Chapter 15 - Sampling for Tests of Transactions

Chapter 15 Audit Sampling for Tests of Transactions

Key objectives:

1. Explain the concept of representative sampling. 2. Distinguish between statistical and nonstatistical sampling. 4. Define and describe audit sampling for exception rates. 5. Use nonstatistical sampling in tests of controls and substantive tests of transactions. 6. Define and describe attribute sampling and a sampling distribution. 7. Use attribute sampling in tests of controls and substantive tests of transactions

1. Introduction

Now that we have determined the procedures to be performed, we are ready to select and evaluate a sample.

2. Objective of tests (review)

a. Tests of control - determine if control is effective for purposes of assessing control risk and determining likelihood of error in financial statements.

b. STOT - Determine existence of monetary errors in transactions for purposes of determining likelihood of material error in financial statements.

Both are tests of transactions, and are evaluated the same way, in a yes/no fashion for each transaction. (That is, the control either worked or did not; the transaction is correct or is not, without regard to the size of the monetary error). However, it is also important to consider the qualitative aspects of any errors.

Such procedures test for attributes of the population. Note that attribute sampling does not apply to certain tests, such as footing journals or observation, that do not involve testing individual transactions (see problem 15-27 (a)).

c. Evaluation of Results

TOC - decide whether we can support assessed level of control risk, leading to lower STOT and/or TODB.

STOT - assess whether transactions are correctly recorded so that TODB can be reduced. Results of STOT will also effect control risk assessment.

3. Sample selection

a. Representativeness - For tests of transactions, our goal will almost always be to select a sample that is representative of the population. The population is generally all transactions of that type for the year.

1. We insure representativeness by selecting a random sample. Today, almost everyone can easily access random numbers by computer.

2. Other forms of sampling have their uses - but infrequently for tests of transactions.

b. Relationship between selection and evaluation

Statistical evaluation Sampling must be random (probabilistic)
Nonstatistical evaluation
Judgmental selection allowed, but random selection is still preferable

Point: Random selection is always desirable.

Random - each item has an equal chance of selection. This is normally desired for TOT.

Probabilistic - each item has a known probability of being selected. This is often desired when testing balances.

4. Sampling and Nonsampling Risk

a. Sampling Risk - Risk the sample is not representative. Some sampling risk is inherent whenever testing less than the full population. The auditor can reduce by:  Using appropriate sampling (statistical methods allow the quantification of sampling risk)  Increasing sample size (why?)

b. Nonsampling risk is the risk the auditor fails to uncover existing errors in the sample. Reduce by:

1. Carefully defining and performing the test. For example:

Pricing test - one penny deviation is an error. Existence - invoice without shipper attached = error. But if client finds shipper, states it accidentally became detached, it is probably not an error (But be careful that shipper is authentic).

2. Control over sample (prevents client alteration).

Question: Is a random sample representative?

5. Sampling for Exception Rates

In sampling for tests of transactions, we are interested in the exception rate for the attribute of interest.

 The sample exception rate is the best estimator of the population exception rate (recall the concept of expectation or an unbiased estimator from statistics).

 The auditor is primarily interested in estimating the maximum rate in the population, given the sample results. This is called the Computed Upper Exception Rate (CUER), and equals the sample exception rate plus an allowance for sampling risk.

CUER = SER + ASR (allowance for sampling risk) Recall the concept of a confidence interval from statistics.

Assuming a normal distribution, the following relations exist (taller curve is for larger sample size (n)):

______________________________|______________________________

Sample exception rate CUER (SER) Given a sample exception rate, the sample size (n) and the Acceptable risk of assessing control risk too low (ARACR) determine the width of the confidence interval and associated Computed Upper Exception Rate (CUER).

We can say with the given confidence level that the true population exception rate is less than the CUER given our sample.

A. Determining Sample Size

Sample size is determined by three factors:

Variable
Description
Definition
EPER
Expected population exception rate
Last year's results or auditor expectations
TER
(precision)
Tolerable exception rate
Maximum rate of exception in population that auditor will allow and rely on result of test. Accordingly, it is lower for important objectives.
ARACR
(confidence level)
Acceptable risk of assessing control risk too low.
Risk of relying on a control/STOT when it is ineffective. Lower ARACR implies greater reliance on test, and therefore lower control risk. 1. EPER - expected population exception rate. The size of our sample will be related to the number of errors expected. This will usually be low, and can be based on the results of the test in the prior year, or auditor expectations. You would use expectations, rather than prior year results, if the controls have changed. 2. TER - Tolerable exception rate. The percentage of errors we will allow and still conclude that the test results were satisfactory. Remember, this amount is lower for important objectives that we wish to place high reliance upon.

3. ARACR - Acceptable risk of assessing control risk too low. This is the degree of confidence we desire in our results. Lower ARACR for lower levels of control risk (i.e., if we wish to rely on controls, we want to minimize the risk of overreliance).

4. Population size - Surprisingly, this is not an important factor in determining sample size.

B. Changes in variables on sample size

To understand changes in factors on sample size, it is important to understand that increases in sample size lower sampling risk.

The initial sample size is one in which (called initial sample size because if the sample exception rate exceeds EPER, a larger sample may be necessary):

TER = EPER + ASR (planning relation) Effect of Changes in Variables on Sample Size

Change in Factor
Effect
on n

Reason
EPER increase
Increase
More expected exceptions, therefore less ASR for a given TER.
TER increase
Decrease
Willing to tolerate more exceptions, therefore more allowance for sampling risk for given EPER.
ARACR increase
Decrease
Less reliance on controls; less evidence needed.

The biggest effect on sample size is TER - EPER. This should seem intuitive. If the number of expected errors is low relative to TER, you have a greater margin for error (ASR), and therefore require a smaller sample size.

Multiple choice 15-22 (a)

If all other factors remain constant, changing ARACR from 10% to 5% would cause sample size to:

1. Increase.
2. Remain the same.
3. Decrease.
4. Become indeterminate.

B. Evaluating the sample

1. TER, ARACR, and sample size are as before.

2. Sample exceptions actual exception rate in sample

SER = (# of exceptions/sample size)

Identifying errors involves considerable judgment (see problem 15-32 (a)).

It is also important to analyze the qualitative nature of individual exceptions to determine the cause of the exception and its audit implications.

Evaluation decision

Approach 1: compute CUER = SER + ASR

Accept if CUER < TER

Under this approach, the auditor adds an estimated allowance for sampling risk to computer CUER.

Approach 2: compute ASR = TER - SER

Accept if ASR is sufficiently large (SER sufficiently less than TER)

Determination whether ASR is sufficiently large is a function of auditor judgment and sample size.

C. Options when results are not acceptable

1. Increase sample size - lowers CUER, even if SER is the same, by decreasing the allowance for sampling risk.

You would elect to increase sample size if you thought your sample was not representative, and considered it important to rely upon the TOC or STOT. Decision is based on cost of increasing sample (and probability that you get good results) versus benefit in reduced substantive tests. 2. Revise CR assessment and increase other substantive tests. Logical decision if results for initial sample are not close to acceptable.

TOC fails - Can increase related STOT or TODB; usually the STOT.

STOT fails - Usually increase related TODB only.

3. Adjust TER and/or ARACR - Not advisable. You must have valid reasons for revising initial assessment.

Whatever course is followed, the auditor normally also advises the client of the exceptions found. This can be done orally, or in writing in the management letter. If the exceptions are excessive or severe in nature, they may be considered significant deficiencies.

Effect of Changes in Variables on CUER (CUER = SER +ASR)

Change in Factor
Effect
on CUER

Reason
SER increase
Increase
CUER = SER + ASR n increase
Decrease
Lower sampling risk
ARACR increase
Decrease
Less reliance on controls; threshold for acceptance is lower.

Multiple choice 15-23 (a)

An auditor estimates with 10% ARACR that CUER is between 4% and 6%. The auditor's major concern is that there is a one chance in 20 that the true exception rate is:

1. More than 6%.
2. Less than 6%.
3. More than 4%.
4. Less than 4%.

6. Statistical Attribute Sampling - Attribute sampling is a statistical method for quantifying sampling risk.

A. Sample size = f (ARACR, EPER, TER)

 see Table 15-8 on p. 504

B. CUER = f (ARACR, SER, n)

 see Table 15-9 on p. 505

Homework Problems

15-27 (a,b)
15-28 (b,c,d,e,f)
15-29 (b,c,d)

Sample Multiple Choice (Text)

15-22 (a), (b), (d)
15-23 (a), (b), (c), (d)
15-24 (a), (b), (c)
Problem 15-27 (a,b)

Audit Procedures: 1. Review cash receipts journal for large and unusual transactions. 2. Trace entries from prelisting of cash receipts to the cash receipts journal to determine each one is recorded. 3. Compare customer name, date, and amount on the prelisting with the cash receipts journal. 4. Examine related remittance advice for entries selected from the prelisting to determine if cash discounts were approved. 5. Trace entries from prelisting to the deposit slip to determine if each one has been deposited.

a. Which can be tested using attributes sampling? All except audit procedure 1, which is an analytical procedure for which the auditor is testing 100% of the cash receipts journal.

b. What is the appropriate sampling unit? The primary objective for procedure #2 is completeness, for which sampling should be done from the prelisting. The remaining procedures are non-directional tests and can be adequately performed using the prelisting.

The specific sampling unit can be the day of a prelisting, a line item on the prelisting, or a combination of both.

Problem 15-28 part b.

1
2
3
4
5
6
7
ARACR
TER
EPER
Population size

Sample size
10
6
2
1,000

88
5
6
2
100,000

127
5
5
2
6,000

181
5
6
2
1,000

127
10
20
8
500

25
10
20
2
500

18
5
2
0
1,000,000

149

part c.

Change in factor
Effect on
Sample size

Illustrated
1.
Increase in ARACR
Decrease
Compare 4 to 1
2.
Increase in TER
Decrease
Compare 3 to 2
3.
Increase in EPER
Increase
Compare 6 to 5
4.
Increase in population size
None
Slight increase if finite correction factor applies

Problem 15-28 (cont.)

d. The main difference between 3 and 6 is the large increase in TER, resulting in a much smaller sample size.

e. The greatest effect in sample size is the difference between TER and EPER. Population size has the smallest effect on sample size. The increase in ARACR had a small effect on the sample size.

f. The sample size is called the initial sample size because it is based on an estimated SER. If the actual SER is higher, the auditor will need a larger sample to achieve acceptable results. (Note the auditor may decide not select a larger sample, and elect not to rely on the test if the results for the initial sample are unacceptable).

Problem 15-29 part b.

1
2
3
4
5
6
7
8
ARACR
Population size
Sample size
Number of exceptions
CUER
10
5,000
200
4
4.0
5
5,000
200
4
4.6
5
5,000
50
1
9.2
5
50,000
200
4
4.6
5
500
100
2
6.2
5
900
100
10
16.4
5
5,000
100
0
3.0
5
500
25
0
11.3

part c.

Change in factor
Effect on CUER
Illustrated
1.
Decrease in ARACR
Increase
Compare 1 to 2
2.
Decrease in population size
No effect
Small decrease only when finite correction factor is used.
3.
Decrease in sample size
Increase
Compare 2 to 3
4.
Decrease in # of exceptions
Decrease
Compare 6 to 7

d. The sample exception rate (SER) has the largest effect on CUER (compare columns 5 and 6). Population size has the least effect.

e. The auditor compares the CUER and TER because the CUER represents the actual results of the sample, while TER represents what the auditor will allow. The results of the test are acceptable only if CUER is less than or equal to TER.