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What is Audit Sampling?
 Applying
a procedure to less than 100%
of a population
 To estimate some characteristic of the
population


Qualitative
Quantitative
9-2
Risk

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
 Non-sampling risk
 risk pertaining to non-sampling errors
 Can be reduced to low levels through
effective planning and supervisions of audit
engagements
9-3
Non-statistical 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
9-4
Advantages of Statistical Sampling
 Allows
auditors to measure and control
sampling risk which helps:



Design efficient samples
Measure sufficiency of evidence
Objectively evaluate sample results
9-5
Selection 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
9-6
Random Number Table
9-7
Other Methods of Sample 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
9-8
An Illustration of Stratification
9-9
Types of Statistical Sampling Plans
 Attributes
sampling
 Discovery sampling
 Classical variables sampling



Mean-per-unit estimation
Ratio estimation
Difference estimation
 Probability-proportional-to-size
sampling
9-10
Dual 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
9-11
Allowance 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
9-12
Sample 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
9-13
Requirements of Audit
Sampling 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
9-14
Sampling Risks--Tests of Controls
Actual Extent of Operating Effectiveness
of the Control Procedure is
Adequate
Inadequate
The Test of Controls
Sample Indicates:
Extent of Operating
Effectiveness is
Adequate
Extent of Operating
Effectiveness
Inadequate
Correct
Decision
Incorrect
Decision
(Risk of Assessing
Control Risk
Too High)
Incorrect
Decision
(Risk of Assessing
Control Risk
Too Low)
Correct
Decision
9-15
Audit Sampling Steps for Tests 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
9-16
Attributes Sampling: Relationship Between the
Planned Assessed Level of Control Risk and the
Tolerable Deviation Rate
9-17
Illustration of Attributes Sampling-Determining Sample Size
 Risk
of Assessing Control Risk Too Low—
5 percent
 Tolerable Deviation Rate—9 percent
 Expected Population Deviation Rate—2
percent
9-18
Figure 9.4: Statistical Sample Sizes for Tests of Controls
at 5 Percent Risk of Assessing Control Risk Too Low
9-19
Sample 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.
9-20
Attributes Sampling Evaluation of
Results
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.
9-21
Example A--No Deviations Identified (Evaluating
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 2
You 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.
9-22
Figure 9.5 Statistical Sampling Results Evaluation Table for
Tests of Controls: Achieved Upper Deviation Rate at
5 Percent Risk of Assessing Control Risk Too Low
9-23
Example B--3 Deviations Identified (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.
9-24
Other Statistical Attributes Sampling
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
9-25
Sampling Risks--Substantive Tests
The Population Actually is
Not Materially
Materially
Misstated
Misstated
The Substantive
Procedure Sample
Indicates
Misstatement in
Account Exceeds
Tolerable Amount
Misstatement in
Account Is Less
Than Tolerable
Amount
Correct
Decision
Incorrect
Decision
(Risk of Incorrect
Acceptance)
Incorrect
Decision
(Risk of Incorrect
Rejection)
Correct
Decision
9-26
Audit Sampling Steps for Substantive
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
9-27
Population Variability—Why it Matters
Item
1
2
3
4
5
Population A Population B
2,100
2,100
2,100
2,100
2,100
8,000
25
2,000
400
75
Mean
2,100
2,100
Standard
deviation
-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.
9-28
Factors Affecting Sample Size
9-29
Mean Per Unit (MPU) Illustration
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 %
9-30
MPU Risk Coefficients
Acceptable
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
9-31
Determining Sample Size--MPU
(1 of 2)



1
+
(Incorrect
acceptance
coefficien
t
/
Incorrect
rejection
coefficien
t)


Tolerable misstatement
Planned ASR = 


 = $200,000
 1 + (1.64 / 2.00) 
Planned ASR = 
$364,000
9-32
Determining Sample Size--MPU
(2 of 2)
 Population size * Incorrect rejection coefficien t * Est. std. dev. 
Sample Size  

Planned
allowance
for
sampling
risk


 100,000 * 2.00 * $15 
Sample Size  

$200,000


2
2
= 225 Accounts
9-33
Variables Sampling Illustration--MPU
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.
9-34
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
audited value
$6,100,000
[$5,910,933
+
+
to
Adjusted allowance
for sampling risk
$189,067
$6,289,067]
9-35
Acceptance Interval - Figure 9-12
9-36
Difference Estimation
 Difference

Use sample to estimate the avg. difference
between the audited value and book value of
items in population
Projected
Misstatement

=
Sample Net Misstatement * Pop. Items
Sample items
Most appropriate when size of misstatements
does not vary significantly in comparison to
book value
9-37
Ratio Estimation
 Use
a sample to estimate the ratio of
misstatement in a sample to its book
value and project it to population
Projected
Misstatement


=
Sample Net Misstatement * Pop. Book Value
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
9-38
Nonstatistical Variables Sampling
Illustration

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
9-39
Nonstatistical Sampling--Determination of
Sample Size
Sample size = Population book value X Reliability factor
Tolerable misstatement
= $200,000 X 2.0
$10,000
= 40 items
9-40
Nonstatistical Sampling--Evaluation of
Sample Results
• Sample results:
40 accounts in sample
$350 net overstatement
$60,000 book value of sample items

Projected misstatement:
= [Sample net misstatement]
[ Book value of sample ]
X Book value of population
= [ $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.
9-41
PPS 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
9-42
PPS Sampling Reliability and
Expansion Factors
9-43
PPS Sample Size Computation
Sample
size =
Recorded amount of population * Reliability factor
Tolerable misstatement - (Expected misstatement * Expansion factor)
=
$6,250,000 * 3.0
$364,000 - ($50,000 * 1.6)
=
66
Sampling interval = Book value of the population
Sample size
= $6,250,000
66
= $95,000 (approximately)
9-44
Figure 9.16 PPS Sample
Selection Process
9-45
PPS Evaluation of Results
Upper Limit on misstatement =
Projected misstatement
+ Basic precision (Rel. factor x interval)
+ Incremental allowance
9-46
Calculation of Upper Limit on Misstatement
9-47
Comparison of statistical sampling
techniques for substantive procedures
9-48
Audit 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.


9-49