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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