Audit Sampling Concepts

CAS 530

Importance of Sampling

• Auditor does not look at everything– How does this affect the opinion?

• Auditor CANNOT look at everything– Why?

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Purpose of Sampling

• The auditor examines only a portion of the population in order to estimate

• How much is a portion?

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When to Do SamplingWhen:

1. The nature and materiality of the balance or class of transactions does not demand a 100% audit

2. A decision must be made about the balance or class of transactions.

3. The time and cost to audit 100% of the population would be too great

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When is Sampling Used?

To conduct:1. Walk through tests

2. For tests of controls

3. Tests of details

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Representative Sampling

• Having a representative sample is important

• What does representative mean?

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• Non-sampling risk:

• Sampling risk:

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Statistical Sampling

• Uses the laws of probability for selecting and evaluating a sample from a population

• Selected at random

• Statistical calculations are used

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Statistical vs. Non-Statistical

Similarities• Both require a

structured process

• The use of stratification

Differences• Sampling risk cannot be

quantified

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Non-Probabilistic Sample Selection Methods

• Directed sample selection

– When used?• Auditor often able to identify items likely to contain errors

• Items containing selected characteristics

• Large dollar item coverage

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• Block sample selection• A selection of several items in a sequence

• Reasonable number of blocks must be chosen

• Haphazard sample selection• Auditor goes through the population and haphazardly selects items

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Probabilistic Sample Selection Methods

• Sampling risk requires

• Simple random sample selection• Every member of the population has an equal chance of being

selected

• Systematic sample selection– Auditor calculates an interval and use the interval to select sample

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– Major problem with systematic sampling is bias• Once the first item is chosen

• No problem if

• But some characteristics

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Probability Proportionate-to-Size Sampling Methodology

• A key statistical methodology– also known as

– the sampling unit is

– MUS allows the result to be stated

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Attribute Sampling Methodology

• Another key statistical methodology– very useful for tests of controls

– The main question to be answered is

– If the auditor can allow 5% deviations

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Advantages of Statistical Sampling

Provides:

• for quantitative evaluation of the sample results

• a more defensible expression of the test results

• It is more objective

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Disadvantages of Statistical Sampling

• Requires random sample selection which may be more costly and time consuming.

• Might require additional training costs for staff members to use statistics or specialized software

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Advantages of non-statistical sampling

• Allows the auditor to inject his or her subjective judgment in determining the sample size

• May be designed so that it is equally effective and efficient as statistical sampling while being less costly

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Disadvantages of non-statistical sampling

• Cannot draw objectively valid statistical inferences from the sample results

• Cannot quantitatively measure and express sampling risk.

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The Main Phases of the Sampling Process

Both statistical and non-statistical methods

1. Planning the sample

2. Selecting the sample

3. Performing the tests

4. Evaluating the results

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Sampling Process

• Fourteen steps in the sampling process.

• Look at tests of controls versus tests of details

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1. State the Objectives of the Test

Test of control:• Are the controls working as

specified?

• Are there monetary errors or fraud or other irregularities

Test of detail:• Auditor wants to determine the

maximum amount of overstatement and understatement that could exist based on the sample

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2. Decide if Audit Sampling Applies

• Test of control:• some controls can be

sampled

• others cannot be

• Test of detail:• sampling test of details

depends on the nature of the population

• high volume can be

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3. Define attributes and exception or error conditions

Planning term: Test of control

Test of detail

Define the item of interest

Identify the characteristic or attribute of interest

Individual dollars

Define exceptions or errors

Define the control deviation or exception

Normally, any monetary difference or error

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4. Define the population

• Population can be defined in a way to suit the audit tests

• Must sample from the entire population as defined

• In testing controls over sales, what is the population?

• In testing details in accounts receivable it is the recorded dollar population

• Most populations can be stratified, if needed.

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5. Define the sampling unit

Tests of controls:• Usually a physical unit

Test of detail:• If MUS

• If non-statistical sampling

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6. Specify tolerable exception rate (TER) or specify materiality

• Test of control• TER`

• As TER increases

• Test of detail• Materiality is used

• These decisions require the use of

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7. Specify ARACR or ARIA

• Test of control• What is ARACR?• Acceptable Risk of Assessing

Control Risk Too Low

• Test of detail• What is ARIA?• Acceptable Risk of Incorrect

Acceptance

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• Test of control• Assume– TER 6%

– ARACR 10%

– But unknown to the auditor the true error rate is 8%

• Test of detail• If ARIA changes from

10% to 5%– Since assurance required

increases

– When controls are good

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8. Estimate population exception rate or misstatements

• Test of control• Estimated population error rate

(EPER)

• The lower the EPER, the smaller the sample size

• Test of detail• Provide an advance estimate

of the total dollar error, i.e. misstatements, in the population

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9. Determine the initial sample size

• For non-statistical or judgmental sampling, professional judgment is used to calculate the sample size

• For statistical sampling, mathematical formulae are used, either in specially prepared tables or using software designed for audit sampling

• For stratified sampling, the sample is allocated among the strata

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10. Select the sample

• Using the number of items determined in Step #9, choose the items from the population using the sampling unit defined in Step #5

• Use probabilistic or non-probabilistic methods

• To enable quantification of sampling risk, probabilistic, i.e. statistical, methods must be used

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11. Perform the audit procedures

• For test of controls, examine each item for the attribute defined in Step #3, recording all exceptions found

• For test of details, apply the audit procedures designed in the audit program

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12. Generalize from the sample to the population

• Test of controls sample error rate (SER)

• But that is not necessarily equal to the actual population rate

• In practice, auditors tend to test controls when they expect no exceptions

• But ultimately, the method of generalization depends on the sampling methodology used

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• When generalizing tests of details, auditors deal with

• Misstatements found are projected from the sample results to the population

• Auditor must calculate a point estimate

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• To calculate the point estimate:– (Client Misstatement / Recorded Value of Sample) x Recorded Book Value of the Population

– Thus for a misstatement of $500 in A/R with a sample value of $10,000 and a total book value of $25,000

– Note that if the population is divided into strata

– The total point estimate may not be an adequate result for the population

– The auditor must consider this fact

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Example of Point Estimate Calculation

StratumClient Misstatement /Recorded Value from Sample

xRecorded Book Value for Stratum

= Point Estimate of Misstatement

1 $(2,740)/$88,955 $88,955 $(2,470)

2 971/43,995 71,235 1,572

3 2,158/13,105 47,105 7,757

Total $6,589

Example of Errors Found

Dollars Audited

Stratum Sample Size

Book Value of Stratum

Recorded Value

Audited Value

Client Misstatement

1 3 $88,955 $88,955 $91,695 $(2,740)

2 6 71,235 43,995 43,024 971

3 6 47,105 13,105 10,947 2,158

Total 15 $207,295 $146,055 $145,666 $389

Calculating Point Estimate for a population

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13. Analyze exceptions or misstatements

• Test of control• What breakdown in internal

controls caused the exceptions?

• Should additional substantive testing be conducted because of these results?

• Test of detail• Were the misstatements

caused by control exceptions?

• Is additional substantive testing required?

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14. Decide the acceptability of the population

• Test of control• If TER is sufficiently

larger than SER

• If TER – SER is too small

• Test of detail• Compare the difference

between the projection to the population

• If projection is greater than materiality level

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• What if the auditor decides the population is NOT acceptable? What to do?– 1. Revise TER (tolerable error rate), ARACR, or ARIA (the

risks of accepting incorrect populations)

– 2. Expand the sample size.

– 3. Revise assessed control risk.

– 4. Report weaknesses in management letter.

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Problem 11-22, p. 357• For the examination of the financial statements of Scotia Inc., Rosa Schellenberg, a public accountant, has

decided to apply non-statistical audit sampling in the tests of sales transactions. Based on her knowledge of Scotia’s operations in the area of sales, she decides that the estimated population deviation rate is likely to be 3 percent and that she is willing to accept a 5 percent risk the true population rate is not greater than 6 percent. Given this information, Rosa selects a random sample of 150 sales invoices from the 5,000 prepared during the year and examines them for exceptions. She notes the following exceptions in her working papers. There is no other documentation.

• REQUIREDa. Which of the invoices in the table should be defined as an exception?b. Explain why it is inappropriate to set a single acceptable TER and EPER for the combined exceptions.c. State the appropriate analysis of exceptions for each of the exceptions in the sample.

Invoice No:

Comment

5028 Sales invoice had incorrect price, but a subsequent credit not was sent out as a correction.

6791 Voided sales invoice examined by auditor.

6810 Shipping document for a sale of merchandise could not be located.

7364 Sales invoice for $2,875 has not been collected and is six months past due.

7625 Client unable to locate the printed duplicate copy of the sales invoice.

8431 Invoice was dated three days later than the date of the shipping document.

8528 Customer purchase order is not attached to the duplicate sales invoice.

8566 Billing is for $100 less than it should be due to a pricing error.

8780 Client unable to locate the printed duplicate copy of the sales invoice.

9169 Credit is not authorized, but the sale was only for $7.65.Sampling-41

Problem 11-25, page 358

You have just completed the accounts receivable confirmation process in the audit of Danforth paper Company Ltd., a paper supplier to retail shops and commercial users.

Following are the data related to the process:

Accounts receivable recorded balance $2,760,000Number of accounts 7,320

A non-statistical sample was taken as follows:All accounts over $10,000 (23 accounts) $465,00077 accounts under $10,000 $81,500Materiality $100,000Inherent and control risk are both highNo relevant analytical procedures were performed.

The table below gives the results of the confirmation proceduresREQUIREDEvaluate the results of the non-statistical sample. Consider both the direct implications of the misstatements found and the effect of using a sample.

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Recorded Value

Audited Value

Items over $10,000 $465,000 $432,000

Items under $10,000 81,500 77,150

Individual misstatements for items under $10,000

Item 12 5,120 4,820

Item 19 485 385

Item 33 1,250 250

Item 35 3,975 3,875

Item 51 1,850 1,825

Item 59 4,200 3,780

Item 74 2,405 0

19,285 14,935

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Problem 11-23 (p.357)

You have been asked to do planning for statistical testing of the audit of cash receipts. Following is a partial audit program for the audit of cash receipts.1.Review the cash receipts journal for large and unusual transactions.2.Trace entries from the prelisting of cash receipts to the cash receipts journal to determine whether each is recorded.3.Compare customer name, date, and amount on the prelisting with the cash receipts journal.4.Examine the related remittance advice for entries selected from the prelisting to determine whether cash discounts were approved.5.Trace entries from the prelisting to the deposit slip to determine whether each has been deposited.REQUIRED

a. Identify which audit procedures can be tested using attribute sampling. Justify your response.b. State the appropriate sampling unit for each of the tests in part (a).c. Define the attributes that you would test for each of the tests in part (a). State the audit object associated

with each of the attributes.d. Define exception conditions for each of the attributes that you have described in part (c).e. Which of the exceptions would be indicative of potential fraud? Justify your response.

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Lenter Supply Corp. is a medium sized distributor of wholesale hardware supplies in southern Manitoba. It has been a client of yours for several years and has instituted excellent internal control for the control of sales, at your recommendation.

In providing control over shipments, the client has prenumbered “warehouse removal slips” that are used for every sale. It is company policy never to remove goods from the warehouse without an authorized warehouse removal slip. After shipment, two copies of the warehouse removal slip are sent to billing for the computerized preparation of a sales invoice. One copy is stapled to the duplicate copy of the prenumbered sales invoice, and the other copy is filed numerically. In some cases more than one warehouse removal slip is used for billing one sales invoice. The smallest warehouse removal slip number for the year is 14682 and the largest is 37521. The smallest invoice number is 47821 and the largest is 68507.

In the audit of sales, one of the major concerns is the effectiveness of the control in making sure all shipments are billed. The auditor has decided to use attribute sampling in testing internal control.

Problem 12-24, page 404

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(a) State an effective audit procedure for testing whether shipments have been billed. What is the sampling unit for the audit procedure?

(b) Assuming the auditor expects no deviations in the sample but is willing to accept a TDR of 3%, at a 10% ARACR, what is the appropriate sample size?

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EXPECTED POPULATION DEVIATION RATE (IN PERCENTAGE)

TOLERABLE DEVIATION RATE

(IN PERCENTAGE)

2 3 4 5 6 7 8 9 10 15 20

5 PERCENT RISK OF OVER RELIANCE (ARACR)

0.00 149 99 74 59 49 42 36 32 29 19 14

0.25 236 157 117 93 78 66 58 51 46 30 22

0.50 . 157 117 93 78 66 58 51 46 30 22

0.75 . 208 117 93 78 66 58 51 46 30 22

1.00 . . 156 93 78 66 58 51 46 30 22

1.25 . . 156 124 78 66 58 51 46 30 22

1.50 . . 192 124 103 66 58 51 46 30 22

1.75 . . 227 153 103 88 77 51 46 30 22

2.00 . . . 181 127 88 77 68 46 30 22

2.25 . . . 208 127 88 77 68 61 30 22

2.50 . . . . 150 109 77 68 61 30 22

2.75 . . . . 173 109 95 68 61 30 22

3.00 . . . . 195 129 95 84 61 30 22

3.25 . . . . . 148 112 84 61 30 22

3.50 . . . . . 167 112 84 76 40 22

3.75 . . . . . 185 129 100 76 40 22

4.00 . . . . . . 146 100 89 40 22

5.00 . . . . . . . 158 116 40 30

6.00 . . . . . . . . 179 50 30

7.00 . . . . . . . . . 68 37

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2 3 4 5 6 7 8 9 10 15 2010 PERCENT RISK OF OVER RELIANCE (ARACR)

0.00 114 76 57 45 38 32 28 25 22 15 11

0.25 194 129 96 77 64 55 48 42 38 25 18

0.50 194 129 96 77 64 55 48 42 38 25 18

0.75 265 129 96 77 64 55 48 42 38 25 18

1.00 . 176 96 77 64 55 48 42 38 25 18

1.25 . 221 132 77 64 55 48 42 38 25 18

1.50 . . 132 105 64 55 48 42 38 25 18

1.75 . . 166 105 88 55 48 42 38 25 18

2.00 . . 198 132 88 75 48 42 38 25 18

2.25 . . . 132 88 75 65 42 38 25 18

2.50 . . . 158 110 75 65 58 38 25 18

2.75 . . . 209 132 94 65 58 52 25 18

3.00 . . . . 132 94 65 58 52 25 18

3.25 . . . . 153 113 82 58 52 25 18

3.50 . . . . 194 113 82 73 52 25 18

3.75 . . . . . 131 98 73 52 25 18

4.00 . . . . . 149 98 73 65 25 18

4.50 . . . . . 218 130 87 65 34 18

5.00 . . . . . . 160 115 78 34 18

5.50 . . . . . . . 142 103 34 18

6.00 . . . . . . . 182 116 45 25

7.00 . . . . . . . . 199 52 25

7.50 . . . . . . . . . 52 25

8.00 . . . . . . . . . 60 25

8.50 . . . . . . . . . 68 32

EPDR

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Effect of population size

-Initial sample size only

-Possible to make adjustment to initial sample size based on overall population size

-Finite correction factor

n = revised sample size

n’ = initial sample size

N = population size

n = n’

1 + n’/N

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From the problem 12-24

Population is

n’ =

Thus revised sample size is

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c) Use of a random number table

– A one-to-one correspondence between warehouse removal slip

– How is this correspondence achieved?

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37039 97547 64673 31546 99314 66854 97855

25145 84834 23009 51584 66754 77785 5235798433 54725 18864 65866 76918 78825 5821097965 68548 81545 82933 93545 85959 6328278049 67830 14624 17563 25697 07734 4824350203 25658 91478 08509 23308 48130 65047

40059 67825 18934 64998 49807 71126 7781884350 67241 54031 34535 04093 35062 5816330954 51637 91500 48722 60988 60029 6087386723 36464 98305 08009 00666 29255 1851450188 22554 86160 92250 14021 65859 16237

50014 00463 13906 35936 71761 95755 8700266023 21428 14742 94874 23308 58533 2650704458 61862 63119 09541 01715 87901 9126057510 36314 30452 09712 37714 95482 3050743373 58939 95848 28288 60341 52174 11879

61500 12763 64433 02268 57905 72347 4949878938 71312 99705 71546 42274 23915 3840564257 93218 35793 43671 64055 88729 1116856864 21554 70445 24841 04779 56774 9612935314 29631 06937 54545 04470 75463 77112

40704 48823 65963 39359 12717 56201 2281107318 44623 02843 33299 59872 86774 0692694550 23299 45557 07923 75126 00808 0131234348 81191 21027 77087 10909 03676 9772392277 57115 50789 68111 75305 53289 39751

56093 58302 52236 65756 50273 61566 6196216623 17849 96701 94971 94758 08845 3226050848 93982 66451 32143 05441 10399 1777548006 58200 58367 66577 68583 21108 4136156640 27890 28825 96509 21363 53657 60119

Random Number

Table

Random Stab

Population of Warehouse Removal Slips

14,682 – 37,521

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Computed Upper Exception Rate

• Sample size =• TER = • ARACR = • Number of deviations =

• Using the following tables:

• CUER =

• Are the controls working?

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SAMPLE SIZEACTUAL NUMBER OF DEVIATIONS FOUND

0 1 2 3 4 5 6 7 8 9 10

5 PERCENT RISK OF OVER RELIANCE

25 11.3 17.6 . . . . . . . . .

30 9.5 14.9 19.5 . . . . . . . .

35 8.2 12.9 16.9 . . . . . . . .

40 7.2 11.3 14.9 18.3 . . . . . . .

45 6.4 10.1 13.3 16.3 19.2 . . . . . .

50 5.8 9.1 12.1 14.8 17.4 19.9 . . . . .

55 5.3 8.3 11.0 13.5 15.9 18.1 . . . . .

60 4.9 7.7 10.1 12.4 14.6 16.7 18.8 . . . .

65 4.5 7.1 9.4 11.5 13.5 15.5 17.4 19.3 . . .

70 4.2 6.6 8.7 10.7 12.6 14.4 16.2 18.0 19.7 . .

75 3.9 6.2 8.2 10.0 11.8 13.5 15.2 16.9 18.4 20.0 .

80 3.7 5.8 7.7 9.4 11.1 12.7 14.3 15.8 17.3 18.8 .

90 3.3 5.2 6.8 8.4 9.9 11.3 12.7 14.1 15.5 16.8 18.1

100 3.0 4.7 6.2 7.6 8.9 10.2 11.5 12.7 14.0 15.2 16.4

125 2.4 3.7 4.9 6.1 7.2 8.2 9.3 10.3 11.3 12.2 13.2

150 2.0 3.1 4.1 5.1 6.0 6.9 7.7 8.6 9.4 10.2 11.0

200 1.5 2.3 3.1 3.8 4.5 5.2 5.8 6.5 7.1 7.7 8.3

Table 14-10

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Sample size ACTUAL NUMBER OF DEVIATIONS FOUND

0 1 2 3 4 5 6 7 8 9 10

10 PERCENT RISK OF OVER RELIANCE

20 10.9 18.1 . . . . . . . . .

25 8.8 14.7 19.9 . . . . . . . .

30 7.4 12.4 16.8 . . . . . . . .

35 6.4 10.7 14.5 18.1 . . . . . . .

40 5.6 9.4 12.8 15.9 19.0 . . . . . .

45 5.0 8.4 11.4 14.2 17.0 19.6 . . . . .

50 4.5 7.6 10.3 12.9 15.4 17.8 . . . . .

55 4.1 6.9 9.4 11.7 14.0 16.2 18.4 . . . .

60 3.8 6.3 8.6 10.8 12.9 14.9 16.9 18.8 . . .

70 3.2 5.4 7.4 9.3 11.1 12.8 14.6 16.2 17.9 19.5 .

80 2.8 4.8 6.5 8.3 9.7 11.3 12.8 14.3 15.7 17.2 18.6

90 2.5 4.3 5.8 7.3 8.7 10.1 11.4 12.7 14.0 15.3 16.6

100 2.3 3.8 5.2 6.6 7.8 9.1 10.3 11.5 12.7 13.8 15.0

120 1.9 3.2 4.4 5.5 6.6 7.6 8.6 9.6 10.6 11.6 12.5

160 1.4 2.4 3.3 4.1 4.9 5.7 6.5 7.2 8.0 8.7 9.5

200 1.1 1.9 2.6 3.3 4.0 4.6 5.2 5.8 6.4 7.0 7.6

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Problem 13-21, Page 439, Canadian 11th. Edition

a. For each of the following independent problems, design an unbiased random sampling plan using an electronic spreadsheet or a random number generator. The plan should include defining the sampling unit and establishing a numbering system for the population. After the plan has been designed, select the sample using the computer. Assume that the sample size is 50 for each od (1) thorough (4).

1. Prenumbered sales invoices in a sales journal where the lowest number is 1 and the highest is 6,211.

2. Prenumbered bills of lading where the lowest document number is 21,926 and the highest is 28,511

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.

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 the 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.

b. Using systematic sampling, select the first five sample items for population (1).

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Problem 11-21, Page 356, Canadian 12th. Edition

Lam, PA, is auditing the financial statements of his client, Harvesters Ltd., a company that sells and distributes agricultural equipment across Canada. Lam has performed a preliminary evaluation of the company’s internal control over sales transactions, and has concluded that the quality of system design is very good. The system was developed for the client and installed by a well-respected consulting firm, and the system relies heavily on automated information systems. Lam decides that performing tests of control using computer-assisted audit techniques would likely be cost-effective. In addition, after completing his assessment of control risk over revenue transactions, Lam plans to use monetary-unit sampling to verify the client’s recorded accounts receivable at year end. In planning the engagement, Lam has assessed materiality to be $175,000.

Required:Explain the basic principles of sample selection for monetary unit (dollar unit) sampling.Also discuss how computer-assisted audit techniques could be used to assist in sample selection, assuming that the population of year-end accounts receivable is available to Lam as a data file compatible with his software.Assume that the client’s recorded accounts receivable total $2,000,000 at year end and that Lam examines a valid random sample of 50 dollar units, and finds two errors as follows:

Account Number 26751 Account Number 87523

Recorded Amount $20,000 $10,000

Amount conformed by customer

$10,000 Nil

Both errors were caused by the client’s failure to record equipment returned by customers, where the equipment was deemed to be defective. The client agrees with the customer’s decision in both cases.

What further action is required on the part of the auditor with respect to these errors?

(Extract from CGA Canada Examinations.)