©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Audit Sampling for Tests of Details of Balances Chapter 16

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Learning Objective 1 Differentiate audit sampling for tests of details of balances and for tests of controls and substantive tests of transactions.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Tests of Details of Balances, Controls, and Substantive Transactions Tests of Controls Substantive tests of transactions Sampling for tests of details of balances

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Learning Objective 2 Apply nonstatistical sampling to tests of details of balances.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Nonstatistical Sampling There are 14 steps required in audit sampling for tests of details of balance. These steps parallel the 14 steps used for sampling for tests of controls and substantive tests of transactions. There are a few differences because of the different objectives of the tests.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Tests of Controls and Substantive Tests of Transactions Step 1State the objectives of the audit test. Step 2Decide whether audit sampling applies. Step 3Define attributes and exception conditions. Step 4Define the population. Step 5Define the sampling unit.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Audit Sampling for Tests of Details of Balances Step 1State the objectives of the audit test. Step 2Decide whether audit sampling applies. Step 3Define misstatement conditions. Step 4Define the population. Step 5Define the sampling unit. Difference

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Tests of Controls and Substantive Tests of Transactions Specify acceptable risk of assessing control risk too low. Estimate the population exception rate. Determine the initial sample size. Step 7 Step 8 Step 9 Specify the tolerable exception rate.Step 6

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Audit Sampling for Tests of Details of Balances Specify acceptable risk of incorrect acceptance. Estimate misstatements in the population. Determine the initial sample size. Step 7 Step 8 Step 9 Specify tolerable misstatement.Step 6 Difference

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Tests of Controls and Substantive Tests of Transactions Select the sample. Perform the audit procedures. Step 10 Step 11 This is the same as in “Audit Sampling for Tests of Details of Balances.”

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Tests of Controls and Substantive Tests of Transactions Generalize from the sample to the population. Analyze the exceptions. Decide the acceptability of the population. Step 12 Step 13 Step 14

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Audit Sampling for Tests of Details of Balances Generalize from the sample to the population. Analyze the misstatements. Decide the acceptability of the population. Step 12 Step 13 Step 14 Difference

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Action When a Population Is Rejected Take no action until tests of other audit areas are completed Perform expanded audit tests in specific areas Increase the sample size Adjust the account balance Request the client to correct the population Refuse to give an unqualified opinion

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Learning Objective 3 Apply monetary unit sampling.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Monetary Unit Sampling MUS is an innovation in statistical sampling methodology that was developed specifically for use by auditors.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Differences between MUS and Nonstatistical Sampling The definition of the sampling unit is an individual dollar. The population size is the recorded dollar population. Preliminary judgment of materiality is used for each account instead of tolerable misstatement.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Differences between MUS and Nonstatistical Sampling Sample size is determined using a statistical formula. A formal decision rule is used for deciding the acceptability of the population. Sample selection is done using probability proportional to size sample selection (PPS).

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley The Auditor Generalizes from the Sample to the Population Using MUS Attributes sampling tables are used to calculate the results. The attributes results must be converted to dollars. The auditor must make an assumption about the percentage of misstatement for each population item that is misstated. The statistical results when MUS is used are called misstatement bounds.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Generalizing from the Sample to the Population Overstatement amounts equal 100 percent; understatement amounts equal 100 percent; misstatement bounds at a 5 percent ARIA are: Upper misstatement bound = $1,200,000 × 3% × 100% = $36,000 Lower misstatement bound = $1,200,000 × 3% × 100% = $36,000

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Generalizing from the Sample to the Population All amounts have to be overstatements. All population items misstated have to be 100 percent misstated.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Generalizing from the Sample to the Population Overstatement amounts equal 10 percent; understatement amounts equal 10 percent; misstatement bounds at a 5 percent ARIA are: Upper misstatement bound = $1,200,000 × 3% × 10% = $3,600 Lower misstatement bound = $1,200,000 × 3% × 10% = $3,600

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Generalizing from the Sample to the Population Overstatement amounts equal 20 percent; understatement amounts equal 200 percent; misstatement bounds at a 5 percent ARIA are: Upper misstatement bound = $1,200,000 × 3% × 20% = $7,200 Lower misstatement bound = $1,200,000 × 3% × 200% = $72,000

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Appropriate Percent of Misstatement Assumption The appropriate assumption for the overall percent of misstatement in those population items containing a misstatement is an auditor’s decision.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Generalizing When Misstatements Are Found 1. Overstatement and understatement amounts are dealt with separately and then combined. 2. A different misstatement assumption is made for each misstatement, including the zero misstatements.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Generalizing When Misstatements Are Found 3. The auditor must deal with layers of the computed upper exception rate (CUER) from the attributes sampling table. 4. Misstatement assumptions must be associated with each layer.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Illustration of the Auditor’s Decision Rule for MUS LMB – Tolerable Misstatement $0 Misstatement + Tolerable Misstatement UMB LMBUMB #4 LMBUMB LMB UMB #5 #3 #2 #1 #2

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Determining Sample Size Using MUS Materiality Assumption of the average percent of misstatement for population items that contain a misstatement Acceptable risk of incorrect acceptance Recorded population value

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Determining Sample Size Using MUS Estimate of the population exception rate Relationship of the audit risk model to sample size for MUS PDR = AAR ÷ (IR × CR)

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Learning Objective 4 Describe variables sampling.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Frequency Distribution of Sample Means Value of in dollars x Frequency of values in percent

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Sampling Distribution A corollary is that the sample mean value with the highest frequency of occurrence is also equal to the population mean. The mean value of all the sample means is equal to the population mean. (X)(X) (x)(x)

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Sampling Distribution The shape of the frequency distribution of the sample means is that of a normal distribution (curve), as long as the sample size is sufficiently large, regardless of the distribution of the population.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Sampling Distribution The percentage of sample means between any two values of the sampling distribution is measurable.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Sampling Distribution for a Population Distribution Value of in dollars x Frequency of values in percent Sampling distribution – Normal Population distribution – Skewed Mean

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Variables Methods Difference estimation Ratio estimation Mean-per-unit estimation

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Stratified Statistical Methods All of the elements of the population are divided into two or more subpopulations. Each subpopulation is independently tested.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Sampling Risks (ARIA and ARIR) Conclude that theCorrect conclusionIncorrect conclusion population is – no risk – risk is ARIR materially misstated Conclude that theIncorrect conclusionCorrect conclusion population is – risk is ARIA – no risk not materially misstated Actual State of the Population Actual AuditMaterially Not Materially DecisionMisstated Misstated

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Learning Objective 5 Use difference estimation in tests of details of balances.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Plan the Sample and Calculate the Sample Size State the objectives of the audit test. Decide whether audit sampling applies. Define misstatement conditions. Define the population. Define the sampling unit. Specify tolerable misstatement.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Plan the Sample and Calculate the Sample Size Specify acceptable risk. ARIAARIR

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Estimate Misstatement in the Population Estimate an expected point estimate. Make an advance population standard deviation estimate – variability of the population.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Calculate the Initial Sample Size n = {[SD*(Z A + Z B )N] ÷ (TM – E*)} 2 n= initial sample size SD* = advance estimate of the standard deviation Z A = confidence coefficient for ARIA Z B = confidence coefficient for ARIR N= population size TM= tolerable misstatement for the population (materially) E*= estimated point estimate of the population misstatement

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Select the Sample and Perform the Procedures Select the sample. Perform the audit procedures.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Evaluate the Results Compute the point estimate of the total misstatement. Compute an estimate of the population standard deviation. Compute the precision interval. Compute the confidence limits.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Effect of Changing Factors Effect on the Computed Type of Change Precision Interval Increase ARIADecrease Increase the point estimate of the misstatementsIncrease Increase the standard deviationIncrease Increase the sample sizeDecrease

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Analyze the Misstatements The auditor must evaluate misstatements to determine the cause of each misstatement and decide whether modification of the audit risk model is needed.

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley Auditor’s Decision Rule for Difference Estimation LCL – Tolerable Misstatement $0 Misstatement + Tolerable Misstatement UCL LCLUCL #4 LCLUCL LCL UCL #5 #3 #2 #1 #2

©2003 Prentice Hall Business Publishing, Auditing and Assurance Services 9/e, Arens/Elder/Beasley End of Chapter 16