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Variables Sampling 689
I have edited a portion of Module G from your textbook so that it more closely follows my lecture. I
need to acknowledge that this is not my original work and much of it is taken word for word from the
2nd edition of Auditing & Assurance Services by Louwers, Ramsay, Sinason and Strawser.
Tad Miller
Classical Variables Sampling
LEARNING OBJECTIVE Understand the basic process underlying classical variables
sampling in an audit examination.
When performing substantive procedures, one approach is classical variables sampling.
Classical variables sampling methods use normal distribution theory and the Central Limit Theorem to
provide a range estimate of the account balance. The auditor uses the sample estimates to determine
whether the account balance is fairly stated. The Central Limit Theorem indicates larger sample sizes
provide a sampling distribution that more closely reflects a normal distribution. Therefore, larger
sample sizes will yield a lower level of sampling risk.
In this section, we briefly illustrate mean-per-unit classical variables sampling. We illustrate
the manual calculations necessary to determine sample size and evaluate sample results. However, if
clients maintain records in electronic format, auditors typically use computer software to perform these
tasks.
Classical Variables Sampling: Planning
In the planning stages of classical variables sampling, the auditor determines the objective of
sampling, defines the attribute of interest, and defines the population. We will utilize the basic
information from Rice, Inc.'s accounts receivable introduced in the previous section. Recall that Rice's
accounts receivable are comprised of 1,505 individual customer accounts which are recorded at
$416,000. Also recall that the auditor is interested in evaluating the existence or occurrence assertion
and the valuation and allocation assertion. The following assessments or judgments have been made
prior to selecting individual customer accounts for confirmation:
Overall exposure to audit risk = 5 percent (goal or objective)
Inherent risk for the assertions = 100 percent
Control risk related to the assertions = 50 percent
β risk (incorrect acceptance) = 10 percent (based on above risks).
α risk (incorrect rejection) = 15 percent (based on above risks).
Tolerable error = $6,000
AR = IR * CR * DR
DR = AR / ( IR * CR)
DR = 1.00 / ( 1.00 *0.50 )
AR = 5%
IR = 100%
CR = 50%
β = 10%
α = 15%
DR = 0.10
Assume that Rice's five largest customers have balances totaling $120,000 and that each of
these balances is larger than the $6,000 tolerable error. Also, assume that no other customer had a
balance larger than $2,000. Auditors should always examine each item that exceeds the tolerable
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error. In this case, the auditor will audit each of the accounts for the five largest customers. The
auditor would then sample from among the remaining 1,500 accounts which have a balance of
$296,000.
This is a form of stratification. When using classical variables sampling, we can reduce
variability and ensure the selection of individually significant items by subdividing the population into
different (more homogenous) groups based on account size. Stratification can reduce the necessary
sample size. In the above scenario, the auditor divides the population into two strata: the five largest
dollar accounts with a balance of $120,000 and the remaining 1,500 accounts with a balance of
$296,000.
Assuming we find no errors in the five largest account balances, then our objective is to
gather evidence that the remaining accounts are not materially overstated. Although we cannot prove
the recorded balance is correct using with statistics, we can provide persuasive evidence that the
account balance is not materially overstated. If tolerable error is $6,000, then we want persuasive
evidence that the balance in the account exceeds $410,000 ($416,000 - $6,000) or that the balance of
the 1,500 remaining accounts exceeds $290,000. If we are 90% confident that the true balance of
these accounts exceeds $290,000, then we would be 90% confident that the true balance of the
account exceeds $410,000 and the account is not materially overstated.
Classical Variables Sampling: Determining Sample Size
The formula for calculating sample size using mean-per-unit estimation is:
book value
critical value
-3
-2
-1.44
hypothetical
mean
0
0
292,824
296,000
195.216
197.333
2
1.28
290,000
292,824
193.333
195.216
TE = [α/2+ β] * σ/√n
N
n
N
α
β
σ
TE
EE
1
-1
OR
average
3
average
n=
risk of incorrect acceptance
β risk = 10%
one-tail
risk of incorrect rejection
α risk = 15%
two-tail => 7.5% in each ta
N*[α/2+ β] * σ) 2
TE
= Sample size
= Population size (number of accounts)
= Reliability factor for the risk of incorrect rejection (alpha Type I)
α = 15%
= Reliability factor for the risk of incorrect acceptance (beta Type II)
β = 10%
= Standard deviation
from page 4
S = 31
= Tolerable error
$6,000
= Expected error
The remainder of this section focuses on the two new factors that are considered in classical
variables sampling: the risk of incorrect rejection and standard deviation.
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Risk of Incorrect Rejection
Classical variables sampling explicitly considers both β risk (incorrect acceptance) and α risk
(incorrect rejection) to determine the sample size. α risk is the probability that the auditor concludes
that the account balance is materially misstated when, in fact, it is fairly stated. As with any sampling
risk, achieving lower levels of α risk would increase the necessary sample size. That is, α risk has an
inverse relationship with sample size.
Prior to proposing an adjustment to the financial statements based on our sample results, we
would ordinarily expand the sample to include additional customers. This greatly reduces the likelihood
that we will actually reject a balance that is correct. So, the issue of interest related to α risk (incorrect
rejection) is efficiency; what does it cost the auditor to expand the sample?
EXHIBIT G.11
Reliability Factors for Different Levels of Sampling Risk in Classical Variables Sampling
no. of
level of risk
level
No. of std dev
std dev
α 2-tail
β 1-tail
of risk
α 2-tail
β 1-tail
2.58
.01
.005
0.01
2.58
2.33
2.33
.02
.01
.012
2.25
1.96
.05
.025
0.05
1.96
1.65
1.65
.10
.05
0.10
1.65
1.28
1.44
.15
.075
0.15
1.44
1.04
1.28
.20
.10
0.20
1.28
0.84
1.04
.30
.15
0.84
.40
.20
0.56
.288
In contrast, if the cost sample is relatively high, the auditor would be concerned about these
costs and would ordinarily choose a lower level of α risk (incorrect rejection). This lower risk would, in
turn, result in an increased sample size.
How are sampling risks incorporated in the determination of sample size? Exhibit G .11
provides a listing of reliability factors for various levels of α/2 risk (incorrect rejection) and β risk
(incorrect acceptance). These factors represent the proportions of observations that fall within a certain
number of standard deviations in a normally distributed population.
For Rice, Inc. the auditor assesses α risk at 15 percent. β risk was previously established at 10
percent.
Standard Deviation
The standard deviation represents the variability of the population being examined; it is
calculated by summing the squared differences between each item in the population and the population
mean. As the population is more variable (i.e., the items comprising the population differ more widely
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with respect to dollar amount), the standard deviation increases. When the standard deviation of dollar
amounts is larger, it is more difficult for the auditor to select a representative sample. To do so, he or
she needs to increase the necessary sample size. Thus, the standard deviation has a direct relationship
with sample size. That is, as the standard deviation is higher, the sample size Increases.
How can the auditor determine the standard deviation? In mean-per-unit estimation, the auditor
is interested in knowing the standard deviation of audited values of customer accounts. The auditor can
either rely on experience from prior years' audits or use a small pilot sample in the current year.
Assume that the sample standard deviation for Rice, Inc.'s accounts receivable is $31.
Calculating Sample Size
At this point, the sample size can be determined as follows:
n = (N * [α/2 + β] * σ) 2
TE
(1,500 * [1.44 + 1.28] * $31) 2
$6,000
= 444.37 or 445 accounts
N (the number of accounts) can be readily determined from the client's records; in this case,
Rice's accounts receivable were comprised of 1,505 customer accounts, of which we have already
audited each of the five individual accounts that exceeded tolerable error. The factors for α risk and β
risk correspond to 15 percent α risk and 10 percent β risk are drawn from Exhibit G.11. Based on
prior years' audits, the standard deviation was estimated as $31. Tolerable error was determined
based on the recorded account balance and overall financial statement materiality and was established
at $6,000.
If the auditor had not decided to examine the five individually significant customer account
balances the variability of the population would have been significantly larger.
Review Checkpoints
G.21 How does classical variables sampling provide the auditor with evidence as to the fairness of an
account balance or class of transactions?
G.23 What is the standard deviation? How does it affect the necessary sample size?
Classical Variables Sampling: Selecting the Sample
One of the basic tenants of statistical sampling is that each sampling unit has an equal
probability of selection. The sample of Rice, Inc.'s customer accounts could be selected in either of the
following ways:
1. Identify 445 random numbers and select the corresponding customer accounts for confirmation
(unrestricted random selection).
2. Randomly select a starting point (or a number of starting points) in the population and select every
nth customer account thereafter for confirmation (systematic random selection).
Classical variables sampling defines the sampling unit as the customer account balance. As a
result, the auditor will select 445 of Rice's remaining 1,500 customer account balances for examination.
Without stratification, classical variables sampling does not provide the auditor with assurance that the
largest dollar account balances are selected for examination.
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Classical Variables Sampling: Measuring Sample Items
Once the sample size has been determined and the sample has been selected, the auditor
measures the sample items. In the audit of accounts receivable, this will require the auditor to determine
the audited value of the customers' accounts receivable. This will be done using accounts receivable
confirmation procedures as well as additional procedures necessary to follow up on any discrepancies
revealed by the confirmation procedures. This is the stage where the auditor may be exposed to nonsampling risk if his or her substantive procedures fail to detect misstatements.
Assume that the auditor's examination of the 445 customer accounts receivable revealed a total
audited value of $87,220. Therefore, the mean audited value per unit is $196 ($87,220 / 445 = $196).
In addition, the standard deviation of the audited values is $25, which is different than we used for
planning our tests.
Classical Variables Sampling: Evaluating Sample Results using critical value approach
If the standard deviation of the sample was the same as that used to plan the tests, then we
would compare the sample mean ($196) to the critical value ($195.22). Because the sample mean
exceeds the critical value the auditor could assume that there is less than a 10% probability that the
sample mean of $196 would come from a population with a true mean of $193.33.
However, the sample standard deviation was $25. Therefore, the auditor must evaluate the
sample results using this new information. The auditor could calculate a new critical value using the
standard deviation from the sample.
critical value = μ + β * σ /√n
Or
critical value = (BV – TE) +N* β * σ
/√n
193.333 + 1.28*25/√445
290,000 + 1,500*1.28*25/√445
$ 194.85
$ 292,275.42
n = Sample size
N = Population size (number of accounts)
β = Reliability factor for the risk of incorrect acceptance (beta risk Type II error)
σ = Standard deviation (for items selected for examination)
Because the sample mean of $196 exceeds the critical value, there is less than a 10% probability that a
sample mean greater than $194.85 would come from a population with a mean of $193.333. Therefore
there is less than a 10% probability that the true mean is less than $193.33. There is less than a 10%
probability that the recorded book value is materially overstated.
Evaluating Sample Results using achieved precision approach
Another approach would be to use the sample information to calculate the achieved precision. With
this approach, the auditor compares the sample mean ($196) to the hypothetical mean ($193.333) and
calculates the probability of obtaining such results. We would divide the $2.667 difference between our
sample mean and the hypothetical mean by the standard deviation ($25/√445), or $2.667/$1.185, to
determine the sample mean is 2.25 standard deviations greater than the hypothetical mean (or tolerable
mean). From table G1.11, we can see that there is a 1.2% probability of a sample mean being 2.25
standard deviations greater. The achieved precision would be greater than 98.8%. We can be more than
98.8% confident that the recorded book value is not materially overstated.
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Repeat the exercise assuming a sample mean of $194
Evaluating Sample Results using critical value approach
If the standard deviation of the sample was $31, the same as that used to plan the tests, a
comparison of the sample mean ($194) to the critical value ($195.22) would not allow the auditor to
state that the account balance is not materially overstated. However, the sample standard deviation
was $25. Therefore, when the auditor evaluates the sample results using this new information, (s)he
would calculate a new critical value using the standard deviation from the sample.
critical value = μ + β * σ /√n
Or
critical value = (BV – TE) +N* β * σ
/√n
193.333 + 1.28*25/√445
290,000 + 1,500*1.28*25/√445
$194.85
$292,275.42
n = Sample size
N = Population size (number of accounts)
β = Reliability factor for the risk of incorrect acceptance (beta risk Type II error)
σ = Standard deviation (for items selected for examination)
Because the sample mean of $194 is less than the revised critical value, there is still greater than a 10%
probability that a sample mean greater than $194 could come from a population with a mean of
$193.333. Therefore, there is more than a 10% probability that the recorded book value is materially
overstated.
Achieved precision approach
Another approach would be to use the sample information to calculate the achieved precision. With
this approach, the auditor would compare the sample mean ($194) to the hypothetical mean ($193.333)
and calculate the probability of obtaining such results. We would divide the $0.667 difference between
our sample mean and the hypothetical mean by the standard deviation ($25/√445), or $0.667/$1.185 to
determine that the sample mean is 0.56 standard deviations greater than the hypothetical mean (or
tolerable mean). From table G1.11, we can see that there is greater than a 29% probability of a sample
mean being 0.56 standard deviations greater. The achieved precision would be 71%. There is an
unacceptable probability that the recorded book value might be materially overstated.
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