Chapter 17 Audit Sampling for Tests of Details of Balances Review Questions 17-1 The most important difference between (a) tests of controls and substantive
tests of transactions and (b) tests of details of balances is in what the auditor wants to measure. In tests of controls and substantive tests of transactions, the primary concern is testing the effectiveness of internal controls and the rate of monet monetary ary miss missta tate teme ment nts. s. When When an audi audito torr perfo perform rmss test testss of contr control olss and and substantive tests of transactions, the purpose is to determine if the exception rate in the population is sufficiently low to justify reducing assessed control ris to reduce substantive tests. When statistical sampling is used for tests of controls and substantive tests of transactions, attributes sampling is ideal because it measures the fre!uency of occurrence (exception rate). In tests of details of bala balance nces, s, the the conce concern rn is dete determ rmin inin ing g whet whethe herr the the mone moneta tary ry amoun amountt of an account balance is materially misstated. "ttributes sampling, therefore, is seldom useful for tests of details of balances. 17- #tratified sampling is a method of sampling in w hich all the elements in the total
population are divided into two or more subpopulations. $ach subpopulation is then independently sampled, tested and the results projected to the population. "fter the results of the individual parts have been computed, they are combined into one overall overall popula populatio tion n measur measureme ement. nt. #trati #tratifie fied d sampli sampling ng is import important ant in auditi auditing ng in situations where the misstatements are liely to be either large or small. In order for an auditor to obtain a stratified sample of %& items from each of three strata in the confirmation of accounts receivable, he or she must first divide the population into three mutually exclusive strata. " random sample of %& items is then selected independently for each stratum. 17-! The point estimate is an estimate of the total amount of misstatement in the
population as projected from the nown misstatements found in the sample. The projection is based on either the average misstatement in the sample times the population si'e, or the net percent of misstatement in the sample times the population boo value. The true value of misstatements in the population is the net sum of all misstatements in the population and can only be determined by a && audit. 17-" The statement illustrates how the misuse of statistical estimation can impair
the use of an otherwise valuable audit tool. The auditor*s mistae is that he or she treats the point estimate as if it is the true population value, instead of but one possible value in a statistical distribution. +ather than judge whether the point estimate is material, the auditor should construct a statistical confidence -
17-" #continued$
interval interval around the point estimate, estimate, and consider consider whether whether the interval indicates indicates a material misstatement. "mong other factors, the interval will reflect appropriate levels of ris and sample si'e. 17-% onetary unit sampling is a method whereby the population is defined as
the individual dollars (or other currency) maing up the account balance. " random sample is drawn of these individual monetary units and the physical audit units containing them are identified and audited. The results of auditing the physical audit units are applied, pro rata, to the random monetary units, and a statistical conclusion about all population monetary units is derived. onetary unit sampling is now the most commonly used method of statistical sampling for tests of details of balances. This is because it uses the simplicity of attributes sampling yet still provides a statistical result expressed in dollars. It does this by using attribute tables to estimate the total proportion of population population dollars misstated, based on the number of sample dollars dollars misstated, misstated, and then modifies this amount by the amounts of misstatements found. This latter aspect gives monetary unit sampling its /variables/ dimension, although normal distribution theory is not used0 rather an arbitrary rule of thumb is applied to mae the adjustment. 1717-& #ampling ris is the ris that the characteristics in the sample are not
representative of those in the population. The two types of sampling ris faced by the auditor testing an account balance are1 a. b.
The The ris ris of incor incorre rect ct acce accept ptan ance ce ("+I ("+I") ")2t 2thi hiss is the the ris ris that that the the sample supports the conclusion that the recorded account balance is not materially misstated when it is materially misstated. The The ris ris of incor incorre rect ct rej reject ection ("+I ("+I+) +)2t 2thi hiss is the ris ris that hat the the sample supports the conclusion that the recorded account balance is materially misstated when it is not materially misstated.
#ampling ris occurs whenever a sample is taen from a population and therefore applies to all sampling methods. While "+I" applies to all sampling methods, "+I+ is only used in variables sampling sa mpling and difference estimation. 17-7 The steps in nonstatistical nonstatistical sampling sampling for tests of details of balances balances and for
test testss of cont control rolss are are almo almost st iden identitica cal,l, as illu illust strat rated ed in the the text text.. The The majo major r differences are that sampling for tests of controls deals with exceptions and sampling for tests of details of balances concerns dollar amounts. This results in differences in the application of the two methods, but not the steps. 17-' The two methods of selecting a monetary unit sample are random sampling and
systematic sampling. 3nder random sampling, in this situation, 4 random numbers would be obtained (the sample si'e in -5) between and 6,764,&&&. These would be sorted into ascending se!uence. The physical audit units in the
-6
17-' #continued$
inventory listing containing the random monetary units would then be identified by cumulating amounts with an adding machine or spreadsheet if the data is in machine-readable form. "s the cumulative total exceeds a successive random number, the item causing this event is identified as containing the random dollar unit. When When syst system emat atic ic sampl samplin ing g is used, used, the the popu popula latition on tota totall amou amount nt is divided by the sample si'e to obtain the sampling interval. " random number is chosen between and the amount of the sampling interval to determine the starting point. The dollars to be selected are the starting point and then the starting point plus the interval amount applied successively to the population total. The items on the inventory listing containing the dollar units are identified using the cumulative method described des cribed previously. In applying the cumulative method under both random sampling and systematic sampling, the page totals can be used in lieu of adding the detailed items if the page totals are considered to be reliable. 17-( 17-( " uni!ue aspect of monetary unit sampling is the use of the preliminary
judgment about materiality, materiality, as discussed in 8hapter 9, to directly determine the tolerable misstatement amount for the audit of each account. ost sampling techni!ues re!uire the auditor to determine tolerable misstatement for each account by allocating the preliminary judgment about materiality. This is not re!uired when monetary unit sampling is used. The preliminary preliminary judgment judgment about materiality is used. 17-1) 17-1) "cceptable ris of incorrect acceptance ("+I") is the ris the auditor is
willing to tae of accepting a balance as correct when the true misstatement in the balance is greater than tolerable misstatement. "+I" is the e!uivalent term to acceptable ris of assessing control ris too low for audit sampling for tests of controls and substantive tests of transactions. The primary factor affecting the auditor*s decision about "+I" is control ris in the audit ris model, which is the extent to which the auditor relies on intern internal al cont control rols. s. When When inte intern rnal al contr control olss are are effe effect ctiv ive, e, contr control ol ris ris can can be reduced, which permits the auditor to increase "+I", which in turn reduces the re!uired sample si'e. :esides control ris, "+I" is also affected directly by acceptable audit ris and inversely by inherent ris and other substantive tests already already perform performed ed on the account account balanc balance, e, assumin assuming g effec effectiv tive e results results.. ;or example, if acceptable audit ris is reduced, "+I" must also be reduced. If analytical procedures were performed and there is no indication of problem areas, there is a lower lielihood of misstatements in the account being tested, and "+I" can be increased. 17-1 17-11 1 The statement reflects a misunderstanding of the statistical inference
process. The process is based on the long-run probability that the process will produce correct results in a predictable proportion of the times it is applied. Thus,
-%
17-11 #continued$
a random sampling process that produces a 9& confidence interval will produce intervals that do, in fact, contain the true population value 9& of the time.
controls and substantive tests of transactions. If internal controls are considered to be effective, control ris can be reduced. " lower control ris re!uires a lower "+"8+, which re!uires a larger sample si'e for testing. If controls are determined to be effective after testing, control ris can remain low, which permits the auditor to increase "+I". "n increased "+I" allows the auditor to reduce sample si'es for tests of details of balances. 17-1! In using the binomial distribution, monetary unit sampling estimates the
proportion of all population dollars misstated by some amount . ;or the sample items actually misstated, the amounts of those misstatements are used.
Tolerable misstatement
4&&,&&&
> "verage misstatement percent assumption
÷
> +ecorded population value ? Tolerable exception rate
6,764,&&&
.&& 4&&,&&& 5
3sing the table for a & "+"8+ with an expected population exception rate of 'ero and a tolerable exception rate of 5, the preliminary sample si'e is 4.
-5
17-1%
*isstatement +ounds using the attri+utes ta+les
*,SSTAT*.T
6 %
RC0RDD 2A3
@9.7 5.&6 ,76.7@
AD,TD 2A3
*,SSTAT*.T/ *,SSTAT- RC0RDD *.T A*0.T
7&9.7 & ,466.7@
6@@.&& 5.&6 99.&&
.%6 .&&& .&7
3sing the attributes sampling table for a sample si'e of &&, and an "+I" of &, the 83$+ is1
.06 05 *,SSTAT*.TS
CR
,.CRAS ,. B0.D RS3T,.4 5R0* A. ADD,T,0.A3 *,SSTAT*.T
&
.&6%
.&%9
.&7
6
.&4%
.&5
%
.&77
.&%
In order to calculate the upper and lower misstatement bounds, it will be assumed that for a 'ero misstatement rate the percent of misstatement is &&. The upper misstatement bound1 *,S-
.06 05 *,SSTAT-
STAT*.T
&
.,T RC0RDD CR *,SSTAT 2A3 -*.T 90RT,0. 6,764,&&& .&&& .&6%
6,764,&&&
.&7
.&&&
6&6,&&&
6
6,764,&&&
.&5
.%6
47,%
%
6,764,&&&
.&%
.&7
&,&6
3pper isstatement :ound
449,65
*.TS
-4
8
B0.D 90RT,0.
69&,%4
17-1% #continued$
The lower misstatement bound1 :efore adjustment1
.06 05 *,SSTAT*.TS
6,764,&&&
&
.,T
RC0RDD CR *,SSTAT 2A3 -*.T 90RT,0.
.&6%
.&&&
*,SSTAT*.T
8
B0.D 90RT,0.
69&,%4
"djustment1 Aoint estimate for overstatements ? sum of misstatement percents x recorded value B sample si'e =
(.%6 C .&&& C .&7) x (6,764,&&& B &&)
=
.%@6 x 67,64&
=
5,5@
"djusted lower misstatement bound ? initial bound - point estimate for overstatements =
69&,%4 - 5,5@
=
4,@9
:ased on this calculation method, the population is not acceptable as stated since the upper misstatement bound exceeds the D4&&,&&& materiality limit. 17-1& The difficulty in determining sample si'e lies in estimating the number and
amount of misstatements that may be found in the sample. The upper bound of a monetary unit sample is sensitive to these factors. Thus, sample si'e varies a great deal with differing assumptions about them. Eenerally, the auditor will determine sample si'e by maing reasonable but conservative assumptions about the sample exception rate and average misstatement amount. In the absence of information about misstatement amount, which is most difficult to anticipate, a && assumption is often used.
-7
17-17
The decision rule for difference estimation is1 If the two-sided confidence interval for the misstatements is completely within plus or minus tolerable misstatements, accept the hypothesis that the boo value is not misstated by a material amount. =therwise, accept the hypothesis that the boo value is misstated by a material amount. ;or example, assume the F8F is -&,&&&, the 38F is 5&,&&& and tolerable misstatement is D54,&&&. The following illustrates the decision rule1 - T - 54,&&&
& - &,&&& F8F
C T C 54,&&& C 5&,&&& 38F
The auditor can conclude that the population is not materially misstated since both F8F and 38F are within the tolerable misstatement limits. 17-1' When a population is not considered acceptable, there are several
possible courses of action1 .
6.
%. 5. 4.
Aerform expanded audit tests in specific areas. If an analysis of the misstatements indicates that most of the misstatements are of a specific type, it may be desirable to restrict the additional audit effort to the problem area. Increase the sample si'e. When the auditor increases the sample si'e, sampling error is reduced if the rate of misstatements in the expanded sample, their dollar amount, and their direction are similar to those in the original sample. Increasing the sample si'e, therefore, may satisfy the auditor*s tolerable misstatement re!uirements. Increasing the sample si'e enough to satisfy the auditor*s tolerable misstatement standards is often costly, especially when the difference between tolerable misstatement and projected misstatement is small. "djust the account balance. When the auditor concludes that an account balance is materially misstated, the client may be willing to adjust the boo value. +e!uest the client to correct the population. In some cases the client*s records are so inade!uate that a correction of the entire population is re!uired before the audit can be completed. +efuse to give an un!ualified opinion. If the auditor believes the recorded amount in accounts receivable or any other account is not fairly stated, it is necessary to follow at least one of the above alternatives or to !ualify the audit opinion in an appropriate manner.
-
17-1( The population standard deviation is a measure of the difference between
the individual values and the mean of the population. It is calculated for all variables sampling methods but not for monetary unit sampling. ;or the auditor, it is usually estimated before determining the re!uired sample si'e, based on the previous year*s results or on a preliminary sample. The population standard deviation is needed to calculate the sample si'e necessary for an acceptable precision interval when variable sampling methods are used. "fter the sample is selected and audited, the population standard deviation is estimated from the standard deviation calculated from the values in the sample. The re!uired sample si'e is directly proportional to the s!uare of the population standard deviation. 17-)
This practice is improper for a number of reasons1 .
6.
%. 5.
Go determination was made as to whether a random sample of && inventory items would be sufficient to generate an acceptable precision interval for a given confidence level. In fact, a confidence limit was not even calculated. The combined net amount of the sample misstatement may be immaterial because large overstatement amounts may be offsetting large understatement amounts resulting in a relatively small combined net amount. "lthough no misstatement by itself may be material, other material misstatements might not have exhibited themselves if too small of a sample was taen. +egardless of the si'e of individual or net amounts of misstatements in a sample, the effect on the overall population cannot be determined unless the results are evaluated using a statistically valid method.
17-1 Hifference estimation is a method for estimating the total misstatement in a
population by multiplying the average misstatement (the audited value minus the recorded value) in a random sample by the number of items in the entire population. +atio estimation is !uite similar to difference estimation.
17-1 #continued$
The following are examples where each method could be used1 a.
b. c. d.
Hifference estimation can be used in computing the balance in accounts receivable by using the misstatements discovered during the confirmation process, where a significant number of misstatements are found. +atio estimation can be used to determine the amount of the FI;= reserve where internal inventory records are maintained on a ;I;= basis but reporting is on FI;=. ean-per-unit estimation can be used to determine total inventory value where the periodic inventory method is employed. #tratified mean-per-unit estimation can be used to determine total inventory value where there are several locations and each is sampled separately.
onetary unit sampling would generally be preferable to any of these where few or no misstatements are expected. Hifference and ratio estimation are not reliable where the exception rate is low, and mean-per-unit is generally not as efficient.
materiality allocated to each individual account. It is the amount of misstatement the auditor believes can be present in an account and the account balance still be acceptable for audit purposes. #ince hypothesis testing re!uires a decision rule based on materiality, that amount should be tolerable misstatement for an individual account balance. If test results provide a confidence limit greater than tolerable misstatement, the auditor would conclude the account is misstated. This would result in one or more of several actions1 . 6. %. 5. 4.
Aerform expanded audit tests in specific areas. Increase the sample si'e. "djust the account balance. +e!uest the client to correct the population. +efuse to give an un!ualified opinion.
In addition, it may be possible to adjust tolerable misstatement (upward) and remae the decision. The basis for this would be a reconsideration of the original judgment concerning determining overall materiality and allocation to the accounts. ;or example, audit wor completed on another account may indicate that a much lower tolerable misstatement exists for that account then originally planned. This would allow a reallocation providing a larger tolerable misstatement to the subject account.
-9
17-! Hifference estimation can be very effective and very efficient where () an
audited value and a boo value is available for each population item, (6) a relatively high fre!uency of misstatements is expected, and (%) a result in the form of a confidence interval is desired. In those circumstances, difference estimation far outperforms both 3# and mean-per- unit estimation. It may or may not outperform ratio estimation, depending on the relationship of misstatement amounts to recorded amounts, but it does re!uire less computational effort than ratio estimation in any case. If focus on large dollar value items is re!uired, difference estimation can be used with stratification. 17-" $xamples of audit conclusions resulting from the use of attributes,
monetary unit, and variables sampling are as follows1 3se of attributes sampling in a test of sales transactions for internal verification1 We have examined a random sample of && sales invoices for indication of internal verification0 two exceptions were noted. :ased on our sample, we conclude, with a 4 ris, that the proportion of sales invoices to which internal verification has not been applied does not exceed 7.6. 3se of monetary unit sampling in a test of sales transactions for existence1
We have examined a random sample of && dollar units of sales transactions for existence. "ll were supported by properly prepared sales orders and shipping documents. :ased on our sample, we conclude, with a 6& ris, that invalid sales do not exceed D5&,&&&. 3se of variables sampling in confirmation of accounts receivable (in the form of an interval estimate and a hypothesis test)1 We have confirmed a random sample of && accounts receivable. We obtained replies or examined satisfactory other evidence for all sample items. " listing of exceptions is attached. :ased on our sample, we estimate, with & ris, that the true population misstatement is between D6&,&&& understatement and D5&,&&& overstatement. #ince tolerable misstatement for accounts receivable is judged to be D4&,&&&, we conclude, with a ris of 4, that accounts receivable are not materially misstated. *ultiple Choice Questions from C9A aminations 17-% a. 17-& a.
(5) (5)
b. b.
(%) (6)
c. c.
(%) (6) -&
Discussion Questions and 9ro+lems 17-7
a. If random selection is performed using $xcel (A
[email protected]), the command to select numbers randomly from the population is1 ?+"GH:$TW$$G(,@59%) The & random numbers selected using this approach will vary for each student. The command for selecting the random numbers can be entered directly onto the spreadsheet, or can be selected from the function menu (math K trig) functions. It may be necessary to add the analysis tool pac to access the +"GH:$TW$$G function. =nce the formula is entered, it can be copied down to select additional random numbers. G=T$1 +andom dollar items are matched with population item numbers where the cumulative boo value of the population includes the random dollar selected. b. Interval
?
Aopulation total Total dollars in the population B Gumber of items selected for testing
=
@,59% &
=
,@59 Interval
3sing @4 as a starting point, we have1
6 % 5 4 7 @ 9 &
#L#T$"TI8 H=FF"+ ,@4 9&7 444 645&5 %%64% 5&6 5@94 47@&& 75759 659@
-
A=A3F"TI=G IT$ G=. 6 @ & 4 @ 66 64 % %4 %@
G=T$1 #ystematic dollar items are related to population item numbers in the same manner as for part a above. "ll items larger than the interval will be automatically included. In this case there are no items larger than the interval of ,@59..
d.
e.
The same is not necessarily true for random number selection, but the probability is high. There is no significant difference in ease of selection between computer generation of random numbers and systematic selection. #ome auditors prefer the use of random numbers because they believe this helps ensure an unbiased sample. onetary unit sampling would be used because () it is efficient and (6) it focuses on large dollar items.
-6
17-'
a.
The following summari'es the confirmation responses1 Recorded 2alue
"cct. %
Confirmation Response *isstatement
D@%,69
D%,69
&
"cct. 69 6%,54 "cct. 67 @,5%9 "cct. 57 ,55% "cct. 4% ,546 "cct. 7@9 5,%@ "cct. @5 %5,4@% Total misstatement
7,9% ,@7 & 7,@%6 & 6%,759
7,46& 46 ,55% 76& & &,9%5 D%7,&@9
b.
#tratum #tratum 6 #tratum % Totals
17-( a.
8utoff error $rror in !uantity shipped
8utoff error Aricing error Timing difference 8utoff error
$stimate of total misstatement Sample 2alue
c.
Timing difference
Sample *isstatements
D 9%9,9
D
Boo: 2alue
&
D 9%9,9
,5,47
%5,@9
5,7@,@@7
,6%9
,96
D 6,@5,99
D %7,&@9
@96,46 D7,49,7&5
9ro;ected *isstatement
D
& %9,6@&
5,9%5 D45,65
The population is not acceptable since the projected misstatement of D45,65 exceeds tolerable misstatement of D&&,&&&. The auditor is liely to propose an adjustment andBor increase testing. In this situation, many of the errors involved cutoff, so the auditor could expand testing in this area and propose an adjustment for the errors found. :ecause the cutoff errors were isolated and testing expanded in this area, the cutoff errors would not be included in the projection of error for each stratum. The differences that were uncovered include only five misstatements rather than seven. Items 6 and are not misstatements, but only timing differences. Therefore, only the five misstatements are summari'ed in order to compute the upper and lower misstatement bounds. These misstatements are summari'ed below.
-%
17-( #continued$
*,SSTAT-
,T*
% 5 4 7
RC0RDD 2A3
AD,TD 2A3
D6,6@.&& %,@9&.&& 9.&& 45@.&& %,4.&&
*,SSTAT*.T
D6,59@.&& ,9&.&& @4.&& ,&%.&& %,9&.&&
*.T/ RC0RDD 2A3
D 6%&.&& 6,&&.&& (65.&&) (5@9.&&) (4.&&)
.&@5 .795 (.&%&) (.@96) (.&65)
3pper misstatement bound before adjustment1 *,S-
.06 05 *,SSTAT*.TS & 6
RC0RDD 2A3
D,94,&&& ,94,&&& ,94,&&&
CR 90RT,0.
.&6% .&7 .&5 .&4%
*,SSTAT*.T < ASS*9T,0.
STAT8 *.T
B0.D
.&&& .795 .&@5
D54,564 6,9%& 6,%6% D 79,7@
3ower misstatement +ound +efore ad;ustment= *,S-
.06 05 *,SSTAT*.TS & 6 %
RC0RDD 2A3
D,94,&&& ,94,&&& ,94,&&& ,94,&&&
CR 90RT,0.
.&6% .&7 .&5 .&% .&77
-5
*,SSTAT*.T < ASS*9T,0.
.&&& .@96 .&%& .&65
STAT8 *.T
B0.D D54,564
6@,@ @%& 77
D4,&4@
17-( #continued$
"djustment of upper misstatement bound1 Aoint estimate for understatement amounts ? sum of misstatement percents x recorded value B sample si'e =
(.@96 C .&%& C .&65) x (,94,&&& B &&)
=
.957 x 9,4&
=
@,7@5
"djusted bound ? initial bound - point estimate for understatement amounts =
79,7@ - @,7@5
=
4&,995
"djustment of lower misstatement bound1 Aoint estimate for overstatement amounts ? sum of misstatement percents x recorded valueBsample si'e =
(.795 C .&@5) x (,94,&&& B &&)
=
.@ x 9,4&
=
4,%77
"djusted bound ? initial bound - point estimate for overstatements =
4,&4@ - 4,%77
=
49,796
b.
The population is not acceptable as stated because both the lower misstatement bound and upper misstatement bound exceed materiality. In this situation, the auditor has the following options1 .
6.
#egregate a specific type of misstatement and test it separately (for the entire population). The sample would then not include the specified type of misstatement since it is being tested separately. Increase the sample si'e. -4
17-( #continued$
%.
"djust the account balance (i.e., propose an adjustment). 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 with additional audit wor in areas outside of accounts receivable. =f these options, segregating a specific type of misstatement may prove to be the most beneficial. In this problem, items % and 4 are cutoff misstatements. #egregating these items, testing cutoff more extensively, and eliminating them from the sample would result in the following bounds1
3pper misstatement bound1 *,S-
.06 05 *,SSTAT*.TS &
RC0RDD CR 2A3 90RT,0. D,94,&&& .&6%
,94,&&&
*,SSTAT*.T < ASS*9T,0.
8
.&&& .&@5
.&7 .&%9
STAT *.T B0.D
D54,564 6,745 D5@,&9
Fess adjustment M(.&%& C .&65) (9,4&)N
(,&7) D 5,&6
Fower misstatement bound1 *,S-
.06 05 *,SSTAT*.TS & 6
RC0RDD CR 2A3 90RT,0. D,94,&&& .&6%
,94,&&& ,94,&&&
.&7 .&5 .&4%
Fess adjustment M(.&@5) (9,4&)N
*,SSTAT*.T < ASS*9T,0.
.&&& .&%& .&65
8
STAT *.T B0.D
D54,564 95@ 775 D5,&% (,749) D54,%@
It can be seen that both misstatement bounds are now within materiality after cutoff misstatements were segregated. These misstatements were significant in two ways. Their existence increased the overall estimated population exception rate, and their magnitude contributed to the amount of estimated misstatements in the portion of the population represented by the misstatements in the sample.
-7
17-!) a. The differences that were uncovered include only five
misstatements rather than seven. Items 6 and are not misstatements, but only timing differences. Therefore, only the five misstatements are summari'ed in order to compute the upper and lower misstatement bounds. These misstatements are summari'ed below. *,SSTAT*.T/ ,T*
RC0RDD 2A3
AD,TD 2A3
D6,6@.&&
D6,59@.&&
% 5 4 7
%,@9&.&& 9.&& 45@.&& %,4.&&
,9&.&& @4.&& ,&%.&& %,9&.&&
*,SSTAT- RC0RDD *.T 2A3
D 6%&.&& 6,&&.&& (65.&&) (5@9.&&) (4.&&)
.&@5 .795 (.&%&) (.@96) (.&65)
pper misstatement +ound +efore ad;ustment=
.06 05 *,SSTAT- RC0RDD CR 90RT,0. *.TS 2A3
*,S STAT*,SSTAT*.T < 8 *.T ASS*9T,0. B0.D
&
D,94,&&&
.&6%
.&&&
D54,564
,94,&&& ,94,&&&
.&7
.&5 .&4%
.795 .&@5
6,9%& 6,%6% D79,7@
6
3ower misstatement +ound +efore ad;ustment=
.06 05 *,SSTAT- RC0RDD CR 90RT,0. *.TS 2A3
*,S STAT*,SSTAT*.T < 8 *.T ASS*9T,0. B0.D
&
D,94,&&&
.&6%
.&&&
D54,564
,94,&&& ,94,&&& ,94,&&&
.&7
.&5
.&% .&77
.@96 .&%& .&65
6@,@ @%& 77 D4,&4@
6 %
-
17-!) #continued$ Ad;ustment of upper misstatement +ound=
Aoint estimate for understatement amounts ? sum of misstatement percents x recorded value B sample si'e =
(.@96 C .&%& C .&65) x (,94,&&& B &&)
=
.957 x 9,4&
=
@,7@5
"djusted bound ? initial bound - point estimate for understatement amounts =
79,7@ - @,7@5
=
4&,995
Ad;ustment of lower misstatement +ound=
Aoint estimate for overstatement amounts ? sum of misstatement percents x recorded valueBsample si'e =
(.795 C .&@5) x (,94,&&& B &&)
=
.@ x 9,4&
=
4,%77
"djusted bound ? initial bound - point estimate for overstatements
b.
=
4,&4@ - 4,%77
=
49,796
The population is not acceptable as stated because both the lower misstatement bound and upper misstatement bound exceed materiality. In this situation, the auditor has the following options1 .
6.
#egregate a specific type of misstatement and test it separately (for the entire population). The sample would then not include the specified type of misstatement since it is being tested separately. Increase the sample si'e. -@
17-!) #continued$
%. 5. 4.
"djust the account balance (i.e., propose an adjustment). +e!uest the client to review and correct the population. 8onsider !ualifying the opinion is the client refuses to correct the problem.
7.
8onsider the criteria used in the test, possibly in connection with additional audit wor in areas outside of accounts receivable.
=f these options, segregating a specific type of misstatement may prove to be the most beneficial. In this problem, items % and 4 are cutoff misstatements. #egregating these items, testing cutoff more extensively, and eliminating them from the sample would result in the following bounds1 pper misstatement +ound=
.06 05 *,SSTAT*.TS
RC0RDD CR 2A3 90RT,0.
*,SSTAT*.T < ASS*9T,0.
*,S STAT8 *.T B0.D
&
D,94,&&&
.&6%
.&&&
D54,564
,94,&&&
.&7 .&%9
.&@5
6,745 D5@,&9
Fess adjustment M(.&%& C .&65) (9,4&)N
(,&7) D5,&6
3ower misstatement +ound= *,S.06 05 *,SSTAT*.TS
*,SSTATCR *.T < RC0RDD 2A3 90RT,0. ASS*9T,0.
STAT*.T 8 B0.D
&
D,94,&&&
.&6%
.&&&
D54,564
,94,&&& ,94,&&&
.&7 .&5 .&4%
.&%& .&65
95@ 775 D5,&%
6
Fess adjustment M(.&@5) (9,4&)N
-9
(,749) D54,%@
17-!) #continued$
It can be seen that both misstatement bounds are now within materiality after cutoff misstatements were segregated. These misstatements were significant in two ways. Their existence increased the overall estimated population exception rate, and their magnitude contributed to the amount of estimated misstatements in the portion of the population represented by the misstatements in the sample. 17-!1 a.
Computer Solution. This is an excellent problem to demonstrate
the use of the computer in auditing, as it re!uires a great deal of computational wor. " solution prepared using $xcel is included on the 8ompanion Website and on the InstructorOs +esource 8H-+=, which is available upon re!uest (;ilename A%6.xls). Important points to stress are1 . 6.
%.
The spreadsheet program is set up in two sections1 one for data entry and one for computations. 8ells are set up for variables by name, and the values for the ). variables are then entered in those cells (e.g., sample si'e ? 8omputations are then done by reference to the cells rather than by entering values in the formulas. This allows the worsheet to be used as a general program for similar problems. "lthough the program assures computational accuracy, the formulas must be correct . They should always be reviewed and double checed, and test data should be processed to assure accuracy. a.8alculating the point estimate1 e ;
8.>?
n
∧
8 1'") >
17! 6&(
') 8 !((" 6'7
-6&
17-!1 #continued$
:efore computing the computed precision interval, we must compute the standard deviation1 e ; SD 8
D(6.&&) 74.& 5.& %7.& 4.@& (.6) %&.&& 6. %.79
? #e ; $ @ n #e$ n@1 173.69 16,521.79
2
P80
80
P 80
P1
8 1"6!)
#e ;$
4,@5.&& 5,%7.59 ,
[email protected] ,%&%.% 6,7@%.65 .& 9&&.&& 554.7% 7,46.9
8omputed precision interval1 C9, 8 . A > SD >
n
.@n
.
C9, 8 1'") > 16&" > 1" 6!) >
1'") @ ')
')
1'")
C9, 8 "71' 6"&
The confidence interval is expressed as %,995.@ C 5,@.57. To compute the confidence limits, 38F ? Q C 8AI ? %,995.@ C 5,@.57 ? @,%.%% F8F ? Q - 8AI ? %,995.@ - 5,@.57 ? -6%.49 b.
c.
The auditor should not accept the boo value of the population since the maximum misstatement in the population that she was willing to accept, D7,&&&, at a ris level of 4, is less than the possible amount of true misstatement indicated by the 38F of D@,%.%%. The options available to the auditor at this point are1 . Aerform expanded audit tests in specific areas. 6. Increase the sample si'e. %. "djust the account balance. 5. +e!uest the client to correct the population. 4. +efuse to give an un!ualified opinion. -6
17-!
. (a)
6. (c)
%. (a)
5. (d)
4. (d)
17-!! Computer Solution . This is an excellent problem to demonstrate the use of the
computer in auditing, as it re!uires a great deal of computational wor. " solution prepared using $xcel is included on the 8ompanion Website (;ilename A%%.xls). Important points to stress are1 . 6.
%.
The spreadsheet program is set up in two sections1 one for data entry and one for computations. 8ells are set up for variables by name, and the values for the variables are then entered in those cells (e.g., sample si'e ? ). 8omputations are then done by reference to the cells rather than by entering values in the formulas. This allows the worsheet to be used as a general program for similar problems. "lthough the program assures computational accuracy, the formulas must be correct . They should always be reviewed and double checed, and test data should be processed to assure accuracy. a.8alculating the point estimate1
:efore computing the computed precision interval, we must compute the standard deviation1
-66
e j
(e j)6
D(6.&& ) 74.& 5.& %7.& 4.@&
4,@5.&& 5,%7.59 ,
[email protected] ,%&%.% 6,7@%.65 (.6 ) .& %&.&& 9&&.&& 6. 554.7% D%.79 7,46.9
17-!! #continued$
8omputed precision interval1
The confidence interval is expressed as %,995.@ C 5,@.57.
To compute the confidence limits, 38F ?
Q C 8AI ? %,995.@ C 5,@.57 ? @,%.%%
F8F ?
Q - 8AI ? %,995.@ - 5,@.57 ? -6%.49
b.
The auditor should not accept the boo value of the population since the maximum misstatement in the population that she was willing to accept, D7,&&&, at a ris level of 4, is less than the possible amount of true misstatement indicated by the 38F of D@,%.%%.
c.
The options available to the auditor at this point are1 . 6. %. 5. 4.
Aerform expanded audit tests in specific areas. Increase the sample si'e. "djust the account balance. +e!uest the client to correct the population. +efuse to give an un!ualified opinion.
-6%
17-!"
a. It would be desirable to use unstratified difference estimation when the auditor believes that there is not a small number of misstatements in the population that are in total material, and the population has a large number of small misstatements that in total could be material. 3nstratified difference estimation would not be appropriate when either of the above characteristics is not present. ;or example, if the auditor believes that certain large accounts payable may contain large misstatements that are material, they should be tested separately. " significant consideration in this situation is whether the auditor can identify the entire population. This consideration applies whether using stratified or unstratified difference estimation. The auditor in this instance is identifying the population based upon an accounts payable list. If this list includes only those accounts with an outstanding balance, the sample is ignoring those accounts that have 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 si'e may be computed as follows1
where T - $R
c.
?
54,&&& - 6&,&&& ? D64,&&&
In order to determine whether the population is fairly stated, the computed precision interval must be calculated.
8I 8I
? Q C 8AI ? 6,&&& C 66,%5
38F
? 5%,%5 -65
17-!" #continued$
F8F
?
-,%5
#ince both 38F and F8F are less than tolerable misstatement, the auditor can conclude that the population is fairly stated. The primary reasons the population is acceptable is that () the actual point estimate is reasonably close to the expected misstatement, and (6) the actual sample standard deviation is less than the estimated standard deviation.
d.
8onsidering the "+I+, the sample si'e may be computed from the following formula1
e.
The sample si'e increases significantly with the inclusion of the "+I+ because by including it the auditor is establishing the ris he or she will tae of rejecting an acceptable population, as well as considering the ris of accepting an unacceptable population. It taes more effort (sample items) to control two riss, rather than just one. The effect can be seen from reviewing the formula for calculating the sample si'e.
f.
The approach described will only result in an appropriate sample si'e by chance. This would occur when the 64 increment is e!ual to the increase in the sample si'e re!uired when the "+I+ is considered. This is not a liely occurrence. This approach is not desirable because it is inefficient in terms of time and cost. 3nless by chance the sample si'e is approximately e!ual to the sample si'e re!uired by considering "+I+, the sample si'e will be either too 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 it is even possible. 8onversely, too large a sample results in the auditor performing more wor than is re!uired.
-64
Cases 17-!%
a. Determination of ARIA - Gote that there are many ways to estimate "+I". =ne method is as follows1 "+I" ? ""+ B (I+ x 8+ x "A+) .&4 B (.@ x .4 x .&) .&4 B .5 .% rounded to .& (to be conservative) = = =
Tolerable misstatement as a percent 1
T$+ ? = =
T B Aopulation @&&,&&& B 6,&&&,&&& .&7 rounded to .&7 (to be conservative)
#ample si'e determined using Ta+le 1%-' (assumes an expected misstatement of 'ero and a misstatement percent of &&)1 n ? %@ b.
Determination of ARIA - Gote that there are many ways to estimate
"+I". =ne method is as follows1 "+I" ? = = =
""+ B (I+ x 8+ x "A+) .&4 B M.& x .@ x ( - .7)N .&4 B .%6 .7 rounded to .4
Tolerable misstatement as a percent 1
T$+ ? = =
T B Aopulation @&&,&&& B 6%,&&&,&&& .&%4 rounded to .&% (to be conservative)
There is no table available for an "+I" of 4. Inherent ris and control ris for inventory are greater than for accounts receivable.
17-!% #continued$
separate tests, (i.e. & from the examples shown in re!uirements a and b). Tolerable misstatement as a percent 1
T$+ ? T B Aopulation @&&,&&& B (6,&&&,&&& C 6%,&&&,&&&) @&&,&&& B %4,&&&,&&& .&6% (rounded to .&6) = = =
#ample si'e computed using Ta+le 1%-' (allows for a .&&4 exception rateS an average of the expected misstatements for accounts receivable and inventory2and assumes misstatement percent of &&)1 n ? 95 d.
The generation of random numbers using $xcel (A%5.xls) to obtain the sample of %@ accounts receivable for confirmation would be obtained as follows1 Aopulation boo value ? D6,&&&,&&& 8ommand to obtain each random number1 ?+"GH:$TW$$G(,6&&&&&&) =nce the formula is entered, it can be copied down to select additional random numbers. To obtain a sorted list, the list of random numbers should be copied to a separate column, and pasted as a value (use the Aaste #pecialJ command and select valueJ). Then use the Hata #ortJ command to obtain a sorted list. The command for selecting the random numbers can be entered directly onto the spreadsheet, or can be selected from the function menu (math K trig) functions. It may be necessary to add the analysis tool pac to access the +"GH:$TW$$G function. "n example prepared using $xcel is included on the 8ompanion Website and on the InstructorOs +esource 8H-+=, which is available upon re!uest (filename A%5.xls).
-6
17-!&
a. This nonstatistical (i.e., nonprobabilistic or judgmental) sample is a stratified sample. "ll 6% items over D&,&&& were examined &&. The remaining ,69 items were tested with a sample of items. "lthough this was not a probabilistic sample, auditing standards re!uire that in the auditor*s judgment, it is a representative one. "ccordingly, the results must be projected to the population and a judgment made about sampling ris, although sampling ris and precision cannot be measured. Arojection of the total population misstatement would be as follows1 Items over D&,&&&1 Arojected isstatement ? "udited value - +ecorded value ? 5%6,&&& - 574,&&& ? (%%,&&&) overstatement Items under D&,&&& - average misstatement amount method1
Arojected isstatement ? "verage sample misstatement x population si'e ? M(5,%4&) B N x (,%6& - 6%) ? (47.59) x 69 ? (56,6&) overstatement Items under D&,&&& - proportional amount method1 Arojected isstatement ? #ample misstatement ratio x population boo value ? M(5,%4&) B @,4&&N x (6,7&,&&& 574,&&&) ? (.&4%) x 6,694,&&& ? (6,7%4) overstatement
-6@
17-!& #continued$
Where sample misstatements are1 ,T*
AD,TD 2A3
6
5,@6&
4,6&
(%&&)
9
%@4 64& %,@4 ,@64 %,@& & 5,9%4
5@4 ,64& %,94 ,@4& 5,6&& 6,5&4 9,6@4
(&&) (,&&&) (&&) (64) (56&) (6,5&4) (5,%4&)
%% %4 4 49 5
Totals
RC0RDD 2A3
*,SSTAT*.T
Gote that the sample misstatements are divided by the sample boo value of D@,4&& to calculate the sample misstatement ratio. The projected misstatement is significantly lower using the proportional amount method because the average account si'e in the sample is large than the average account si'e in the population. Total misstatement is either1 (%%,&&&) C (56,6&) ? (554,6&) overstatement
or (%%,&&&) C (6,7%4) ? (45,7%4) overstatement In either case, the following can be said1 There are a significant number of misstated items in the sample, and the amount is !uite large. #ince the sample is representative, it is clear that there is a material misstatement of the population. The amount of misstatement is not easily estimable from the sample. It could be significantly higher or lower than either point estimate. "t this point, the best course of action would be to as the client to mae a study of their records for all population items to identify more accurately the misstatements that exist and correct them. b.
If this were a AA# sample, the sampled portion would be evaluated as follows1
-69
17-!& #continued$
isstatement taintings1 ,T*
AD,TD 2A3
RC0RDD 2A3
*,SSTAT*.T
9RC.T
6
5,@6&
4,6&
(%&&)
(.&49)
9 %% %4 4 49 5
%@4 64& %,@4 ,@64 %,@& &
5@4 ,64& %,94 ,@4& 5,6&& 6,5&4
(&&) (,&&&) (&&) (64) (56&) (6,5&4)
(.6&7) (.@&&) (.&64) (.&5) (.&&) (.&&&)
8alculation of overstatement bound1 *,S-
02RSTAT*.T
& () 6 % 5 4 7
.,T *,SSTAT*.T ASS*9T,0.
STAT*.T B0.D 90RT,0.
93
RC0RDD 2A3
.&5&
6,694,&&&
.&
9,@&&
.&66 .&6& .&9 .& .&@ .&7 .&
6,694,&&& 6,694,&&& 6,694,&&& 6,694,&&& 6,694,&&& 6,694,&&& 6,694,&&&
.& .@&& .6&7 .&& .&49 .&64 .&5
4&,59& %7,6& @,9@% %,9&6 6,5% 9@ 457
() ;rom Table 4-9 using an "+I" of 4 percent and a sample si'e of 4. =verstatement bound from sample isstatement of && items Total overstatement bound
-%&
94,97 %%,&&&
66@,97
17-!& #continued$
"n adjusted understatement bound is calculated as follows1 Initial understatement bound ? .&5& x 6,694,&&& ? 9,@&& Aoint estimate for overstatements ? sum of unit misstatement assumptions B sample si'e x recorded population amount
? 6.6&5 B x 6,694,&&& ? 74,79 "djusted understatement bound ? initial bound - point estimate for overstatements ? 9,@&& - 74,79 ? 67,&9 "s would be expected, this is very small. #ince all misstatements were overstatements, one wouldn*t expect a net understatement to occur. The results of a AA# sample indicate that the accounts receivable balance is overstated by as much as D66@,97. This is about @ percent of the recorded boo amount. It is significantly greater than tolerable misstatement, indicating that the population is unacceptable and must be subject to more scrutiny either by the client andBor the auditor.
c.
" template for the AA# portion of the problem is prepared using $xcel on the 8ompanion Website (;ilename A%4.xls). This template is a complete worsheet for 3#, including appropriate tables for various exception rates and ris levels. Lou will note that the results are very similar to those computed manually, the differences being due to rounding.
,nternet 9ro+lem Solution= *onetar nit Sampling Considerations 17-1 onetary unit sampling (3#) is the most commonly used statistical
method of sampling for tests of details because of its simplicity and its ability to provide statistical results in dollars. "n article about using 3# appeared in the ay 6&&4 issue of The CPA Journal . #ee the following1 Mhttp1BBwww.nysscpa.orgBcpajournalB6&&4B4&4BessentialsBp%7.htmN.
-%
