ACCOUNTING 441
WINTER 2002
PPS OR MUS SAMPLING
1.
DEFINE THE SAMPLING UNIT.
-THE SAMPLING UNIT IS EACH DOLLAR IN THE POPULATION
HOWEVER,
-THE AUDITOR DOES NOT TEST INDIVIDUAL DOLLARS, BUT THE
TRANSACTION CONTAINING THE DOLLAR SELECTED
ACCOUNT OR
-THE ACCOUNT OR TRANSACTION SELECTED IS CALLED THE LOGICAL UNIT
2.
SELECT THE SAMPLE
-EACH DOLLAR IN THE POPULATION HAS AN EQUAL CHANCE OF BEING SELECTED.
-THE POPULATION IS DIVIDED INTO EQUAL GROUPS OF DOLLARS.
-EACH GROUP OF DOLLARS IS CALLED A SAMPLING INTERVAL.
-THE SAMPLING INTERVAL IS DETERMINED BY DIVIDING THE BOOK VALUE OF THE
POPULATION BY THE SAMPLE SIZE (SEE SAMPLE SIZE SELECTION BELOW)
-EXAMPLE:

TOTAL ACCOUNTS RECEIVABLE OF $2,500,000 AND AUDITOR DECIDES TO
SAMPLE 100 ITEMS (SEE BELOW FOR DETERMINING SAMPLE SIZE). MUST
GET THE ACCOUNTS IN ORDER AND GO FROM TOP OF LIST DOWN

THE SAMPLE INTERVAL WILL BE $2,500,000 DIVIDED BY 100 (SAMPLE SIZE)
OR $25,000

USE RANDOM NUMBER BETWEEN 1 AND 25,000 AS THE STARTING POINT.

FOR OUR EXAMPLE ASSUME THE STARTING POINT IS $3,000
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3.

FIRST ITEM IS THE ACCOUNT THAT CONTAINS DOLLAR NUMBER 3,000;
SECOND ITEM CONTAINS DOLLAR NUMBER 28,000; THE THIRD DOLLAR
53,000; ETC.

NOTE THAT AN ACCOUNT WITH AN AMOUNT OF $25,000 WILL ALWAYS BE
SELECTED. AN ACCOUNT WITH A BALANCE OF $50,000 OR MORE WILL BE
SELECTED TWICE OR MORE (BUT WILL ONLY BE TESTED (E.G.
CONFIRMED) ONCE

LOGICAL UNITS (SEE DEFINITION ABOVE) CONTAINING MORE DOLLARS
HAVE A GREATER CHANCE OF SELECTION. ALSO, EACH DOLLAR HAS AN
EQUAL CHANCE OF BEING SELECTED (E.G. EVERY 25,000TH ITEM. IT ALL
DEPENDS ON THE RANDOM STARTING POINT.)
DETERMINING THE SAMPLE SIZE
-COULD USE SAMPLE SIZE FORMULA PAGE 71OF INTERNAL AUDIT SAMPLING TOO
DIFFICULT FOR TEACHER, SO GO TO NEXT ALTERNATIVE.
-NEXT ALTERNATIVE-- DETERMINE SAMPLE SIZE FROM ATTRIBUTE SAMPLING
TABLES. (TABLES 1 And 2.)

FIRST DETERMINE RISK OF INCORRECT ACCEPTANCE (5% -TABLE 1 -OR
10% -Table 2)

THEN, CONVERT THE TOLERABLE MISSTATEMENT AND THE EXPECTED
MISSTATEMENT TO PERCENTAGES. THESE FACTORS ARE BASED ON
AUDITORS JUDGEMENT

EXAMPLE:
AUDITOR ESTABLISHES TOLERABLE MISSTATEMENTS OF $125,000 AND
EXPECTED MISSTATEMENT OF $25,000.
CONVERT TO PERCENTAGES:
-TOLERABLE $125,000/$2,500,000 = 5%
-EXPECTED $25,000/2,5000,000
= 1%
-ASSUME 5% RISK - TABLE 1 - SAMPLE EQUALS 93
(IF 10% RISK-TABLE 2- SAMPLE WOULD BE 77)
THE TWO TABLE, PAGES 113 AND 114 AND ATTACHED TABLES 1 And 2,
RELATE TO THE RISK OF ASESSING CONTROL RISK TOO LOW.
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4.
EVALUATING SAMPLE RESULTS
-MUST PROJECT MISSTATEMENTS IN THE SAMPLE TO THE POPULATION. THIS IS
CALLED THE UPPER LIMIT OF MISSTATEMENT.
TWO ASSUMPTIONS ARE MADE;
NO MISSTATEMENTS DETECTED IN THE SAMPLE
OR
SOME OR ONE MISSTATEMENTS DETECTED IN THE SAMPLE
A.
NO MISSTATEMENTS DETECTED

THE PROJECTION OF THE SAMPLE RESULTS EQUALS "0" DOLLARS PLUS
AN ALLOWANCE FOR SAMPLING RISK (BASIC PRECISION)
BASIC PRECISION (FOR 0 ERRORS) WOULD BE EQUAL TO OR LESS THAN
THE TOLERABLE MISSTATEMENT USED TO DESIGN THE SAMPLE. (5%
ABOVE)
TO GET THE UPPER LIMIT OF MISSTATEMENT - MULTIPLY THE
RELIABILITY FACTOR FOR THE RISK OF INCORRECT ACCEPTANCE BY THE
SAMPLING INTERVAL. TABLE 5 OF THESE NOTES OR PAGE 123 OF THE
BOOK MAY BE USED TO FIND THE RELIABILITY FACTOR FOR ERRORS OF
OVERSTATEMENT (NOT AS CONCERNED WITH UNDERSTATEMENTS IN
VARIABLE SAMPLING).

REMEMBER FROM ABOVE WE SELECTED A SAMPLE OF 93 AT THE 5%
LEVEL. THE INTERVAL IS THEN $2,500,000/93 = $26,882.

THE UPPER LIMIT OF MISSTATEMENT IS THEN $26,882 X 3.0 (THE
RELIABILITY RACTOR) OR $80,646. (THIS IS BELOW THE $125,000 SET IN 3.
ABOVE.)
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B.
ASSUME MISSTATEMENTS ARE DETECTED (ERRORS FOUND)
MUST CALCULATE
(1)
A PROJECTED MISSTATEMENT
AND
(2)
AN ALLOWANCE FOR SAMPLING RISK
MISSTATEMENTS)
(AS ABOVE FOR "0"
THREE POSSIBILITIES EXIST:
(A)
THE LOGICAL UNIT (ACCOUNT SELECTED) IS EQUAL TO OR
GREATER THAT THE SAMPLING INTERVAL ($26,882)
EXAMPLE:
BOOK VALUE OF LOGICAL UNIT = $35,000
ERROR DETECTED
= $ 1,500
THE PROJECTED MISSTATEMENT IS $1,500 BECAUSE THE ENTIRE
SAMPLE INTERVAL ($26,882) WAS AUDITED (NO SAMPLING RISK
ASSOCIATED WITH THIS SAMPLE ITEM)
(B)
THE BOOK VALUE OF THE LOGICAL UNIT (E.G. $250) IS LESS
THAN THE SAMPLING INTERVAL ($26,882) AND IS 100%
MISSTATED. EXAMPLE; ASSUME THAT THE AUDITED VALUE IS
"0" AND THAT THE BOOK VALUE IS $250.
THE PROJECTED ERROR FOR THIS LOGICAL UNIT IS $26,882 (THE
TOTAL SAMPLING INTERVAL) DETERMINED BY;
PERCENTAGE MISSTATEMENT
TIMES SAMPLE INTERVAL
EQUALS
THEN NEED TO ADD AN ALLOWANCE
AFTER (C) BELOW)
4
100%
$26,882
$26,882
FOR SAMPLING RISK (SEE
(C)
THE LOGICAL UNIT'S BOOK VALUE IS LESS THAN THE SAMPLING
INTERVAL AND IT IS MISSTATED BY LESS THAN 100 PERCENT:
TAINTING FACTOR - THE PERCENTAGE OF MISSTATEMENT IN
THE LOGICAL UNIT
THE TAINTING FACTOR IS = BOOK VALUE OF THE
LOGICAL UNIT MINUS AUDIT VALUE DIVIDED BY BOOK
VALUE.
E.G. LOGICAL UNIT = 5,000; AUDIT VALUE = 3,000
TAINTING FACTOR IS (5,000-3,000= 2,000)
2,000/5,000 40%
SEE BELOW AND ON PAGE 74 ON INTERNAL AUDIT SAMPLING
FOR TABLE TO CALCULATE PROJECTED ERROR WITH TAINTING.
THE PROJECTED MISSTATEMENT FOR THE UNIT WOULD BE 40%
X $26,882 = $10,783
ALSO NEED TO ADD AN ALLOWANCE FOR SAMPLING RISK.
ALLOWANCE FOR SAMPLING RISK RELATED TO (B) AND (C) ABOVE
(SEE BOTTOM OF PAGE 75 OF INTERNAL SAMPLING REPRODUCED BELOW.)
GET THE INCREMENTAL CHANGE IN RELIABILITY FACTOR (P123)
EXAMPLE TO GO FROM 0 ERRORS TO 1 ERROR THE CHANGE IS
(4.75-3.0 = 1.75)
NO
OF
ERRORS
0
1
2
3
RELIABILITY CHANGE CHANGE
INCREMENTAL
FACTOR (RF)
IN IN RF
PROJECTED ALLOW FOR
TABLE 5
RF -1.00
ERROR
PROJECTED ERROR
3.00
4.75
6.30
7.76
1.75
1.55
1.46
.75
.55
.46
750
375
300
TOTAL INCREMENTAL ALLOWANCE FOR PROJECTED ERROR
$563
$206
$138
$907
SEE NEXT PAGE OR PAGE 74 FOR CALCULATION OF PROJECTED ERROR.
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CALCULATION OF PROJECTED ERROR
SAMPLING.
SEE ALSO P 74 OF INTERNAL AUDIT
FIRST MUST CALCULATE THE TAINTING %, THEN MULTIPLY THE TAINTING %
TIMES THE SAMPLING INTERVAL.
BOOK AUDITED
SAMPLING
PROJECTED
VALUE
VALUE
TAINTING %
INTERVAL
ERROR
$ 500
$ 400
100/500 = 20%
$ 8,000
$1,600
$ 900
$ 855
45/900 = 5%
$8,000
400
$3,000
$2,700
300/3000 = 10%
$8,000
800
$9,500
$9,050 (1)
450
TOTAL PROJECTED ERROR
$3,250
(1) BOOK VALUE GREATER THAN SAMPLING INTERVAL = NO TAINTING.
SUMMARY OF STEPS TO EVALUATE SAMPLE RESULTS IN PPS SAMPLING (P73…)
WHEN ERRORS ARE PRESENT IN THE SAMPLE:
1.
COMPUTE PROJECTED ERROR
THE PROJECTED ERROR IS DETERMINED BY MULTIPLYING THE ERROR RATE IN THE
LOGICAL UNIT TESTED (I.E. BOOK VALUE IS $1,000, AUDIT VALUE IS $200 = 20%
ERROR RATE) TIMES THE SAMPLING INTERVAL. IF SAMPLING INTERVAL IS $12,000,
THE PROJECTED ERROR FOR THE INTERVAL WILL BE $2,400 (20% X $12,000). THE
20% IS REFERRED TO AS THE TAINTING FACTOR.
NOTE THAT IF THE LOGICAL UNIT (SAMPLE ITEM) IS EQUAL TO OR LARGER THAN
THE SAMPLE INTERVAL, THE PROJECTED ERROR WILL BE EQUAL TO THE ACTUAL
AMOUNT OF ERROR FOUND IN THE LOGICAL UNIT. EXAMPLE: LOGICAL UNIT IS
$15,000 AND THE ERROR IS $2,500 (15%) THE PROJECTED ERROR OF THE INTERVAL
WILL BE $2,500. THERE WILL BE NO TAINTING FACTOR.
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2.
COMPUTE THE BASIC PRECISION AND INCREMENTAL CHANGE IN THE
RELIABILITY FACTOR WHEN MORE ERRORS ARE FOUND (E.G. GO FROM "0"
ERRORS TO "1" ERROR OR FROM "1" TO 2.)
BASIC PRECISION IS AN AMOUNT OF ASSUMED ERROR THAT IS EITHER UNKNOWN
OR UNDISCOVERED IN THE POPULATION DUE TO SAMPLING RATHER THAN
EXAMINING EVERY ITEM IN THE POPULATION. BASIC PRECISION IS COMPUTED
EVEN WHEN THERE ARE NO DETECTED ERRORS.
THIS AMOUNT MAY ALSO BE KNOW AS AN ALLOWANCE FOR SAMPLING RISK. IT IS
ADDED TO THE PROJECTED ERROR (OR MISSTATEMENT) FROM 1 ABOVE BY
MULTIPLYING THE INCREMENTAL CHANGE (DON'T DO LIKE YOUR DUMB
TEACHER AND TAKE THE RELIABILITY FACTOR FOR THE SPECIFIC ERROR. USE THE
CHANGE FROM THE LAST FACTOR) IN THE RELIABILITY FACTOR (TABLE ON PAGE
123) FOR THE RISK OF INCORRECT ACCEPTANCE (5% OR 10%). YOU DO NOT NEED
TO DETERMINE THE INCREMENTAL CHANGE WHEN ZERO ERRORS ARE DETECTED.
3.
COMPUTE THE UPPER ERROR LIMIT (ALSO KNOWN AS THE UPPER LIMIT OF
MISSTATEMENT (ULM)
THE UPPER ERROR LIMIT IS THE PROJECTED OVERSTATEMENT ERRORS IN THE
POPULATION.
THE UPPER ERROR LIMIT IS DETERMINED BY THE FOLLOWING:
THE PROJECTED ERROR
ERROR/BOOK VALUE (TAINTING %)
MULTIPLIED BY THE SAMPLE INTERVAL
PLUS
BASIC PRECISION
PLUS
INCREMENTAL ALLOWANCE FOR PROJECTED ERROR
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Extended Accounts Receivable Example:
Assume the following information:
1.
Book value of the clients account……………
$2,500,000
2.
Tolerable Misstatement……………………….
$ 125,000
3.
Expected Misstatement……………………….
$
4.
Risk of Incorrect Acceptance………………..
5%
5
Sample Size (Table Page 113)……………….
93
6
Sampling Interval……………………………..
$
25,000
26,882
$2500,000/93 = 26,882
Case 1.
Assume the auditor detects the following overstatement errors;
Book
Error # Value
800
Audit
Value
Tainting Factor
$ 680
.15
(800-680)/800
1.
$
2.
$29,000
$26,500
No tainting
BV equal or greater than SS
3.
$15,000
$ 8,250
.45
(15000-8250)/15000
Determine the (ULM) Upper Limit of Misstatement for overstate errors:
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Rank Tainting Factors from largest to smallest:
Tainting
No. Factor
Sample
Interval
3.
.45
X 26,882 =
1.
.15
X 26,882 =
2.
Projected
Misstatement
12,097 X
4,032
X
Page 124
Incremental
Change
(4.75)-(3.0)
1.75
Projected Misstatement
Plus Allowance
for Sampling
_______Risk_________
(6.3)-(4.75)
1.55
Add known error when BV>Sample Interval
$ 21,170
$ 6,250
$ 2,500
$ 29,920
Basic Precision = the assumed amount of possible unknown
And undiscovered error in the population due to Sampling (P75)
Sample Interval X Reliability Factor (p123)
$26,882
X
3.0
ULM = Upper Limit of Misstatement
$ 80,646
$110,566
ACCEPT SAMPLE BECAUSE $110, 586<$125,000 OR 5%.
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Case 2
Same as case one except the audit value of error 3 is “0” rather that $8,250.
Book
Error #
Value
3. $15,000
Audit
Value
-0-
Tainting Factor
1.0
(15000-0)/15000
Determine the (ULM) Upper Limit of Misstatement for overstate errors:
Rank Tainting Factors from largest to smallest:
Tainting
No. Factor
3.
1.
2.
1.0
.15
Sample
Interval
X 26,882 =
X 26,882 =
Projected
Misstatement
26,882 X
4,032
X
Page 124
Incremental
Change
(4.75)-(3.0)
1.75
(6.3)-(4.75)
1.55
Add known error when BV>Sample Interval
Projected Misstatement
Plus Allowance
for Sampling
_______Risk_________
$47,044
$ 6,250
$ 2,500
$55,794
Basic Precision = the assumed amount of possible unknown
And undiscovered error in the population due to Sampling (P75)
Sample Interval X Reliability Factor (p123)
$26,882
X
3.0
ULM = Upper Limit of Misstatement
$80,646
$136,440
Auditor concludes that the Accounts Receivable Account is materially misstated.
(136,440>125,000)
Options:
1. Increase sample size. (Expensive also which sample do you believe?)
2. Use other substantive tests. (subsequent collections; determine if there is a
common problem e.g. all result from an error in pricing)
3. Have client adjust account by at least $11,440 (136,440- 125,000)
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