Audit Sampling
Chapter 9
McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
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What is Audit Sampling?What is Audit Sampling?
Applying a procedure to less than 100% of a population
To estimate some characteristic of the population Qualitative Quantitative
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RiskRisk
Sampling risk risk that the auditors’ conclusions based on a
sample may be different from the conclusion they would reach if they examined every item in the population
Nonsampling risk risk pertaining to nonsampling errors Can be reduced to low levels through
effective planning and supervisions of audit engagements
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Nonstatistical samplingNonstatistical sampling
The auditor estimates sampling risk by using professional judgment rather than statistical techniques
Provides no means of quantifying sampling risk
Sample may be larger than necessary or auditors may unknowingly accept a higher than acceptable degree of sampling risk
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Advantages of Statistical SamplingAdvantages of Statistical Sampling
Allows auditors to measure and control sampling risk which helps: Design efficient samples Measure sufficiency of evidence Objectively evaluate sample results
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Selection of Random SampleSelection of Random Sample
Random sample results in a statistically unbiased sample that may not be a representative sample
Random sample techniques Random number tables Random number generators Systematic selection
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Random Number Table
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Other Methods of Other Methods of Sample SelectionSample Selection
Other methods Haphazard selection
• Select items on an arbitrary basis, but without any conscious bias
Block selection• Block sample consists of all items in a selected
time period, numerical sequence or alphabetical sequence
Stratification Technique of dividing population into relatively
homogeneous subgroups
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An Illustration of Stratification An Illustration of Stratification
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Types of Statistical Sampling PlansTypes of Statistical Sampling Plans
Attributes sampling Discovery sampling Classical variables sampling
Mean-per-unit estimation Ratio estimation Difference estimation
Probability-proportional-to-size sampling
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Dual Purpose TestDual Purpose Test
Tested used both as a test of control and substantiating the dollar amount of an account balance Ex. Test to evaluate the effectiveness of a
control over recording sales transactions and to estimate the total overstatement or understatement of the sales account
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Allowance for Sampling RiskAllowance for Sampling Risk
Amount used to create a range, set by + or – limits from the sample results, within which the true value of the population characteristic being measured is likely to lie
Precision Wider the interval, more confident but less
precise conclusion Can be used to construct a dollar interval
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Sample SizeSample Size
Significant effect on allowance for sampling risk and sampling risk Sample size increase -> sampling risk and
allowance for sampling risk decrease Sample size affected by characteristics of
population Generally as Population increases -> sample
size increase
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Requirements of AuditRequirements of AuditSampling PlansSampling Plans
When planning the sample consider: The relationship of the sample to the relevant audit objective Materiality or the maximum tolerable misstatement or
deviation rate Allowable sampling risk Characteristics of the population
Select sample items in such a manner that they can be expected to be representative of the population
Sample results should be projected to the population Items that cannot be audited should be treated as
misstatements or deviations in evaluating the sample results
Nature and cause of misstatements or deviations should be evaluated
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Actual Extent of Operating Effectiveness
of the Control Procedure is
Adequate Inadequate
The Test of ControlsSample Indicates:
Extent of Operating Effectiveness is Adequate
Extent of Operating Effectiveness Inadequate
Sampling Risks--Tests of ControlsSampling Risks--Tests of Controls
CorrectDecision
Incorrect Decision
(Risk of Assessing Control Risk
Too Low)
IncorrectDecision
(Risk of AssessingControl Risk
Too High)
Correct Decision
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Audit Sampling Steps for Audit Sampling Steps for Tests of ControlsTests of Controls
Determine the objective of the test Define the attributes and deviation conditions Define the population to be sampled Specify:
The risk of assessing control risk too low The tolerable deviation rate
Estimate the population deviation rate Determine the sample size Select the sample Test the sample items Evaluate the sample results Document the sampling procedure
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Attributes Sampling: Relationship Between the Attributes Sampling: Relationship Between the Planned Assessed Level of Control Risk and the Planned Assessed Level of Control Risk and the
Tolerable Deviation RateTolerable Deviation Rate
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Illustration of Attributes Sampling--Illustration of Attributes Sampling--Determining Sample SizeDetermining Sample Size
Risk of Assessing Control Risk Too Low—5 percent
Tolerable Deviation Rate—9 percent Expected Population Deviation Rate—2
percent
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Figure 9.4: Statistical Sample Sizes for Tests of Controls Figure 9.4: Statistical Sample Sizes for Tests of Controls at 5 Percent Risk of Assessing Control Risk Too Lowat 5 Percent Risk of Assessing Control Risk Too Low
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Sample SizeSample Size
Sample size using Figure 9-4 (next slide)
=68 (2)
This means the auditor should select a sample of 68 items. We will discuss the (2) in a few slides.
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Attributes Sampling Evaluation of Attributes Sampling Evaluation of ResultsResults
2 possible approaches:
1. Use the bracketed number from Table 9.4. If you find that number or less deviations, conclude that you have accomplished your audit objective.
2. Use Table 9.5 for a more precise conclusion.
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Example A--No Deviations Identified (Evaluating Example A--No Deviations Identified (Evaluating
Attributes Sampling Results)Attributes Sampling Results) Approach 1—You have met your audit objective (because the bracketed number was (2),
you meet objective when you identify 0, 1 or 2 deviations). What can you say?
“I believe that the deviation rate in the population is less than 9 percent.” You will be wrong 5 percent of the time when the deviation is exactly 9 percent. If the deviation rate is in excess of 9 percent you will be wrong even less than 5 percent of the time. The planned assessed level of control risk is achieved.
Approach 2You have tested 68 items, a number not on Table 9-5 (next slide
To be conservative go to next lowest number on table (65) and use it for your conclusions (we could, but won't interpolate for a more precise answer).
You have met your audit objective. Table 9-5 gives us an answer of 4.6 percent. What can you say?
"I believe that the deviation rate in the population is less than 4.6 percent.” You will be wrong 5 percent of the time when the deviation rate is exactly 4.6 percent. If the deviation rate is in excess of 4.6 percent you will be wrong even less than 5 percent of the time. The planned assessed level of control risk is achieved.
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Figure 9.5Figure 9.5 Statistical Sampling Results Evaluation Table for Statistical Sampling Results Evaluation Table for Tests of Controls: Achieved Upper Deviation Rate atTests of Controls: Achieved Upper Deviation Rate at5 Percent Risk of Assessing Control Risk Too Low5 Percent Risk of Assessing Control Risk Too Low
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Example B--3 Deviations Identified Example B--3 Deviations Identified (Evaluating Attributes Sampling Results(Evaluating Attributes Sampling Results
Approach 1—You have not met your audit objective. What can you say?“The achieved upper deviation rate is higher than 9 percent.” The planned assessed level of control risk is not achieved. You need to consider increasing the assessed level of control risk above the planned assessed level.
Accordingly, you may not “rely” on internal control to the extent planned. Thus, the auditor will need to increase the scope of substantive procedures (the nature, timing, and/or extent).
Approach 2—You have not met your audit objective. Table 9-5 provides us an answer of 11.5 percent
“I believe that the deviation rate in the population is less than 11.5 percent.” You will be wrong 5 percent of the time when the deviation rate is exactly 11.5 percent. But this is not good enough as you wanted 9 percent rather than 11.5 percent. The planned assessed level of control risk is not achieved. You need to consider increasing the assessed level of control risk above the planned assessed level.
As per Approach 1, an increase in the scope of substantive procedures is appropriate.
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Other Statistical Attributes Other Statistical Attributes Sampling ApproachesSampling Approaches
Discovery sampling Purpose is to detect at least one deviation,
with a predetermined risk of assessing control risk too low if the deviation rate in population is greater than specified tolerable deviation rate
Useful in suspected fraud Sequential (Stop-or-Go) Sampling
Audit sample taken in several stages
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Sampling Risks--Substantive TestsSampling Risks--Substantive Tests
The Population Actually is Not Materially Materially
Misstated MisstatedThe Substantive Procedure SampleIndicates
Misstatement in Account Exceeds Tolerable Amount
Misstatement in Account Is Less Than Tolerable Amount
CorrectDecision
Incorrect Decision
(Risk of Incorrect Rejection)
IncorrectDecision
(Risk of Incorrect Acceptance)
Correct Decision
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Audit Sampling Steps for Audit Sampling Steps for Substantive TestsSubstantive Tests
Determine the objective of the test Define the population and sampling unit Choose an audit sampling technique Determine the sample size Select the sample Test the sample items Evaluate the sample results Document the sampling procedure
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Population Variability—Why it MattersPopulation Variability—Why it Matters
Item Population A Population B
1 2,100 8,000 2 2,100 25 3 2,100 2,000 4 2,100 400 5 2,100 75
Mean 2,100 2,100
Standarddeviation -0- 3,395
The variability determines how much information each of the items in the population tells you about the other items in the population.
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Factors Affecting Sample Size
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Mean Per Unit (MPU) Mean Per Unit (MPU) IllustrationIllustration
Population Size = 100,000 accounts
Book value = $6,250,000
Other information:
Tolerable misstatement = $364,000
Sampling risk
Incorrect Acceptance = 5%
Incorrect Rejection = 4.6 %
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MPU Risk CoefficientsMPU Risk CoefficientsAcceptable Level of Risk (%)
Incorrect Acceptance Coefficient
Incorrect Rejection Coefficient
1.0 2.33 2.58
4.6 1.68 2.00
5.0 1.64 1.96
10.0 1.28 1.64
15.0 1.04 1.44
20.0 .84 1.28
25.0 .67 1.15
30.0 .52 1.04
40.0 .25 .84
50.0 .00 .67
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Determining Sample Size--MPUDetermining Sample Size--MPU(1 of 2) (1 of 2)
t)coefficien rejectionIncorrect t / coefficien acceptance (Incorrect + 1
ntmisstateme Tolerable = ASR Planned
000,00$2 = )00.2/64(1. + 1
000,364$ = ASR Planned
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Determining Sample Size--MPU Determining Sample Size--MPU (2 of 2)(2 of 2)
2
risk samplingfor allowance Planned
dev. std. Est.*t coefficien rejectionIncorrect * size Population Size Sample
2
000,200$
$15 * 2.00 * 100,000Size Sample
= 225 Accounts
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Adjusted allowance for sampling risk =
Tolerable _ (Population size * Incorrect acceptance coef. * Sample stan. dev.) misstatement Sample size
This formula “adjusts” the allowance for sampling risk to consider the standard deviation of the audited values in the sample. It holds the risk of incorrect acceptance at its planned level.
Variables Sampling Illustration--MPU Variables Sampling Illustration--MPU
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Variables Sampling Illustration--MPU
Using the text example with a standard deviation of audited values of $16
Adjusted allowance for sampling risk =
Tolerable _ (Population size * Incorrect acceptance coef. * Sample stan. dev.)
misstatement Sample size
= $364,000 _ ($100,000 * 1.64 * $16)
225
= $189,067
We would still “accept” the book balance because the $6,250,000 (book value) falls within this interval
Estimate of total + Adjusted allowanceaudited value for sampling risk $6,100,000 + $189,067 [$5,910,933 to $6,289,067]
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Acceptance IntervalAcceptance IntervalFigure 9-12Figure 9-12
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Difference EstimationDifference Estimation
Difference Use sample to estimate the avg. difference
between the audited value and book value of items in population
Projected = Sample Net Misstatement * Pop. Items
Misstatement Sample items
Most appropriate when size of misstatements does not vary significantly in comparison to book value
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Ratio EstimationRatio Estimation
Use a sample to estimate the ratio of misstatement in a sample to its book value and project it to population
Projected = Sample Net Misstatement * Pop. Book Value
Misstatement Book Value of Sample
Preferred when the size of misstatements is nearly proportional to the book values of the items
Large accounts have large misstatements
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Nonstatistical Variables Sampling Nonstatistical Variables Sampling IllustrationIllustration
Plan Sample: Population:
• Size = 363 items• Book value = $200,000
Tolerable misstatement = $10,000 Risk assessments:
• Inherent and control risk = Slightly below maximum• Other substantive tests = Moderate
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Nonstatistical Sampling--Nonstatistical Sampling--Determination of Sample SizeDetermination of Sample Size
Sample size = Population book value X Reliability factor Tolerable misstatement
= $200,000 X 2.0 = 40 items $10,000
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Nonstatistical Sampling--Evaluation Nonstatistical Sampling--Evaluation of Sample Resultsof Sample Results
Sample results: 40 accounts in sample $350 net overstatement $60,000 book value of sample items
Projected misstatement:
= [Sample net misstatement] X Book value of population [ Book value of sample ] = [ $350 ] X $200,000 [$60,000]
= $1,167
Since the projected misstatement is only 11.7 percent ($1,167/$10,000) of tolerable misstatement, it is likely that the auditors would conclude that the account balance is materially correct.
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PPS Sampling IllustrationPPS Sampling Illustration
Population book value = $6,250,000 Other Information:
Tolerable misstatement = $364,000 Sampling risk--Incorrect acceptance = 5% Expected misstatement = $50,000
Use Figures 9-14 and 9-15 to obtain a “reliability factor” and an “expansion factor”--next slide
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PPS Sampling Reliability and PPS Sampling Reliability and Expansion FactorsExpansion Factors
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PPS Sample Size ComputationPPS Sample Size Computation
Sample size =
Recorded amount of population * Reliability factor
Tolerable misstatement - (Expected misstatement * Expansion factor)
= $6,250,000 * 3.0 = 66 $364,000 - ($50,000 * 1.6)
Sampling interval = Book value of the population Sample size
= $6,250,000 = $95,000 (approximately) 66
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Figure 9.16 PPS Sample Figure 9.16 PPS Sample Selection ProcessSelection Process
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PPS Evaluation of ResultsPPS Evaluation of Results
Upper Limit on misstatement =
Projected misstatement
+ Basic precision (Rel. factor x interval)
+ Incremental allowance
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Calculation of Upper Limit on Misstatement
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Comparison of statistical sampling Comparison of statistical sampling techniques for substantive procedurestechniques for substantive procedures
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Audit RiskAudit Risk
AR = IR x CR x DR
where AR=The allowable audit risk that a material misstatement might
remain undetected for the account balance and related assertions.
IR= Inherent risk, the risk of a material misstatement in an assertion, assuming there were no related controls.
CR= Control risk, the risk that a material misstatement that could occur in an assertion will not be prevented or detected on a timely basis by internal control.
DR= Detection risk, the risk that the auditors’ procedures will fail to detect a material misstatement if it exists.
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