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Module 6: Audit sampling
Overview
In this module, you learn about the methods that can be used to gather evidence by obtaining representative
samples of transactions and balances. Attributes sampling is presented in the context of tests of controls, while
dollar-unit sampling (DUS) is explained as it is used in performing substantive tests. (You may need to reviewstatistical sampling from earlier accounting courses to ensure that you understand the statistical concepts
underlying the material in this module.)
As you work through each topic, you build your understanding of audit sampling. You also apply what you have
learned by using a spreadsheet to perform dollar-unit sampling, and evaluate the merits and limitations of this
type of sampling.
Assignment reminder: Assignment 2 in Module 7 is due at the end of Week 7 (see Course Schedule). You
may wish to take a look at the assignment before working on Module 6 to familiarize yourself with the
requirements and to prepare for any work that may be required in advance.
Test your knowledge
Begin your work on this module with a set oftest-your-knowledge questions designed to help you gauge the
depth of study required.
Learning objectives
6.1 Audit sampling: Introduction Explain why auditors use sampling, and describe the two
applications of audit sampling. (Level 1)
6.2 Statistical and non-statistical sampling Explain statistical and non-statistical sampling, describe the
advantages of each method, and explain when each method
should be used. (Level 1)
6.3 Sampling and non-sampling error Explain sampling error, including errors arising from alpha
risk (Type 1 error), beta risk (Type II error), and non-
sampling error. (Level 2)
6.4 Attribute sampling and tests of controls Outline the seven-step framework for conducting attribute
sampling for tests of control. (Level 2)
6.5 Determining sample size and selecting the
sample
Describe the factors that influence sample size
determination in attribute sampling, and determine the
sample size for a test of controls using statistical sampling.
(Level 2)
6.6 Evaluating test results for attribute sampling Describe how the auditor evaluates the results of a test inthe contexts of non-statistical and statistical sampling for
attribute sampling. (Level 2)
6.7 Audit sampling for substantive testing Describe the nature of audit risk, sampling risk, and
materiality in the context of substantive testing. (Level 1)
6.8 Sampling procedures for substantive testing Outline the seven-step framework for audit sampling in
substantive testing, and explain how an auditor can use
stratification to reduce sample sizes in audit sampling. (Level
2)
6.9 Determining sample size in substantive testing Describe the factors that influence sample size
determination in substantive testing. (Level 2)
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6.10 Evaluating test results for substantive testing Explain how the auditor evaluates the results of a test in
substantive testing. (Level 2)
6.11 Dollar-unit sampling Describe and demonstrate the dollar-unit sampling process
to test an account balance. (Level 2)
Module summary
Print this module
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6.1 Audit sampling: Introduction
Learning objective
Explain why auditors use sampling, and describe the two applications of audit sampling. (Level 1)
Required reading
Chapter 10, pages 367-370
CAS 530, Audit Sampling
Note: This module uses a variety of acronyms. Here is a table that summarizes some of these terms. The
definitions and terms on page 367 of the text are also important for understanding this module.
LEVEL 1
Auditors usually do not test 100% of transactions or items in account balances because the cost of doing so
would be prohibitive. Also, auditors seek only reasonable assurance.Audit sampling occurs whenever an
auditor draws a conclusion about an entire class of transactions or account balance based on the results of a
(representative) sample from the class or the balance.
Audit sampling addresses the sufficiency aspect of evidence as required under GAAS, insofar as it relates to the
extent of audit procedures used.
Why do auditors use sampling?
Auditors use sampling
to test controls (compliance tests) for assessing control risk
to test balances (substantive tests) to determine whether balances are materially misstated
Exhibit 6.1-1 summarizes how the type of test performed relates to the focus of the audit sampling. You can
print out this table for later reference.
Exhibit 6.1-1: Comparison of audit sampling tests
Compliance tests Substantive tests
Focus To obtain evidence that the client complies
with the internal control procedures
designed and implemented to achieve
specific control objectives
To obtain evidence to support
managements assertions regarding
account balances
Population from
which a sample is
drawn
Class of transactions Items in an account balance
Example Testing control procedures relating to the
acquisition and payment cycle for
authorization for purchasing
proper classification of asset
expenditure
evidence that the extensions on
invoices were checked and that the
payment was authorized
Auditing the
1. existence of accounts receivable
2. valuation of inventory (expressed at
cost)
Sample Sample of payments from the cheque 1. Existence of accounts receivable:
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register or cash disbursement journal
(random sample of cheque numbers)
Sample from customers with
outstanding accounts receivable
balances
2. Valuation of inventory (expressed at
cost): Sample from the year-end
itemized inventory listing
Method of testing The auditor would obtain supporting
documentation for each cheque and test for
appropriate controls, such as the existence
of an approved purchase order matched to
the invoice, a supervisors initials on the
invoice to authorize payment, and so on.
1. The auditor would support existence of
accounts receivable by choosing a
sample from the customers with
outstanding balances and obtain
confirmations directly from those
customers.
2. The auditor would support inventory
valuation (expressed at cost) by
selecting a sample from the itemized
inventory listing and vouching the cost
to supporting documentation such as
the purchase invoice.
Conclusion Based on the results of the testing, the
auditor would conclude whether or not the
control is operating effectively and if the
control risk assessment is appropriate.
Based on the results of the testing, the
auditor would conclude whether or not the
account balance being tested is materially
misstated.
For a better understanding of the sampling concepts, review What you really need to know, posted under
Chapter 10 on McGraw-Hill Connect. (Registration is required. Each copy of the AU1 text includes a McGraw-Hill
Connect access card. When registering, select CGA as the school and Self Study as the course.)
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Acronyms
Attribute sampling
Substantive test sampling: dollar-unit sampling
(DUS)
ARO = acceptable risk of over-reliance ARIA = acceptable risk of incorrect acceptance
ARACR = acceptable risk of assessing control risk toolow
RIA = risk of incorrect acceptance
ARO = ARACR = sampling risk ARIA = RIA = sampling risk
Note: Sampling risk refers to both RIA and RIR or risk
of incorrect rejection (the decision to accept a balance
as being materially misstated when it is not). The
auditor, however, will focus on RIA.
TDR = tolerable deviation rate Tolerable misstatement (TM)/
Recorded population value
(Tolerable misstatement can be expressed either as adollar amount or as a percentage of the book or
recorded value of the population from which the
sample is selected.)
EPDR = expected population deviation rate EE = estimated error rate in population
(The estimated error can be expressed either as a
dollar amount or as a percentage of the book, or
recorded, value of the population from which the
sample is selected.)
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6.2 Statistical and non-statistical sampling
Learning objective
Explain statistical and non-statistical sampling, describe the advantages of each method, and
explain when each method should be used. (Level 1)
Required reading
Chapter 10, pages 370-371 (Begin at Why Auditors Sample.)
LEVEL 1
Statistical sampling involves the random selection of a sample from a population. Non-statistical sampling does
not use statistical calculations in determining the sample or expressing the results.
Exhibit 6.2-1 provides a comparison of statistical as opposed to non-statistical sampling.
Exhibit 6.2-1: Statistical and non-statistical sampling
Statistical Non-statistical
Sample selection Sample mustbe randomly selected. No requirement for random selection.
Documentation More extensive documentation of decision
process because of the greater rigour of
statistical sampling. Auditors provide a
better source for review, planning, and
evidence should the audit work be called
into question.
Auditors would be wise to fully document
the audit work to provide background for
review, planning, and evidence.
(The importance of documentation cannot
be too strongly emphasized; if the audit
work is called into question when non-
statistical sampling is used, only good
documentation can provide proof of the
soundness of the auditors judgment.)
Quantification of
sampling risk
Requires the auditor to quantify the
sampling risk that influences the testing
decision in determining the sample size.
No quantification of sampling risk.
Inference of
results
(Whether using
statistical or non-
statistical sampling,
the auditor infers the
results of the sample
to the population.)
The measurement of the sample results
must be based on statistical methods
(permits statistical evaluation). With a
statistical sample, the inference is
formalized mathematically (that is, a point
estimate is calculated and the confidence
interval is constructed).
With a non-statistical sample, the inference
must be made informally (that is, the
auditor must use judgment to estimate the
population balance or error).
Exhibit 6.2-2 illustrates the additional information obtainable from the use of statistical sampling, which is
determining sampling risk and the achieved (computed) upper error limit (LM + sampling risk adjustment). The
upper error limit, in turn, would be used for comparing with tolerable misstatement or materiality. Note the
illustration still does not include non-sampling risk; the auditor needs to consider the possibility of non-
sampling risk when evaluating the account balance or financial statements.
Exhibit 6.2-2: Estimating upper error limit for comparison with tolerable misstatement
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Non-statistical sampling Statistical sampling
The auditor is not required by generally accepted auditing standards to base evaluations on statistically based
tests, but only to apply professional judgment. As a result, both statistical and non-statistical sampling are used
in public practice. Review the boxes on pages 371 (Use Statistical Sampling When) and 372-373 (Use Non-
statistical Sampling When) to review the circumstances in which it is best to use each method.
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6.3 Sampling and non-sampling error
Learning objective
Explain sampling error, including errors arising from alpha risk (Type I error), beta risk (Type II
error), and non-sampling error. (Level 2)
Required reading
Chapter 10, pages 372-374 (to Test of Controls for Assessing Control Risk)
LEVEL 2
Sampling risk and sampling error
Whenever auditors make inferences about a population based on a sample drawn from that population, they
hope the sample will be truly representative of that population. However, whenever a sample is selected from a
population, it may or may not be representative. The risk of the sample not being representative is called
sampling risk. The error that ensues from sampling risk is called sampling error.
The following exhibit summarizes the concepts of sampling risk and sampling error. Read the table now and
print it out for review later.
Exhibit 6.3-1: Sampling risk and sampling error
Sampling
risk
What is sampling risk?
The risk of the sample not being
representative is called sampling
risk.
Can sampling risk be
reduced?
Sampling risk can be reduced by
increasing the sample size
(which means that more of the
population is being examined).
Can sampling risk be
quantified?
When statistical sampling
methods are used, sampling risk
can be quantified it is the
probability that the sample is notan accurate reflection of the
population.
Sampling
error
What is sampling error? How
does it occur?
The error that ensues from
sampling risk is called sampling
error. A sampling error occurs
when an auditor forms an
incorrect conclusion about a
population based on the results
of examining a sample drawn
from it because the sample was
not representative of thepopulation.
Why does sampling error
occur?
Sampling error occurs because
the auditor has examined less
than 100% of the population.
How many types of sampling
errors are there?
Sampling risk can lead to two
types of sampling error: Type I
error and Type II error.
Type I error
A Type I error is when an auditor incorrectly rejects an account balance that is not actually materially
misstated, or rejects a control as ineffective when it is actually effective.
Scenario 6.3-1: Testing for a specific control procedure
Dean, an auditor, is interested in testing transactions for a specific control procedure. He finds that out of a
population of 5,000 transactions, only four population units are missing the control; that is, the deviation rate
for the population is 0.08% (4/5,000).
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Suppose Dean chooses a sample of 20 units from this population, and that sample turns out to contain all four
population units with the missing control. What population deviation rate do you think Dean would estimate
based on the sample results? What would Dean be likely to conclude?
Solution
Type II error
A Type II error is when an auditor incorrectly accepts an account balance that is actually materially misstated,
or concludes that a control is effective when it is actually ineffective.
Scenario 6.3-2: True deviation rate
Suppose that in Deans situation, the true deviation is 20%; that is, the population of 5,000 transactions
contains 1,000 population units with the missing control. What if Deans sample of 20 contains none of the
units with the missing control? What do you think Deans conclusion will be based on the sample results?
Solution 1
The sampling risk associated with a Type I error is usually referred to as alpha risk, while the risk associated
with a Type II error is called betarisk. Alpha risk is not as much of a concern to the auditor as beta risk.
a. Why is beta risk more of a concern than alpha risk?
b. How can an auditor reduce sampling risk?
Solution 2
Non-sampling error
The auditor is also concerned about non-sampling error. Non-sampling error arises from non-sampling
risk, which occurs when
the auditor is not careful in examining the sample and fails to identify an exception or error in a
selected sample item
the audit test is not appropriate (for example, the sample was selected from an inappropriate
population for the purposes of the assertion being tested)
the auditor has misjudged the relevant risks (that is, the inherent risk and/or control risk)
the auditor has made an error in the evaluation of the sampling results
Any or all of the above audit or auditor weaknesses could lead the auditor to perform less work than is required
to adequately support a conclusion.
Scenario 6.3-3: Testing for controls over disbursements
Lina, an auditor, forgets to look at the signatures on a sample of cheques to test for controls over
disbursements and, as a result, she would fail to reassess control risk if there were incorrect signatures in the
sample. Is this an example of a sampling error?
Solution
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Scenario 6.3-1 solution
Dean would estimate that the population deviation rate is 20% (4/20) and conclude that the control is not
effective, when, in fact, it is. In this case, Dean would conclude that the population is not validwhen, in fact, it
is. This is called Type I error.
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Scenario 6.3-2 solution 1
Dean would likely conclude that the control is effective, when, in fact, it is not. In this situation, Dean would
conclude that a population is valid when, in fact, it is not. This is called Type II error.
The sampling risk associated with a Type I error is usually referred to as alpha risk, while the risk associated
with a Type II error is called betarisk. Alpha risk is not as much of a concern to the auditor as beta risk.
a. Why is beta risk more of a concern than alpha risk?
b. How can an auditor reduce sampling risk?
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Scenario 6.3-2 solution 2
a. In the case of a Type I error (alpha risk) when using sampling for test of controls, the greatest
risk is of performing an inefficient audit. For example, if an auditor concludes that controls are
ineffective when they are effective, the auditor might decide to reassess control risk at maximum
and perform additional substantive procedures. If a Type I error occurs during substantive
testing, there is a risk that an unnecessary adjustment may result. However, when an error is
discovered during substantive testing, further work is usually done to determine the adjustment,which may result in reversing the error. A Type I error results in additional work that is not
necessary.
For the risk of a Type II error (beta risk), however, the auditor would not be motivated to do
additional work beyond what has already been planned. Unless other planned procedures or
analysis reveal a deviation or misstatement that was not detected in sampling, the deviation or
misstatement will go uncorrected, and wrong decisions will be made regarding adjustments and
the audit opinion.
If an auditor determines, based on substantive audit procedures, that an adjustment is not
necessary when in fact the account balance is materially misstated, this will result incorrectly
issuing an unqualified audit report.
Consequently, the auditor is very concerned about beta risk; therefore, when deciding on an
acceptable sampling risk for audit purposes, the focus is on beta risk, not alpha risk.
b. An auditor can reduce sampling risk by increasing sample size and increasing the randomness
of the selection process (in non-statistical sampling, where selection need not be random or
systematic).
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Scenario 6.3-3 solution
This is an example of non-sampling error. The incorrect control risk assessment would lead the auditor to
perform fewer audit procedures than required by the circumstances.
Through proper instructions and supervision and by careful design of audit procedures, the auditor can reduce
the chance of non-sampling error. The general standard of GAAS requires adequate technical training and
proficiency in auditing, while the first examination standard requires proper supervision and adequate planning.In short, non-sampling risk should be of relatively little concern to a competent auditor.
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6.4 Attribute sampling and tests of controls
Learning objective
Outline the seven-step framework for conducting attribute sampling for tests of controls. (Level
2)
Required reading
Chapter 10, pages 375-380 (up to Calculating Sample Size) and 383-388 (to Substantive
Procedures for Auditing Account Balances)
LEVEL 2
In the next three topics, you learn to use statistical sampling for tests of controls using attribute sampling.
Statistical sampling helps quantify sampling risk and evaluate the results of testing. For tests of controls,
sampling risk consists of the probability that the auditor will incorrectly assess control risk because the sample
is not representative of the population.
What is an attribute?
An attribute is a characteristic in which the auditor is interested, and the attribute of interest in testing controls
is whether or not a deviation from the specified controls has occurred (for example, whether or not the control
failed). An attribute supporting the control objective of validity, for example, is the existence of a matching
shipping document for each invoice. Other examples of attributes include the department supervisors initials on
each invoice payable (authorization) and correct account coding on each invoice (classification).
The text describes the tests of controls steps (beginning on page 376) relevant to attribute sampling as a
seven-step framework using the sales and collection cycle to illustrate this.
Step 1 (Specifying the objectives of the test) and Step 6 (Performing the test of controls audit procedures)
from the seven-step framework relate mainly to the performance of tests themselves; these are explained in
more detail in Modules 8 and 9, which deal with specific transaction cycles. This topic describes steps 2 and 3.
Steps 4 and 5 are covered in Topic 6.5; step 7 is covered in Topic 6.6.
Step 2: Defining the conditions for deviations
A clear understanding of what constitutes a deviation is important. Deviations need to be defined in advance so
that they can be recognized and treated consistently, and so that they relate to the objectives of the test of
control. Auditors must also fully understand the attributes being tested in order to know when a deviation
occurs. That is, the auditor must understand the controls and how the controls should function in order to be
able to identify cases in which the control has not functioned correctly. Normally, the absence of an attribute
constitutes a deviation, but that is not always the case. In certain situations, there will be justifiable reasons for
the absence of an attribute.
Scenario 6.4-1: Audit of the acquisitions and payments cycle
Lee is auditing the acquisitions and payments cycle and selecting a sample of payments to test the controls.
One of the procedures may be to check that for each invoice, there is a corresponding purchase order (PO)
properly approved. If Lee finds that one of the invoices in the sample is missing a purchase order, is this a
deviation?
Solution
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Step 3: Defining the population
When conducting a test of controls, the auditor aims to support control risk assessments assertion by assertion.
This means that the population from which the sample is drawn must lend itself to testing the controls that will
support a given assertion, either directly or indirectly through a control objective relevant to the assertion.
When studying the direction of a test, you saw that sources of evidence would be used in different ways
(different directions) to test for different assertions. You must ensure that the population that you select from
has attributes consistent with the assertions you wish to test.
Scenario 6.4-2: Verifying invoices
If you are verifying that paid invoices are supported by purchase orders, would you select the sample from the
purchase order issued or from the paid invoices?
Solution
Scenario 6.4-3: Year end and tests of controls
Padma is performing an audit for a client whose year end is December 31. She is planning to conduct tests of
controls at an interim date, commencing October 1. What are the implications for Padma and on her audit
work?
Solution
Scenario 6.4-4: Concerns for the population to be sampled
When defining the population from which to draw a sample, there are two more factors auditors should be
aware of:
physical proximity
physical characteristics of the population
What is the auditors concern when documentation defined as the intended population is kept in a remote off-
site location?
Solution
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Scenario 6.4-1 solution
It depends. The invoice for which the PO is missing may have a standing order, in which case a regular PO is
not required. An example of this would be an electricity bill; Lee would not have a purchase order for each
electricity invoice received during the period. In this situation, the missing PO does not constitute a deviation.
Of course, Lee would have to document this situation and specify the compensating audit procedure to be
used.
If the invoice was for the purchase of inventory, for which the appropriate control procedure is the creation of a
PO that is properly approved, then this would constitute a deviation.
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Scenario 6.4-2 solution
You would select the sample from the population of paid invoices so that you can then vouch it to the purchase
order, which would provide evidence that each paid invoice has a related purchase order. It would not be
appropriate to select from the purchase orders issued because this would not indicate any missing purchase
orders.
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Scenario 6.4-3 solution
Depending on the type and volume of transactions that occur between October 1 and December 31, Padma
must consider that a significant portion of the population may not yet exist.
For a control to be relied on, it must be consistently applied throughout the period under audit. Thus, Padma
may need to perform additional tests at year end to determine whether the control assessment made during
the interim audit continues to be valid for the balance of the year. As a summary, an auditor must also beaware of the completeness of the population at the time the samples are selected.
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Scenario 6.4-4 solution
If the population is kept in a remote off-site location, it could be difficult or may be even impossible to select a
sample from this population. Thus, the auditor must ensure that a sample can actually be drawn from the
population and the chosen population must have physical characteristics that allow an auditor to sample from
it.
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6.5 Determining sample size and selecting the sample
Learning objective
Describe the factors that influence sample size determination in attribute sampling, and determine
the sample size for a test of controls using statistical sampling. (Level 2)
Required reading
Chapter 10, pages 378-381 and 383-384
CAS 530, Audit Sample, Appendix 2 and Appendix 4
LEVEL 2
This topic examines steps 4 and 5 of the seven-step framework: determining the sample size for tests of
controls and selecting the sample.
Step 4: Determining the sample size
The relationships between the factors and sample size are the same regardless of whether you are using non-
statistical or statistical sampling. When auditors use statistical sampling, they must quantify sampling risk
(ARACR) to enable them to use statistical tables for determining sample size.
The text correctly points out that sampling risk includes the risk that the control risk assessment will be too
low, as well as the risk that the control risk assessment will be too high. However, the relationship between
sample size and sampling risk as previously described assumes that the auditor is concerned with the former,
not the latter. This is consistent with the auditor focusing on beta risk (see Topic 6.3).
The following exhibit shows a statistical sample size that you can use to see how these factors interact in a
statistical sampling approach.
Exhibit 6.5-1: Statistical sample size table 5% sampling risk (RIA/ARACR/ARO)
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Note: This table assumes a large population.
* Sample size is too large to be cost-effective for most audit applications.
Source: Lemon, Arens, and Loebbecke,Auditing: An Integrated Approach, Canadian Fifth Edition
(Scarborough, ON: Prentice Hall, 1993), page 409
The table in Exhibit 6.5-1 is developed using sampling (probability) distributions and allows auditors to estimate
the probability of representativeness (sampling risk) of a sample. It depicts the sample sizes for a sampling risk
set at 5%. The percentage figures down the left side of the table represent EPDR, while those across the top
represent TDR. The numbers in the table are the sample sizes required for each combination of TDR and EPDR.
Activity 6.5-1: Determining the sample size
Work through this activity to test your understanding of how to determine the sample size.
Suppose an auditor chooses a sampling risk of 5% and has assessed preliminary control risk at medium (about
35%). On the basis of this preliminary control risk assessment, the auditor feels comfortable with a TDR of 7%
(see the table in the text on page 379, which exemplifies the relationship between control risk and TDR).Based on the prior years experience, the auditor also determines that EPDR is about 2%. (In other words, the
auditor is willing to take a 5% risk of concluding that the control is effective when it is not. The auditor will
tolerate deviations of up to 7% and still assess CR at 35%, even though the auditor only expects 2% deviations
in the population.)
Using these figures and the table in Exhibit 6.5-1, what sample size would the auditor choose?
Hint: Look at the intersection of the row for EPDR of 2% and a tolerable deviation rate of 7% in Exhibit 6.5-1.
Solution
Step 5: Selecting the sample
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Once the auditor has determined the sample size, the next step when conducting attribute sampling is to select
the sample. Sampling risk is reduced by increasing sample size and increasing the randomness of the
selection process (in non-statistical sampling where selection need not be random or systematic).
What are the risks in selecting a non-random sample?
Non-random sampling includes both haphazard selection and block (or cluster) sampling. A non-random sample
can contain biases; the auditor is naturally attracted to non-routine transactions and, therefore, might be
biased in choosing only unusual items. Even though a randomly selected sample may not be representative, at
least the probability of this happening (sampling risk) is measured and controlled. Block or cluster sampling
occurs when the auditor selects a number of blocks of consecutive transactions rather than selecting
transactions individually. Although this is sometimes mandated by difficulties in locating selected items, such
selection should be avoided where possible, even for non-statistical samples.
The text describes various methods used to select a random sample. They include selection based on assigning
random numbers to a population (from a random number table or from computer-generated random numbers),
as well as by using systematic random selection. Another way to select a random sample is to use audit
software where the program chooses the sample based on population characteristics, such as cheque numbers,
invoice numbers, and account numbers, keyed in by the auditor. (InAdvanced External Auditing [AU2], you will
learn to use a software package calledAudit Command Language[ACL] to select samples.)
The text also describes non-statistical sampling methods and explains some of the problems associated with
them.
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Activity 6.5-1 solution
The auditor would choose a sample size of 88.
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6.6 Evaluating test results for attribute sampling
Learning objective
Describe how the auditor evaluates the results of a test in the contexts of non-statistical and
statistical sampling for attribute sampling. (Level 2)
Required reading
Chapter 10, pages 385-388 (except the section on Sample Evaluation)
LEVEL 2
The final step of the seven-step framework for attribute sampling is evaluation of the test results.
Step 7: Evaluating the test results
The results of the test will either support or refute the auditors preliminary assessment of control risk. The
number of deviations found in the sample will help the auditor decide whether the controls are reliable by
inferring the proportion of the population that contains deviations.
Remember that the auditor is concerned about sampling risk insofar as it relates to the risk of assessing control
risk too low. (On pages 386 and 387, the section on Sample Evaluation is not required reading.)
One way to evaluate the results of your test in a statistical context is to use a table like the one shown in
Exhibit 6.6-1. This table computes the upper deviation or error limit (UEL) for a given ARACR, sample size, and
number of deviations found in tests of controls. The computed UEL is a statistical calculation that factors in
sampling error, and is used to estimate the deviation rate of the entire population. The sample deviation rate
(actual sample deviations sample size) may be lower or higher than the actual population deviation rate.
Because auditors are mainly concerned with the risk of assessing CR too low, the higher/upper limit is
calculated to estimate how high the estimated population deviation rate might be.
Exhibit 6.6-1: Statistical sampling results evaluation computed upper deviation rates at 5%
sampling risk
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Note: This table presents computed upper deviation rates (CUDR) as percentages. This table assumes a large
population.
* Over 20%.
Source: Lemon et al, page 412.
Activity 6.6-1: Determining the upper error limit (UEL)
Using a sampling risk of 5%, TDR of 7%, and EPDR of 2%, it was determined earlier that the sample size
would be 88. If the auditor found one deviation in the sample, what would be the UEL?
Using Exhibit 6.6-1, at a sample size of 90 (which is the closest to 88 in the table), one deviation gives a UEL
of 5.2%. When comparing UEL to TDR, the UEL must be less than or equal to TDR for the population (the CRassessment) to be considered acceptable. Because this UEL is less than the auditors TDR of 7%, the auditor
would conclude that the results of the test do support the preliminary control risk assessment.
Suppose the auditor encounters three deviations instead of one. What would be the UEL in this case? What
would the auditor conclude?
Solution
When using non-statistical sampling for the evaluation of tests of controls, the auditor does not quantify the
calculation of UEL using statistical tables. The auditor uses professional judgment to consider sample error and
generalize from the sample deviation rate to the population deviation rate.
By contrast, using statistical sampling for the evaluation of tests of controls enables the auditor to compute
UEL (and sampling risk), given the sample size and number of deviations found. Using a sample size of 90:
Number of deviations found
0 1 3
Sample deviation rate 0/90 = 0% 1/90 = 1.1% 3/90 = 3.3%
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UEL (per Exhibit 6.6-1) 3.3 5.2 8.4
Sampling risk
(UEL sample deviation rate)
3.3 0 = 3.3% 5.2 1.1 = 4.1% 8.4 3.3 = 5.1%
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Activity 6.6-1 solution
The UEL would be 8.4%, which is greater than the 7% TDR. In this case, the auditor could decide to increase
the sample size and do more testing to see if the deviations persist, or the auditor would conclude that the test
does not support the preliminary control risk assessment, and therefore would reassess control risk as high or
very high and design substantive audit procedures accordingly.
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6.7 Audit sampling for substantive testing
Learning objective
Describe the nature of audit risk, sampling risk, and materiality in the context of substantive
testing. (Level 1)
Required reading
Chapter 10, pages 388-390 (to Sampling Steps For Account Balance Audit)
LEVEL 1
As mentioned in the text, substantive procedures include both analysis and tests of details of balances. (Topic
3.4 describes how analysis is used to provide audit evidence.) Analysis is not subject to sampling because
analytical procedures are applied to overall balances and financial relationships. Therefore, this topic on audit
sampling for substantive testing will focus only on tests of details of account balances. Details of an account
balance are the items and transactions that make up the account balance.
For example, three customers, Mr. A, Ms. B, and Mrs. C owe the company $500, $800, and $1,500 respectively,
and make up the total accounts receivable balance of $2,800. A test of details for accounts receivable could
consist of verifying (testing) the individual customer balances (details making up the total balance). (Note,
however, that testing 100% of the accounts does not constitute audit sampling.)
Audit sampling for substantive procedures is much more related to audit risk (that is, the risk that the auditor
will issue an unqualified opinion when the financial statements are materially misstated) than is audit sampling
for tests of controls. This is because substantive testing provides direct evidence about the financial statement
assertions, and it is on those assertions that the audit opinion is based.
In substantive testing, sampling risk refers to the probability that the auditor will form an incorrect conclusion
about an account balance based on the results of a sample from the population of items that make up the
account balance. In other words, it is the probability that the auditor will accept an account balance based on
the results of the sample, when in fact the account balance is materially misstated. This would occur because
the sample selected was not representative of the population.
Sampling error will still be Type I or Type II and arise from alpha risk and beta risk respectively, as was the
case for attribute sampling for tests of controls. This is why the expanded audit risk model on page 393 only
considers RIA, not RIR. Also note that in this model, detection risk consists of the risk that a material
misstatement will not be detected by either analysis (analytical procedures risk = APR) or tests of details; that
is, DR = APR RIA. Because this topic focuses on tests of details, for the remainder of this module, assume
that no substantive tests other than tests of details are performed, and hence APR = 1 (that is, DR = RIA).
In substantive testing, the auditor tests whether or not an account balance is materially misstated. But how
many dollars constitute an error? This is where materiality enters into the audit sampling process. More
specifically, the auditor considers tolerable misstatement, which is usually the same as overall materiality.
Determining tolerable misstatement is subject to the auditors professional judgment, but in some cases, the
auditor may choose to systematically allocate materiality among balance sheet accounts. For example, suppose
overall materiality has been estimated at $177,000 based on estimated after-tax income, and also assume the
auditor wishes to test accounts receivable, which amount to $235,000. If total assets are $3,550,000, then the
auditor could calculate tolerable misstatement for accounts receivable as follows:
$235,000 $177,000 = $11,717
$3,550,000
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However, in many cases, such a systematic approach will not be appropriate, and the auditor will have to rely
more on qualitative factors to arrive at tolerable misstatement for each balance. The primary basis for
materiality decisions is always the auditors judgment.
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6.8 Sampling procedures for substantive testing
Learning objective
Outline the seven-step framework for audit sampling in substantive testing, and explain how an
auditor can use stratification to reduce sample sizes in audit sampling. (Level 2)
Required reading
Chapter 10, pages 390-402 (excluding the paragraphs starting from When the population is
stratified on page 399 to take the sampling risks into account on page 400)
LEVEL 2
The main objective of substantive testing is to detect a material misstatement in an account balance. To
achieve this objective, the auditor performs audit sampling within a seven-step framework similar to the one
described in the topics for tests of controls. This topic explains steps 2 and 3 of the framework: defining the
population and choosing an audit sampling method. (Steps 4, 5, and 7 are covered in the following two topics.
Steps 1 and 2: specifying the objectives of the test and performing the substantive audit procedures, relate
mainly to the performance of tests themselves; these are explained in more detail in Modules 8 and 9, which
deal with specific transaction cycles.)
This framework, described on pages 390-400, differs from that for tests of controls in two ways:
First, there is no need to identify conditions for deviations. An error condition in substantive
testing is fairly obvious it is whenever the recorded dollar amount differs from the audited
amount.
Second, in substantive testing, the auditor needs to choose a sampling method, which was not
the case for tests of controls. For tests of controls, the sampling method is attribute sampling,
whereas the sampling method for substantive testing can be dollar-unit sampling which is a form
of attribute sampling, or variables sampling (not covered in this course).
Step 2: Defining the population
The population in substantive testing must be consistent with the objective of the test. The population must
also be complete and have physical characteristics that will allow the auditor to select a sample from it.
Scenario 6.8-1: Testing the completeness assertion
Pat, an auditor, designs a substantive test to provide evidence that accounts payable are not understated; that
is, to test the completeness assertion. She uses a population consisting of an accounts payable listing made up
of vendors with recorded balances, especially large ones. Is Pats choice of population appropriate in this case?
Solution 1
What type of population would be more appropriate to test the completeness assertion?
Solution 2
The main difference between defining a population for tests of controls and for substantive testing relates to
the issue of materiality. In substantive testing, the auditor focuses on amounts, because substantive
procedures are designed and performed to support the account balances as stated in the financial statements.
Because of materiality, the auditor will often segregate individually material amounts from other population
units and test these amounts to a greater extent. This is known as stratification, and it is explained on pages
391-392.
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The degree of stratification will have an impact on evaluating the results of the evidence because each stratum
becomes a sub-population that may be significantly different from the others. This implies that extrapolating
the results of a large stratum to all the population (all strata) may be misleading, because the sample from that
stratum is not representative of the whole population.
Several other factors, aside from size, can be considered in stratification, such as location, risk, and the length
of time an item such as a receivable or payable has been outstanding. The auditor has to exercise considerable
judgment in stratifying a population, but the ultimate criteria are always to efficiently and effectively meet the
audit objectives.
Step 3: Choosing an audit sampling method
The auditor can use statistical or non-statistical sampling for substantive testing because the base concepts
apply to both. If statistical sampling is chosen, however, the auditor must choose between a number of
sampling methods such as variables or dollar-unit sampling. The most popular approach is dollar-unit
sampling (DUS). Simply stated, DUS considers each individual dollar in a population as a separate unit
making up the account balance.
If the auditor chooses a non-statistical sampling approach, then the sampling method is based on judgment
and does not require that all population units (physical items) have an equal chance of being selected. In fact,
in judgmental sampling, the auditor usually targets larger dollar amounts and/or unusual transactions.
Statistical tables are not used to project the sample misstatements to the population. Therefore, non-statisticalsampling may not always provide a good defensible basis for projecting a likely misstatement when evaluating
the evidence.
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Scenario 6.8-1 solution 1
In Pats case, the chosen population is not appropriate, because understatement of accounts payable implies
that there may be vendors with no recorded balance, and these vendors will not be in the population.
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Scenario 6.8-1 solution 2
A more appropriate population type would be a list of regular vendors showing small or zero balances in the
year-end accounts payable.
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6.9 Determining sample size in substantive testing
Learning objective
Describe the factors that influence sample size determination in substantive testing. (Level 2)
Required reading
Chapter 10, pages 392-395
CAS 530 Appendix 3
LEVEL 2
This topic explains steps 4 and 5 of the seven-step framework for substantive testing.
Step 4: Determining the sample size
Similar to choosing the sample size in tests of controls, determining the sample size for substantive testing is
affected by a number of factors that apply to both statistical and non-statistical sampling. Sampling risk refers
to both RIA and RIR, and as for tests of controls, the auditor focuses on beta risk (RIA). In substantive testing,
the sampling risk is chosen by the auditor based on the relationship described by the audit risk model. In other
words, the auditors choice of RIA is a function of AR, IR, and CR.
The auditor also chooses the level of tolerable misstatement for a given account based on professional
judgment (with reference to the amount of misstatement considered material for that audit). After choosing
sampling risk and tolerable misstatement, the auditor will determine sample size based on the relationships
outlined. (You have an opportunity to work through an example of sample size selection in the context of
dollar-unit sampling in Computer activity 6.11-1.)
Step 5: Selecting the sample
When using non-statistical sampling, the auditor may select a random or non-random sample. Judgmentally
selecting the sample has similar drawbacks for tests of details as for tests of controls. If the auditor uses
statistical sampling, then the sample must be selected so that it is truly random. Inferences about achieved RIA
or likely misstatement based on statistical methods will be valid only if the sample is randomly selected.
The methods used to select a random sample in substantive testing are similar to those described for tests of
controls. In addition, it is acceptable to use systematic sampling to select the sample when using statistical
sampling.
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6.10 Evaluating test results for substantive testing
Learning objective
Explain how the auditor evaluates the results of a test in substantive testing. (Level 2)
Required reading
Chapter 10, pages 397-401 (excluding the paragraphs starting from When the population is
stratified on page 399 and ending before Consider Sampling Risks, on page 400)
LEVEL 2
Step 7: Evaluating the results of the evidence obtained
In Topic 4.5 you learned that identified misstatements are extrapolated to determine the likely misstatement in
the population. Whenever a misstatement is found in a representative sample, its likely effects on the
population must be evaluated. There are several methods, both statistical and non-statistical, that can be used
to make these inferences. This is step 7 in the seven-step framework for substantive testing.
In order to evaluate the results obtained, the auditor calculates the upper error limit, discussed in Topic 6.6,
and the likely misstatement (LM), discussed in Topic 4.5. The difference between the upper error limit and the
likely misstatement is a result of sampling risk.
Pages 398-399 explain the average difference method for calculating likely misstatements in a stratified
and non-stratified population. This method is relatively simple and can be applied in both statistical and non-
statistical sampling.
You should be aware that the evaluation of sample results also requires a consideration ofqualitative factors
(see page 400). The best example, as described in the text, is that of finding missing controls during a
substantive test. Auditors cannot ignore finding control deviations simply because they are performing a
substantive test as opposed to a test of control. Deviations would affect the control risk assessment, which, in
turn, affects the amount of substantive work required to support the assertions.
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6.11 Dollar-unit sampling
Learning objective
Describe and demonstrate the dollar-unit sampling process to test an account balance. (Level 2)
Required reading
Reading 6 -1: Dollar-unit sampling
Reading 6 -2: Random number table
Note: Page 5 of Reading 6-1 refers to Table 12-2 on page 396. This table has been reproduced as Reading
6-2. Also, Reading 6-1 sometimes refers to an attributes sampling table on page 412. This table has been
reproduced as Exhibit 6.6-1 in Topic 6.6.
LEVEL 2
Dollar-unit sampling (or sampling with probability proportionate to size) is an attribute sampling method that
can be used for sampling for tests of controls and for sampling for tests of details of balances. On pages 1 and
2 of Reading 6-1, the steps in attributes sampling are compared with DUS. To better outline the sequence of
events in substantive testing, read the steps in Reading 6-1, which are more detailed than the seven-step
framework presented in the navigation pane.
The population in DUS is a pool of dollars not a pool of transactions, a pool of individual accounts receivable
balances, or a pool of inventory items. Imagine that the dollars represented by a single account receivable are
laid out end-to-end and that the dollars in each account are laid out sequentially so that the general ledger
account accounts receivable is represented by a long line of dollars. For example, a company has a $600
accounts receivable balance that is represented by three customer accounts. The balances in the customer
accounts are $100, $200, and $300. The population in DUS would be the pool of dollars beginning at $1 to
$600.
Scenario 6.11-1: Dollar-unit sampling
Sam, the auditor for Fine Music Ltd., finds that the accounts receivable balance in the general ledger is
$239,800 and that it represents 42 individual accounts receivable. The dollars are numbered from 1 (the first
dollar of the first account receivable) to 239,800 (the last dollar of the 42nd account receivable).
To select a sample, Sam selects dollars from the line on a random basis and uses a random number table to do
this. If the dollar in a particular account receivable is selected, then that account is selected. Using this method,
could Sam select a particular account more than once?
Solution
Scenario 6.11-2: Reliability of dollar-unit sampling
Lisa, one of your co-workers, insists that dollar-unit sampling is both reliable and efficient in sampling for tests
of detail of balances, works well with any population type, and has no known limitations. Do you agree with
her statement?
Solution
Auditing activity 6.11-1 Dollar-unit sampling
The methodology and the steps for using dollar-unit sampling are described in Reading 6-1. Now click on the
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icon and work through the Dollar-unit sampling auditing activity to see how to compute the upper and lower
error bounds when performing statistical tests of details of balances.
Computer activity 6.11-1
Work through Computer activity 6.11-1 to see how a worksheet can be designed to assist the auditor in
performing dollar-unit sampling.
Assignment reminder
Assignment 2 in Module 7 is due at the end of Week 7 (see Course Schedule). Be sure to read it through nowto familiarize yourself with the requirements and to prepare for any work that may be required in advance.
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Scenario 6.11-1 solution
Using the random selection process does result in a particular account being selected more than once if more
than one dollar from that account receivable is selected. When that occurs, Sam counts each dollar as a
separate sample unit and examines fewer accounts receivable than if each dollar had come from a separate
account.
One of the advantages of DUS is that the larger items in the population have a correspondingly greaterprobability of being selected. Why is this so?
In Sams situation, an account with a balance of $72,900 has a likelihood of 72,900/239,800 of being selected,
whereas an account with a balance of $1,500 has a likelihood of 1,500/239,800 of being selected. Under
classical sampling, where the sampling unit would be the individual account receivable, each of the two
accounts would have a 1/42 chance of selection.
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Scenario 6.11-2 solution
Dollar-unit sampling is reliable. However, it does have certain limitations:
It is not as efficient as some of the variable sampling techniques (such as difference estimation
and ratio estimation) in that it requires selection of larger sample sizes in populations with a large
number of misstatements.
Selecting a sample is difficult if the system is not computerized.
Adjustments have to be made to include zero and negative balances.
Accounts that are understated are less likely to be selected than if they were stated at their
correct balances.
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Computer activity 6.11-1: Dollar-unit sampling with the computer
Learning objectives
Explain how a worksheet can be designed to assist the auditor to perform dollar-unit sampling.
(Level 2)
Evaluate the merits and limitations of worksheets for this type of sampling. (Level 2)
LEVEL 2
The manual selection of the sample for dollar-unit sampling can be a tedious task. In this computer activity,
you will find examples of how a spreadsheet program can be used to perform dollar-unit sampling. It uses
spreadsheet commands to assist in sample selection for a batch of invoices in a continuous sequence.
Material provided
The file AU1M6P1 contains two worksheets:
M6P1 the summarized accounts receivable data
M6P1S the solution worksheet for M6P1
Description
The auditor has exported all the accounts receivable information from an accounts receivable system to a
worksheet, in file AU1M6P1. The auditor wants to select a random sample of the invoices for confirmation,
using dollar-unit sampling technique.
The auditor has determined the following:
1. Population. The population is made up of the total dollars recorded on all outstanding invoices
and credit notes drawn from the last four months of the fiscal year.
2. Sampling unit. The sampling unit will be an individual dollar.
3. Sample size. Using the technique described in Reading 6-1, pages 15 to 16, the auditor
determined the appropriate sample size from the following information:
Required
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Retrieve the file AU1M6P1. Using the RAND function, construct formulas to select a random sample using
dollar-unit sampling techniques. Work through the steps provided in the next section to perform random
sampling for a population of invoices that are in a continuous sequence.
Procedure
1. Open the file AU1M6P. (Before you begin working on the data files in this course, you must first
download them and save them to your hard drive. Click the data files link in the course
introduction, then follow the instructions for downloading and saving the files.) Click the sheet tab
M6P1.
Caution: In this activity, it is critical to save the file under a different filename because the
activity includes a sorting procedure; you must keep the original data file intact in case of error in
the sorting procedure.
2. Examine columns A to H of the worksheet. Columns A to C are the accounts receivable data
exported to the worksheet by the auditor; columns D to H will be used by the auditor to perform
dollar-unit sampling. The following describes the content of each column:
Sorting the data by invoice number
1. Check column A and observe that the invoices are in no specific order. If you manually perform
the dollar-unit sampling, this data set will work well without any sorting. However, to do the
sampling in a worksheet, you must first sort the data in ascending order by invoice number
(column A).
Note: The command functions provided below may vary depending on the version of the Excel
software being used by each individual student.
2. To sort the data by invoice number, perform the following steps:
a. With the range A6 to C55 highlighted, select Data Sort and the Sort dialog box will
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appear.
b. In the My list has or My data has headers box, make sure that Header Row has
been chosen. In the Sort by box, the header of column A invoice should be
displayed. Click Ascending (or Smallest to Largest) and OK.
With this command, you specify that you want to sort the complete data based on column A
(invoice number). Notice that the data range contains the invoice numbers, customer names, and
invoice amounts. When you specify the data range for sorting, be careful not to include the
heading of each column. You also need to specify that the invoice number is to be sorted in
ascending order.
3. Columns A6 to C55 should now be sorted in ascending order by invoice number. Move the cell
pointer and examine the first and last invoices. The first record (in row 6) should be:
and the last record in row 55 should be:
If your results do not correspond, you may have sorted the data incorrectly. Erase the currentworksheet without saving it. Then retrieve the worksheet and repeat steps 1 and 2 carefully.
Converting the amounts to absolute values
1. In performing dollar-unit sampling, you will normally construct a column containing the
cumulative amount for the invoices. However, due to the negative values of the credit notes,
simply adding all the invoice amounts in column C will not provide correct cumulative amounts.
To obtain the proper amount, you must convert credit amounts in column C to positive values.
There are several ways to do this, the simplest being to create a new column (column D) which
contains absolute values, using the ABS function, corresponding to the values in column C.
Move the cell pointer to cell D6. Type the formula:
= ABS (C6)
The resulting value should be 5,000.00, the same as the value in cell C6, because the absolute
value of a positive amount is the same as the amount.
2. To obtain the absolute values of cells C7 to C55, copy the formula from cell D6 to cells D7 to D55.
Column D should be filled with the corresponding absolute values of column C. Note that cell D11
shows positive 2,300.00, which is the absolute value of (2,300.00) in cell C11. Compare the cells
in columns C and D and correct any errors before proceeding to the next step.
Setting up columns for cumulative amount and invoice number
1. One of the key components of dollar-unit sampling is the creation of the cumulative column,
which contains a running cumulative total. The SUM function is used to compute this amount in
each row.
Move the cell pointer to cell E6 to create the formula for the cumulative amount. Type the
formula:
= ROUND (SUM ($D$6:D6), 0)
This formula creates a cumulative total, rounded to dollars, of the amount in column D, starting
with cell D6. For example, if this formula is copied to cell E8, it calculates the cumulative total of
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invoice amounts in cells D6 to D8, resulting in the value of 63,081.
2. Copy the formula in cell E6 to cells E7 to E55. To confirm that you have set up the formulas
correctly, move the cell pointer to cell E55 and check that the cumulative amount is 887,746. If
your answer is not the same, carefully repeat the previous steps and correct the errors before
continuing.
3. Once the cumulative amounts are set up, you must copy the invoice numbers in column A to
column F. This is to make it possible to use the VLOOKUP function. Column F will be used by the
VLOOKUP function to perform the lookup of the invoice number corresponding to random
numbers you will generate in column G. Copy the invoice numbers from column A to column F.
4. Move the cell pointer to column F and review the sorted invoice numbers. They should be the
same as those in column A. The first five records in columns E and F should be as follows:
5. Notice that the first amount in column E is 5,000. To manage the situation in which the random
number selected is between 1 and 5,000, you must insert a row between rows 5 and 6. After
inserting the new row 6, in cell E6, type the start dollar value of0, and in cell F6, type the
invoice number of120100. (This invoice will never be selected since it corresponds to the dollar
amount of zero.) After you have completed the change, the first six records in columns E and F
should be as follows:
Generating the set of random numbers
1. Now, you need to generate 42 random numbers to select the sample of invoices for
confirmation. The RAND function generates a random number between 0 and 1.
The population of dollars in this activity is between 1 and 887,746 (in cell E56). Togenerate a set of 42 random numbers from this population, type the following
formula in cell G6:
= ROUND(RAND()*($E$56 1),0)+1
2. Copy the formula to the next 41 cells below cell G6 to make up the required sample
size of 42.
3. After entering the formulas, press F9 several times to recalculate the worksheet and
to achieve a randomized selection.
4. Suppose one of the random numbers generated is 43,207. In this case, cell E8
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contains the exact cumulative amount of 43,207, and the corresponding invoice to
be sampled is 120102. Now suppose that a random number generated is 51,205.
This amount cannot be exactly matched to entries in column E. Instead, this
amount falls between the two values in cells E8 and E9. In this case, the invoice to
be sampled is 120103. That is, the 51,205th dollar belongs to invoice number
120103.
In the next section, you will work through these two examples, starting with the simple (unusual)
case of an exact match (using the number 43,207), then the more usual case of no match (using
the number 51,205).
Selecting the required invoices
1. The function VLOOKUP may be used to automatically select the invoices
corresponding to the random numbers generated. To see how a VLOOKUP formula
can be constructed, replace the random number formula in cell G6 with the number
43207. Then, in cell H6, type the formula:
= VLOOKUP(G6,$E$6:$F$56,2)
This formula looks up the value in cell G6 from the lookup table in E6 to F56. That
is, the value in cell G6 (43,207) is compared with each of the cells in column E,
starting with cell E6. If an exact match is found, the VLOOKUP function returns thecorresponding value in column F. In other words, it looks to the right of the
column where the match was found; this is the meaning of the 2 at the end of the
VLOOKUP formula. (For Excel, the data column E is considered column 1 and F is
column 2.) Thus, the invoice number of 120102 from cell F8 is returned as the
result of the VLOOKUP function in cell H6.
2. What happens when there is no exact match? Replace the formula in cell G7 with
the number 51,205. Copy the formula from cell H6 to H7. Notice the use of absolute
cell references so E6 and F56 are not adjusted. After copying, the formula in cell H7
should be:
= VLOOKUP(G7,$E$6:$F$56,2)
3. By visual inspection of the worksheet, you can see that the correct result in cell H7
should be 120103. However, the VLOOKUP function yields the invoice number of
120102. Clearly, the VLOOKUP function does not work where no exact match exists
between the value to be looked up (in column G) and the values in column E. Thus,
it is necessary to modify the formula so that it can cope with either an exact match
(as in step 1 of this section) or a no-exact-match situation (as in step 2). Type
this formula in cell H7:
=IF(G7=VLOOKUP(G7,$E$6:$F$56,1),
VLOOKUP(G7,$E$6:$F$56,2),
VLOOKUP(G7,$E$6:$F$56,2)+1)
This formula is not as overwhelming as it may appear! It actually has three parts.
The first part of the formula compares the random number in cell G7 with the result
of VLOOKUP of column E to determine whether these two values are identical. (The
1 at the end of the first line of the formula means look in column E.) If they are
identical, the IF function returns the second part of the formula, which looks up the
invoice number in column F. If, however, the random number does not exactly
match any of the values in column E, the formula returns the third part, which
looks up the invoice number in column F and adds one to it (in this case, correctly
yielding the value of 120103). Note that this formula will only work when the invoice
number sequence is unbroken, that is, exactly one number apart.
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4. Copy the VLOOKUP formula from cell H7 to cell H6 and to cells H8 to H47.
5. Replace the random numbers entered in cells G6 and G7 in steps 1 and 2 with the
correct random number formula (the simplest way is to copy the formula in cell G8
to cells G6 to G7).
Freezing the random numbers and printing the results
The random numbers generated by the formulas in cells G6 to G47 change each time the
worksheet is modified, or if F9 is pressed. However, once you have generated a set of random
numbers, you need to freeze them so they can be used for sample selection. To freeze therandom numbers in cells G6 to G47, use the following procedure:
1. Select the range you want to freeze G6 to G47.
2. After the range G6 to G47 has been highlighted, select Edit Copy.
3. Click cell G6 and select Edit Paste Special.
4. In the Paste Special box, click Values in the Paste box. Then Click OK. The formulas
in cells G6 to G47 will now be replaced with the calculated values. Press F9 to
confirm that these values are frozen.
5. After the random numbers are frozen, and the formulas in cells G6 to G47 are
replaced by values, save a copy of your worksheet.
6. Print a copy of the worksheet.
7. If you notice that your worksheet contains errors, repeat the previous steps to make
any necessary corrections. The solution is provided for your comparison on sheet
tab M6P1S. However, notice that the random numbers in the solution may not be
the same as on your own worksheet. Print a copy of the formulas in the solution
worksheet and compare it with your worksheet. Reconcile and correct any
discrepancies in the formulas.
Commentary
Because Excel performs random number generation with replacement, it is possible to generate
duplicate invoice numbers in the sample. For example, the invoice number 120130 shows up
twice in column H. In dollar-unit sampling, duplicated sample items are permitted, but they will
be treated as two sample items statistically, and if this sample is found to be misstated, it will be
counted as two errors (see bottom of page 5 and top of page 6 of Reading 6-1).
Attribute sampling: Seven-step framework
1. Specifying the objectives of the test
2. Defining the conditions for deviation3. Defining the population from which a sample can be drawn
4. Determining the sample size
5. Selecting the sample
6. Performing the tests of control audit procedures
7. Evaluating the evidence of the test
Substantive testing: Seven-step framework
1. Specifying the objectives of the test
2. Defining the population from which a sample can be drawn
3. Choosing an audit sampling method
4. Determining the sample size
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Module 6 self-test
Question 1
What are the primary distinctions between statistical and non-statistical sampling?
Solution
Question 2
What are the two specific purposes for which audit sampling is used?
Solution
Question 3
Sheila Mackenzie, CGA, tested sales transactions for the month of June in the audit of the financial statements
of Coast Co. for the year ended December 31, 20X1. Based on her assessment of control risk below maximum
and the excellent results of tests of controls, she decided to significantly reduce her direct tests of details ofthe financial balances at the year end. Evaluate this decision.
Solution
Question 4
In applying tests of controls, why is it necessary to define a compliance deviation in advance? Give an example
of a compliance deviation for sales for each of the seven internal control objectives.
Solution
Question 5
What criterion must be met if a sample is to be considered random?
Solution
Question 6
When auditing account balances, why is an incorrect acceptance decision considered more serious than an
incorrect rejection decision?
Solution
Question 7
What major difference between tests of controls and tests of details of balances makes physical (or population)
unit attributes sampling inappropriate for tests of details of balances?
Solution
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Self-test 6
Solution 1
The primary distinctions between statistical and non-statistical sampling are:
Non-statistical sampling does not use statistical calculations to express the results.
Statistical samples must be random.Statistical sampling involves using mathematical calculations to express the results.
Statistical sampling enables the measurement/computation of sampling risk and the calculation of
the upper error limit.
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Self-test 6
Solution 2
Audit sampling is used to assess control risk, for example, in tests of control to audit compliance
with internal control procedures.
Audit sampling can also be used to obtain direct evidence about financial statement assertions.For instance, in substantive tests (tests of balances), sampling can be used to audit the dollar
amounts and therefore determine if they are materially misstated.
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Self-test 6
Solution 3
Generally, assessment of control risk at below maximum and a successful test of controls allow for a reduction
of tests of details of balances at year end. However, Sheila chose the month of June, which only represents
one-twelfth of the year, for her test period. With such a short test period, Sheila cannot conclude that she has
selected a representative sample from the total population; therefore, without testing additional months (theconsensus of public accounting firms is nine months, but no specific standard exists), Sheila cannot change the
scope of her tests of details of financial balances at year end.
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Self-test 6
Solution 4
Compliance deviations should be defined in advance so auditors will know what to look for and
will know one when they see it.
Seven examples based on seven internal control objectives:
Objective Example
1. Validity 1. Sales recorded without supporting shipping orders
2. Authorization 2. Lack of credit manager approval for a credit sale
3. Accuracy 3. Mathematical errors in sales invoice calculations
4. Classification 4. Sales classified in wrong product l ine revenue account
5. Proper period 5. Sales recorded in month (quarter, year) before the actual shipment
6. Accounting 6. Sales charges fail to be posted to a customers account
7. Completeness 7. Sales charges fail to be billed to customers and recorded as sales and
receivables
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Self-test 6
Solution 5
A sample is considered random if each unit in the population has an equal likelihood (probability) of being
included in the sample.
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Self-test 6
Solution 6
An incorrect acceptancedecision directly impairs the effectiveness of an audit. Auditors decide
that account balances are materially accurate and the material misstatement appears in the
financial statements.
An incorrect rejectiondecision impairs the efficiency of an audit by causing unnecessary work.
Further investigation of the cause and amount of misstatement provides a chance to reverse the
initial decision error.
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Self-test 6
Solution 7
The most important difference between tests of controls and tests of details of balances is in what the auditor
wants to measure.
In tests of controls, the primary concern is testing the effectiveness of internal controls. Whenstatistical sampling is used for tests of controls, physical attributes sampling is ideal because it
measures the maximum deviation rate for a population of physical units such as individual
transactions or items. The deviation rate is the frequency of occurrence or the error rate in these
physical units.
In tests of details of balances, the concern is determining whether the monetary amount of an
account balance is materially misstated. Physical unit attributes sampling, which measures the
error rate in population units, therefore will not likely be useful for tests of details of balances.
Dollar-unit or monetary-attributes sampling is more appropriate because it measures the deviation
in dollar amounts directly since the population is the recorded dollar value of the population (for
example, the monetary amount of the account balance).
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Module 6 summary
Explain why auditors use sampling, and describe the two applications of
audit sampling.
Auditors use sampling because it is seldom possible to obtain absolute assurance about an audit
population and, in any case, it is not economical to examine every transaction or item making upan account balance.
Audit sampling can be applied when testing controls to assess control risk. It can also be used
during substantive testing of the balances appearing in financial statements.
Explain statistical and non-statistical sampling, describe the advantages of
each method, and explain when each method should be used.
Statistical sampling uses the laws of probability when selecting the sample and extrapolates the
sample result to the population. Non-statistical sampling is audit sampling not based on statistical
calculations.
The advantage of statistical sampling is that the sampling risk is known. It requires a precise and
definite approach to the audit problem. The advantage of non-statistical sampling is that it
permits the auditor to use greater judgment.
Statistical sampling is used when random numbers can be assigned to population items and an
objectively defensible result is desired. Non-statistical samples can be used when the auditor has
additional knowledge about the population, when strictly defensible results are not required, and
when assignment of random numbers to population items is difficult or impossible.
Explain sampling error, including errors arising from alpha risk (Type I
error), beta risk (Type II error), and non-sampling error.
Sampling error arises because there is always a risk that the sample will not be representative of
the population.
The risk of incorrect rejection of the population is referred to as alpha risk (or the risk of a Type I
error); the risk of incorrect acceptance is referred to as beta risk (or the risk of a Type II error).
The auditor is mainly concerned about beta risk, which could lead to accepting a population that
contains material misstatements.
Non-sampling risk arises because the auditor fails to detect an error when verifying selected items
or because the test is inappropriately designed for the purpose for which it is conducted (for
example, selecting from the wrong population).
Outline the seven-step framework for conducting attribute sampling for
tests of controls.
The seven steps in attribute sampling are
1. Specify the audit objectives.
2. Define the deviation conditions.
3. Define the population.
4. Determine the sample size.
5. Select the sample.
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6. Perform the test of controls procedures.
7. Evaluate the evidence.
Describe the factors that influence sample size determination in attribute
sampling, and determine the sample size for a test of controls using
statistical sampling.
The factors that influence sample size in attribute sampling are
acceptable risk of overreliance
tolerable deviation rate
expected population deviation rate
population size
Sample size varies directly with the expected population deviation rate and the population size.
Sample size varies inversely with the acceptable risk of overreliance and the tolerable deviation
rate. The influence of the population size is minimal, except for relatively small populations.
The sample size is determined using a table for the appropriate acceptable risk of overreliance
and selecting the value on that table corresponding to the tolerable deviation rate and the
expected population deviation rate.
The table corresponding to an acceptable risk of overreliance of 5% is found in Exhibit 6.5-1 of
Topic 6.5.
Describe how the auditor evaluates the results of a test in the contexts of
non-statistical and statistical sampling for attribute sampling.
If using statistical sampling, the auditor can look up the computed upper deviation rates on the
table corresponding to the predetermined acceptable risk of overreliance. The auditor finds the
value for the sample size used and the number of deviations found. The auditor can accept or
reject the population by comparing the upper deviation rate with the tolerable deviation rate used
in determining the sample size.
If using non-statistical sampling, the auditor must use judgment to decide whether to accept or
reject the population based on the sample results.
Describe the nature of audit risk, sampling risk, and materiality in the
context of substantive testing.
Sampling risk in substantive testing is related to audit risk as both denote the risk that the auditor
will accept a population containing a material misstatement.
The tolerable misstatement for such sampling is usually set as the overall financial statement
materiality determined by the auditor at the planning stage of the audit.
Outline the seven-step framework for audit sampling in substantive testing,and explain how an auditor can use stratification to reduce sam