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1Chapter 17
Audit Sampling forTests of Details of Balances
Review Questions
17-1 The most important difference between (a) tests of controls
and substantivetests of transactions and (b) tests of details of
balances is in what the auditorwants to measure. In tests of
controls and substantive tests of transactions, theprimary concern
is testing the effectiveness of internal controls and the rate
ofmonetary misstatements. When an auditor performs tests of
controls andsubstantive tests of transactions, the purpose is to
determine if the exception ratein the population is sufficiently
low to ustify reducing assessed control ris! toreduce substantive
tests. When statistical sampling is used for tests of controls
and
substantive tests of transactions, attributes sampling is ideal
because it measuresthe fre"uency of occurrence (exception rate). In
tests of details of balances, theconcern is determining whether the
monetary amount of an account balance ismaterially misstated.
#ttributes sampling, therefore, is seldom useful for tests
ofdetails of balances.
17- $tratified sampling is a method of sampling in which all the
elements in thetotal population are divided into two or more
subpopulations. %ach subpopulationis then independently sampled,
tested and the results proected to the population.
#fter the results of the individual parts have been computed,
they are combinedinto one overall population measurement.
$tratified sampling is important in
auditing in situations where the misstatements are li!ely to be
either large or small.In order for an auditor to obtain a
stratified sample of &' items from eachof three strata in the
confirmation of accounts receivable, he or she must firstdivide the
population into three mutually exclusive strata. # random sample of
&'items is then selected independently for each stratum.
17-! The point estimate is an estimate of the total amount of
misstatement inthe population as proected from the !nown
misstatements found in the sample.The proection is based on either
the average misstatement in the sample timesthe population sie, or
the net percent of misstatement in the sample times thepopulation
boo! value.
The true value of misstatements in the population is the net sum
of allmisstatements in the population and can only be determined by
a 1'' audit.
17-" The statement illustrates how the misuse of statistical
estimation can impairthe use of an otherwise valuable audit tool.
The auditor*s mista!e is that he orshe treats the point estimate as
if it is the true population value, instead of butone possible
value in a statistical distribution. +ather than udge whether
thepoint estimate is material, the auditor should construct a
statistical confidence
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17-" #continued$
interval around the point estimate, and consider whether the
interval indicates amaterial misstatement. #mong other factors, the
interval will reflect appropriatelevels of ris! and sample sie.
17-% onetary unit sampling is a method whereby the population is
definedas the individual dollars (or other currency) ma!ing up the
account balance. #random sample is drawn of these individual
monetary units and the physical auditunits containing them are
identified and audited. The results of auditing thephysical audit
units are applied, pro rata, to the random monetary units, and
astatistical conclusion about all population monetary units is
derived.
onetary unit sampling is now the most commonly used method
ofstatistical sampling for tests of details of balances. This is
because it uses thesimplicity of attributes sampling yet still
provides a statistical result expressed indollars. It does this by
using attribute tables to estimate the total proportion
ofpopulation dollars misstated, based on the number of sample
dollars misstated,
and then modifies this amount by the amounts of misstatements
found. Thislatter aspect gives monetary unit sampling its
/variables/ dimension, althoughnormal distribution theory is not
used0 rather an arbitrary rule of thumb is appliedto ma!e the
adustment.
17-& $ampling ris! is the ris! that the characteristics in
the sample are notrepresentative of those in the population. The
two types of sampling ris! faced bythe auditor testing an account
balance are
a. The ris! of incorrect acceptance (#+I#)2this is the ris! that
thesample supports the conclusion that the recorded account
balance
is not materially misstated when it is materially misstated.b.
The ris! of incorrect reection (#+I+)2this is the ris! that
thesample supports the conclusion that the recorded account
balanceis materially misstated when it is not materially
misstated.
$ampling ris! occurs whenever a sample is ta!en from a
population andtherefore applies to all sampling methods. While #+I#
applies to all samplingmethods, #+I+ is only used in variables
sampling and difference estimation.
17-7 The steps in nonstatistical sampling for tests of details
of balances andfor tests of controls are almost identical, as
illustrated in the text. The maor
differences are that sampling for tests of controls deals with
exceptions andsampling for tests of details of balances concerns
dollar amounts. This results indifferences in the application of
the two methods, but not the steps.
17-' The two methods of selecting a monetary unit sample are
random samplingand systematic sampling. 3nder random sampling, in
this situation, 4 randomnumbers would be obtained (the sample sie
in 1-15) between 1 and 16,764,'''.These would be sorted into
ascending se"uence. The physical audit units in the
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17-' #continued$
inventory listing containing the random monetary units would
then be identifiedby cumulating amounts with an adding machine or
spreadsheet if the data is inmachine-readable form. #s the
cumulative total exceeds a successive randomnumber, the item
causing this event is identified as containing the random
dollar
unit.When systematic sampling is used, the population total
amount is divided
by the sample sie to obtain the sampling interval. # random
number is chosenbetween 1 and the amount of the sampling interval
to determine the startingpoint. The dollars to be selected are the
starting point and then the starting pointplus the interval amount
applied successively to the population total. The itemson the
inventory listing containing the dollar units are identified using
thecumulative method described previously.
In applying the cumulative method under both random sampling
andsystematic sampling, the page totals can be used in lieu of
adding the detaileditems if the page totals are considered to be
reliable.
17-( # uni"ue aspect of monetary unit sampling is the use of the
preliminaryudgment about materiality, as discussed in 8hapter 9, to
directly determine thetolerable misstatement amount for the audit
of each account. ost samplingtechni"ues re"uire the auditor to
determine tolerable misstatement for eachaccount by allocating the
preliminary udgment about materiality. This is notre"uired when
monetary unit sampling is used. The preliminary udgment
aboutmateriality is used.
17-1) #cceptable ris! of incorrect acceptance (#+I#) is the ris!
the auditor iswilling to ta!e of accepting a balance as correct
when the true misstatement in
the balance is greater than tolerable misstatement. #+I# is the
e"uivalent term toacceptable ris! of assessing control ris! too low
for audit sampling for tests ofcontrols and substantive tests of
transactions.
The primary factor affecting the auditor*s decision about #+I#
is controlris! in the audit ris! model, which is the extent to
which the auditor relies oninternal controls. When internal
controls are effective, control ris! can bereduced, which permits
the auditor to increase #+I#, which in turn reduces there"uired
sample sie. :esides control ris!, #+I# is also affected directly
byacceptable audit ris! and inversely by inherent ris! and other
substantive testsalready performed on the account balance, assuming
effective results. ;orexample, if acceptable audit ris! is reduced,
#+I# must also be reduced. If
analytical procedures were performed and there is no indication
of problemareas, there is a lower li!elihood of misstatements in
the account being tested,and #+I# can be increased.
17-11 The statement reflects a misunderstanding of the
statistical inferenceprocess. The process is based on the long-run
probability that the process willproduce correct results in a
predictable proportion of the times it is applied. Thus,
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17-11 #continued$
a random sampling process that produces a 9' confidence interval
will produceintervals that do, in fact, contain the true population
value 9' of the time.
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17-1% *isstatement +ounds using the attri+utes ta+les
*,SSTAT-*.T
RC/RDD0A2
A2D,TD0A2
*,SSTAT-*.T
*,SSTAT-*.T3
RC/RDDA*/2.T
1
6
&
?9.17
5.'6
1,761.7?
7'9.17
'
1,466.7?
6??.''
5.'6
99.''
.&61
1.'''
.'71
3sing the attributes sampling table for a sample sie of 1'', and
an #+I#of 1', the 83%+ is
./4 /5*,SSTAT*.TS C2R
,.CRAS ,. B/2.DRS2T,.6 5R/* A.
ADD,T,/.A*,SSTAT*.T
'
1
6
&
.'6&
.'&9
.'4&
.'77
.'17
.'15
.'1&
In order to calculate the upper and lower misstatement bounds,
it will beassumed that for a ero misstatement rate the percent of
misstatement is 1''.
The upper misstatement bound
./4 /5*,SSTAT-
*.TSRC/RDD
0A2 C2R
8/RT,/.
2.,T*,SSTAT
-*.T 9
*,S-STAT-*.T
B/2.D8/RT,/.
'
1
6
&
16,764,'''
16,764,'''
16,764,'''
16,764,'''
.'6&
.'17
.'15
.'1&
1.'''
1.'''
.&61
.'71
69',&4
6'6,'''
47,&
1','16
3pper isstatement :ound 449,165
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17-1% #continued$
The lower misstatement bound
:efore adustment
./4 /5*,SSTAT-
*.TSRC/RDD
0A2 C2R
8/RT,/.
2.,T*,SSTAT
-*.T 9
*,S-STAT-*.T
B/2.D8/RT,/.
' 16,764,''' .'6& 1.''' 69',&4
#dustment
@oint estimate for overstatements > sum of misstatement
percents
x recorded value A sample sie
> (.&61 B 1.''' B .'71) x (16,764,''' A 1'')
> 1.&?6 x 167,64'
> 15,5?
#dusted lower misstatement bound > initial bound - point
estimatefor overstatements
> 69',&4 - 15,5?
> 114,?9
:ased on this calculation method, the population is not
acceptable asstated since the upper misstatement bound exceeds the
C4'',''' materiality limit.
17-1& The difficulty in determining sample sie lies in
estimating the number andamount of misstatements that may be found
in the sample. The upper boundof a monetary unit sample is
sensitive to these factors. Thus, sample sie variesa great deal
with differing assumptions about them.
Denerally, the auditor will determine sample sie by ma!ing
reasonablebut conservative assumptions about the sample exception
rate and averagemisstatement amount. In the absence of information
about misstatement amount,which is most difficult to anticipate, a
1'' assumption is often used.
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17-17 The decision rule for difference estimation is
If the two-sided confidence interval for the misstatements is
completelywithin plus or minus tolerable misstatements, accept the
hypothesis thatthe boo! value is not misstated by a material
amount. =therwise, acceptthe hypothesis that the boo! value is
misstated by a material amount.
;or example, assume the E8E is -1',''', the 38E is 5',''' and
tolerablemisstatement is C54,'''. The following illustrates the
decision rule
- T B T- 54,''' ' B 54,'''
- 1',''' B 5','''E8E 38E
The auditor can conclude that the population is not materially
misstatedsince both E8E and 38E are within the tolerable
misstatement limits.
17-1' When a population is not considered acceptable, there are
several possiblecourses of action
1. @erform expanded audit tests in specific areas. If an
analysis of themisstatements indicates that most of the
misstatements are of aspecific type, it may be desirable to
restrict the additional audit effortto the problem area.
6. Increase the sample sie. When the auditor increases the
samplesie, sampling error is reduced if the rate of misstatements
in theexpanded sample, their dollar amount, and their direction are
similar
to those in the original sample. Increasing the sample sie,
therefore,may satisfy the auditor*s tolerable misstatement
re"uirements.Increasing the sample sie enough to satisfy the
auditor*s
tolerable misstatement standards is often costly, especially
whenthe difference between tolerable misstatement and
proectedmisstatement is small.
&. #dust the account balance. When the auditor concludes
that anaccount balance is materially misstated, the client may be
willing toadust the boo! value.
5. +e"uest the client to correct the population. In some cases
the client*srecords are so inade"uate that a correction of the
entire population
is re"uired before the audit can be completed.4. +efuse to give
an un"ualified opinion. If the auditor believes therecorded amount
in accounts receivable or any other account is notfairly stated, it
is necessary to follow at least one of the abovealternatives or to
"ualify the audit opinion in an appropriate manner.
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17-1( The population standard deviation is a measure of the
difference betweenthe individual values and the mean of the
population. It is calculated for all variablessampling methods but
not for monetary unit sampling. ;or the auditor, it isusually
estimated before determining the re"uired sample sie, based on
theprevious year*s results or on a preliminary sample.
The population standard deviation is needed to calculate the
sample sie
necessary for an acceptable precision interval when variable
sampling methodsare used. #fter the sample is selected and audited,
the population standarddeviation is estimated from the standard
deviation calculated from the values inthe sample.
The re"uired sample sie is directly proportional to the s"uare
of thepopulation standard deviation.
17-) This practice is improper for a number of reasons
1. Fo determination was made as to whether a random sample of
1''inventory items would be sufficient to generate an
acceptable
precision interval for a given confidence level. In fact, a
confidencelimit was not even calculated.6. The combined net amount
of the sample misstatement may be
immaterial because large overstatement amounts may be
offsettinglarge understatement amounts resulting in a relatively
small combinednet amount.
&. #lthough no misstatement by itself may be material, other
materialmisstatements might not have exhibited themselves if too
small of asample was ta!en.
5. +egardless of the sie of individual or net amounts of
misstatementsin a sample, the effect on the overall population
cannot be determined
unless the results are evaluated using a statistically valid
method.
17-1 Gifference estimation is a method for estimating the total
misstatement ina population by multiplying the average misstatement
(the audited value minusthe recorded value) in a random sample by
the number of items in the entirepopulation.
+atio estimation is "uite similar to difference estimation.
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17-1 #continued$
The following are examples where each method could be used
a. Gifference estimation can be used in computing the balance
inaccounts receivable by using the misstatements discovered
during
the confirmation process, where a significant number of
misstatementsare found.
b. +atio estimation can be used to determine the amount of the
EI;=reserve where internal inventory records are maintained on a
;I;=basis but reporting is on EI;=.
c. ean-per-unit estimation can be used to determine total
inventoryvalue where the periodic inventory method is employed.
d. $tratified mean-per-unit estimation can be used to determine
totalinventory value where there are several locations and each
issampled separately.
onetary unit sampling would generally be preferable to any of
thesewhere few or no misstatements are expected. Gifference and
ratio estimation arenot reliable where the exception rate is low,
and mean-per-unit is generally not asefficient.
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17-! Gifference estimation can be very effective and very
efficient where (1)an audited value and a boo! value is available
for each population item, (6) arelatively high fre"uency of
misstatements is expected, and (&) a result in the formof a
confidence interval is desired. In those circumstances, difference
estimationfar outperforms both 3$ and mean-per-unit estimation. It
may or may notoutperform ratio estimation, depending on the
relationship of misstatement amounts
to recorded amounts, but it does re"uire less computational
effort than ratioestimation in any case. If focus on large dollar
value items is re"uired, differenceestimation can be used with
stratification.
17-" %xamples of audit conclusions resulting from the use of
attributes,monetary unit, and variables sampling are as follows
3se of attributes sampling in a test of sales transactions for
internalverification
We have examined a random sample of 1'' sales invoices for
indication of internal verification0 two exceptions were noted.
:asedon our sample, we conclude, with a 4 ris!, that the proportion
ofsales invoices to which internal verification has not been
applieddoes not exceed 7.6.
3se of monetary unit sampling in a test of sales transactions
for existence
We have examined a random sample of 1'' dollar units of
salestransactions for existence. #ll were supported by properly
preparedsales orders and shipping documents. :ased on our sample,
weconclude, with a 6' ris!, that invalid sales do not exceed
C5','''.
3se of variables sampling in confirmation of accounts receivable
(in theform of an interval estimate and a hypothesis test)
We have confirmed a random sample of 1'' accounts receivable.We
obtained replies or examined satisfactory other evidence forall
sample items. # listing of exceptions is attached. :ased onour
sample, we estimate, with 1' ris!, that the true
populationmisstatement is between C6',''' understatement and
C5','''overstatement. $ince tolerable misstatement for accounts
receivableis udged to be C4',''', we conclude, with a ris! of 4,
that accounts
receivable are not materially misstated.
*ultiple Choice Questions from C8A aminations
17-% a. (5) b. (&) c. (&)
17-& a. (5) b. (6) c. (6)
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Discussion Questions and 8ro+lems
17-7 a. 96 (:oo! valueAtolerable misstatement) x assurance
factor >(7,9'','''A14',''') x 6
b. If poor results were obtained for tests of controls and
substantive tests
of transactions for sales, sales returns and allowances, and
cashreceipts, the re"uired sample sie for tests of details of
balanceswould need to be increased. 3sing the formula in the
problem, theauditor would increase sample sie by increasing the
assurancefactor. This has the same effect has specifying a lower
acceptableris! of incorrect acceptance (#+I#).
c. # systematic sample can be selected based on the number
ofaccounts, or the dollar value of the population. To select a
systematicsample based on the number of accounts, the total number
ofaccounts in the population is divided by the re"uired sample sie
to
determine the interval. # random number is then selected
betweenone and the interval as the starting point. :ecause each
accounthas an e"ual li!elihood of selection, this method is
appropriate if allthe accounts are similar in sie, or if the
population is stratified intotwo or more samples.
To select a systematic sample based on the dollar value ofthe
population, the population value is divided by the re"uired
samplesie to obtain the appropriate interval. # random number is
thenselected between one and the interval as the starting point.
Theinterval is added to the starting point to determine the dollar
unitsselected. #ccounts are selected for testing where the
cumulative
total of accounts receivable includes the random number.
Thismethod of selection is similar to monetary unit selection,
andaccounts greater than the amount of the interval are
automaticallyselected using this method.
d. The direct proection of error for the sample can be computed
asfollows
(%rrors in sampleAsample boo! value) x population boo! value
>(1,4''A6&',''') x 7,9'',''' > C54,''' overstatement
The proected error of C54,''' is well below tolerable
misstatementof C14',''' and provides an allowance for sampling ris!
of C1'4,'''.#ccordingly, the population is deemed to be fairly
stated.
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17-' a. The following summaries the confirmation responses
Recorded Confirmation
0alue Response *isstatement
#cct. 11& C1?&,619 C1&,619 ' Timing difference
#cct. 619 6&,54 17,9& 7,46' 8utoff error#cct. 67
?,5&9 ,?7 46 %rror in "uantity shipped
#cct. 57 1,55& ' 1,55& 8utoff error
#cct. 4& ,546 7,?&6 76' @ricing error
#cct. 7?9 5,&?1 ' ' Timing difference
#cct. ?5 &5,4?& 6&,759 1',9&5 8utoff error
Total misstatement C&7,'?9
b. %stimate of total misstatement
Sample Sample Boo: 8ro;ected0alue *isstatements 0alue
*isstatement
$tratum 1 C 9&9,19 C ' C 9&9,19 C '
$tratum 6 1,15,471 &5,?9 5,7?,??7 1&9,6?'
$tratum & 1,6&9 1,196 ?96,461 15,9&5
Totals C6,1?5,99 C&7,'?9 C7,419,7'5 C145,615
c. The population is not acceptable since the proected
misstatementof C145,615 exceeds tolerable misstatement of C1'','''.
Theauditor is li!ely to propose an adustment andAor increase
testing. In
this situation, many of the errors involved cutoff, so the
auditorcould expand testing in this area and propose an adustment
for theerrors found. :ecause the cutoff errors were isolated and
testingexpanded in this area, the cutoff errors would not be
included in theproection of error for each stratum.
17-( a. If random selection is performed using %xcel (@169.xls),
the commandto select numbers randomly from the population is
>+#FG:%TW%%F(1,6'694)
The 1' random numbers selected using this approach will vary
foreach student.
The command for selecting the random numbers can be
entereddirectly onto the spreadsheet, or can be selected from the
functionmenu (math J trig) functions. It may be necessary to add
the analysistool pac! to access the +#FG:%TW%%F function. =nce the
formulais entered, it can be copied down to select additional
random numbers.
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17-( #continued$
./T< +andom dollar items are matched with population
itemnumbers where the cumulative boo! value of the population
includesthe random dollar selected.
b.Interval >
@opulation totalFumber of items selected
>6',694 1'
> 6',69 Interval
3sing 1?4 as a starting point, we have
S=ST*AT,C
D/AR
8/82AT,/.
,T* ./4
16&547?9
1'
1,?466,4?75&,&1475,'55?5,&
1'4,4'6167,6&1157,97'17,7?91??,51?
67??
146'67&'&'&4
./T< $ystematic dollar items are related to population
itemnumbers in the same manner as for part a above.
c. #ll items larger than the interval will be automatically
included. If theinterval is 6',69 item &' will be included at
least once, and item ?at least twice.
The same is not necessarily true for random number selection,but
the probability is high. Fote that for item ?, there is a
probabilityof approximately 66 (55,11'A6',694) of its being
included in agiven sample draw. It was included twice in a sample
of 1'.
d. There is no significant difference in ease of selection
betweencomputer generation of random numbers and systematic
selection.$ome auditors prefer the use of random numbers because
theybelieve this helps ensure an unbiased sample.
e. onetary unit sampling would be used because (1) it is
efficientand (6) it focuses on large dollar items.
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17-!) #continued$
Ad;ustment of upper misstatement +ound sum of
misstatementpercents x recorded value A sample sie
> (.?96 B .'&' B .'65) x (1,94,''' A 1'')
> .957 x 19,4'
> 1?,7?5
#dusted bound > initial bound - point estimate for
understatementamounts
> 79,7? - 1?,7?5
> 4',995
Ad;ustment of lower misstatement +ound sum of
misstatementpercents x recorded valueAsample sie
> (.795 B .'?5) x (1,94,''' A 1'')
> .? x 19,4'
> 14,&77
#dusted bound > initial bound - point estimate for
overstatements
> 4,'4? - 14,&77
> 49,796
b. The population is not acceptable as stated because both
thelower misstatement bound and upper misstatement bound exceed
materiality.In this situation, the auditor has the following
options
1. $egregate a specific type of misstatement and test it
separately(for the entire population). The sample would then not
includethe specified type of misstatement since it is being
testedseparately.
6. Increase the sample sie.
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17-!) #continued$
&. #dust the account balance (i.e., propose an adustment).5.
+e"uest the client to review and correct the population.4. 8onsider
"ualifying the opinion is the client refuses to correct
the problem.
7. 8onsider the criteria used in the test, possibly in
connection withadditional audit wor! in areas outside of accounts
receivable.
=f these options, segregating a specific type of misstatementmay
prove to be the most beneficial. In this problem, items & and
4are cutoff misstatements. $egregating these items, testing cutoff
moreextensively, and eliminating them from the sample would result
inthe following bounds
2pper misstatement +ound
ASS2*8T,/. 9
*,S-STAT-*.T
B/2.D
'
1
C1,94,'''
1,94,'''
.'6&
.'17
.'&9
1.'''
.'?5
C54,564
6,745
C5?,'9
Eess adustment K(.'&' B .'65) (19,4')L (1,'7)
C5,'16
ower misstatement +ound
ASS2*8T,/. 9
*,S-STAT-*.T
B/2.D
'
1
6
C1,94,'''
1,94,'''
1,94,'''
.'6&
.'17
.'15
.'4&
1.'''
.'&'
.'65
C54,564
95?
775
C5,'&
Eess adustment K(.'?5) (19,4')L (1,749)
C54,&?
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17-!) #continued$
It can be seen that both misstatement bounds are now
withinmateriality after cutoff misstatements were segregated.
Thesemisstatements were significant in two ways. Their existence
increasedthe overall estimated population exception rate, and their
magnitude
contributed to the amount of estimated misstatements in the
portionof the population represented by the misstatements in the
sample.
17-!1 a. The audit approach of testing all three account
balances is acceptable.This approach is also desirable when the
following conditions arepresent
1. The auditor can obtain valid, reliable information to
performthe re"uired tests in all of the areas.
6. The internal controls for each of the three areas are
comparable.&. isstatements are expected to occur evenly over
the entire
population. ;or instance, the auditor does not expect a
largenumber of misstatements in accounts receivable and few, ifany,
in inventory.
b. The re"uired sample sie for all three accounts is
Tolerable misstatement 1'',''' M+ecorded population value
1',''',''' > .'1
;rom the attributes table sample sie n cannot be determined,
butusing interpolation it is approximately 115 B (115 - 7)>146.
(This is
not an appropriate method to determine sample sie in
practice.)
c. The re"uired sample sies if each account is tested separately
are
ACC/2.T 5ACT/R
A88R/?4SA*8
S,@
#ccounts receivable n > 1'','''M&,7'','''
> .'& 7
Inventory n >
1'','''M
5,?'',''' > .'6 115
ar!etable securities n > 1'','''M1,7'','''
> .'7 &?
The important point is that sample sie under b is much
smallerthan for the combined samples in c.
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17-!1 #continued$
d. The population would be arranged so that all accounts
receivablewould be first, followed by inventory and mar!etable
securities. Theitems would be identified by the cumulative totals.
In the example,the number 5,76,?1 would relate to an inventory item
since it is
between the cumulative totals of C&,7'',''' and
C?,5'','''.#ccordingly, for this number the inventory audit
procedures wouldbe performed.
e. The misstatement data are as follows
RC/RDDA*/2.T
A2D,TDA*/2.T D,55R.C
*,SSTAT*.T3RC/RDD A*/2.T
C9?.16 C??.16 C1''.'' 1'.1
#ssuming a 1'' average misstatement in the population whenthere
are no misstatements found and an #+I# of 1', themisstatement
bounds are
3pper misstatement bound
C1',''',''' x .'16 x 1.' > C16','''
C1',''',''' x .''? x .1'1 > ?,'?'C16?,'?'
Eower misstatement bound
:efore adustment
C1',''',''' x .'16 x 1.' > C16','''
#dustment
C1',''','''.1'1 x 6'' > (4,'4')
C115,94'
:ased on the sample results and the stated combined
acceptablemisstatement of C1'',''', the population (i.e., accounts
receivable,inventory, and mar!etable securities combined) should
not beaccepted as stated without further testing.
17-! 1. (a) 6. (c) &. (a) 5. (d) 4. (d)
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17-!!
Computer Solution. This is an excellent problem to demonstrate
the use of thecomputer in auditing, as it re"uires a great deal of
computational wor!. # solutionprepared using %xcel is included on
the 8ompanion Website (;ilename @1&&.xls).Important points
to stress are
1. The spreadsheet program is set up in two sections one for
data entryand one for computations.
6. 8ells are set up for variables by name, and the values for
thevariables are then entered in those cells (e.g., sample sie >
). 8omputations are then done by reference to the cellsrather than
by entering values in the formulas. This allows thewor!sheet to be
used as a general program for similar problems.
&. #lthough the program assures computational accuracy, the
formulasmust be correct. They should always be reviewed and double
chec!ed,and test data should be processed to assure accuracy.
a. 8alculating the point estimate
:efore computing the computed precision interval, we mustcompute
the standard deviation
e (e)6
C(6.'')74.'51.1'&7.1'41.?'
(.16)&'.''
61.11C1&.79
4,1?5.''5,&17.591,7?9.611,&'&.&16,7?&.65
.'19''.''
554.7&17,461.9
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17-!! #continued$
8omputed precision interval
The confidence interval is expressed as &,995.? B
5,1?.57.
To compute the confidence limits,
38E > N B 8@I > &,995.? B 5,1?.57 >
?,1&.&&
E8E > N - 8@I > &,995.? - 5,1?.57 > -6&.49
b. The auditor should not accept the boo! value of the
population sincethe maximum misstatement in the population that she
was willing toaccept, C7,''', at a ris! level of 4, is less than
the possible amountof true misstatement indicated by the 38E of
C?,1&.&&.
c. The options available to the auditor at this point are
1. @erform expanded audit tests in specific areas.6. Increase
the sample sie.
&. #dust the account balance.5. +e"uest the client to
correct the population.4. +efuse to give an un"ualified
opinion.
17-!" a. It would be desirable to use unstratified difference
estimation whenthe auditor believes that there is not a small
number ofmisstatements in the population that are in total
material, and thepopulation has a large number of small
misstatements that in totalcould be material.
3nstratified difference estimation would not be appropriatewhen
either of the above characteristics is not present. ;or
example,
if the auditor believes that certain large accounts payable
maycontain large misstatements that are material, they should be
testedseparately.
# significant consideration in this situation is whether
theauditor can identify the entire population. This consideration
applieswhether using stratified or unstratified difference
estimation. Theauditor in this instance is identifying the
population based upon an
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17-!" #continued$
accounts payable list. If this list includes only those accounts
withan outstanding balance, the sample is ignoring those accounts
thathave a recorded balance of ero.
Thus, many accounts could be understated but not considered
in the sample or the statistical inferences drawn from the
sample.
b. Ignoring the #+I+, the re"uired sample sie may be computed
asfollows
where
T - %O > 54,''' - 6',''' > C64,'''
c. In order to determine whether the population is fairly
stated, thecomputed precision interval must be calculated.
8I > N B 8@I
8I > 61,''' B 66,&5
38E > 5&,&5
E8E > -1,&5
$ince both 38E and E8E are less than tolerable misstatement,the
auditor can conclude that the population is fairly stated.
The primary reasons the population is acceptable is that (1)the
actual point estimate is reasonably close to the
expectedmisstatement, and (6) the actual sample standard deviation
is lessthan the estimated standard deviation.
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17-!" #continued$
d. 8onsidering the #+I+, the sample sie may be computed from
thefollowing formula
e. The sample sie increases significantly with the inclusion of
the#+I+ because by including it the auditor is establishing the
ris! heor she will ta!e of reecting an acceptable population, as
well asconsidering the ris! of accepting an unacceptable
population. Itta!es more effort (sample items) to control two
ris!s, rather than
ust one. The effect can be seen from reviewing the formula
forcalculating the sample sie.
f. The approach described will only result in an appropriate
samplesie by chance. This would occur when the 64 increment is
e"ualto the increase in the sample sie re"uired when the #+I+
isconsidered. This is not a li!ely occurrence. This approach is
notdesirable because it is inefficient in terms of time and cost.
3nlessby chance the sample sie is approximately e"ual to the
samplesie re"uired by considering #+I+, the sample sie will be
eithertoo small or too large. Too small a sample will re"uire the
sample to
be increased. This may be both time consuming and expensive, if
itis even possible. 8onversely, too large a sample results in
theauditor performing more wor! than is re"uired.
Cases
17-!% a. Determination of ARIA- Fote that there are many ways to
estimate#+I#. =ne method is as follows
#+I# > ##+ A (I+ x 8+ x #@+)
> .'4 A (.? x .4 x 1.')> .'4 A .5> .1& rounded to
.1' (to be conservative)
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17-!% #continued$
Tolerable misstatement as a percent
T%+ > T A @opulation> ?'',''' A 16,''','''> .'7 rounded
to .'7 (to be conservative)
$ample sie determined using Table 14-? (assumes anexpected
misstatement of ero and a misstatement percent of 1'')
n > &?
b. Determination of ARIA- Fote that there are many ways to
estimate#+I#. =ne method is as follows
#+I# > ##+ A (I+ x 8+ x #@+)> .'4 A K1.' x .? x (1 -
.7)L> .'4 A .&6> .17 rounded to .14
Tolerable misstatement as a percent
T%+ > T A @opulation> ?'',''' A 6&,''','''>
.'&4 rounded to .'& (to be conservative)
There is no table available for an #+I# of 14. Inherent ris!and
control ris! for inventory are greater than for accounts
receivable. T A @opulation
> ?'',''' A (16,'''.''' B 6&,''',''')> ?'',''' A
&4,''','''> .'6& (rounded to .'6)
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17-!% #continued$
$ample sie computed using Table 14-? (allows for a .''4exception
ratePan average of the expected misstatements foraccounts
receivable and inventory2and assumes misstatementpercent of
1'')
n > 195
d. The generation of random numbers using %xcel (@1&4.xls)
to obtainthe sample of &? accounts receivable for confirmation
would beobtained as follows
@opulation boo! value > C16,''','''
8ommand to obtain each random number
>+#FG:%TW%%F(1,16'''''')
=nce the formula is entered, it can be copied down to
selectadditional random numbers. To obtain a sorted list, the list
ofrandom numbers should be copied to a separate column, andpasted
as a value (use the H@aste $pecial command and selectHvalue). Then
use the HGata $ort command to obtain a sorted list.
The command for selecting the random numbers can beentered
directly onto the spreadsheet, or can be selected from thefunction
menu (math J trig) functions. It may be necessary to addthe
analysis tool pac! to access the +#FG:%TW%%F function.
#n example prepared using %xcel is included on the8ompanion
Website (filename @1&4.xls).
17-!& a. This nonstatistical (i.e., nonprobabilistic or
udgmental) sample is astratified sample. #ll 6& items over
C1',''' were examined 1''.The remaining ,69 items were tested with
a sample of items.
#lthough this was not a probabilistic sample, auditing
standardsre"uire that in the auditor*s udgment, it is a
representative one.
#ccordingly, the results must be proected to the population and
audgment made about sampling ris!, although sampling ris!
andprecision cannot be measured.
@roection of the total population misstatement would be as
follows
Items over C1','''
@roected isstatement > #udited value - +ecorded value>
5&6,''' - 574,'''> (&&,''') overstatement
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17-!& #continued$
Items under C1',''' - average misstatement amount method
@roected isstatement > #verage sample misstatementx
population sie
> K(5,&4') A L x (,&6' - 6&)> (47.59) x 69>
(516,6') overstatement
Items under C1',''' - proportional amount method
@roected isstatement > $ample misstatement ratio xpopulation
boo! value
> K(5,&4') A ?1,4''L x (6,7',''' -574,''')
> (.'4&) x 6,694,'''> (161,7&4) overstatement
Where sample misstatements are
,T*A2D,TD0A2
RC/RDD0A2 *,SSTAT*.T
16
19
&&
&4
41
495
Totals
5,?6'
&?4
64'
&,?4
1,?64
&,?' '
15,9&4
4,16'
5?4
1,64'
&,94
1,?4'
5,6'' 6,5'4
19,6?4
(&'')
(1'')
(1,''')
(1'')
(64)
(56')(6,5'4)
(5,&4')
Fote that the sample misstatements are divided by the sample
boo!value of C?1,4'' to calculate the sample misstatement ratio.
Theproected misstatement is significantly lower using the
proportionalamount method because the average account sie in the
sample islarge than the average account sie in the population.
Total misstatement is either
(&&,''') B (516,6') > (554,6') overstatementor
(&&,''') B (161,7&4) > (145,7&4)
overstatement
In either case, the following can be said There are a
significantnumber of misstated items in the sample, and the amount
is "uitelarge. $ince the sample is representative, it is clear that
there is a
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17-!& #continued$
material misstatement of the population. The amount of
misstatementis not easily estimable from the sample. It could be
significantlyhigher or lower than either point estimate. #t this
point, the bestcourse of action would be to as! the client to ma!e
a study of their
records for all population items to identify more accurately
themisstatements that exist and correct them.
b. If this were a @@$ sample, the sampled portion would be
evaluatedas follows
isstatement taintings
,T*A2D,TD0A2
RC/RDD0A2
*,S-STAT*.T 8RC.T
16
19&&
&4
41
49
5
5,?6'
&?464'
&,?4
1,?64
&,?'
'
4,16'
5?41,64'
&,94
1,?4'
5,6''
6,5'4
(&'')
(1'')(1,''')
(1'')
(64)
(56')
(6,5'4)
(.'49)
(.6'7)(.?'')
(.'64)
(.'15)
(.1'')
(1.''')
8alculation of overstatement bound
/0R-STAT-*.T 28
RC/RDD0A2
2.,T*,SSTAT-
*.TASS2*8T,/.
*,S-STAT-*.T
B/2.D8/RT,/.
' (1)
1
6
&
5
4
7
.'5'
.'66
.'6'
.'19
.'1
.'1?
.'17
.'1
6,694,'''
6,694,'''
6,694,'''
6,694,'''
6,694,'''
6,694,'''
6,694,'''6,694,'''
1.'
1.'
.?''
.6'7
.1''
.'49
.'64
.'15
91,?''
4',59'
&7,6'
?,9?&
&,9'6
6,5&
91?457
(1) ;rom Table 14-9 using an #+I# of 4 percent and a samplesie
of 4.
=verstatement bound from sample 194,97isstatement of 1'' items
&&,'''
Total overstatement bound 66?,97
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17-!& #continued$
#n adusted understatement bound is calculated as follows
Initial understatement bound > .'5' x 6,694,'''>
91,?''
@oint estimate for overstatements > sum of unit
misstatementassumptions A sample sie x recorded population
amount
> 6.6'5 A x 6,694,'''> 74,791
#dusted understatement bound > initial bound - point
estimatefor overstatements
> 91,?'' - 74,791
> 67,1'9
#s would be expected, this is very small. $ince all
misstatementswere overstatements, one wouldn*t expect a net
understatementto occur.
The results of a @@$ sample indicate that the accounts
receivablebalance is overstated by as much as C66?,97. This is
about ? percentof the recorded boo! amount. It is significantly
greater than tolerablemisstatement, indicating that the population
is unacceptable andmust be subect to more scrutiny either by the
client andAor the auditor.
c. # template for the @@$ portion of the problem is prepared
using %xcelon the 8ompanion Website (;ilename @1&4.xls). This
template isa complete wor!sheet for 3$, including appropriate
tables forvarious exception rates and ris! levels. Qou will note
that the resultsare very similar to those computed manually, the
differences beingdue to rounding.
,nternet 8ro+lem Solution< *onetar 2nit Sampling
Considerations
17-1 onetary unit sampling (3$) is the most commonly used
statisticalmethod of sampling for tests of details because of its
simplicity and its ability toprovide statistical results in
dollars. #n article about using 3$ appeared in theay 6''4 issue of
The CPA Journal. $ee the following
KhttpAAwww.nysscpa.orgAcpaournalA6''4A4'4AessentialsAp&7.htmL.
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,nternet 8ro+lem 17-1 #continued$
1. The authors suggest that there are three critical steps in
applying3$. What are these stepsR
Answer
