1. Audit Sampling 1

2. Audit Sampling Defined
SAS No. 39 defines audit sampling as the application of an audit procedure to less than 100 percent of the items within an account balance or class of transactions for the purpose of evaluating some characteristic of the balance or class (AU ).

3. Advantages of Statistical Sampling
Design efficient samples
Measure sufficiency of evidence
Objectively evaluate sample results 2

4. Requirements of Audit Sampling Plans
When planning the sample consider:
The relationship of the sample to the relevant audit objective
Materiality or the maximum tolerable misstatement or deviation rate
Allowable sampling risk
Characteristics of the population
Select sample items in such a manner that they can be expected to be representative of the population
Sample results should be projected to the population
Items that cannot be audited should be treated as misstatements or deviations in evaluating the sample results
Nature and cause of misstatements or deviations should be evaluated 5 5

5. Selection of Random Sample
Random number tables
Random number generators
Systematic selection
Haphazard Selection
Note that these methods are often used in conjunction with a stratification process. 3

6. Terminology Sampling risk Precision (allowance for sampling risk)
Risk of assessing CR too high / Risk of incorrect rejection
Risk of assessing CR too low / Risk of incorrect acceptance
Precision (allowance for sampling risk)

7. Types of Statistical Sampling Plans
Attributes sampling
Discovery sampling
Classical variables sampling
Probability-proportional-to-size sampling 4

8. Attribute Sampling Applied To Tests Of Controls
Attribute sampling is a statistical method used to estimate the proportion of a characteristic in a population.
The auditor is normally attempting to determine the operating effectiveness of a control procedure in terms of deviations from the prescribed internal control.

9. Sampling Risk for Tests of Controls
True State of Population
Deviation Rate Deviation Rate Exceeds Is Less Than
Auditors’ Conclusion Tolerable Rate Tolerable Rate
From the Sample Is:
Deviation Rate
Exceeds
Tolerable Rate
Is Less Than
Correct
Decision
Incorrect
Decision
(Risk of Assessing
Control Risk
Too High)
Incorrect
Decision
(Risk of Assessing
Control Risk
Too Low)
Correct
Decision 6 6

10. Attribute Sampling for Tests of Controls
Determine the objective of the test
Define the attributes and deviation conditions
Define the population to be sampled
Specify:
The risk of assessing control risk too low
The tolerable deviation rate
The estimated population deviation rate
Determine the sample size
Select the sample
Test the sample items
Evaluate the sample results
Document the sampling procedure
Planning
Performance
Evaluation
Documentation 7 7

11. Discovery Sampling A modified case of attributes sampling
Purpose is to detect at least one deviation (i.e. critical deviations)
Useful in fraud detection
Auditor risk and deviation assessments:
Risk of assessing control risk too low (i.e. 5%)
Tolerable rate (normally set very low, i.e. < 2%)
Expected deviation rate is generally set at 0

12. Nonstatistical Attributes Sampling
Determination of required sample size
Must consider risk of assessing control risk too low and tolerable deviation rate
Need not quantify the risks
Evaluation of results
Compare tolerable deviation rate to sample deviation rate. Assuming appropriate n:
If SDR somewhat less than TDR, then conclude that risk of assessing control risk too low is set appropriately.
If SDR approaches TDR it becomes less likely that PDR < TDR
Must use professional judgment

13. Audit Sampling for Substantive Tests
Determine the objective of the test
Define the population and sampling unit
Choose an audit sampling technique
Determine the sample size
Select the sample
Test the sample items
Evaluate the sample results
Document the sampling procedure
Planning
Performance
Evaluation
Documentation 9 9

14. Audit Sampling for Substantive Tests Sampling Risk
True State of Population
Misstatement in Misstatement in Account Exceeds Account Is Less
Auditors’ Conclusion Tolerable Amount Than Tolerable
From the Sample Is: Amount
Misstatement in
Account Exceeds
Tolerable Amount
Account Is Less
Than Tolerable
Amount
Correct
Decision
Incorrect
Decision
(Risk of Incorrect
Rejection)
Incorrect
Decision
(Risk of Incorrect
Acceptance)
Correct
Decision 8 8

15. Risk of Incorrect Acceptance (RIA)
Modification of audit risk model:
AR = IR x CR x DR
DR comprised of two types of substantive procedures, each with an associated type of risk:
Risk associated with AP and other procedures that do not involve audit sampling (AP)
Risk associated with procedures involving audit sampling (RIA)
AR = IR x CR x AP x RIA
RIA = AR /(IR x CR x AP)

16. Classic Variables Sampling
Mean per-unit estimation
Difference and Ratio Estimation
Appropriate when differences between audited and book values are frequent
Difference estimation is most appropriate when the size of the misstatements does not vary significantly in comparison to book value
Ratio estimation is most appropriate when the size of misstatements is nearly proportional to the book values of the items.

17. Mean Per-unit (MPU) Estimation Determining the Sample Size
N = population size
Ur = incorrect rejection coefficient (Table 9-8)
SDE = estimated population standard deviation
A = planned allowance for sampling risk 10

18. Mean Per-unit (MPU) Estimation Determining the Sample Size
Standard deviation
Population SD
Sample SD 2 11

19. MPU Estimation Determining the Sample Size
Calculation of planned allowance for sampling risk (A):
TM = tolerable misstatement
Ua = Incorrect acceptance coefficient (Table 9-8)
Ur = incorrect rejection coefficient (Table 9-8)

20. MPU Estimation Adjusted Allowance for Sampling Risk
Calculation of adjusted allowance for sampling risk (A´):
TM = Tolerable misstatement
Ua = Incorrect acceptance coefficient (Table 9-8)
SDC = Sample (calculated) standard deviation
n = sample size 4 13

21. MPU Estimation = Mean audited value x Number of accounts
Estimated total audited value
= Mean audited value x Number of accounts
Acceptance interval
= Estimated total audited value +/- Adjusted allowance for sampling risk
Projected misstatement
= Estimated total audited value – Book value of population

22. Nonstatistical Variables Sampling
PBV = population book value
RF = reliability factor (based on auditors’ combined assessment of inherent and control risk and the risk that other substantive procedures will fail to detect misstatements) (Table 9-13).
TM = tolerable misstatement 8 15

23. Nonstatistical Variables Sampling
PM = projected misstatement
SNM = sample net misstatement
SBV = sample book value
PBV = population book value
Test: compare PM to TM.
Rule-of-thumb: if PM exceeds 1/3 of TM, PM “becoming too high”

24. Probability-proportional-to-size (PPS) Sampling
Applies the theory of attributes sampling to estimate the total dollar amount of misstatement in a population.
Population is defined by the individual dollars comprising the population’s book value ($1 = 1 item).
Relatively easy to use and often results in smaller sample sizes than classical variables approaches.
Assumptions underlying PPS sampling:
Expected misstatement rate in the population is small.
Amount of misstatement in physical unit should not exceed recorded BV of the item.
PPS focuses on overstatements.

25. PPS Sampling Determination of Sample Size
PBV = population book value
RF = reliability factor (Table 9-14)
TM = tolerable misstatement
EM = expected misstatement
EF = expansion factor (Table 9-15) 6 18

26. PPS Sampling Sample Selection
Systematic selection is generally used with PPS sampling:
SI = sampling interval
PBV = population book value
n = sample size 7 19

27. PPS Sampling Evaluation of Sample Results
Allowance for sampling risk
ULM = upper limit on misstatement
PM = projected misstatement
BP = basic precision
IA = incremental allowance

28. PPS Sampling Evaluation of Sample Results
Projected misstatement (PM)
If BV < SI, PM = TF x SI
TF = tainting factor = (BV – AV) / BV
BV = book value
AV = audit value
If BV > SI, PM = actual misstatement

29. PPS Sampling Evaluation of Sample Results
Allowance for sampling risk
Basic precision = SI x RF0
Incremental allowance
If no misstatements in sample found, IA = 0
If misstatements found:
For misstatements in which BV < SI, rank order projected misstatements from largest to smallest, multiply by corresponding incremental factor (from Table 9-14) and sum to calculate IA.

30. PPS Sampling Evaluation of Sample Results
Compare ULM to TM:
If ULM < TM, conclude that population is not misstated by more than TM at the specified level of sampling risk.
If ULM > TM, conclude that the sample results do not provide enough assurance that the population misstatement is less than the TM and balance adjustment may be warranted.