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9-1


Applying a procedure to less than 100% of a
population
To estimate some characteristic of the
population


Qualitative
Quantitative
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9-2

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
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9-3



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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9-4

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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9-5


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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9-6
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9-7

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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9-8
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9-9



Attributes sampling
Discovery sampling
Classical variables sampling




Mean-per-unit estimation
Ratio estimation
Difference estimation
Probability-proportional-to-size sampling
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9-10

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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9-11




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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9-12

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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9-13

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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9-14
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
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9-15




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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9-16
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9-17



Risk of Assessing Control Risk Too Low—5
percent
Tolerable Deviation Rate—9 percent
Expected Population Deviation Rate—2 percent
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9-18
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9-19
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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9-20
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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9-21
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.
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9-22
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9-23
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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9-24

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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9-25
The Substantive
Procedure Sample
Indicates
The Population Actually is
Not Materially
Materially
Misstated
Misstated
The population is Not
Materially Misstated
Correct
Decision
The Population is
Materially Misstated
Incorrect
Decision
(Risk of Incorrect
Rejection)
Incorrect
Decision
(Risk of Incorrect
Acceptance)
Correct
Decision
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9-26








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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9-27
Item Population A Population B
1
2
3
4
5
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.
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9-28
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9-29
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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9-30
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
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9-31



1
+
(Incorrect
acceptance
coefficien
t
/
Incorrect
rejection
coefficien
t)


Tolerable misstatement
Planned ASR = 


 = $200,000
 1 + (1.64 / 2.00) 
Planned ASR = 
$364 ,000
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9-32
 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
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9-33
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.
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9-34
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]
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9-35
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9-36

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
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9-37

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
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9-38

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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9-39
Sample size = Population book value X Reliability factor
Tolerable misstatement
= $200,000 X 2.0
$10,000
= 40 items
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9-40
 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.
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9-41


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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9-42
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9-43
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)
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9-44
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9-45
Upper Limit on misstatement =
Projected misstatement
+ Basic precision (Rel. factor x interval)
+ Incremental allowance
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9-46
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9-47
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9-48
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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9-49