.21 Obtaining the Required Sample Size to Meet the Classical Variables Sampling Objectives

3. Combined ratio

4. Separate ratio

5. Combined regression

6. Separate regression

By clicking on each of the types, the graph will display a report of each statistical evaluation. Since the mean-per-unit estimation method has the largest error, we will click on the word mean to open the mean-estimation report displayed in Figure 4.25 .

Based on this report, it has been determined that it is 95 percent confident that the errors in voided transactions are in the range between -$15,144.76 and -$5,189.96. The most likely total error of $10,167.36 is the difference between the sample total of $52,104.18 and the projected audited amount (based on the 21 errors) of $41,936.82. Therefore the void amounts recorded are likely to be overstated by $10,167.36 either by error or as a result of fraud.

FIGURE 4.22 Extracting the Required Sample Size

FIGURE 4.23 Entering the Results of the Audit of the Classical Variables Samples

to Be Evaluated

FIGURE 4.24 Statistical Evaluations of Classical Variables Sampling

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