9-1
Chapter Nine
Audit Sampling:
An Application to Substantive Tests
of Account Balances
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9-2
Substantive Tests of Details of Account
Balances
The statistical concepts we discussed in the last
chapter apply to this chapter as well. Three important
determinants of sample size are
1. Desired confidence level.
2. Tolerable misstatement (error).
3. Estimated misstatement (error).
Misstatements discovered in the audit sample must
be projected to the population, and there must be an
allowance for sampling risk.
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9-3
Substantive Tests of Details of Account
Balances
Consider the following information about the inventory
account balance of an audit client:
Book value of inventory account balance
Book value of items sampled
Audited value of items sampled
Total amount of overstatement observed in audit sample
€ 3,000,000
€
100,000
98,000
€
2,000
The ratio of misstatement in the sample is 2%
(€2,000 ÷ €100,000)
Applying the ratio to the entire population produces a best
estimate of misstatement of inventory of €60,000.
(€3,000,000 × 2%)
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9-4
Substantive Tests of Details of Account
Balances
The results of our audit test depend
upon the tolerable error associated with
the inventory account. If the tolerable
error is €50,000, we cannot conclude
that the account is fairly stated because
our best estimate of the projected error
is greater than the tolerable error.
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9-5
Monetary-Unit Sampling (MUS)
MUS uses attribute-sampling theory to
express a conclusion in monetary amounts
(e.g. in euros or other currency) rather than
as a rate of occurrence. It is commonly used
by auditors to test accounts such as
accounts receivable, loans receivable,
investment securities and inventory.
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9-6
Monetary-Unit Sampling (MUS)
MUS uses attribute-sampling theory to
estimate the percentage of monetary units in
a population that might be misstated and then
multiplies this percentage by an estimate of
how much the euros are misstated.
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9-7
Monetary-Unit Sampling (MUS)
Advantages of MUS
1. When the auditor expects no misstatement, MUS
usually results in a smaller sample size than classical
variables sampling.
2. The calculation of the sample size and evaluation of
the sample results are not based on the variation
between items in the population.
3. When applied using the probability-proportional-to-size
procedure, MUS automatically results in a stratified
sample.
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9-8
Monetary-Unit Sampling (MUS)
Disadvantages of MUS
1. The selection of zero or negative balances generally
requires special design consideration.
2. The general approach to MUS assumes that the
audited amount of the sample item is not in error by
more than 100%.
3. When more than one or two misstatements are
detected, the sample results calculations may
overstate the allowance for sampling risk.
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9-9
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Planning
1. Determine the test objectives.
2. Define the population characteristics.
• Define the population.
• Define the sample unit.
• Define a misstatement.
3. Determine the sample size, using the following inputs:
• The desired confidence level or risk of incorrect acceptance.
• The tolerable misstatement.
• The expected population misstatement.
• Population size.
Performance
4. Select sample items.
5. Perform the auditing procedures.
Evaluation
6. Calculate the projected misstatement and the upper limit on misstatement.
7. draw final conclusions.
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9-10
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Planning
1. Determine the test objectives.
2. Define the population characteristics.
• Define the population.
• Define the sample unit.
• Define a misstatement.
Sampling may be used for substantive testing to:
1. Test the reasonableness of assertions about a
financial statement amount.
2. Develop an estimate of some amount.
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9-11
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Planning
1. Determine the test objectives.
2. Define the population characteristics.
• Define the population.
• Define the sample unit.
• Define a misstatement.
For MUS the population is defined as the
monetary value of an account balance,
such as accounts receivable, investment
securities or inventory.
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9-12
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Planning
1. Determine the test objectives.
2. Define the population characteristics.
• Define the population.
• Define the sample unit.
• Define a misstatement.
An individual euro represents the sampling unit.
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9-13
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Planning
1. Determine the test objectives.
2. Define the population characteristics.
• Define the population.
• Define the sample unit.
• Define a misstatement.
A misstatement is defined as the difference between
monetary amounts in the client’s records and
amounts supported by audit evidence.
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9-14
Steps in MUS Sampling Application
Steps in MUS Sampling Application
3. Determine the sample size, using the following inputs:
• The desired confidence level or risk of incorrect acceptance.
• The tolerable misstatement.
• The expected population misstatement.
• Population size.
Factor
Relationship
to Sample Size
Desired confidence level
Direct
Tolerable mistatement
Inverse
Expected mistatement
Direct
Population size
Direct
McGraw-Hill/Irwin
Change
in Factor
Lower
Higher
Lower
Higher
Lower
Higher
Lower
Higher
Effect on
Sample
Decrease
Increase
Increase
Decrease
Decrease
Increase
Decrease
Increase
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9-15
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Performance
4. Select sample items.
5. Perform the auditing procedures.
Evaluation
6. Calculate the projected misstatement and the upper limit on misstatement
7. Draw final conclusions.
The auditor selects a sample for MUS by using a
systematic selection approach called probabilityproportionate-to-size selection. The sampling interval
can be determined by dividing the book value of the
population by the sample size. Each individual euro in
the population has an equal chance of being selected.
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9-16
Steps in MUS Sampling Application
Assume a client’s book value of accounts receivable is €2,500,000, and the
auditor determined a sample size of 93. The sampling interval will be
€26,882 (€2,500,000 ÷ 93). The random number selected is €3,977 the
auditor would select the following items for testing:
Account
1001 Ace Emergency Center
1002 Admington Hospital
1003 Jess Base
1004 Good Hospital Corp.
1005 Jen Mara Corp.
1006 Axa Corp.
1007 Green River Mfg.
1008 Bead Hospital Centers
•
•
1213 Andrew Call Medical
1214 Lilly Health
1215 Janyne Ann Corp.
Total Accounts Receivable
McGraw-Hill/Irwin
Balance
€
2,350
15,495
945
21,893
3,968
32,549
2,246
11,860
•
•
26,945
1,023
€ 2,500,000
Cumulative
Euros
€
2,350
17,845
18,780
40,673
44,641
77,190
79,436
91,306
•
•
2,472,032
2,498,977
€ 2,500,000
Sample
Item
€
3,977
(1)
30,859
(2)
57,741
(3)
84,623
•
•
(4)
2,477,121
€ 3,977
26,882
€ 30,859
(93)
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9-17
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Performance
4. Select sample items.
5. Perform the auditing procedures.
Evaluation
6. Calculate the projected misstatement and the upper limit on misstatement
7. Draw final conclusions.
After the sample items have been selected,
the auditor conducts the planned audit
procedures on the logical units containing
the selected euro sampling units.
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9-18
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Evaluation
6. Calculate the projected misstatement and the upper limit on misstatement
7. Draw final conclusions.
The misstatements detected in the sample
must be projected to the population.
Example Information
Book value
Tolerable misstatement
Sample size
Desired confidence level
Expected amount of misstatement
Sampling interval
McGraw-Hill/Irwin
€ 2,500,000
€ 125,000
93
5%
€
25,000
€
26,882
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9-19
Steps in MUS Sampling Application
Basic Precision
If no misstatements are found in the sample,
the best estimate of the population
misstatement would be zero euros.
Sample
Size
65
70
85
80
90
100
125
Actual Number of Deviations Found
0
1
2
3
4.6
7.1
9.4
11.5
4.2
6.6
8.8
10.8
4.0
6.2
8.2
10.1
3.7
5.8
7.7
9.5
3.3
5.2
6.9
8.4
3.0
4.7
6.2
7.6
2.4
3.8
5.0
6.1
€26,882 × 3.0 = €80,646 upper misstatement limit
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9-20
Steps in MUS Sampling Application
Misstatements Detected
In the sample of 93 items the following misstatements
were found:
Customer
Good Hospital
Marva Medical Supply
Axa Corp.
Learn Heart Centers
Book Value
€
21,893
6,705
32,549
15,000
Audit Value
€
18,609
4,023
30,049
-
Difference
€
3,284
2,682
2,500
15,000
Tainting
Factory
15%
40%
NA
100%
Because the Axa balance of €32,549 is greater than the
€3,284
÷ €21,893
15% all
interval of €26,882, no sampling
risk
is added.=Since
the euros in the large accounts are audited, there is no
sampling risk associated with large accounts.
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9-21
Steps in MUS Sampling Application
Compute the Upper Misstatement Limit
We compute the upper misstatement limit by calculating basic
precision and ranking the detected misstatements based on
the size of the tainting factor from the largest to the smallest.
Tainting
Customer
Factor
Basic Precision
1.00
Learn Heart Centers
1.00
Marva Medical
0.40
Good Hospital
0.15
Add misstatments greater
that the sampling interval:
Axa Corp.
NA
Sample
Interval
€ 26,882
26,882
26,882
26,882
Projected
Misstatement
NA
(26,882)
(10,753)
(4,032)
26,882
NA
Upper Misstatement Limit
95% Upper
Limit
3.0
1.7 (4.7 - 3.0)
1.5 (6.2 - 4.7)
1.4 (7.6 - 6.2)
Upper
Misstatement
€
80,646
45,700
16,130
5,645
€
2,500
150,621
(0.15 × €26,882 × 1.4 = €5,645)
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9-22
Steps in MUS Sampling Application
Steps in MUS Sampling Application
Evaluation
6. Calculate the projected misstatement and the upper limit on misstatement
7. Draw final conclusions.
In our example, the final decision is
whether the accounts receivable balance
is materially misstated or not.
We compare the tolerable misstatement to the upper
misstatement limit. If the upper misstatement limit is less
than or equal to the tolerable misstatement, we conclude
that the balance is not materially misstated.
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9-23
Steps in MUS Sampling Application
In our example the upper misstatement limit of €150,621
is greater than the tolerable misstatement of €125,000, so
the auditor concludes that the accounts receivable
balance is materially misstated.
When faced with this situation, the auditor may:
1. Increase the sample size.
2. Perform other substantive procedures.
3. Request the client adjust the accounts receivable balance.
4. If the client refuses to adjust the account balance, the
auditor would consider issuing a qualified or adverse
opinion.
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9-24
Risk When Evaluating Account Balances
True State of Financial Statement Account
Auditor's Decision Based
on Sample Evidence
Supports the fairness of
the account balance
Does not support the
fairness of the account
balance
McGraw-Hill/Irwin
Not Materially Misstated
Correct decision
Risk of incorrect
rejection (Type I)
Materially Misstated
Risk of incorrect
acceptance (Type II)
Correct Decision
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9-25
Why is Sample Size Not Used in
Evaluating MUS Results?
Most MUS evaluation approaches use the misstatement
factors and increments associated with a sample size of 100,
regardless of the actual sample size used by the auditor.
Number of
Errors
0
1
2
3
4
McGraw-Hill/Irwin
95% Confidence Level
Misstatement
Incremental
Factor
Increase
3.0
4.7
1.7
6.2
1.5
7.6
1.4
9.0
1.4
90% Confidence Level
Misstatement
Incremental
Factor
Increase
2.3
3.9
1.6
5.3
1.4
6.6
1.3
7.9
1.3
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9-26
Effect of Understatement Misstatements
MUS is not particularly effective at detecting
understatements. An understated account is less likely to be
selected than an overstated account.
Customer
Wayne County Medical
Book
Value
€ 2,000
Audit
Value
€ 2,200
Difference
€
(200)
Tainting
Factor
-10%
The most likely error will be reduced by €2,688
(– 0.10 × €26,882)
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9-27
Non-statistical Sampling for Tests of
Account Balances
The sampling unit for non-statistical sampling is normally
a customer account, an individual transaction, or a line
item on a transactions. When using non-statistical
sampling, the following items must be considered:
o Identifying individually significant items.
o Determining the sample size.
o Selecting sample items.
o Calculating the sample results.
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9-28
Identifying Individually Significant Items
The items to be tested individually are items that may
contain potential misstatements that individually exceed
the tolerable misstatement. These items are tested
100% because the auditor is not willing to accept any
sampling risk.
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9-29
Determining the Sample Size
Sample
=
Size
Population book value
Tolerable misstatement
Combined Assessment of
Inherent and Control Risk
Maximum
Slightly below maximum
Moderate
Low
McGraw-Hill/Irwin
× Assurance factor
Risk That Other Substantive Procedures Will Fail to
Detect Material Misstatements
Slightly Below
Maximum
Maximum
Moderate
Low
3.0
2.7
2.3
2.0
2.7
2.4
2.0
1.6
2.3
2.1
1.6
1.2
2.0
1.6
1.2
1.0
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9-30
Selecting Sample Items
Auditing standards require that the sample items be
selected in such a way that the sample can be expected
to represent the population.
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9-31
Calculating the Sample Results
One way of projecting the sampling results to the
population is to apply the misstatement ratio in the
sample to the population.
Assume the auditor
finds €1,500 in
misstatements in a
sample of €15,000.
The misstatement
ratio is 10%.
McGraw-Hill/Irwin
If the population
total is €200,000,
the projected
misstatement would
be €20,000
(€200,000 × 10%)
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9-32
Calculating the Sample Results
A second method is the difference estimation. This
method projects the average misstatement of each item
in the sample to all items in the population.
Assume
misstatements in a
sample of 100 items
total €300, and the
population contains
10,000 items.
McGraw-Hill/Irwin
The projected
misstatement would
be €30,000, (€300 ÷
100 = €3 × 10,000).
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9-33
Non-statistical Sampling Example
The auditor’s of Calabro Paging Service have decided
to use non-statistical sampling to examine the accounts
receivable balance. Calabro has 11,800 accounts with
a balance of €3,717,900. The auditor’s stratify the
accounts as follows:
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9-34
Non-statistical Sampling Example
The auditor’s decide . . .
o There is a low assessment for inherent and control risk.
o The tolerable misstatement is €40,000, and the expected
misstatement is €15,000.
o There is a moderate risk that other auditing procedures
will fail to detect material misstatements.
o All customer account balances greater than €25,000 are
to be audited.
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9-35
Non-statistical Sampling Example
Sample
=
Size
Population book value
Tolerable misstatement
× Assurance factor
€3,717,900 – €550,000
Sample
=
Size
€3,167,900
€40,000
Combined Assessment of
Inherent and Control Risk
Maximum
Slightly below maximum
Moderate
Low
McGraw-Hill/Irwin
× 1.2 = 95 rounded
Risk That Other Substantive Procedures Fail to
Detect Material Misstatement
Slightly Below
Maximum
Maximum
Moderate
Low
3.0
2.7
2.3
2.0
2.7
2.4
2.0
1.6
2.3
2.1
1.6
1.2
2.0
1.6
1.2
1.0
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9-36
Non-statistical Sampling Example
The auditor sent positive confirmations to each of the 110
(95 + 15) accounts selected. Either the confirmations were
returned or alternative procedures were successfully
used. Four customers indicated that their accounts were
overstated and the auditors determined that the
misstatements were the result of unintentional error by
client personnel. Here are the results of the audit testing:
Stratum
>€25,000
>€3,000
<€3,000
McGraw-Hill/Irwin
Book Value
€
550,000
850,500
2,317,400
Book Value
of Sample
€
550,000
425,000
92,000
Audit Value
of Sample
€
549,500
423,000
91,750
Amount of
OverStatement
€
500
2,000
250
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9-37
Non-statistical Sampling Example
As a result of the audit procedures, the following
projected misstatement was prepared:
Amount of
Stratum
Misstatement
>€25,000
€
500
>€3,000
2,000
<€3,000
250
Total projected misstatement
Ratio of Misstatement
in Stratum Tested
100%
€2,000 ÷ 425,000 × €850,500
€250 ÷ 92,000 × €2,317,400
Projected
Misstatement
€
500
4,002
6,298
€
10,800
The total projected misstatement of €10,800 is less than
the expected misstatement of €15,000, so the auditors
may conclude that there is a low risk that the true
misstatement exceeds the tolerable misstatement.
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9-38
Why Did Statistical Sampling Fall Out Of
Favor?
1.Firms found that some auditors were
over relying on statistical sampling
techniques to the exclusion of good
judgment.
2.There appears to be poor linkage
between the applied audit setting and
traditional statistical sampling
applications.
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9-39
Classical Variable Sampling
Classical variables sampling uses normal distribution
theory to evaluate the characteristics of a population
based on sample data. Auditors most commonly use
classical variables sampling to estimate the size of
misstatement.
Sampling distributions are formed by plotting the
projected misstatements yielded by an infinite
number of audit samples of the same size taken
from the same underlying population.
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9-40
Classical Variables Sampling
A sampling distribution is useful because it allows us
to estimate the probability of observing any single
sample result.
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9-41
Classical Variables Sampling
In classical variables sampling, the sample mean is
the best estimate of the population mean.
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9-42
Classical Variables Sampling
Advantages
1. When the auditor expects a large number of
differences between book and audited values, this
method will result in smaller sample size than
MUS.
2. The techniques are effective for both
overstatements and understatements.
3. The selection of zero balances generally does not
require special sample design considerations.
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9-43
Classical Variables Sampling
Disadvantages
1. To determine sample size, the auditor must
estimate the standard deviation of the audited
value or differences.
2. If few misstatements are detected in the sample
data, the true variance tends to be
underestimated, and the resulting projection of the
misstatements to the population is likely not to be
reliable.
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9-44
Applying Classical Variables Sampling
Defining the Sampling Unit
The sampling unit can be a customer account,
an individual transaction, or a line item. In
auditing accounts receivable, the auditor can
define the sampling unit to be a customer’s
account balance or an individual sales invoice
included in the account balance.
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9-45
Applying Classical Variables Sampling
Determining the Sample Size
Population size × ZIA × SD
Sample
=
Tolerable misstatement – Estimated misstatement
Size
2
where
ZIA = One-tailed Z value for the specified
level of the risk of incorrect acceptance.
SD = Estimated standard deviation.
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9-46
Applying Classical Variables Sampling
The risk of incorrect acceptance is the risk that the
auditor will mistakenly accept a population as fairly
stated when the true population misstatement is greater
than tolerable misstatement.
Risk of Incorrect
Acceptance
2.5%
5.0%
10.0%
15.0%
20.0%
McGraw-Hill/Irwin
Z Value
1.96
1.65
1.28
1.04
0.84
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9-47
Applying Classical Variables Sampling
The year-end balance for accounts receivable contains 5,500
accounts with a book value of €5,500,000. The tolerable
misstatement for accounts receivable is set at €50,000. The
expected misstatement has been judged to be €20,000. The
risk of incorrect acceptance is 2.5%. Based on work
completed last year, the auditor estimates the standard
deviation at €31. Let’s calculate sample size.
Sample
=
Size
McGraw-Hill/Irwin
5,500 × 1.96 × €31
€50,000 – €20,000
2
= 125
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9-48
Applying Classical Variables Sampling
Calculating the Sample Results
The sample selection usually relies on randomselection techniques. Upon completion, 30 of the
customer accounts selected contained misstatements
that totaled €330.20. Our first calculation is the mean
misstatement in an individual account which is
calculated as follows:
Mean
Total audit difference
misstatement
=
Sample size
per sampling
item
= €330.20 = €2.65
125
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9-49
Applying Classical Variables Sampling
The mean misstatement must be
projected to the population.
Projected
population = Population size × Mean misstatement
(in sampling units)
per sampling item
misstatement
= 5,500 × €2.65 = €14,575
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9-50
Applying Classical Variables Sampling
Point estimate of accounts receivable balance . . .
Accounts receivable
Book
Projected population
=
–
point estimate
value
misstatement
= €5,500,000 – €14,575 = €5,485,425
The sum of the audited differences squared is
equal to €36,018.32. We will use this value to
calculate the standard deviation.
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9-51
Applying Classical Variables Sampling
The formula for the standard deviation is . . .
SD =
Total audit
–
differences squared
Sample
Mean difference
×
Size
per sampling item2
Sample size – 1
=
McGraw-Hill/Irwin
€36,018.32 – (125 × 2.652)
124
= €16.83
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
9-52
Applying Classical Variables Sampling
SD
Confidence
Population
=
× ZIA ×
bound
size
16.83
= 5,500 × 1.96 × √ 125
Population
Confidence
=
point estimate
interval
Sample size
= €16,228
±
Confidence
bound
= €5,485,425 ± €16,228
McGraw-Hill/Irwin
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
9-53
Applying Classical Variables Sampling
Book
value
€5,500,000
Lower
bound
Point
estimate
Upper
bound
€5,469,197
€5,485,425
€5,501,652
Confidence interval
If the precision interval includes the book value, the
evidence supports the conclusion that the account is not
materially misstated.
McGraw-Hill/Irwin
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
9-54
Applying Classical Variables
Sampling
Book
value
€5,508,000
Lower
bound
Point
estimate
Upper
bound
€5,469,197
€5,485,425
€5,501,652
Confidence interval
When the evidence indicates that the account may be materially
misstated the auditor might consider (1) increasing sample size, (2)
performing additional substantive procedures, (3) adjusting the
account, or (4) issue a qualified or adverse opinion.
McGraw-Hill/Irwin
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.
9-55
End of Chapter 9
McGraw-Hill/Irwin
Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved.