474
CHAPTER 20
Statistical Sampling Concepts for Tests of Controls and Tests of Balances
LEARNING OBJECTIVES
PART I
Review
Checkpoints
Exercises
Problems
Cases
1. Explain the role of professional
judgment in assigning numbers to
risk of assessing control risk too
low, risk of assessing control risk
too high, and tolerable deviation
rate.
1, 2, 3, 4,
5, 6, 7, 8
38, 44
21, 45, 46
2. Use statistical tables or
calculations to determine test of
controls sample sizes.
9, 10, 11,
12
39
21, 45, 46
3. Calculate the effect an test of
controls sample sizes of subdividing
a population into two relevant
populations.
13
4. Use your imagination to overcome
difficult sampling unit selection
problems.
14, 15
40, 41
5. Use evaluation tables or
calculations to compute statistical
results (CUL, the computed upper
limit) for evidence obtained with
detail test of controls procedures.
16, 17
42
6. Use the discovery sampling
evaluation table for assessment of
audit evidence.
18
43
7. Choose a test of controls sample
size from among several equally
acceptable alternative.
19, 20
45
21, 46
Ashton Behavioral Cases continues
before problem 20.21
PART II
22, 45
36, 37
Review
Checkpoints
Exercises
and Problems
Cases
8. Calculate a risk of incorrect
acceptance, given judgments about
inherent risk, control risk and
analytical procedures risk using the
audit risk model.
21,22,23,24
70
9. Explain the considerations
determining a risk of incorrect
rejection.
25, 26, 27
85
10. Explain the characteristics of
dollar-unit sampling and its
28, 29, 29,
DUS: 41
475
relationships to attribute sampling.
30, 31
11. Calculate a dollar-unit sample size
for the audit of the details of an
account balance.
32, 33, 34,
35
12. Describe a method for selecting a
dollar-unit sample, define a
"logical unit," and explain the
stratification effect of dollar-unit
selection.
36, 37, 38
13. Calculate an upper error limit for
the evaluation of dollar-value
evidence, and discuss the relative
merits of alternatives for
determining an amount by which a
monetary balance should be adjusted.
39, 40, 41,
42, 43
70 84
82
86
70 71
POWERPOINT SLIDES
PowerPoint slides are included on the website. Please take special note of:
*
*
Risk and Materiality in Sampling
Illustration of Dollar Unit Sampling as a Hook
SOLUTIONS FOR REVIEW CHECKPOINTS
20.1
Use the model AR = IR x CR x DR to solve for different values of Audit Risk
(AR) when internal control risk (CR) is given different values. In all cases
IR = 0.90 and DR = 0.10, therefore, AR = 0.90 x CR x 0.10
When CR is
0.10
0.50
0.70
0.90
1.00
20.2
AR is
0.009
0.045
0.063
0.081
0.090
or
or
or
or
or
.9
4.5
6.3
8.1
9.0
percent
percent
percent
percent
percent
Roberts' method in equation form is:
( RIA at assessed
RIA at maximum )
Incremental RIA = RACRTL x ( control risk
- control risk
)
The method produces low RACRTL at the low control risk levels and high RACRTL
at the higher control risk levels.
The logic of the method is: "At the lower control risk levels RACRTL should be
small because assessing control risk quite low makes a big difference in the
substantive sample size and hence in the risk of incorrect acceptance in the
substantive balance-audit work, but at the higher control risk levels the
RACRTL can be high because assessing control risk slightly too low does not
affect the substantive sample size and risk of incorrect acceptance very
much."
20.3
Assessing the control risk too low causes auditors to assign less control risk
(CR) in planning procedures than proper evaluation would cause them to assign.
476
The result could be (1) inadvertently conducting less audit work than properly
necessary and taking more audit risk (AR) than originally contemplated,
perhaps to the unpleasant results of failing to detect material misstatements
(damaging the effectiveness of the audit) or (2) discovering in the course of
the audit work that control is not as good as first believed, causing an
increase in the audit work, perhaps at a time when doing to is very costly
(damaging the efficiency of the audit).
The important considerations when auditing a particular account are questions
related to (1) How sensitive is the final substantive audit work to assessing
control risk too low?, and (2) Is "recovery"--increasing the substantive audit
work at a later date upon discovery of the decision error--more expensive and
time consuming than planning more work at the outset (i.e., planning to
"overaudit")?
20.4
Assessing the Control Risk Too High
The important consideration involved in judging an acceptable risk of
assessing the control risk too high is the efficiency of the audit. Assessing
control risk too high causes auditors to think they need to perform a level of
substantive work which is greater than a proper evaluation of control would
suggest. Assessing control risk too high leads to overauditing.
Some auditors may be willing to accept high risks of assessing the control
risk too high because they intend to overaudit anyway, and the audit budget
can support the work.
Other auditors want to minimize their work (within acceptable professional
bounds of audit risk) and thus want to minimize the risk (probability) of
overauditing by mistake.
Technically, the risk of assessing control risk too high in relation to an
attribute sample is the probability of finding in the sample (n) one deviation
more than the "acceptable number" for the sampling plan. For example, if the
plan called for a sample of 100 units and a tolerable rate of 3 percent at a
.10 risk of assessing control risk too low, the "acceptable number" is zero
deviations. (Appendix 13-B.3 shows CUL = 3 percent when zero deviations are
found in a sample of 100 units.)
The probability of finding 1 or more deviations when the population rate is
actually 2 percent is:
20.6
Probability (x > 0 : n = 100, r = .02)
= 1 - (1 - r)n
= 1 - (1 - 0.2)100
= .867
or 86.7 percent
Probability (x > 0 : n = 100, r = .005 )
= (1 - r)n
= 1 - (1 - .005)100
= 0.394
or 39.4 percent
The "connection" is a direct relationship between control risk and the
tolerable deviation rate. (1) When larger values are planned for control risk
(say, 0.95, 0.90) in an audit plan, more analytical procedure and test of
detail work will be done. Auditors will not rely very much on internal
controls. Therefore, not much help is expected from the controls anyway, so
the tolerable deviation rate can be larger. The direct relation is: The higher
he control risk, the higher the tolerable deviation rate can be. (2) When
477
lower values are assigned to control risk (say, 0.10, 0.20) in an audit plan,
less analytical procedure and test of detail work will be done. Auditors
intend to rely on internal accounting controls. Therefore, effective
compliance with control policies and procedures is important, and the
tolerable deviation rate ought to be low. The direct relation is: The higher
the planned control risk, the higher the tolerable deviation rate can be.
20.7
The connection between tolerable dollar misstatement assigned for the
substantive audit of a balance and tolerable deviation rate used in a test of
controls sample is the smoke/fire multiplier judgment. It is the factor by
which auditors believe transactions can be exposed to control deviations yet
not create dollar-for-dollar misstatements in the related account balance.
20.8
Professional Judgments and Estimates in Test of Controls Attribute Sampling
20.9
a.
Risk of Assessing Control Risk Too Low (RACRTL) is a matter of judgment
about the importance ("key") characteristic of a particular client
control procedure. An auditor can take more risk of assessing control
risk too low on unimportant controls than on important ("key") ones.
Alternatively, the risk of assessing control risk too low can be
considered a constant (say, .10) and the importance of a control can be
measured in terms of a smaller or larger tolerable rate. A more logical
approach is to use Robert's method to derive RACRTL from the separate
judgment of an "incremental risk of incorrect acceptance" for the related
substantive audit.
b.
Risk of Assessing Control Risk Too High is a matter of judgment abut the
efficiency of an audit engagement. The risk can be quite high when the
audit team is willing to do extensive substantive work anyway. If the
work budget is tight, auditors need to find objective ways (e.g., larger
test of controls audit samples) to mitigate the risk.
c.
Tolerable Deviation Rate is a judgment about how many control deviations
can exist in the population, yet the control can still be considered
effective. Auditors need to be careful about brushing aside findings of
deviations. The smoke/fire multiplier is a judgmental connection of the
tolerable misstatement in the substantive sample with the tolerable
deviation rate in the test of controls sample.
d.
Expected Deviation Rate in the Population is an estimate, usually based
on assumptions or sketchy information, of the imbedded incidence of
control deviations. The only use of this estimate in classical attribute
sampling is to figure a sample size in advance. The statistical
evaluation (CUL calculation) does not use it.
e.
Population Definition might be called a judgment about identification of
the population of control performances that correspond to an audit
objective. For example, an auditor would want to be sure he is sampling
from a file of recorded documents if his objective is to audit the
controls over transaction validity.
To figure a test of controls sample size using the Appendix 20-A tables, you
need to know:
FACT: population size, for finite correction if it is fewer than 1000.
JUDGMENT: tolerable deviation rate
JUDGMENT: risk of assessing control risk too low (RACRTL)
ESTIMATE: estimated deviation number or rate.
478
The risk of assessing control risk too high (RACRTH) is not used in the
calculation.
20.10 Enter Appendix 20A for BETA = 5 percent (confidence level = 95 percent)
And use N = R/P with R(K=.3, BETA=.05), N= 7.76/(.09)=87
20.11 To figure a test of controls sample size using the Appendix 20.A poisson risk
factors, you need to know:
FACT: population size, for finite correction if it is fewer than 1000.
JUDGMENT: tolerable deviation rate.
*
These two are needed to get the correct poisson risk factor:
* JUDGMENT: risk of assessing control risk too low (RACRTL)
* ESTIMATE: estimated deviation number or rate.
The risk of assessing control risk too high (RACRTH) is not used in the
calculation.
20.12 Enter Appendix 20-A for BETA = 5 percent.
N= R(k=2, BETA=.05)/ P= 6.31/.09= 70
20.13 Based on the specifications of risk of assessing control risk too low,
tolerable deviation rate and expected population deviation rate, sample sizes
would be determined independently for the two populations in the subdivision.
If the criteria are at least as stringent for each of the two as they would be
for the undivided population, the sum of the two sample sizes would be at
least twice the size of the sample figured for the single population (provided
both subdivided populations have 1,000 or more units). This is because the
formula is used twice, once for each population.
20.14 No, with any random number table arrangement, you can use 1,2,3,4,5,6, or more
random digits wherever they appear in the printed table.
20.15 1.
2.
3.
Divide the population size by the sample size. obtaining a quotient k.
Obtain a random start in the file and select every kth item for inclusion
in the sample.
If the end of the file is reached, return to the beginning of the file to
complete the selection.
*
When multiple random starts are taken (say, 5), the sampling
interval is 5 x k instead of k.
20.16 The UEL is the estimated worst likely deviation rate in the population, with
the probability = risk of assessing control risk too low that the actual
population deviation rate is even higher.
20.17 Using the Poisson risk factor equation:
UEL = Poisson risk factor for 3 errors, BETA=35% = 4.45 = 9.7%
Sample Size
46
20.18 The discovery sampling table probability is the probability of finding at
least one defined deviation in a sample of a given size, provided the
population deviation rate is equal to the critical rate of occurrence.
20.19 The links that connect test of controls sample planning with substantive
balance-audit sample planning are these:
479
(1)
(2)
(3)
the smoke/fire multiplier judgment that relates tolerable dollar
misstatement in the substantive balance-audit sample to the anchor
tolerable deviation rate in the test of controls sample.
Roberts' method of calculating RACRTL that relates an audit judgment of
incremental risk of incorrect acceptance for the substantive balanceaudit sample to the risk of incorrect acceptance consequences of
assessing control risk too low.
considering the cost of the substantive balance-audit sample to decide
the test of controls sample size and the planned control risk assessment.
20.20 Choose the one that produces the lowest total cost of test of controls
sampling and substantive balance-audit sampling.
SOLUTIONS FOR REVIEW CHECKPOINTS (Dollar-Unit Sampling in Chapter)
20.21 The objective of test of control with attribute sampling is to produce
evidence about the rate of deviation from company control procedures for the
purpose of assessing the control risk. Measuring the dollar effect of control
deviations is a secondary consideration.
The objective of a test of a balance with dollar-value sampling is to produce
direct evidence of dollar amounts of error in the account. This is called
dollar-value sampling to indicate that the important unit of measure is dollar
amounts. Sometimes, dollar-value sampling is called variables sampling just to
distinguish it from attributes sampling and the control risk assessment
objective.
20.22 Use of the audit risk model to calculate RIA does not remove audit judgment
from the risk determination process because all the elements of the model are
auditors' judgments and assessments.
20.23 Yes, the benefit from using the audit risk model to calculate RIA is that it
captures the independent nature of different audit considerations in its
multiplication form, and it enables different auditors who have the same
judgments to produce the same RIA.
20.24 TD =
AR
=
0.015
= 0.20
IR x CR x AP
0.50 x 0.30 x 0.50
You can ask students to illustrate why this comes out the same as the one in
the textbook illustration where
TD =
AR
=
0.03
= 0.20
IR x CR x AP
1.0 x 0.30 x 0.50
20.25 An incorrect acceptance decision directly impairs the effectiveness of an
audit. Auditors wrap up the work and the material misstatement appears in the
financial statements.
An incorrect rejection decision impairs the efficiency of an audit. Further
investigation of the cause and amount of misstatement provides a chance to
reverse the initial decision error.
20.26 The important considerations are cost/benefit and audit efficiency. The
"model" is unique to each audit engagement and to each account because costs
and relationships will differ from client to client.
20.27 Generally accepted auditing standards define and mention the risk of incorrect
rejection, but GAAS takes no "position" on it. GAAS offers no model or method
480
for thinking about RIR. GAAS concentrates attention on the risk of incorrect
acceptance and the effectiveness of audit sampling.
20.28 Some of the other names for types of dollar-unit sampling are: combined
attributes-variables sampling (CAV), cumulative monetary amount sampling
(CMA), monetary unit sampling (MUS), and sampling with probability
proportional to size (PPS).
20.29 The unique feature of dollar-unit (DUS) sampling is its definition of the
population as the number of dollars in an account balance or class of
transactions. Thus, for any given account balance (recorded amount, book
balance) the population is defined as the number of dollars. With this
definition of the population, the audit is theoretically conducted on a sample
of dollar units, and each of these sampling units is either right or wrong.
20.30 Advantages:
1.
2.
3.
4.
An estimate of a normal distribution standard deviation is not required.
A minimum number of errors is not required for statistical accuracy.
Sample sizes are generally small (efficient).
Stratification is accomplished automatically.
Disadvantages:
1.
2.
3.
The DUS assignment of dollar amounts to errors is conservative (high)
because rigorous mathematical proof of DUS upper error limit calculations
has not yet been accomplished.
DUS is not designed to evaluate financial account understatement very
well. (No sampling estimator is considered very effective for
understatement error, however.)
Expanding a DUS sample is difficult when preliminary results indicate a
decision that a balance is materially misstated.
20.31 DUS resembles attribute sampling for control deviations in the definition of
an error--a dollar is either right or wrong (modified later in connection with
tainting calculations). Also, the population is defined as units (dollar) of a
uniform size, instead of the varied sizes of logical units, such as customer
account balances.
20.32 Larger. DUS sample size varies directly with population size (recorded amount)
20.33 Smaller. DUS sample size varies inversely with the amount of risk of incorrect
acceptance. (The RF becomes smaller.)
20.34 Larger. Larger expected misstatement reduces the planned precision P which is
the denominator in the sample size planning formula.
20.35 Smaller. DUS sample size varies inversely with the amount of tolerable
misstatement.
20.36 The identification of individually significant logical units in an account
balance has the effect of reducing the size of the recorded amount population
for dollar-unit (DUS) sampling. The individually significant units are removed
for 100% audit, and the remainder is the population to be sampled.
20.37 When a $1 unit is selected at random, it "hooks" the logical unit that
contains it, making the logical unit the object of the audit. Since larger
logical units have more $1-units in them, they are more likely to be chosen in
the sample, thus producing a dollar total for the sample larger than the
481
dollar total that would be obtained in an unrestricted random sample in which
the logical unit was the sampling unit (because in the latter case, the
smaller logical units would have an equally likely chance of selection).
20.38 When two dollar units for the sample fall in the same logical unit, the
logical unit is "selected twice." When results are evaluated, any error taint
in this logical unit is counted twice.
20.39 Audit of $600,000 with sample of 100. ASI = $6,000. Calculate the UEL at
various RIA:
RIA
Risk Factor
UEL
3.00
2.31
1.39
0.70
$ 18,000
$ 13,860
$ 8,340
$ 4,200
0.05
0.10
0.25
0.50
20.40 Audit of $600,000 with sample of 100. ASI = $6,000. Calculate the UEL at
various RIA: What is the interpretation of each UEL?
RIA
Risk Factor
UEL
3.00
2.31
1.39
0.70
$ 18,000
$ 13,860
$ 8,340
$ 4,200
0.05
0.10
0.25
0.50
Interpretation: Probability is
5% that actual error exceeds $18,000
10% that actual error exceeds $13,860
25% that actual error exceeds $ 8,340
50% that actual error exceeds $ 4,200
20.41 UEL CALCULATION
(RIA = 0.48)
Basic Error
Likely Error
and PGW
Factors x
Tainting
Percentage
0.73
100.00%
$ 3,125
1.00
1.00
1.00
90.00%
80.00%
75.00%
$ 3,125
$ 3,125
$ 3,125
1. Basic error (0)
2. Most likely error:
First error
Second error
Third error
Average
Sampling
Dollar
x Interval = Measurement
$ 2,281
$2,813
2,500
2,344
Projected likely error
3. Precision gap widening;
First error
0.01
Second error
0.02
Third error
0.01
$ 7,657
90.00%
80.00%
75.00%
$ 3,125
$ 3,125
$ 3,125
$
28
50
23
$
Total upper error limit (0.48 risk of incorrect acceptance)
101
$10,039
Interpretation: The probability is 48% that the error in the account exceeds
$10,039.
UEL CALCULATION
(RIA = 0.05)
Basic Error
Likely Error
Average
482
and PGW
Factors
1. Basic error (0)
2. Most likely error:
First error
Second error
Third error
Projected likely error
x
Tainting
Percentage
Sampling
Dollar
x Interval = Measurement
3.00
100.00%
$ 3,125
1.00
1.00
1.00
90.00%
80.00%
75.00%
$ 3,125
$ 3,125
$ 3,125
3. Precision gap widening;
First error
0.75
Second error
0.55
Third error
0.46
$ 9,375
$2,813
2,500
2,344
$ 7,657
90.00%
80.00%
75.00%
$ 3,125
$ 3,125
$ 3,125
$2,109
1,375
1,078
$ 4,562
Total upper error limit (0.05 risk of incorrect acceptance)
$21,594
Interpretation: The probability is 5% that the error in the account exceeds $21,594.
20.42 The one best measure of a sample-based amount for adjustment, arguably, is the
projected likely error (provided the projection is made from a sufficiently
large sample). The PLM assumes that the error found in the sample is
representative of the error in the remainder of the population, and estimates
the amount of error that might be found if the entire population were audited.
20.43 The auditing profession members cannot even agree on where to go for lunch,
much less agree on a tough concept like sample-based measurement of
misstatement amounts in a population of data. Audit situations are all
different. The errors in one account must be considered in combination with
errors in other accounts. After all, the first pass at an overall materiality
criterion is itself not well defined, and the assignment of tolerable
misstatement to individual accounts is even less well defined. Reportedly,
auditors don't even assess tolerable misstatement anyway. They make ad hoc
adjustment decisions in each individual audit and its circumstances.
SOLUTIONS FOR KINGSTON CASE
20.44 Determine a Test of Controls Sample Size
using Roberts Method Described in solution to 20.2
The test of controls sample sizes were calculated with the Poisson risk factor
equation. The tables do not give all the RACRTL's for the problem.
(1) Required (for basic analysis):
Smoke/fire multiplier = 8.5
TDRCR=.05 = (8.5 x 10,000) / 8,500,000 = .01
Incremental risk of incorrect acceptance = .01
Each RACRTL = .01 / (RIA - RIACR=1.0)
for example RACRTLCR=.20 = 01 / (.25 - .05) = .05
Lowest-cost combination ($873) at CR = .20, indicates a test of controls
sample of 75 (zero deviations expected) followed by a substantive sample of 81
if no deviations are found.
483
EXHIBIT 20.44-1
KINGSTON COMPANY TEST OF CONTROL SAMPLE SIZE ANALYSIS
Test of Control Balance-Audit
Control Risk
Categories
Low control
risk
Moderate
control risk
Control risk
below maximum
Maximum risk
CR
TDR
RIA RACRTL
n[c]
Cost
n[s]
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
.02
.04
.06
.08
.10
.12
.14
.16
.18
.20
.50
.25
.17
.13
.10
.08
.07
.06
.06
.05
196
75
42
26
16
9
5
4
4
4
$ 588
$ 225
$ 126
$ 78
$ 48
$ 27
$ 15
$ 12
$ 12
$ 12
51
81
96
107
117
125
130
136
136
143
.02
.05
.08
.13
.20
.33
.50
.50
.50
.50
Cost
$ 408
$ 648
$ 768
$ 856
$ 936
$1,000
$1,040
$1,088
$1,088
$1,144
TOTAL
$ 996
$ 873
$ 894
$ 934
$ 984
$1,027
$1,055
$1,100
$1,100
$1,156
RIA = Risk of incorrect acceptance for the substantive balance-audit sample.
RIA = (AR=.05)/[(IR=1.0) x CR x (AP=1.0)]
(2) Required (for alternative analysis):
Smoke/fire multiplier = 17
TDRCR=.05 = (17 x 10,000) / 8,500,000 = .02
Incremental risk of incorrect acceptance = .02
Each RACRTL = .02 / (RIA - RIACR=1.0)
for example RACRTLCR=.20 = .02 / (.25 - .05) = .05
Lowest-cost combination ($729) at CR = .10, indicates a test of controls
sample of 107 (zero deviations expected) followed by a substantive sample of
51 if no deviations are found.
Differences:
Jack's criteria produce smaller test of controls samples at the lower control
risk levels, hence smaller total costs and the lowest cost at a lower control
risk level. The two alternatives are about the same at the higher risk levels
because the test of controls sample reach a minimum size.
EXHIBIT 20.44-1
KINGSTON COMPANY TEST OF CONTROL SAMPLE SIZE ANALYSIS
Test of Control Balance-Audit
Control Risk
Categories
Low control
risk
Moderate
control risk
Control risk
below maximum
Maximum risk
CR
TDR
RIA RACRTL
n[c]
Cost
n[s]
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
.03
.05
.07
.09
.11
.13
.15
.17
.19
.21
.50
.25
.17
.13
.10
.08
.07
.06
.06
.05
107
46
25
15
8
5
5
4
4
3
$ 321
$ 138
$ 75
$ 45
$ 24
$ 15
$ 15
$ 12
$ 12
$ 9
51
81
96
107
117
125
130
136
136
143
.04
.10
.17
.25
.40
.50
.50
.50
.50
.50
Cost
$ 408
$ 648
$ 768
$ 856
$ 936
$1,000
$1,040
$1,088
$1,088
$1,144
TOTAL
$ 729
$ 786
$ 843
$ 901
$ 960
$1,015
$1,055
$1,100
$1,100
$1,153
484
RIA = Risk of incorrect acceptance for the substantive balance-audit sample.
RIA = (AR=.05)/[(IR=1.0) x CR x (AP=1.0)]
20.45 Quantitative Evaluation of Compliance Evidence
This problem takes Problem 7.28 into the statistical evaluation calculations.
The quantitative solutions are entered on 10 sampling data sheets. We have not
written any control risk evaluation on these data sheets so you can use them
for transparencies in classroom discussion. We believe a good class exercise
is to write conclusions determined by students for various samples.
Incidentally, under the criteria suggested by the problem data, the small
sample sizes are not large enough to support conclusions of UEL less than 4
percent, even when no deviations are found.
The next ten pages contain the data sheets for various sample sizes. They are
the same as the data sheets in Chapter 7 (Problem 7.27/28), but some
statistical criteria and UEL data are entered here.
[SEE EXHIBIT 20.45 ON NEXT PAGES)
Test of Controls Sampling Data Sheet]
SOLUTIONS FOR MULTIPLE-CHOICE QUESTIONS
485
Exhibit 20.45-1
A
486
Exhibit 20.45-1
B
487
Exhibit 20.45-1
C
488
Exhibit 20.45-1
D
489
Exhibit 20.45-1
E
490
Exhibit 20.45-1
F
491
Exhibit 20.45-1
G
492
Exhibit 20.45-1
H
493
Exhibit 20.45-1
I
494
Exhibit 20.45-1
J
495
496
20.46 a.
Incorrect.
b.
c.
d.
Incorrect.
Incorrect.
Correct.
20.47 a.
Correct.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
20.48 a.
Incorrect.
b.
c.
Incorrect.
Incorrect.
d.
Correct:
a.
b.
Correct.
Incorrect.
c.
Incorrect.
d.
Correct.
Auditors can think about as many tolerable deviation rates as
there are control risk levels.
same reason as in a.
same reason as in a.
same reason as in a.
Auditors should associate only one tolerable deviation rate
with each possible control risk level. Note this may involve
defining deviations to include many conditions.
Only one tolerable deviation rate per control risk level.
Only one tolerable deviation rate per control risk level.
Only one tolerable deviation rate per control risk level.
The control risk levels can be arranged in any logical manner
an auditor desires.
see d. below
They may be low, but the level is a judgment the auditor is
entitled to make.
Tolerable deviation rates should be higher for higher levels
of control risk.
20.49
Using N=R/P with BETA=.05; k=0,and P=.06-.03
This is the sample for 1% risk of overreliance, where 11
deviations is about 3% of the sample of 360.
This is the sample for 10% risk of overreliance, where 5
deviations is about 3% of the sample of 160.
This is the sample for zero expected deviations and 5% risk
of overreliance, i.e., a discovery sample size. Some firms
budget for a discovery sample size audit and so ignore the
expected error rate. This also simplifies audit planning.
20.50 a.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
Correct.
see d. below
see d. below
see d. below
The risks of overreliance should go from low (1%) to high
(10%), because the consequences of overreliance (auditing a
smaller sample size of customer accounts) becomes less
serious.
20.51 a.
b.
c.
d.
Incorrect.
Correct.
Incorrect.
Incorrect.
UEL=Achieved P= R/n (k=2, BETA=.05) /n =6.30/90=.07
7%
20.52 a.
Correct.
The interpretation should relate to a worst rate and a risk
of overreliance.
This answer relates to a lowest rate.
This answer relates to certainty instead of risk.
This answer relates to certainty instead of risk.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
20.53 a., b., and c.
d.
Correct.
should be considered incorrect for the reason given in d,
below. However, for the 1%, 5%, and 10% risks of overreliance
shown in the tables in Appendix 20A, the UELs for one
deviation out of 100 in the sample all exceed 0.04, so the
UEL cannot meet any of the tolerable deviations given in the
three choices.
The decision-maker needs to express a risk of overreliance
criterion in order to assess a control risk level.
497
20.54 a.
b.
c.
Incorrect.
Correct.
Incorrect.
d.
Incorrect.
20.55 a.
Incorrect.
b.
Correct.
c.
d.
Incorrect.
Incorrect.
20.56 a.
Incorrect.
UEL is greater than the tolerable deviation rate.
UEL gives the best conservative measure of deviation rate.
Should not jump to the extreme "rejection" conclusion, unless
5% deviation rate is associated with 100% risk.
Should not impose nonstatistical manipulation on the decision
criteria set up at the beginning of the test.
This assessment could be made, but the better conclusion
would be related to the actual CUL.
The assessed level of control risk should be related to the
actual CUL.
This is nonsense because the control "passed the test."
b. is a better answer, and there's no information that 2% CUL
is associated with the minimum level of control risk.
The tolerable rate reduced by the allowance for sampling risk
is not meaningful.
Assess higher than planned control risk because the computed
upper limit (7%) is higher than the tolerable deviation rate
(5%).
The allowance for sampling risk should be added to the actual
sample results (sample deviation rate) and not to the
tolerable deviation rate.
The relation of computed upper limit (7%) greater than
tolerable deviation rate (5%) leads to the higher rather than
lower control risk assessment.
b.
Correct.
c.
Incorrect.
d.
Incorrect.
20.57 a.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
Correct:
2.5% is the expected deviation rate.
4.5% is the actual sample deviation rate (9/200)
3.5% is not related to anything.
deviations in sample of 200 at 1% risk of assessing control
risk too low = 8.0% P=R(k=9, BETA=.05)/200
=15.77/200
=.07855 =.08
20.58 a.
b.
c.
d.
Incorrect.
Correct
Incorrect.
Incorrect.
limit (8% -
1.0%
8% =
4.5%
5.5%
2.5%
is the risk of assessing control risk too low.
4.5% sample rate plus 3.5% allowance for sampling error.
is the actual sample deviation rate (9/200)
is the expected deviation rate minus the computed upper
= 5.5%)
SOLUTIONS FOR EXERCISES AND PROBLEMS
20.59 Behavioral Decision Case: Determining the Best Evidence Representation
This case is one of Bob Ashton's behavioral decision cases (Accounting Review,
January, 1984, pp. 78-97. He give credit to W. Uecker and W. Kinney, "Judgment
Evaluation of Sample Results: A Study of the Type and Severity of Errors Made
by Practicing CPAs," Accounting, Organizations and Society, vol. 2, no. 3
(1977), pp. 269-75. The "answer" below is taken from Ashton.
NOTE TO INSTRUCTOR: Take a look at this answer. You may want to get the
students to discuss cases 1,2, and 3 first, then give them a chance to think
about Cases 4 and 5. See if they can be fooled to change their minds to choose
the larger samples for Cases 4 and 5, then discuss them.
In this exercise, two pieces of information are available for each of the
498
three pairs of sample outcomes: (1) the sample size, and (2) the sample
deviation rate. While sample size is independent of population parameters,
sample deviation rate is representative of the population characteristic of
interest, i.e. the population deviation rate. Use of the representativeness
heuristic could cause one to ignore the size of the sample, and to base
choices solely on the sample deviation rate. Thus one might choose Sample A in
Case 1 and Sample B in Cases 2 and 3, because their sample deviation rates are
lower.
The calculation of achieved precision P=R/n show, however, that none of these
three sample outcomes provides adequate assurance that the population
deviation rate is below five percent, even at 90 percent confidence [10
percent risk of assessing control risk too low]. The other member of each pair
(choices B,A,A) does provide the desired assurance at a 95 percent confidence
level [5 percent risk of assessing control risk too low]. Thus reliance on the
representativeness of the sample outcomes could lead one to choose the weaker
evidence in these cases.
Notice that the correct choice in Cases 1,2, and 3 is the larger sample. It
might be tempting to conclude that this will always be true, i.e. that larger
samples are always superior to smaller samples. But this simplification will
not always work either. Consider Cases 4 and 5. The correct answers are the
smaller samples. Interestingly, use of the representativeness heuristic (i.e.
focusing on the smaller error rates) would lead to the correct choices in
these two instances, but would result in incorrect choices in the first three
pairs of sample outcomes. This illustrates that while use of simplifying
heuristics can lead to good decisions, it can also lead the decision maker
astray.
20.60 Behavioral Decision Case: Estimating a Frequency
This case is one of Bob Ashton's behavioral decision cases (Accounting Review,
January, 1984, pp. 78-97. He gives credit to M. Gibbins, "Human Inference,
Heuristics and Auditors' Judgment Processes," Proceedings of the CICA Auditing
Research Symposium, Laval University (1977). The answer below is taken from
Ashton.
The best answer to this exercise is the smaller department. This department
processes only 15 invoices per day, while the larger department processes 45.
The smaller department is more likely to have more days in which the number of
invoices specifying discounts deviates from the average of 50 percent, since
sampling variability is greater for small samples than for larger samples. [It
takes only 9 invoices in the smaller department to exceed the 60 percent
variation, while it takes 27 in the larger department.]
People who use the representativeness heuristic, however, often do not
consider the size of the sample, because the degree to which a sample
statistic resembles the population does not depend on sample size.
Consequently, the perceived likelihood of a sample statistic will be
independent of sample size, and people may incorrectly choose "about the same"
as the best answer.
20.61 Calculating Risk of Assessing Control Risk Too High
The proper question is: What is the probability of finding 4 or more
deviations in a sample of 80 when the actual rate in the population is exactly
4 percent?
Use the Poisson approximation to calculate the probability of finding no
499
(zero) deviations, then 1 deviation, then 2, then 3. The sum of these
probabilities is the probability of finding 3 or fewer. Thus the probability
of finding 4 or more is 1 minus the sum.
Look at the equation below. The term "np" is 80 x .04 = 3.2. The term "x" is
0, 1, 2, 3.
P (0;3.2) = 0.0408
Probability of zero deviations in sample of 80
P (1;3.2) = 0.1305
Probability of one deviation in sample of 80
P (2;3.2) = 0.2088
P (3;3.2) = 0.2227
Probability of two deviations in sample of 80
Probability of three deviations in sample of 80
Sum
= 0.6028
Probability of finding 3 or fewer when actual rate is 4
percent.
1 - Sum
= 0.3972
Probability of finding 4 or more when the actual rate is
4 percent.
The risk of assessing control risk too high is 39.72 percent.
Using the Poisson approximation formula, the computed risk of finding zero
deviation when the actual deviation rate in the population is 3 percent is:
Computed risk:=x=0 2.718(80*.03)(80*0.03)0 = 0.091
0!
The computerized risk probability of finding no deviations in a sample of 80
when the population deviation rate is 3 percent is 9.1 percent.
Therefore,
the probability is 1-9.1 percent = 90.9 percent of finding 1 or more
deviations in a sample of 80 when the actual population deviation is 3
percent. Therefore, the auditor who assesses a higher control risk when one
deviation is found is accepting a 90.9 percent risk of assessing the control
risk too high.
If the actual population rate were only 2 percent, the
Poisson probability (computed risk) of finding zero deviations in a sample of
80 would be 20.2 percent. So the risk of assessing control risk too high
would be 1-20.2 percent = 79.8 percent.
Turning to the problem of controlling the risk of assessing control risk too
high in the same example, suppose you decide to sample 140 units so finding 1
deviation could still give you UEL of 3 percent at 10 percent risk of
assessing control risk too low.
The Poisson probability of finding two or
more deviations when the actual population deviation rate is less than 3
percent (say, 2 percent) is 0.7689, calculated as follows:
1.
First calculate the probability (risk) of finding 0 and 1 deviations:
Pp(0:140 x 0.02)=[2.718-2.8(2.8)0]/0! = 0.0608
Pp(1:140 x 0.02)=[2.718-2.8(2.8)1]/1! = 0.1703
Pp(x=0,1:140 x 0.02) = 0.2311
2.
Calculate the probability (risk) of finding two or more deviations:
Since the probability of finding zero or one is 0.2311, the probability of
finding two or more is 0.7689 = 1-0.2311.
Thus, the risk of assessing control risk too high when the actual population
deviation rate is a little less than the tolerable rate is improved to 76.89
500
percent with a sample of 140, from the 79.8 percent risk with a sample of 80.
The improvement is not much, but the example points out how a larger sample
size reduces risk.
Most attribute sampling tables contain probabilities calculated using the
binomial equation.
The binomial equation approximates fairly closely the
hypergeometric equation which is mathematically accurate for finite
populations and for sampling-without-replacement methods.
The hypergeometric
equation is even more difficult to solve than the binomial equation.
The Poisson distribution approximates fairly closely the binomial
distribution, and it is easier to calculate using a pocket calculator capable
of raising numbers to a power.
Auditors can use the equation shown below to
calculate risk because the Poisson distribution is a limiting case of the
binomial distribution when the population is large and the deviation rate is
low (commonly found in audit situations).
Note: Even if the deviation rate is high the Poisson distribution provides a
conservative (i.e. overestimates) the actual deviation rate.
Pp(x;np)= e-np(np)x
X!
where:
Pp(x;np)=Poisson probability of finding exactly x number of deviations in a
sample having np expected number of deviations
e= Base of natural logarithms, approximately 2.718
n= Sample size
p= Hypothesized deviations rate
x= Number of deviations
The computed risk of finding a given number of deviations is a cumulative
function:
Computed risk = x e-np(np)x
x!
For example, consider the illustration in Chapter 20 concerning the risk of
assessing control risk too high.
When the procedure of vouching a random
sample of 80 invoices to supporting shipping orders was performed, the auditor
found no deviations (no cases of missing shipping orders).
The example says
the probability (risk) is 10 percent that the actual population deviation rate
is equal to or greater
20.62 Sample Size Relationships
a.
Tolerable deviation rate = 0.05
Expected population deviation rate = zero.
Sample Size
Risk of Assessing Control
Risk Too Low
0.01
0.05
0.10
b.
Population > 1,000
93
60
47
Population = 500
76
54
45
Acceptable risk of assessing control risk too low = 0.10 (BETA RISK)
Expected population deviation rate = 0.01
501
Sample Size
Tolerable Dev. Rate
Population > 1,000
0.10
0.08
0.05
0.03
0.02
c.
Population = 500
26
33
58
116
231
25
31
52
95
159
Acceptable risk of assessing control risk too low = 0.10 (BETA RISK)
Tolerable deviation rate = 0.10
Sample Size
Expected Population
Deviation Rate
Population > 1,000
0.01
0.02
0.04
0.07
0.09
d.
Population = 500
26
29
39
77
231
25
28
37
68
159
The population size-adjusted sample sizes are figured using the formula
n =
n'
1 + (n' /N)
where
n'
is the sample size in using Appendix 20A and R=NP
N = 500 population size
n = sample size adjusted for population size
20.63 Exercises in Sample Selection
a.
The first 5 usable numbers (using the path down the column and then to
the top of next column) are: 1609, 3342, 2287, 3542, and 1421. Note that
14 numbers were reviewed to get 5, and 9 were discarded.
b.
The first 5 usable five-digit numbers (using the path down the column and
then to top of next column) are: 02921, 05303, 08845, 05851, and 09531.
Note that 26 numbers were reviewed to get 5, and 21 five-digit numbers
were discards.
This selection can be made more efficient by converting the random number to 4
digits to correspond with the number of digits in the population size. There
will be fewer discards if you subtract 2220 from the beginning and ending
numbers and use the sequence 0000 to 9099. Using the first 4 digits and
starting at the same place:
Random numbers Add back 2220
2041
+
2220
=
2870
+
2220
=
7457
+
2220
=
6261
+
2220
=
7568
+
2220
=
Random check number
4261
5099
9677
8481
9788
This method is random and there was only one discard to get 5 usable numbers.
502
c.
1.
2.
3.
d.
Choosing a month at random does not generate a random sample of the
year's vouchers. This is a type of block sample and is not
acceptable for statistical validity (but may be acceptable for
judgmental sample).
Random 7-digit numbers would generate a random sample, but there
would be a large number of discards. This method is not efficient.
Ten vouchers from each month is an acceptable choice if and only if
an equal number of vouchers were recorded each month. This method
can be modified by calculating the relative number of vouchers each
month and selecting that proportion of the vouchers from that month.
For example, if 4,520 vouchers (10%) were issued in January, then
select 12 sample vouchers (10% of 120) from January.
In case (a), select every 100th sales invoice (5,000/50 = 100) starting
with number 1609. The next four numbers are 1709, 1809, 1909, 2009. The
selection may also be made by starting at the front of the file with a
random number between 1 and 100, then selecting every 100th item.
In case (b), select every 91st check (9,100/100 = 91) starting with number
02921. The next four are 03012, 03103, 03194, 03285. As in case (a) other
random starts are possible.
In case (c), select every 376th voucher (45,200/120 = 376) starting with
number 03-01102. The second number is 03-01478. Successive numbers may spill
over into the April file. The auditor has had to count to the end and then
continue the count in succeeding months.
20.64 Imagination in Sample Selection
a.
Sample from Checking Accounts with Overlapping Numbers
*
Let checks in Account #2 be represented by the check numbers 0001 6000 (6,000 checks).
*
Let checks in Account #1 be represented by numbers 6001 - 9000
(obtained by adding the constant 2368 to the actual numbering
sequence of 3633 - 6632). The new sequence contains 3,000 numbers,
the same as the original number of checks.
*
When a 4-digit random number between 0001 and 6000 is selected, it
identifies a check in Account #2.
*
When a 4-digit random number between 6001 and 9000 is selected,
subtract the constant 2368, and the remainder identifies a check in
Account #1.
*
Random numbers 9001 - 9999 are discards.
*
Starting in Appendix 20B, row 1, column 2:
Check Selected in
b.
Random
Number
Discard or
Constant
9541
9985
6815
2543
5190
1925
0030
Discard
Discard
-2368
NA
NA
NA
NA
Account #1
Account #2
4447
2543
5190
1925
0030
Sample of Purchase Orders
*
First convert of 5-digit real sequence (09000 - 13999) to the 4digit sequence 000 - 4999 by subtracting the constant 09000.
503
Then let the sequence of numbers from 5000 - 9999 also represent
purchase orders in the sequence.
Now all the 4-digit random numbers represent purchase orders, and
none will be discards.
You will, however, need to convert each 4-digit random number into a
purchase order number.
Starting in Appendix 13-A, row 30, column 3:
*
*
*
*
Random
Number
Conversion for
5000 - 9999
4251
8991
5077
9431
5595
-
Conversion to
P.O. Sequence
NA
5000
5000
5000
5000
+
+
+
+
+
9000
9000
9000
9000
9000
Purchase
Order No.
13251
12991
09077
13431
09595
Note: When sampling without replacement, some numbers will be discards
when two of them identify the same purchase order. For example, random
numbers 4251 and 9251 both identify purchase order 13251.
c.
d.
Sample of Perpetual Inventory Records
Systematic sampling is probably the most efficient method. You know the
list has 3740 item descriptions (74 x 50 + 40). The factor k = 37.4. Take
5 random starts by entering a random number table and choosing a 2-digit
random number between 1 and 75 to represent a page, choosing the next 2digit random number to represent a line, then selecting every 187th
description (187 = 5 x 37.4). Repeat the procedure five times. For
example, start in Appendix 13-A, row 1, column 1:
Page
Line
1st item
32
59
65
14
28
07
46
35
20
02
p.
p.
p.
p.
p.
32,
59,
65,
14,
28,
1.
1.
1.
1.
1.
7
46
35
20
02
2nd item
3rd item...
20th item
p. 35, 1. 44
p. 63, 1. 32
p. 69, 1. 22
p. 39, 1. 31
p. 67, 1. 19
p. 73, 1. 9
Sample of Physical Inventory
The physical selection might involve problems not covered in the
exercise--separate selection of high-value items, the physical size of
items. This solution is simplistic for ignoring these potential
complications.
The inventory physical frame is 3-dimensional. It has width (300 rows of
shelves), length (75 feet each row) and height (10 tiers in each row of
shelves).
Two ways to select the sample are:
1.
Think of the layout as 22,500 linear feet of shelf rows (300 x 75).
Using systematic sampling, take 5 random starts, each time pacing
off 1,125 feet (5 x 22,500/100) in a pre-determined path around the
warehouse. At each stop, select a random number between 1 and 10 to
identify the tier, and select that item.
2.
Think of the layout as 300 2-dimensional coordinates. Select 100 of
the rows, but be careful about any systematic selection because
physical storage may be in some nonrandom pattern. For each selected
row, select a random number between 1 and 75 to locate a position on
504
the row. Then select a random number between 1 and 10 to identify a
tier and the inventory at that location.
20.65 Cases a, b, c: Illustrate effect of different risks of assessing control risk
too low.
Cases d, e, f: Illustrate effect of larger samples with same sample deviation
rate.
Cases g, h, i: Illustrate different sample deviation rates.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Actual sample
deviation rate
2%
2%
2%
2%
2%
2%
10%
6%
0
Computed upper
limit (approx.)
5%
4%
12%
3%
3.6% 6.3% 4.6% 3.6% 17%
20.66 Discovery of Sampling Calculations
Critical rate
of occurrence
Required reliability
Sample size
(minimum)
(a)
(b)
(f)
(g)
(h)
(i)
.4%
.5% 1.0% 2.0% 1.0% .5%
.4%
.4%
.4%
99
(c)
99
1,000 900
(d)
(e)
99
99
91
70
70
85
95
460
240
240
240
300
460
700
SOLUTIONS FOR DISCUSSION CASES
20.67 Mistakes in Sampling Application
Mistake
Explanation
1. The statistical criteria call for a
sample of 160, not 100.
1. He apparently read Appendix 13-B.2
for the 1% expected rate instead of
for 2%.
2. He used two test months.
2. Even a selection of two months does
not make the sample representative
of the year's population.
3. He stratified the population, but
did not adjust the total sample size
accordingly.
3. Nothing is wrong with
stratification, but in this case the
sample size in each stratum would be
about 160.
4. He apparently did not define the
error attribute carefully before
starting the audit work.
4. Indicated by his after-the-fact
rationalization of the two errors
into non-errors. In fact the pay
rate error has dollar-value impact
that he made no effort to recognize
(i.e.), liability for underpayment
of wages).
He did not follow up sufficiently on
the errors that he did find.
5. He improperly combined a stratified
5. When stratification is done
505
sample into a single evaluation.
properly, the two samples should be
evaluated independently.
6. The reviewers (senior and partner)
were not competent to review the
statistical application.
6. This is not Tom's mistake, but it's
worthwhile to point out that
competence is as necessary at the
review level as it is at the
operational level.
20.68 Determine a Test of Controls Sample Size
Test of controls sample size = 75, with plan to assess control risk = .10 and
audit minimum (25) substantive balance-audit sample. This choice gives the
lowest total cost if control risk is actually assessed at .10.
EXHIBIT 20.B-1
GOODWIN MANUFACTURING COMPANY
Test of Control Balance-Audit
Control Risk
Categories
Low control
risk
Moderate
control risk
Control risk
below maximum
Maximum risk
CR
TDR
RIA RACRTL
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
.04
.06
.08
.10
.12
.14
.16
.18
.20
.22
.50
.28
.19
.14
.11
.09
.08
.07
.06
.06
.05
.09
.15
.25
.40
.50
.50
.50
.50
.50
n[c]
75
40
24
14
8
5
4
4
4
3
Cost
$
$
$
$
$
$
$
$
$
$
900
480
288
168
96
60
48
48
48
36
n[s]
25
46
60
71
80
87
91
96
101
101
Cost
$ 625
$1,150
$1,500
$1,775
$2,000
$2,175
$2,275
$2,400
$2,525
$2,525
TOTAL
$1,525
$1,630
$1,788
$1,943
$2,096
$2,235
$2,323
$2,448
$2,573
$2,561
Audit risk = .05
Inherent risk = 1.0
Analytical procedures risk = .90 [detection probability = .10]
RIA = (AR=.05)/[(IR=1.0) x CR x (AP=.90)]
Anchor TDR = 7 x 2,000,000 = .03 for CR = .05
467,000,000
RACRTL =
.02
for each RIA for each CR
RIA - .06
SOLUTIONS FOR KINGSTON CASE
20.69
Kingston Company: Dollar-Unit Sampling Audit of Accounts
Receivable
Even though the requirements are intended to guide students
through the steps, there are several ways to go wrong in this
DUS problem.
a.
The first way to go wrong is in deriving the risk of
incorrect acceptance (RIA). If they miss this, nothing
else will correspond to the solutions below. The
506
auditors said they would set audit risk at 0.05 and
that control risk and inherent risk were jointly
assessed at 0.30. About the analytical procedures, they
said: "Too bad we can't say analytical procedures
reduce out audit risk."
RIA =
AR = .05
= 0.166667 (round to 0.17)
(IR=1.0) x (CR=0.30) x (AP=1.0)
b.
Deciding how many errors to estimate for using the
Poisson risk factor method calculation of the DUS
sample size is not specified in the dialogue. Students
have to remember to calculate a ratio of expected error
($4,000 stated in the dialogue) to the balance under
audit ($300,000, and they might mistakenly use the
$400,000 total). This ratio (4,000/300,000) is 1.33
percent, which suggests a number of deviations of more
than one but less than 2.
The sample sizes can be different:
Based on one error
Based on two errors
n = 300,000 x 3.21 = 96
n = 300,000 x 4.53 = 136
10,000
10,000
The solutions below assume that the same 10 errors were found, no matter
the sample size.
c.
Calculate the projected likely error and the upper error limit based on
the errors in the sample.
Another place to go wrong with "nonsampling error"--calculating the
tainting percentages. Here are the correct taints.
Exhibit 14.24-1: ERRORS DISCOVERED IN THE SAMPLE OF ACCOUNTS RECEIVABLE
ACCT #
25
366
465
623
741*
741*
774
1206
1352
1466
*
d.
BOOK
WRONG WRONG
BALANCE QUANT'Y MATH
$503
$492
$507
$195
$3,698
$3,698
$517
$524
$700
$351
WRONG AUDITED
DATE AMOUNT
$115
$112
$136
$63
$100
$100
$140
$119
$400
$59
$388
$380
$371
$132
$3,598
$3,598
$377
$405
$300
$292
ERROR
TAINT
22.86%
22.76%
26.82%
32.31%
2.70%
2.70%
27.08%
22.71%
57.14%
16.81%
Selected twice for two dollar units.
Decide whether to "accept" the recorded amount without adjustment or to
"reject" the recorded amount as an accurate balance with respect to the
tolerable misstatement the auditors will allow.
When students use the UEL decision rule, the decision is a "rejection"
with sample sizes of 96 and 136. In both cases, the upper error limit is
greater than the $10,000 tolerable misstatement.
507
DOLLAR-UNIT SAMPLING UEL CALCULATION
Population recorded amount $300,000
Risk incorrect acceptance
0.17
Estimated number errors
1
Sample size
96
1. Basic
2. Projected likely error
First
Second
Third
Fourth
Fifth
Sixth
Seventh
Eighth
Ninth
Tenth
UEL,PGW
TAINT
ASI
1.77
100.00%
$3,125
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
57.14%
32.31%
27.08%
26.82%
22.86%
22.76%
22.71%
16.81%
2.70%
2.70%
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
DOLLAR AMOUNT
$5,531
$1,786
$1,010
$846
$838
$714
$711
$710
$525
$84
$84
Projected likely error
3. Precision gap widening
First
Second
Third
Fourth
Fifth
Sixth
Seventh
Eighth
Ninth
Tenth
$7,308
0.44
0.32
0.27
0.23
0.21
0.19
0.18
0.17
0.15
0.15
57.14%
32.31%
27.08%
26.82%
22.86%
22.76%
22.71%
16.81%
2.70%
2.70%
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$3,125
$786
$323
$228
$193
$150
$135
$128
$89
$13
$13
$2,058
Upper Error Limit at risk
0.17
of incorrect acceptance
$14,897
DOLLAR-UNIT SAMPLING UEL CALCULATION
Population recorded amount $300,000
Risk incorrect acceptance
0.17
Estimated number errors
1
Sample size
136
1. Basic
2. Projected likely error
First
Second
Third
Fourth
Fifth
Sixth
Seventh
Eighth
Ninth
Tenth
UEL,PGW
TAINT
ASI
1.77
100.00%
$2,206
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
57.14%
32.31%
27.08%
26.82%
22.86%
22.76%
22.71%
16.81%
2.70%
2.70%
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
DOLLAR AMOUNT
$3,905
$1,261
$713
$597
$592
$504
$502
$501
$371
$60
$60
508
Projected likely error
3. Precision gap widening
First
Second
Third
Fourth
Fifth
Sixth
Seventh
Eighth
Ninth
Tenth
$5,161
0.44
0.32
0.27
0.23
0.21
0.19
0.18
0.17
0.15
0.15
57.14%
32.31%
27.08%
26.82%
22.86%
22.76%
22.71%
16.81%
2.70%
2.70%
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$2,206
$555
$228
$161
$136
$106
$95
$90
$63
$9
$9
$1,452
Upper Error Limit at risk
0.17
of incorrect acceptance $10,518
20.70 Kingston Company: Determine the Amount of a Recommended Adjustment--DollarUnit Sampling
The alternative (a) requirement gives the results of the evaluation of a
sample of 96 dollar units from assignment 14.24. Thus the adjusting journal
entry for the alternative and for 14.24 with a sample of 96 should be
identical.
The 65% cost of goods sold hint comes into play in the adjustment of the sales
recorded with the wrong date (Goods shipped in January should not have been
recorded in the year under audit. When these are adjusted, the cost of goods
sold should be reversed and put back into inventory.)
Students must be careful to include the results of auditing the six large
accounts. These errors are not subject to sampling error.
For instructors' information, the actual population has about $15,000
overstatement error in it. (Refer to the Kingston story at the beginning of
this instructors' manual.) An adjustment keyed on the rather small amount of
actual (known) error will be too small. The adjustment given below is based on
the projected likely error amounts, adding the errors found in the six large
accounts.
Debit
Sales (Returns and Allowances)
$2,448
Accounts Receivable
Adjust for the wrong quantities billed.
Sales (Returns and Allowances)
Accounts Receivable
Adjust for the arithmetic errors.
Credit
$2,448
$ 618
Sales
$5,292
Inventory (65%)
$3,440
Accounts Receivable
Cost of Goods Sold
Adjust to reverse the sales recorded
too early and restore the cost of
goods sold to inventory.
$ 618
$5,292
$3,440
509
If these entries are based on students' audit of a sample of 136 in assignment
14.24, the DUS calculations are as follows:
ACCOUNTS RECEIVABLE DUS CALCULATIONS: Sample = 136
Type of Error
Wrong Quantity:
Six large accounts
Sampled accounts
Known
Error
Upper
Error
Limit
600
199
$ 1,305
$ 5,713
600
199
$ 1,305
$ 5,713
Wrong Arithmetic:
Six large accounts
Sampled accounts
450
100
$ 120
$ 4,070
Wrong Date:
Sampled accounts
945
$ 3,736
$ 8,813
1,050
1,244
$ 5,161
$10,518
Wrong Quantity:
Six large accounts
Sampled accounts
All Error:
Six large accounts
Sampled accounts
$
Projected
Likely
Error
$
The adjusting entries based on these data are:
Debit
Sales (Returns and Allowances)
$1,905
Accounts Receivable
Adjust for the wrong quantities billed.
Sales (Returns and Allowances)
$ 570
Accounts Receivable
Adjust for the arithmetic errors.
Sales
$3,736
Inventory (65%)
$2,428
Accounts Receivable
Cost of Goods Sold
Adjust to reverse the sales recorded
Credit
$1,905
$
570
$3,736
$2,428
510
too early and restore the cost of
goods sold to inventory
PART II
SOLUTIONS FOR MULTIPLE-CHOICE QUESTIONS
20.71 a.
b.
c.
d.
Correct.
Incorrect.
Incorrect.
Incorrect.
0.015/(0.50 x 0.30 x 0.50) = 0.20
see a.
see a. Illogical to have a "risk" greater than 1.00
see a.
20.72 a.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
Correct.
see d. not the "best answer"
see d. not the "best answer"
see d. not the "best answer"
They are all audit judgments. TD is a product of the other
judgments.
20.73 a.
b.
Incorrect.
Incorrect.
c.
d.
Correct.
Incorrect.
Efficiency is considered less important than effectiveness.
This is an explanation of how incorrect rejection is
overcome, producing more work than was necessary.
Effectiveness is considered more important than efficiency.
(This is the throwaway!) The evidence may be sufficient and
competent, but the decision can still be wrong.
20.74 a.
b.
c.
d.
Incorrect.
Correct.
Incorrect.
Incorrect.
This is "underauditing."
"Overauditing" is doing too much work.
"Taking more risk" implies doing too little work.
This is a definition of audit risk at the overall level.
20.75 a.
b.
c.
Correct.
Correct.
Incorrect.
d.
Incorrect.
DUS does not require a variability estimate.
DUS uses the statistics of the binomial distribution.
In DUS the logical unit is audited, same as in classical
sampling.
Both methods utilize calculation of an upper error limit in
analyses of results.
20.76 a.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
Correct.
DUS samples are random.
DUS is "sampling with replacement."
DUS auditors do not ignore the risk of incorrect acceptance.
The population is defined as $1 units and not as the number
of logical units in the population.
20.77 a.
b.
c.
d.
Incorrect.
Incorrect.
Incorrect.
Correct.
Auditors must specify audit risk.
Auditors must assign tolerable misstatement.
Auditors must estimate the misstatement.
An estimate of standard deviation is not needed for DUS.
20.78
RIA
a.
b.
c.
d.
20.79 a.
Correct
Correct.
0.03
0.03
0.06
0.10
Errors
2
1
0
2
Recorded
Amount
$1,000,000
$1,000,000
$1,500,000
$1,500,000
Tolerable
Misstatement
$
$
$
$
50,000
35,000
65,000
65,000
Sample
Size
140
153
65
123
The risk of incorrect acceptance must be specified for sample
size calculation and quantitative evaluation of monetary
error evidence.
511
b.
Incorrect.
c.
Incorrect.
d.
Incorrect.
20.80 a.
Incorrect.
b.
Correct.
c.
Incorrect.
d.
Incorrect.
The opposite is true: Smaller logical units have a lesser
probability of selection in the sample than larger units.
The systematic sampling "skip interval" illustrated in the
text is derived by dividing by n-1, while the "average
sampling interval" is derived by dividing by n.
Projected likely misstatement can be calculated in the
quantitative evaluation when one or more errors are
discovered.
DUS sampling loads the sample with high-value sampling units,
and they are more likely to be overstated than understated.
The sample selection automatically achieves high-dollar
selection and stratification.
The sample selection is biased against including a
representative number of small-value population units.
Expanding the sample for additional evidence is not very
easy.
20.81 Selecting a Dollar-Unit Sample
The solution starts by finding the recorded amount of the total = $38,610. The
skip interval is 38,610 / 10 = 3,861. "Random start" at 1,210.
Whitney Company
Inventory Sample Selection
Sept 30, 20XX
Index_____
Account
Number
Account
Balance
Modified
Accumulator Accumulator
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1,750
1,492
994
629
2,272
1,163
1,255
3,761
1,956
1,393
884
729
937
5,938
- 1,210
2,561
306
323
- 1,266
103
1,152
1,052
853
540
- 2,437
- 1,708
771
5,167
15
16
17
18
19
20
21
22
23
2,001
222
1,738
1,228
2,577
1,126
565
2,319
1,681
-
554
332
1,406
1,227
1,350
1,385
820
1,499
681
Dollar
Selected
Logical
Unit
- 1.300
1st
3,862nd
1,210
1,492
- 3,538
7,723rd
629
- 2,709
- 2,809
11,584th
15,445th
1,255
3,761
- 3,321
19,306th
1,393
1,306
- 2,555
23,167*
27,028*
5,938
- 2,455
30,889th
1,738
- 2,511
34,750th
2,577
- 2,362
38,611th
2,319
38,610
*
Two dollar units in the same logical unit.
Total of logical units in the
Prepared by___Date___
Reviewed by___Date___
512
sample of 10 dollar units
22,312
20.82 When RIA is greater than .5 one has to question the usefulness of the
particular statistical test.
The risk model yields even more problematic
treasures if IR x CR x AP is less than or equal to AR. This makes RIA equal to
or greater than 1 (i.e. RIA is more than 100% !) when RIA = AR/(IR x CR x AP).
The model, therefore, suggests that tests of details may not be necessary. All
the evidence thus rests on internal control, inherent risk (the subjective
probability estimate of misstatement getting into the accounting in the first
place), and the effectiveness of analytical procedures. Auditing theory and
practice maintains that some effective substantive procedures should be
performed--perhaps including a minimum sample size for tests of details in
addition to substantive analytical procedures.
SOLUTIONS FOR DISCUSSION CASES
20.83 Relation of Dollar-Unit Sample Sizes to Audit Risk Model
CONTROL RISK INFLUENCE ON SUBSTANTIVE BALANCE-AUDIT SAMPLE SIZE
AR = .05
IR = 1.00
AP = 1.00
(CR)
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
RIA
0.50
0.25
0.167
0.125
0.10
0.083
0.071
0.0625
0.0556
0.05
AR = .10
IR = 1.00
AP = 1.00
AR = .05
IR = .50
AP = 1.00
AR = .05
IR = 1.00
AP = .50
n(s)
RIA
n(s)
RIA
n(s)
RIA
n(s)
21
42
53
61
69
76
80
84
87
90
.50
.50
.33
.25
.20
.17
.14
.13
.11
.10
21
21
33
42
48
53
59
61
66
69
.50
.50
.33
.25
.20
.17
.14
.13
.11
.10
21
21
33
42
48
53
59
61
66
69
.50
.50
.33
.25
.20
.17
.14
.13
.11
.10
21
21
33
42
48
53
59
61
66
69
Discussion:
Comparing the first two sets at left: Larger audit risk produces smaller
samples throughout the entire range of control risks.
Comparing the three sets at the right: Doubling the audit risk from .05 to .10
has the same effect on RIA and sample size as assessing half the IR or AP.
Comparing the two sets at the left: The same change in IR and AP have the same
effect on RIA and sample size.
20.84 Determining an Efficient Risk of Incorrect Rejection (DUS)
For each of the control risk levels, calculate the expected cost savings from
auditing the initial alternative (minimum) sample. Assume that the action in
the event of a rejection decision is to expand the work by selecting
additional units up to the number in the base sample.
Control
Risk
0.20
"Base"
Sample
Alternative
RIR
Alternative
(Minimum)
Sample
80
.02
41
Cost Savings
$8(nb-na)-$19(nb-na)(RIRa-.01)
$312 - $ 7 = $305
513
0.30
96
.02
53
$344 - $10 = $334
0.40
107
.03
62
$360 - $17 = $343
0.50
116
.03
68
$384 - $18 = $366
0.60
122
.03
74
$384 - $18 = $366
0.70
128
.03
78
$400 - $19 = $381
0.80
133
.03
82
$408 - $19 = $389
0.90
137
.03
86
$408 - $19 = $389
1.00
141
.03
89
$416 - $20 = $396
Discuss the potential audit efficiencies and possible inefficiencies from
beginning the audit work with the alternative (minimum) sample size.
The potential audit efficiency is achieving the cost savings scheduled above.
Depending on the control risk level planned for assessment, the savings could
range from $305 to $396.
The large savings arise from the very small increase in RIR for the
alternative (minimum) sample sizes.
These sample sizes were obtained by a method that is fairly insensitive to RIR
changes. The alternative sample sizes are actually minimum samples that also
fit the criterion of alternative RIR from .02 and .03 to .50. In other words,
in this attribute-type dollar-unit sample, the alternative sample sizes are
the minimum sample sizes, no matter what RIR greater than .02 and .03 are
specified. (Note to instructors: I am not sure that very many students, except
the mathematicians, will be interested in this phenomenon.)
20.84(c) The solutions for different sample sizes will be similar in form, although
the numbers will be different.
20.85 Comparison of Sampling Methods
20.85(a) Unrestricted random sample of 10 accounts
RANDOM UNIT SAMPLE
ACCT #
2
5
7
14
20
28
32
35
42
46
Number
Total
Average
Std Dev
Ratio
BALANCE
$346
$1,555
$1,906
$178
$141
$193
$503
$157
$91
$156
10
$5,226
$522.60
$619.56
WRONG
QUANT'Y
WRONG
MATH
WRONG
DATE
$600
$200
$11
$115
1
$200
1
$11
2
$715
20.85(b) Systematic random selection of 10 accounts
SYSTEMATIC RANDOM SAMPLE
MONETARY
ERROR
AUDIT
AMOUNT
$0
$600
$200
$0
$0
$11
$115
$0
$0
$0
$346
$955
$1,706
$178
$141
$182
$388
$157
$91
$156
$926
$92.60
$181.00
0.177190
$4,300
$430.00
$488.43
514
ACCT #
BALANCE
3
5
3
15
23
25
$1,301
$1,555
$320
$188
$145
$461
33
35
43
45
$500
$157
$65
$470
Number
Total
Average
Std Dev
Ratio
10
$5,162
$516.20
$481.36
WRONG
QUANT'Y
WRONG
MATH
WRONG
DATE
$600
$111
$107
$117
1
$111
0
$0
3
$824
MONETARY
ERROR
AUDIT
AMOUNT
$0
$600
$0
$0
$0
$111
$1,301
$955
$320
$188
$145
$350
$107
$0
$0
$117
$393
$157
$65
$353
$935
$93.50
$176.08
0.181131
$4,227
$422.70
$375.11
20.85(c) Systematic random dollar-unit selection of 10 dollars
SYSTEMATIC DOLLAR-UNIT SAMPLE
ACCT #
3
3
5
7
15
23
28
36
45
50
Number
Total
Average
BALANCE
$1,301
$1,301
$1,555
$1,906
$188
$145
$193
$388
$470
$268
10
$6,414
$641.40
WRONG
QUANT'Y
WRONG
MATH
WRONG
DATE
$600
$200
11
$117
1
$200
1
$11
2
$717
MONETARY
ERROR
AUDIT
AMOUNT
$0
$0
$600
$200
$0
$0
$11
$0
$117
$0
$1,301
$1,301
$955
$1,706
188
$145
$182
$388
$353
$268
$928
$92.80
$5,486
$548.60
TAINTS
0.00%
0.00%
38.59%
10.49%
0.00%
0.00%
5.70%
0.00%
24.89%
0.00%
20.85(d) With a sample of 10, the average sampling interval is $1,752. Account
number 7, with a balance of $1,906, will always be included in a dollarunit sample of 10.
20.85(e) Table comparing sampling methods
TABLE COMPARING RESULTS OF SAMPLES
Population size
Recorded total
Sample size
Random
Unit
Sample
Systematic
Unit
Sample
Dollar
Unit
Sample
50
$17,523.00
10
50
$17,523.00
10
$17,523.00
$17,523.00
10
515
Recorded amount
in sample
$5,226.00
Number of error
accounts in sample
4
Monetary misstatement
in sample
$926.00
$5,162.00
$6,414.00
4
4
$935.00
$928.00
$92.60
$93.50
N/A
Error ratio
0.1772
Projected misstatement
Difference method $4,630.00
Ratio method
$3,105.00
Upper error limit
at 2.0 % risk of
incorrect acceptance
Z(B) = 2.05
(Difference method)
$9,930.50
0.1811
N/A
$1,396.00
Average misstatement
in sample
$4,675.00
$3,173.00
$9,975.50
$9,322.00
20.85(f) Calculation of upper error limit
Dollar-Unit Projected Misstatement
Factor
Taint
Sampling
Interval
Projected
Error
Basic error
3.91
100.00%
$1,752
$ 6,850
First error
Second error
Third error
Fourth error
1.00
1.00
1.00
1.00
38.59%
24.89%
10.49%
5.70%
$1,752
$1,752
$1,752
$1,752
$
Projected error
Gap Widening:
First error
Second error
Third error
Fourth error
676
436
184
100
$ 1,396
0.92
0.69
0.56
0.50
38.59%
24.89%
10.49%
5.70%
$1,752
$1,752
$1,752
$1,752
Upper error limit (.02 risk of incorrect acceptance)
$
622
301
103
50
$ 9,322