2. What is Audit Sampling?
Applying a procedure to less than 100% of a population
To estimate some characteristic of the population
Qualitative
Quantitative

3. Risk
Sampling risk
risk that the auditors’ conclusions based on a sample may be different from the conclusion they would reach if they examined every item in the population
Nonsampling risk
risk pertaining to nonsampling errors
Can be reduced to low levels through effective planning and supervisions of audit engagements

4. Nonstatistical Sampling
The auditor estimates sampling risk by using professional judgment rather than statistical techniques
Provides no means of quantifying sampling risk
Sample may be larger than necessary or auditors may unknowingly accept a higher than acceptable degree of sampling risk

5. Advantages of Statistical Sampling
Allows auditors to measure and control sampling risk which helps:
Design efficient samples
Measure sufficiency of evidence
Objectively evaluate sample results 2

6. Selection of Random Sample
Random sample results in a statistically unbiased sample that may not be a representative sample
Random sample techniques
Random number tables
Random number generators
Systematic selection 3

7. Random Number Table

8. Other Methods of Sample Selection
Haphazard selection
Select items on an arbitrary basis, but without any conscious bias
Block selection
Block sample consists of all items in a selected time period, numerical sequence or alphabetical sequence
Stratification
Technique of dividing population into relatively homogeneous subgroups

9. An Illustration of Stratification

10. Types of Statistical Sampling Plans
Attributes sampling
Discovery sampling
Classical variables sampling
Mean-per-unit estimation
Ratio estimation
Difference estimation
Probability-proportional-to-size sampling 4

11. Dual Purpose Test
Tested used both as a test of control and substantiating the dollar amount of an account balance
Ex. Test to evaluate the effectiveness of a control over recording sales transactions and to estimate the total overstatement or understatement of the sales account

12. Allowance for Sampling Risk
Amount used to create a range, set by + or – limits from the sample results, within which the true value of the population characteristic being measured is likely to lie
Precision
Wider the interval, more confident but less precise conclusion
Can be used to construct a dollar interval

13. Sample Size
Significant effect on allowance for sampling risk and sampling risk
Sample size increase -> sampling risk and allowance for sampling risk decrease
Sample size affected by characteristics of population
Generally as Population increases -> sample size increase

14. 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

15. Sampling Risks--Tests of Controls
Actual Extent of Operating Effectiveness
of the Control Procedure is
Adequate Inadequate
The Test of Controls
Sample Indicates:
Extent of Operating
Effectiveness is
Adequate
Effectiveness
Inadequate
Correct
Decision
Incorrect
Decision
(Risk of Assessing
Control Risk
Too Low)
Incorrect
Decision
(Risk of Assessing
Control Risk
Too High)
Correct
Decision 6 6

16. Audit Sampling Steps 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
Estimate the population deviation rate
Determine the sample size
Select the sample
Test the sample items
Evaluate the sample results
Document the sampling procedure 7 7

17. Attributes Sampling: Relationship Between the Planned Assessed Level of Control Risk and the Tolerable Deviation Rate

18. Illustration of Attributes Sampling--Determining Sample Size
Risk of Assessing Control Risk Too Low—5 percent
Tolerable Deviation Rate—9 percent
Expected Population Deviation Rate—2 percent

19. Figure 9.4: Statistical Sample Sizes for Tests of Controls at 5 Percent Risk of Assessing Control Risk Too Low

20. Sample Size Sample size using Figure 9-4 (next slide) =68 (2)
This means the auditor should select a sample of 68 items. We will discuss the (2) in a few slides.

21. Attributes Sampling Evaluation of Results
2 possible approaches:
Use the bracketed number from Table If you find that number or less deviations, conclude that you have accomplished your audit objective.
Use Table 9.5 for a more precise conclusion.

22. Example A--No Deviations Identified (Evaluating Attributes Sampling Results)
Approach 1—You have met your audit objective (because the bracketed number was (2), you meet objective when you identify 0, 1 or 2 deviations). What can you say?
“I believe that the deviation rate in the population is less than 9 percent.” You will be wrong 5 percent of the time when the deviation is exactly 9 percent. If the deviation rate is in excess of 9 percent you will be wrong even less than 5 percent of the time. The planned assessed level of control risk is achieved.
Approach 2
You have tested 68 items, a number not on Table 9-5 (next slide
To be conservative go to next lowest number on table (65) and use it for your conclusions (we could, but won't interpolate for a more precise answer).
You have met your audit objective. Table 9-5 gives us an answer of 4.6 percent. What can you say?
"I believe that the deviation rate in the population is less than 4.6 percent.” You will be wrong 5 percent of the time when the deviation rate is exactly 4.6 percent. If the deviation rate is in excess of 4.6 percent you will be wrong even less than 5 percent of the time. The planned assessed level of control risk is achieved.

23. Figure 9.5 Statistical Sampling Results Evaluation Table for Tests of Controls: Achieved Upper Deviation Rate at 5 Percent Risk of Assessing Control Risk Too Low

24. Example B--3 Deviations Identified (Evaluating Attributes Sampling Results)
Approach 1—You have not met your audit objective. What can you say?
“The achieved upper deviation rate is higher than 9 percent.” The planned assessed level of control risk is not achieved. You need to consider increasing the assessed level of control risk above the planned assessed level.
Accordingly, you may not “rely” on internal control to the extent planned. Thus, the auditor will need to increase the scope of substantive procedures (the nature, timing, and/or extent).
Approach 2—You have not met your audit objective. Table 9-5 provides us an answer of 11.5 percent
“I believe that the deviation rate in the population is less than 11.5 percent.” You will be wrong 5 percent of the time when the deviation rate is exactly 11.5 percent. But this is not good enough as you wanted 9 percent rather than 11.5 percent. The planned assessed level of control risk is not achieved. You need to consider increasing the assessed level of control risk above the planned assessed level.
As per Approach 1, an increase in the scope of substantive procedures is appropriate.

25. Other Statistical Attributes Sampling Approaches
Discovery sampling
Purpose is to detect at least one deviation, with a predetermined risk of assessing control risk too low if the deviation rate in population is greater than specified tolerable deviation rate
Useful in suspected fraud
Sequential (Stop-or-Go) Sampling
Audit sample taken in several stages

26. Sampling Risks--Substantive Tests
The Population Actually is
Not Materially Materially
Misstated Misstated
The Substantive
Procedure Sample
Indicates
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

27. Audit Sampling Steps 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 9 9

28. Population Variability—Why it Matters
Item Population A Population B
, ,000 ,
, ,000 , ,
Mean , ,100
Standard
deviation ,395
The variability determines how much information each of the items in the population tells you about the other items in the population. 2 11

29. Factors Affecting Sample Size

30. Mean Per Unit (MPU) Illustration
Population Size = 100,000 accounts
Book value = $6,250,000
Other information:
Tolerable misstatement = $364,000
Sampling risk
Incorrect Acceptance = 5%
Incorrect Rejection = 4.6 %

31. MPU Risk Coefficients Acceptable Level of Risk (%)
Incorrect Acceptance Coefficient
Incorrect Rejection Coefficient
1.0
2.33
2.58
4.6
1.68
2.00
5.0
1.64
1.96
10.0
1.28
15.0
1.04
1.44
20.0
.84
25.0
.67
1.15
30.0
.52
40.0
.25
50.0
.00

32. Determining Sample Size--MPU (1 of 2)

33. Determining Sample Size--MPU (2 of 2)
= Accounts

34. Variables Sampling Illustration--MPU
Adjusted allowance
for sampling risk =
Tolerable _ (Population size * Incorrect acceptance coef. * Sample stan. dev.)
misstatement Sample size
This formula “adjusts” the allowance for sampling risk to consider the standard deviation of the audited values in the sample. It holds the risk of incorrect acceptance at its planned level. 4 13

35. Variables Sampling Illustration--MPU
Using the text example with a standard deviation of audited values of $16
Adjusted allowance
for sampling risk =
Tolerable _ (Population size * Incorrect acceptance coef. * Sample stan. dev.)
misstatement Sample size
= $364, _ ($100,000 * 1.64 * $16)
225
= $189,067
We would still “accept” the book balance because the $6,250,000 (book value) falls within this interval
Estimate of total Adjusted allowance
audited value for sampling risk
$6,100, $189,067
[$5,910, to $6,289,067]

36. Acceptance Interval Figure 9-12

37. Difference Estimation
Use sample to estimate the avg. difference between the audited value and book value of items in population
Projected = Sample Net Misstatement * Pop. Items
Misstatement Sample items
Most appropriate when size of misstatements does not vary significantly in comparison to book value

38. Ratio Estimation
Use a sample to estimate the ratio of misstatement in a sample to its book value and project it to population
Projected = Sample Net Misstatement * Pop. Book Value
Misstatement Book Value of Sample
Preferred when the size of misstatements is nearly proportional to the book values of the items
Large accounts have large misstatements

39. Nonstatistical Variables Sampling Illustration
Plan Sample:
Population:
Size = 363 items
Book value = $200,000
Tolerable misstatement = $10,000
Risk assessments:
Inherent and control risk = Slightly below maximum
Other substantive tests = Moderate 8 15

40. Nonstatistical Sampling--Determination of Sample Size
Sample size = Population book value X Reliability factor
Tolerable misstatement
= $200,000 X 2.0 = 40 items
$10,000 9 16

41. Nonstatistical Sampling--Evaluation of Sample Results
40 accounts in sample
$350 net overstatement
$60,000 book value of sample items
Projected misstatement:
= [Sample net misstatement] X Book value of population
[ Book value of sample ]
= [ $350 ] X $200,000
[$60,000]
= $1,167
Since the projected misstatement is only 11.7 percent ($1,167/$10,000) of tolerable misstatement, it is likely that the auditors would conclude that the account balance is materially correct. 10 17

42. PPS Sampling Illustration
Population book value = $6,250,000
Other Information:
Tolerable misstatement = $364,000
Sampling risk--Incorrect acceptance = 5%
Expected misstatement = $50,000
Use Figures 9-14 and 9-15 to obtain a “reliability factor” and an “expansion factor”--next slide 6 18

43. PPS Sampling Reliability and Expansion Factors

44. PPS Sample Size Computation
Recorded amount of population * Reliability factor
Tolerable misstatement - (Expected misstatement * Expansion factor)
= $6,250,000 * = 66
$364,000 - ($50,000 * 1.6)
Sampling interval = Book value of the population
Sample size
= $6,250, = $95,000 (approximately) 66 7 19

45. Figure 9.16 PPS Sample Selection Process

46. PPS Evaluation of Results
Upper Limit on misstatement =
Projected misstatement
+ Basic precision (Rel. factor x interval)
+ Incremental allowance

47. Calculation of Upper Limit on Misstatement

48. Comparison of statistical sampling techniques for substantive procedures

49. Audit Risk AR = IR x CR x DR where
AR=The allowable audit risk that a material misstatement might remain undetected for the account balance and related assertions.
IR= Inherent risk, the risk of a material misstatement in an assertion, assuming there were no related controls.
CR= Control risk, the risk that a material misstatement that could occur in an assertion will not be prevented or detected on a timely basis by internal control.
DR= Detection risk, the risk that the auditors’ procedures will fail to detect a material misstatement if it exists.