au1 external audit - module 6

7/30/2019 AU1 External Audit - Module 6

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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)

Course Schedule Course Modules Review and Practice Exam Preparation Resources


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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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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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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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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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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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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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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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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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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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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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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

au1 external audit - module 6

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