Transcript STATISTICAL SAMPLING FOR AUDITORS

STATISTICAL SAMPLING
FOR AUDITORS
APIPA 2009
Jeanne H. Yamamura
CPA, MIM, PHD
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OBJECTIVES
 Review of sampling concepts
 Types of sampling
 Attribute sampling
 Steps
 Nonstatistical attribute sampling
 Compliance auditing
 Monetary unit sampling
 Steps
 Nonstatistical monetary unit sampling
 Classical sampling
 Ratio estimation
 Difference estimation
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AUDIT SAMPLING
 Application of an audit procedure to less
than 100% of the items in a population
 Account balance
 Class of transactions
 Examination “on a test basis”
 Key: Sample is intended to be
representative of the population.
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SAMPLING RISK
 Possibility that the sample is NOT
representative of the population
 As a result, auditor will reach WRONG
conclusion
 Decision errors
 Type I – Risk of incorrect rejection
 Type II – Risk of incorrect acceptance
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TYPE I – RISK OF
INCORRECT REJECTION
 Internal control: Risk that sample
supports conclusion that control is NOT
operating effectively when it really is
 AKA – Risk of underreliance, risk of
assessing control risk too high
 Substantive testing: Risk that sample
supports conclusion that balance is NOT
properly stated when it really is
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TYPE II – RISK OF
INCORRECT ACCEPTANCE
 Internal control: Risk that sample
supports conclusion that control is
operating effectively when it really isn’t
 AKA – Risk of overreliance, risk of assessing
control risk too low
 Substantive testing: Risk that sample
supports conclusion that balance is
properly stated when it really isn’t
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WHICH RISK POSES THE
GREATER DANGER TO AN
AUDITOR?
 Risk of incorrect rejection
 Efficiency
 Risk of incorrect acceptance
 Effectiveness
 Auditor focus on Type II
 Also provides coverage for Type I
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NONSAMPLING RISK
 Risk of auditor error
 Sample wrong population
 Fail to detect a misstatement when applying audit
procedure
 Misinterpret audit result
 Controlled through
 Adequate training
 Proper planning
 Effective supervision
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SAMPLE SIZE FACTORS
 Desired level of assurance
(confidence level)
 Acceptable defect rate (tolerable
error)
 Historical defect rate (expected
error)
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CONFIDENCE LEVEL
 Complement of sampling risk
 5% sampling risk, 95% confidence level
 How much reliance will be placed on test
results
 The greater the reliance and the more severe
the consequences of Type II error, the higher
the confidence level needed
 Sample size increases with confidence level
(decreases with sampling risk)
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TOLERABLE ERROR AND
EXPECTED ERROR
 “Precision” – the gap between tolerable
error and expected error
 AKA Allowance for sampling risk
 Sample size increases as precision
decreases
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WHEN DO YOU SAMPLE?
 Inspection of tangible assets, e.g.,
inventory observation
 Inspection of records or documents, e.g.,
internal control testing
 Reperformance, e.g., internal control
testing
 Confirmation, e.g., verification of AR
balances
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WHEN IS SAMPLING
INAPPROPRIATE?
 Selection of all items with a particular
characteristic, e.g., all disbursements >
$100,000
 Testing only one or a few items, e.g.,
automated IT controls, walk throughs
 Analytical procedures
 Scanning
 Inquiry
 Observation
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WALKTHROUGHS
 Designed to provide evidence regarding the
design and implementation of controls
 Can provide some assurance of operating
effectiveness BUT
 Depends on nature of control (automated or
manual)
 Depends on nature of auditor’s procedures to test
control (also includes inquiry and observation
combined with strong control environment and
adequate monitoring)
 Walkthough = sample of 1
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STATISTICAL VS
NONSTATISTICAL SAMPLING
 Statistical sampling
 Statistical computation of sample size
 Statistical evaluation of results
 Nonstatistical sampling
 Sample sizes should be approximately the
same (AU 350.22)
 Sample sizes must be sufficient to support
reliance on controls and assertions being
tested
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WHEN IS SAMPLING
NONSTATISTICAL?
 If sample size determined judgmentally
 If sample selected haphazardly
 If sample results evaluated judgmentally
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TYPES OF SAMPLING
 Attribute sampling
 Monetary unit sampling
 Classical variables sampling
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ATTRIBUTE SAMPLING
 Used to estimate proportion of a
population that possesses a specific
characteristic
 Most commonly used for T of C
 Can also be used for dual purpose
testing (T of C and Substantive T of T)
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MONETARY-UNIT
SAMPLING
 AKA probability proportional to size
(PPS) sampling, cumulative monetary
unit sampling
 Used to estimate dollar amount of
misstatement
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CLASSICAL VARIABLES
SAMPLING
 Uses normal distribution theory to identify
amount of misstatement
 Useful when large number of differences
expected
 Smaller sample size than MUS
 Effective for both overstatements and
understatements
 Can easily incorporate zero balances
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IN-CLASS EXERCISE
NO. 1
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IN-CLASS EXERCISE
NO. 1
Test
Involves
Sampling?
Attribute / Variable / MUS / NA
1
2
3
4
5
Yes
No
Yes
No
No
6
No
Attribute (ST of T)
NA
Attribute (T of C)
NA
NA (Could be MUS if large
population)
NA
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IN-CLASS EXERCISE
NO. 1
Test
Involves
Sampling?
Attribute / Variable / MUS / NA
7
8
9
10
11
Yes
Yes
No
Yes
No
Attribute (T of C)
MUS
NA
Attribute (T of C/ST of T)
NA
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STEPS IN STATISTICAL
ATTRIBUTE SAMPLING
APPLICATION

Planning
1. Determine the test objectives
2. Define the population characteristics
3. Determine the sample size

Performance
4. Select sample items
5. Perform the auditing procedures

Evaluation
6. Calculate the results
7. Draw conclusions
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STEP 1: DETERMINE THE
TEST OBJECTIVES
 Objective for T of C: To determine the
operating effectiveness of the internal
control
 Support control risk assessment below
maximum
 Identify controls to be tested and
understand why they are to be tested
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TESTS OF CONTROLS
 Concerned primarily with
 Were the necessary controls performed?
 How were they performed?
 By whom were they performed?
 Appropriate when documentary evidence
of performance exists
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STEP 2: DEFINE THE
POPULATION
CHARACTERISTICS
 Define the sampling population
 Assertion
 Completeness
 Define the sampling unit
 Determined by available records
 Define the control deviation conditions
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STEP 3: DETERMINE THE
SAMPLE SIZE
 Determine factors
 Desired confidence level (direct)
 Tolerable deviation rate (inverse)
 Expected population deviation rate (direct)
 Desired confidence level
 If planning to rely on controls, would be 90 to
95%
 Significance of account and importance of
assertion affected by control being tested
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STEP 3: DETERMINE THE
SAMPLE SIZE
 Tolerable deviation rate
 Maximum deviation rate that auditor willing to
accept and still consider control effective
 Control would be relied upon
 Why any errors acceptable?
 Control deviation = Misstatement
Assessed importance of control
Tolerable
deviation rate
Highly important
3-5%
Moderately important
6-10%
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STEP 3: DETERMINE THE
SAMPLE SIZE
 Expected population deviation rate
 Rate expected to exist in population
 Based on prior years’ results or pilot sample
 If expected population deviation rate >
tolerable rate, DO NOT TEST
 SAMPLE SIZE TABLES
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STEP 3: DETERMINE THE
SAMPLE SIZE
 Testing multiple attributes on the same
sample
 Select largest sample size and audit all of
them for all attributes
 Result is some overauditing BUT may take
less time than trying to remember which
sample items need to be tested for which
attribute
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FINITE POPULATION
CORRECTION FACTOR
 When population size < 500
 Apply finite population correction factor
 √1-(n/N)
 Where n = sample size from table and N =
number of units in population
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STEP 4: SELECT THE
SAMPLE ITEMS
 Sample must be selected to be
representative of the population
 Each item must have an equal
opportunity of being selected
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STEP 4: SELECT THE
SAMPLE ITEMS
 Random number selection
 Unrestricted random sampling without
replacement (once selected cannot be
selected again)
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STEP 4: SELECT THE
SAMPLE ITEMS
 Random number table
 Need to document
 Correspondence: relationship between
population and random number table
 Route: selection path, e.g., up or down columns,
and right to left (must be consistent)
 Starting point: starting row, column, digit
 Stopping point: to enable adding more sample
items if needed
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RANDOM NUMBER TABLE
ILLUSTRATION
 Select a sample of 4 items from prenumbered
canceled checks numbered from 1 to 500.
Start at row 5, column 1, digit starting position
1. Select three-digit numbers. Items selected
are:
 145 (sample item #1)
 516 (discard because checks numbers do not
exceed 500)
 032 (sample item #2)
 246 (sample item #3)
 840 (discard)181 (sample item #4)
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RANDOM NUMBER TABLE
ILLUSTRATION
 To minimize discards, table numbers > 500 can
be reduced by 500 to produce a sample item
within the population boundary of 1 to 500. The
four sample items selected are:





145 (sample item #1)
016 (sample item #2 = 516 – 500 = 016)
032 (sample item #2)
246 (sample item #3)
340 (sample item #4 = 840 – 500 = 340)
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RANDOM NUMBER TABLE
ILLUSTRATION
 Select 4 sales invoices numbered from 5000 to
12000. Start at row 21, column 2, digit starting
point 1. Rather than use a 5-digit number,
which produces a large number of discards,
add a constant to get a population with 4 digits.
If a constant of 3000 is used, the usable
numbers selected from 2000 to 9000 are:




6,043 (sample item #1 = 3043 + 3000)
10,120 (sample item #2 = 7120 + 3000)
10,212 (sample item #3 = 7212 + 3000)
5,259 (sample item #4 = 2259 + 3000)
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STEP 4: SELECT THE
SAMPLE ITEMS - EXCEL
 Excel
 Select Tools
 Select Data Analysis
 Select Sampling
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STEP 4: SELECT THE
SAMPLE ITEMS - EXCEL
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STEP 4: SELECT THE
SAMPLE ITEMS








Input Range
Enter the references for the range of data that contains the population of
values you want to sample. Microsoft Excel draws samples from the first
column, then the second column, and so on.
Labels
Select if the first row or column of your input range contains labels. Clear
if your input range has no labels; Excel generates appropriate data labels
for the output table.
Sampling Method
Click Periodic or Random to indicate the sampling interval you want.
Period
Enter the periodic interval at which you want sampling to take place. The
period-th value in the input range and every period-th value thereafter is
copied to the output column. Sampling stops when the end of the input
range is reached.
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STEP 4: SELECT THE
SAMPLE ITEMS
 Number of Samples
 Enter the number of random values you want in the
output column. Each value is drawn from a random
position in the input range, and any number can be
selected more than once.
 Output Range
 Enter the reference for the upper-left cell of the output
table. Data is written in a single column below the cell.
If you select Periodic, the number of values in the
output table is equal to the number of values in the
input range, divided by the sampling rate. If you select
Random, the number of values in the output table is
equal to the number of samples.
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STEP 4: SELECT THE
SAMPLE ITEMS
 Systematic selection
 Determine sampling interval = Population / Sample
Size
 Ensure population is in random order
 Select random starting number (within first interval)
 Better to use multiple random starting points to reduce risk
of missing systematic deviations
 Select every nth item
 Continue sample selection until population is
exhausted
 (Last sample selected + sampling interval) > Last item in
population
 In other words, don’t stop when desired sample size
reached
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STEP 5: PERFORM THE
AUDITING PROCEDURES
 Conduct planned audit procedures
 What if?
 Voided documents - if properly voided, not a
deviation; replace with new sample item
 Unused or inapplicable documents – replace
with new sample item
 Inability to examine sample item – deviation
 Stopping test before completion – large
number of deviations detected
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STEP 5: PERFORM THE
AUDITING PROCEDURES
 Deviations observed
 Investigate nature, cause, and consequence
of every exception
 Unintentional error? Or fraud?
 Monetary misstatement resulted?
 Cause – misunderstanding of instructions?
Carelessness?
 Effect on other areas?
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STEP 6: CALCULATE
RESULTS
 Summarize deviations for each control
 Calculate sample deviation rate and
computed upper deviation rate
 Sample deviation rate + Allowance for
sampling risk = Computed upper deviation
rate
 Statistical sampling results evaluation tables
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STEP 7: DRAW
CONCLUSIONS
 If Computed Upper Deviation Rate >
Tolerable Rate, control is ineffective and
cannot be relied upon.
 If Computed Upper Deviation Rate <
Tolerable Rate, control is effective
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EVALUATION OF
EXPOSURE
 In a sample of 25 manual control
operations from a population of 3,000
control operations, 1 deviation was
identified. The sample was designed with
an expectation that 0 deviations would be
found.
 Looking up the results (in 90%
confidence level table): Computed upper
error limit = 14.7%
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EVALUATION OF
EXPOSURE
 The sample did not meet its design criteria, so there is
a higher than desired risk that the control will fail to
prevent or detect a misstatement.
 To assess the magnitude of the exposure:
 Identify the gross exposure of the account or
process. This is based on the volume of dollars
processed through the control.
 The upper limit on the control deviations was 14.7%.
 The adjusted exposure is $735,000 (14.7% *
$5,000,000).
 The $735,000 exposure may assist the auditor in
evaluating the severity of the control deficiency.
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IN-CLASS EXERCISES
NO. 2 & NO. 3
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IN-CLASS EXERCISE
NO. 2
Problem 1: Prenumbered sales invoices where the
lowest invoice number is 1 and the highest is 6211.
Sampling unit
Sales invoice
Population numbering
system
1 to 6211
Random number table
correspondence
Use 4 digits with random
start at 0029-05 going down
and then right
First 5 items in sample
3553 0081 4429 0484
4881
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IN-CLASS EXERCISE
NO. 2
Problem 2: Prenumbered bills of lading where the
lowest document number is 21926 and the highest is
28511.
Sampling unit
Bill of lading
Population numbering
system
21926 to 28511
Random number table
correspondence
Use last 4 digits with
random start at 0005-07
First 5 items in sample
7744 7632 8120 3736
4091
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IN-CLASS EXERCISE
NO. 2
Problem 3: Accounts Receivable on 10 pages with 60 lines per
page except the last page, which has only 36 full lines. Each line
has a customer name and an amount receivable.
Sampling unit
Population numbering
system
Random number table
correspondence
First 5 items in sample
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Each line
9 * 60 = 540 + 36 = 576
lines
Add 2000 (2001 to 2576)
Use last 4 digits with
random start at 00040-01
going down and then right
2240 2055 2094 2087
2608
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IN-CLASS EXERCISE
NO. 2
Problem 4: Prenumbered invoices in a sales journal where each month
starts over with number 1. (Invoices for each month are designated by
the month and document number.) There is a maximum of 20 pages
per month with a total of 185 pages for the year. All pages have 75
invoices except for the last page for each month.
Sampling unit
Page of invoices
Population numbering system
Starting with January, first page is
1 (up to 185)
Random number table
correspondence
Random start at 0008-03 going
down then right, subtract random
number from next 1000
First 5 items in sample
4000 – 3982 = 18; 7000 – 6847 =
153; 5000 - 4956 = 44; 6000 –
5985 = 15; 5000 – 4941 = 59
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IN-CLASS EXERCISE
NO. 3
For which of these auditing procedures can
attribute sampling be conveniently used?
1
2
3
4
5a
5b
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No
No
No
Yes
Yes
Yes
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IN-CLASS EXERCISE
NO. 3
For which of these auditing procedures can
attribute sampling be conveniently used?
5c
5d
5e
6
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Yes
Yes
Yes
Yes
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IN-CLASS EXERCISE
NO. 3
2. Considering the audit procedures to be
performed, what is the most appropriate
sampling unit for conducting most of the audit
sampling tests?
Sales invoice
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IN-CLASS EXERCISE
NO. 3
For each T of C or ST of T, identify the attribute
being tested and the exception condition.
Attribute
Exception Condition
4. Existence of the sales No record of the sales
invoice number in the
invoice number in the
sales journal
sales journal
5a. Amount and other
The amount recorded in
data in MF agree with
the MF differs from the
the sales journal entry
amount recorded in the
sales journal.
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IN-CLASS EXERCISE
NO. 3
For each T of C or ST of T, identify the attribute
being tested and the exception condition.
Attribute
Exception Condition
5b. Amount and other
Customer name and
data on the duplicate
account number on the
sales invoice agree with invoice differ from the
the sales journal entry
information recorded in
the sales journal
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IN-CLASS EXERCISE
NO. 3
For each T of C or ST of T, identify the attribute
being tested and the exception condition.
Attribute
Exception Condition
5b. Evidence that
Lack of initials indicating
pricing, extensions, and verification of pricing,
footings are checked
extensions, and footings.
(initials and correct
amounts).
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IN-CLASS EXERCISE
NO. 3
For each T of C or ST of T, identify the attribute
being tested and the exception condition.
Attribute
Exception Condition
5c. Quantity and other
Quantity of goods shipped
data on the bill of lading differs from quantity on
agree with the duplicate sales invoice
sales invoice and sales
journal
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IN-CLASS EXERCISE
NO. 3
For each T of C or ST of T, identify the attribute
being tested and the exception condition.
Attribute
Exception Condition
5d. Quantity and other
Quantity on the sales
data on the sales order
order differs from quantity
agree with the duplicate on the duplicate sales
sales invoice
invoice
5e. Quantity and other
Product number and
data on the customer
description on the
order agree with the
customer order differ from
duplicate sales invoice
information on the
duplicate sales invoice
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IN-CLASS EXERCISE
NO. 3
For each T of C or ST of T, identify the attribute
being tested and the exception condition.
Attribute
Exception Condition
5e. Credit is approved
Lack of initials indicating
credit approval
6. For recorded sales in BL is not attached to the
the sales journal, the file duplicate sales invoice
of supporting documents and the customer order.
includes a duplicate
sales invoice, BL, sales
order, and customer
order.
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IN-CLASS EXERCISE
NO. 3
See Solution
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STEPS IN NONSTATISTICAL
ATTRIBUTE SAMPLING
APPLICATION

Planning
1. Determine the test objectives
2. Define the population characteristics
3. Determine the sample size

Performance
4. Select sample items
5. Perform the auditing procedures

Evaluation
6. Calculate the results
7. Draw conclusions
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STEP 3: DETERMINE THE
SAMPLE SIZE
 Consider desired confidence level,
tolerable deviation rate, and expected
population deviation rate
 Judgmentally determine sample size
 NOTE: Check against statistical sample
size tables to verify adequacy
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STEP 3: DETERMINE THE
SAMPLE SIZE
 Guidelines for nonstatistical sample sizes for tests
of controls
 If any errors found, increase sample size or
increase control risk
Desired level of controls reliance
Sample size
Low
15-20
Moderate
25-35
High
40-60
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STEP 4: SELECT SAMPLE
ITEMS
 Random sample
 Systematic sample (with random start)
 Haphazard selection
 Still desire representative sample
 Avoid unusual, large, first or last
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STEP 6: CALCULATE THE
RESULTS
 No computed upper deviation rate
 If sample deviation rate > expected
population deviation rate, control not
effective
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COMPLIANCE AUDITING
 Performance of auditing procedures to
determine whether an entity is complying with
specific requirements of laws, regulations, or
agreements
 Governmental entities and other recipients of
governmental financial assistance
 Compliance with laws and regulations that
materially affect each major federal assistance
program
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COMPLIANCE AUDITING OF
FEDERAL ASSISTANCE
PROGRAMS
 Definition of population for testing of an
internal control procedure that applies to
more than one program
 Define items from each major program as a
separate population, OR
 Define all items to which control is applicable
as a single population
 Second choice usually more efficient
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COMPLIANCE AUDITING EXAMPLE
 Federal financial assistance for Island
City
 Three major federal financial assistance
programs
 Four nonmajor programs
 Control: Transaction review to ensure
that only legally allowable costs are
charged to each program
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COMPLIANCE AUDITING EXAMPLE
 More efficient to select one sample from
population of all transactions (major and
nonmajor programs)
 Confidence level = 95%
 Tolerable deviation rate = 9%
 Expected population deviation rate = 1%
 Sample size: 51
 1 allowable deviation
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SMALL POPULATIONS AND
INFREQUENTLY OPERATING
CONTROLS
Small Population Sample Size Table
Control Frequency and
Population Size
Sample Size
Quarterly (4)
2
Monthly (12)
2-4
Semimonthly (24)
3-8
Weekly (52)
5-9
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IN-CLASS EXERCISE
NO. 4
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IN-CLASS EXERCISE
NO. 4
Selected Payroll T of C
1. Examine the time card
for approval of a
supervisor
2. Account for a
sequence of payroll
checks in the payroll
journal
3. Recompute hours on
the time card
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Moderately critical –
affects E/O of S& W
Very critical – affects E/O
of S&W
Moderately critical –
affects V of S&W
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IN-CLASS EXERCISE
NO. 4
4. Compare the
employee name in the
payroll journal to
personnel records
5. Review OT charges
for approval of a
supervisor
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Very critical – affects
E/O - affects E/O of S&
W; also an area subject
to fraud
Moderately critical –
affects E/O and V of
S&W
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IN-CLASS EXERCISE
NO. 4
Selected Cash Disbursement T of C
6. Examine voucher for
supporting invoices,
receiving reports, etc.
7. Examine supporting
documents for evidence
of cancellation (“paid”)
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Very critical – affects
E/O of purchase
transactions
Moderately critical –
affects validity of
purchase transactions
and relates to double
payment
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IN-CLASS EXERCISE
NO. 4
Selected Cash Disbursement T of C
8. Ascertain whether
cash discounts were
taken
9. Review voucher for
clerical accuracy
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Least critical – affects V
of purchase
transactions; amounts
usually minor
Moderately critical –
affects V of purchase
transactions
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IN-CLASS EXERCISE
NO. 4
Selected Cash Disbursement T of C
10. Agree purchase
order price to invoice
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Moderately critical –
affects V of purchase
transactions
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MONETARY UNIT
SAMPLING
 Uses attribute sampling theory to express
conclusions in dollar amounts
 Estimates the percentage of monetary units in a
population that might be misstated
 Multiples the percentage by an estimate of how
much the dollars are misstated
 Developed by auditors
 Assumes little or no misstatements
 Designed primarily to test for overstatements
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ADVANTAGES
 When no misstatements expected,
results in smaller (more efficient) sample
size than classical variables sampling
 No need to compute/identify standard
deviation
 Automatically stratifies sample
APIPA 2009
82
DISADVANTAGES
 Zero or negative balances must be tested
separately
 Assumes audited amount of sample
items is not in error by more than 100%
 When more than 1 or 2 misstatements
found, allowance for sampling risk may
be overstated
 Auditor more likely to reject balance and
overaudit
APIPA 2009
83
STEPS IN MONETARY UNIT
SAMPLING APPLICATION

Planning
1. Determine the test objectives
2. Define the population characteristics
3. Determine the sample size

Performance
4. Select sample items
5. Perform the auditing procedures

Evaluation
6. Calculate the results
7. Draw conclusions
APIPA 2009
84
STEP 1: DETERMINE THE
TEST OBJECTIVES
 Substantive testing: To test the
reasonableness of an amount, i.e., that
an amount is fairly stated
 To test the assertion that no material
misstatements exist in an account
balance, class of transactions, or
disclosure component of the financial
statements
APIPA 2009
85
STEP 2: DEFINE THE
POPULATION
CHARACTERISTICS
 Define the sampling population
 Monetary value of an account balance
 Verify completeness of population
 Define the sampling unit - Each individual
dollar
 Define the logical unit - The account or
transaction that contains the sampling units
 Define a misstatement – The difference
between the book value and the audited value
APIPA 2009
86
STEP 3: DETERMINE THE
SAMPLE SIZE
 Determine factors (effect on sample size)
 Desired confidence level (direct)
 To increase confidence, more work is required!
(larger sample size)
 Tolerable misstatement (inverse)
 Expected misstatement (direct)
 Population size (direct)
APIPA 2009
87
STEP 3: DETERMINE THE
SAMPLE SIZE
 Computing sample sizes using the
attribute sampling tables
 Select desired confidence level
 Compute tolerable misstatement as
percentage of book value
 Compute expected misstatement as
percentage of book value
 Look up sample size in attribute sampling
table
APIPA 2009
88
STEP 4: SELECT THE
SAMPLE ITEMS
 Systematic selection approach called
probability proportional to size (PPS)
 Calculate sampling interval
 Book value / sample size
 From random start (within first interval),
select every nth dollar
 Logical unit included only once even if
includes more than one sample unit
APIPA 2009
89
STEP 5: PERFORM THE
AUDITING PROCEDURES
 Conduct planned audit procedures on
logical units
 What if?
 Missing document – consider to be a
misstatement
APIPA 2009
90
STEP 6: CALCULATE
RESULTS
 Projected misstatement: Projection of
the errors to the population
 Upper limit on misstatement: Adds an
allowance for sampling risk to the
projected misstatement
APIPA 2009
91
STEP 6: CALCULATE
RESULTS
 Sort misstatements into two groups
 Group 1: Logical unit equal to or greater
than the sampling interval
 Group 2: Logical unit less than the sampling
interval
 For Group 2, compute the tainting factor
for each misstatement
 Tainting factor = Book value – Audit value
Book value
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92
STEP 6: CALCULATE
RESULTS
 Place the Group 2 items in rank order by
tainting factor (from largest to smallest)
 Compute the projected misstatement
 Calculate the upper limit increments (using the
Monetary Unit Sampling – Confidence
Factors for Sample Evaluation table)
 Calculate upper misstatement for each Group
2 item
 Add differences for Group 1
 Total = Upper misstatement limit
APIPA 2009
93
STEP 6: CALCULATE
RESULTS - EXAMPLE





Book value = $3,100,000
Tolerable misstatement = $150,000
Expected misstatement = $25,000
Desired confidence level = 95%
Tolerable misstatement rate =
4.8%,round to 5%
 Expected misstatement rate = .8%, round
to 1%
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94
STEP 6: CALCULATE
RESULTS - EXAMPLE
 Sample size = 93
 Sampling interval = $33,333
 Expected misstatement = $25,000
APIPA 2009
95
STEP 6: CALCULATE
RESULTS - EXAMPLE
Item
Book Value
Audited Value
Difference
Item 1
12,000
3,120
8,880
Item 2
35,000
32,000
3,000
Item 3
1,400
0
1,400
Item 4
45,200
41,000
4,200
Item 5
740
555
185
APIPA 2009
96
STEP 6: CALCULATE
RESULTS - EXAMPLE
Item
Book Value
Audited Value
Difference
Group 1: BV > SI (33,333)
Item 2
35,000
32,000
3,000
Item 4
45,200
41,000
4,200
7,200
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97
STEP 6: CALCULATE
RESULTS - EXAMPLE
Item
Difference
Book Value
Tainting Factor
Group 2: BV < SI (33,333)
Item 1
8,880
12,000
.74
Item 3
1,400
1,400
1.0
Item 5
185
740
.25
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98
STEP 6: CALCULATE
RESULTS - EXAMPLE
Item
Tainting Factor
Sampling
Interval
Projected
Misstatement
(Tainting
Factor * SI)
Item 3
1.0
33,333
33,333
Item 1
.74
33,333
24,666
Item 5
.25
33,333
8,333
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99
STEP 6: CALCULATE
RESULTS - EXAMPLE
Item
Projected
Misstatement
95% Upper
Limit
Increment
Upper
Misstatement
Item 3
33,333
3.0
99,999
Item 1
24,666
1.7
41,932
Item 5
8,333
1.5
12,500
154,431
APIPA 2009
100
STEP 6: CALCULATE
RESULTS - EXAMPLE
Item
Projected
Misstatement
95% Upper
Limit
Increment
Upper
Misstatement
Group 2
154,431
Group 1
7,200
Upper Misstatement Limit
APIPA 2009
161,631
101
STEP 7: DRAW
CONCLUSIONS
 If Upper Misstatement Limit > Tolerable
Misstatement, balance is materially
misstated.
 If Upper Misstatement Limit > Tolerable
Misstatement, balance is not materially
misstated
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102
IN-CLASS EXERCISES
NO. 5 TO NO. 6
APIPA 2009
103
IN-CLASS EXERCISE
NO. 5
1. Sampling interval:
746,237 / 10 =
74,624
APIPA 2009
Loan #
Recorded
Amount
1
141,100
3
66,600
5
10,230
11
4,350
20
16,530
24
2,950
26
131,200
27
50,370
32
5,900
104
IN-CLASS EXERCISE
NO. 5
2. Sampling items always included:
The loans > the sampling interval
Loan #1 – 141,100
Loan #26 – 131,200
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105
IN-CLASS EXERCISE
NO. 6
Recorded amount of accounts receivable =
$400,000

Tolerable misstatement: $20,000; 20,000 /
400,000 = 5%
Risk of incorrect acceptance: 5%
Expected misstatements: 0




Sample size = 59
Sampling interval = 400,000 / 59 = 6,780
APIPA 2009
106
IN-CLASS EXERCISE
NO. 6
Error
Recorded
Amount
Audit
Amount
1
400
320
80
20%
2
500
0
500
100%
3
7,000
6,500
500
NA
APIPA 2009
Difference Tainting
%
107
IN-CLASS EXERCISE
NO. 6
Error
Tainting % Sampling
Interval
Projected
Misstatement
Upper
Limit
Increment
Upper
Limit
Misstatement
Logical unit BV < Sampling Interval
2
100
6,780
6,780
1.7
11,526
1
20
6,780
1,356
1.5
2,034
APIPA 2009
108
IN-CLASS EXERCISE
NO. 6
Error
Tainting % Sampling
Interval
Projected
Misstatement
Upper
Limit
Increment
Upper
Limit
Misstatement
Logical unit BV > Sampling Interval
3
NA
NA
500
NA
500
Basic Precision: 3.0 * 6,780 = 20,340
APIPA 2009
109
IN-CLASS EXERCISE
NO. 6
Error
Tainting % Sampling
Interval
Projected
Misstatement
Upper
Limit
Increment
Upper
Limit
Misstatement
Logical unit BV < Sampling Interval
13,560
Logical unit BV > Sampling Interval
500
Basic Precision
20,340
Upper Misstatement Limit
34,400
Conclusion: The account is materially misstated. The upper misstatement
limit of 34,400 exceeds the tolerable misstatement of 20,000.
APIPA 2009
110
NONSTATISTICAL SAMPLING
– BALANCE TESTING
 Differences in




Identifying individually significant items
Determining sample size
Selecting sample items
Calculating sample results
APIPA 2009
111
IDENTIFYING INDIVIDUALLY
SIGNIFICANT ITEMS




Selected due to large size
Tested 100%
Results similar to PPS selection
For example, selecting all items >
$100,000
APIPA 2009
112
DETERMINING SAMPLE
SIZE
 Sample size =
Sampling Population BV * Assurance
(Tolerable – Expected
Factor
Misstatement)
where Sampling Population BV excludes
individually significant items
APIPA 2009
113
DETERMINING SAMPLE
SIZE
Desired Level of Confidence – Assurance Factors
Assessment
of RMM
Maximum
Slightly below
maximum
Moderate
Low
Maximum
3.0
2.7
2.3
2.0
Slightly below
maximum
2.7
2.4
2.0
1.6
Moderate
2.3
2.1
1.6
1.2
Low
2.0
1.6
1.2
1.0
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114
DETERMINING SAMPLE
SIZE - EXAMPLE







Book value = $3,100,000
Individually significant items = $1,500,000
Tolerable misstatement = $150,000
Expected misstatement = $25,000
Desired confidence level = Maximum
Risk of MM = Maximum
Sample size = 1,600,000
* 3.0
(150,000 – 25,000)
= 38.4, round to 39
APIPA 2009
115
SELECTING SAMPLE
ITEMS
 Random selection
 Systematic selection
 Haphazard selection
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116
CALCULATING SAMPLE
RESULTS
 Sample misstatement MUST be
projected to population
 Two acceptable methods
 Apply sample misstatement ratio to
population (ratio estimation)
 Apply average misstatement $ of each item
in sample to all items in population
(difference estimation)
APIPA 2009
117
CLASSICAL SAMPLING
 Ratio estimation
 Difference estimation
APIPA 2009
118
RATIO ESTIMATION





Sample misstatements = $19,000
Sample book value = $175,000
Sample error rate = 10.9%, round to 11%
Total population BV = $1,840,000
Projected misstatement = $1,840,000 *
11% = $202,400
 Compare projected misstatement to
tolerable misstatement
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119
DIFFERENCE ESTIMATION
 Sample misstatements = $19,000
 # of sample items with misstatements = 5
 Average misstatement per sample item =
$3,800
 # items in population = 256
 Projected misstatement = $3,800 * 256 =
$972,800
 Compare projected misstatement to tolerable
misstatement
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120
IN-CLASS EXERCISE
NO. 7
APIPA 2009
121
IN-CLASS EXERCISE
NO. 7
Nonstatistical Sample Results:
 Errors in accounts > $10,000
 Errors in accounts < $10,000:

Total errors
$ 4,350

Sample BV
$81,500

Error rate
5.34%

Applied to population:

2,760,000

(465,000)

2,295,000 * 5.34%
 Total estimated error
 Tolerable misstatement
 Conclusion: Account materially misstated
APIPA 2009
33,000
122,553
155,553
81,500
122
IN-CLASS EXERCISE
NO. 7 - PPS
PPS Sample Results:
 Accounts receivable recorded
balance:
$2,760,000
 Accounts > $10,000 (tested
separately)
(465,000)
 Accounts receivable population
– PPS
$2,295,000
 Tolerable misstatement
$ 81,500
APIPA 2009
123
IN-CLASS EXERCISE
NO. 7 - PPS
Sample and sampling interval:
Tolerable rate: 81,500 / 2,295,000 = 3.55%,
round to 4%
Expected rate: 0
5% risk of overreliance (since IR and CR are
both high)
Sample size: 74
Sampling interval: 2,295,000 / 74 = 31,014
APIPA 2009
124
IN-CLASS EXERCISE
NO. 7 - PPS
Recorded
Value
Audited
Value
Difference
Tainting %
Item 12
5,120
4,820
300
5.85
Item 19
485
385
100
20.6
Item 33
1,250
250
1,000
80
Item 35
3,975
3,875
100
25.2
Item 51
1,850
1,825
25
1.4
Item 59
4,200
3,780
420
10
Item 74
2,405
0
2,405
100
APIPA 2009
125
IN-CLASS EXERCISE
NO. 7 - PPS
# of Overstatement
Misstatements
APIPA 2009
5%
Upper Limit Increment
0
3.00
1
4.75
1.75
2
6.30
1.55
3
7.76
1.46
4
9.16
1.40
5
10.52
1.36
6
11.85
1.33
7
13.15
1.30
126
IN-CLASS EXERCISE
NO. 7 - PPS
Tainting
%
Sampling
Interval
Projected
Misstatement
Upper Limit
Factor
Upper
Misstatement
Item 74
100
31,014
31,014
1.75
54,275
Item 33
80
31,014
24,811
1.55
38,457
Item 35
25.2
31,014
7,816
1.46
11,411
Item 19
20.6
31,014
6,389
1.40
8,944
Item 59
10
31,014
3,101
1.36
4,218
Item 12
5.85
31,014
1,814
1.33
2,413
Item 51
1.4
31,014
434
1.30
564
120,282
APIPA 2009
127
IN-CLASS EXERCISE
NO. 7 - PPS





Items < Sampling Interval:
120,282
Items > Sampling Interval: None
Basic precision: 3.0 * 31,014 =
93,042
Upper misstatement limit =
213,324
Conclusion: Account is materially misstated.
Upper misstatement limit 213,324 > tolerable
misstatement 81,500
APIPA 2009
128
RESOURCES
 Audit Sampling: An Introduction, 3rd
Edition, Guy, Carmichael & Whittington
 Audit Guide: Audit Sampling, New Edition
as of May 1, 2008, AICPA
 Auditing & Assurance Services, 6th
Edition, Messier, Glover, & Prawitt
 Auditing & Assurance Services, 12th
Edition, Arens, Elder & Beasley
APIPA 2009
129
THE END!
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130