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Audit Sampling: An Application to

Substantive Tests of Account Balances

Chapter Nine


Substantive Tests of Details of Account Balances

The statistical concepts we discussed in the last chapter apply to this chapter as well. Three important determinants of sample size are:

1. Desired confidence level.

2. Tolerable misstatement.

3. Expected misstatement.

Population plays a bigger role in some of the sampling techniques used for substantive testing.

Misstatements discovered in the audit sample must be projected to the population, and there must be an allowance for sampling risk.



Consider the following information about the inventory account balance of an audit client:

The ratio of misstatement in the sample is 2%(€2,000 ÷ €100,000)

Applying the ratio to the entire population produces a best

estimate of misstatement of inventory of €60,000.(€3,000,000 × 2%)

Book value of inventory account balance 3,000,000€ Book value of items sampled 100,000€

Audited value of items sampled 98,000 Total amount of overstatement observed in audit sample 2,000€



The results of our audit test depend upon the tolerable misstatement

associated with the inventory account. If the tolerable misstatement is

€50,000, we cannot conclude that the account is fairly stated because our

best estimate of the projected misstatement is greater than the

tolerable misstatement.


Monetary-Unit Sampling (MUS)

MUS uses attribute-sampling theory to express a conclusion in monetary amounts (e.g. in euros or other currency) rather than as a rate of occurrence. It is commonly used

by auditors to test accounts such as accounts receivable, loans receivable, investment

securities and inventory.



MUS uses attribute-sampling theory (used primarily to test controls) to

estimate the percentage of monetary units in a population that might be misstated and then multiplies this percentage by an estimate of how

much the euros are misstated.



Advantages of MUS1. When the auditor expects no misstatement,

MUS usually results in a smaller sample size than classical variables sampling.

2. The calculation of the sample size and evaluation of the sample results are not based on the variation between items in the population.

3. When applied using the probability-proportional-to-size procedure, MUS automatically results in a stratified sample.



Disadvantages of MUS1. The selection of zero or negative balances

generally requires special design consideration.

2. The general approach to MUS assumes that the audited amount of the sample item is not in error by more than 100%.

3. When more than one or two misstatements are detected, the sample results calculations may overstate the allowance for sampling risk.


Steps in MUS Sampling

Steps in MUS Sampling ApplicationPlanning1. Determine the test objectives.2. Define the population characteristics. • Define the population. • Define the sample unit. • Define a misstatement.3. Determine the sample size, using the following inputs: • The desired confidence level or risk of incorrect acceptance. • The tolerable misstatement. • The expected population misstatement. • Population size.Performance4. Select sample items.5. Perform the auditing procedures.Evaluation6. Calculate the projected misstatement and the upper limit on misstatement.7. Draw final conclusions.


Steps in MUS SamplingSteps in MUS Sampling Application

Planning1. Determine the test objectives.2. Define the population characteristics. • Define the population. • Define the sample unit. • Define a misstatement.

Sampling may be used for substantive testing to:

1. Test the reasonableness of assertions about a financial statement amount (i.e. is the amount fairly stated). This is the most common use of sampling for substantive testing.

2. Develop an estimate of some amount.




For MUS the population is defined as the monetary value of an account balance,

such as accounts receivable, investment securities or inventory.




An individual euro represents the sampling unit.




A misstatement is defined as the difference between monetary amounts in the client’s records and amounts supported by audit

evidence.



3. Determine the sample size, using the following inputs: • The desired confidence level or risk of incorrect acceptance. • The tolerable misstatement. • The expected population misstatement. • Population size.

Factor Relationship

to Sample Size Change in Factor

Effect on Sample

Lower DecreaseHigher IncreaseLower IncreaseHigher DecreaseLower DecreaseHigher IncreaseLower DecreaseHigher Increase

Desired confidence level

Tolerable mistatement

Expected mistatement

Population size

Direct

Inverse

Direct

Direct



Performance4. Select sample items.5. Perform the auditing procedures.Evaluation6. Calculate the projected misstatement and the upper limit on misstatement7. Draw final conclusions.

The auditor selects a sample for MUS by using a systematic selection approach called probability-

proportionate-to-size selection. The sampling interval can be determined by dividing the book

value of the population by the sample size. Each individual euro in the population has an equal chance of being selected and items or ‘logical units’ greater than the interval will always be

selected.


Steps in MUS SamplingAssume a client’s book value of accounts receivable is €2,500,000, and the auditor determined a sample size of 93. The sampling interval will be €26,882 (€2,500,000 ÷ 93). The random number selected is €3,977 the

auditor would select the following items for testing:Cumulative Sample

Account Balance Euros Item1001 Ace Emergency Centre 2,350€ 2,350€

1002 Admington Hospital 15,495 17,845 3,977€ (1)1003 Jess Base 945 18,780

1004 Good Hospital Corp. 21,893 40,673 30,859 (2)1005 Jen Mara Corp. 3,968 44,641

1006 Axa Corp. 32,549 77,190 57,741 (3)1007 Green River Mfg. 2,246 79,436

1008 Bead Hospital Centres 11,860 91,306 84,623 (4)• • • •• • • •

1213 Andrew Call Medical - 2,472,032 1214 Lilly Health 26,945 2,498,977 2,477,121 (93)

1215 Janyne Ann Corp. 1,023 2,500,000€ Total Accounts Receivable 2,500,000€

3,977€ 26,882 30,859€



Steps in MUS Sampling ApplicationPerformance4. Select sample items.5. Perform the auditing procedures.Evaluation6. Calculate the projected misstatement and the upper limit on misstatement.7. Draw final conclusions.

After the sample items have been selected, the auditor conducts the planned audit procedures on the

logical units containing the selected euro sampling units.



Evaluation6. Calculate the projected misstatement and the upper limit on misstatement.7. Draw final conclusions.

The misstatements detected in the sample must be projected to the

population. Let’s look at the following example:

Book value 2,500,000€ Tolerable misstatement 125,000€ Sample size 93 Desired confidence level 95%Expected amount of misstatement 25,000€ Sampling interval 26,882€

Example Information



Basic Precision using the Table If no misstatements are found in the sample, the

best estimate of the population misstatement would be zero euros.

€€26,882 26,882 × 3.0 = × 3.0 = €80,646€80,646 upper misstatement limit upper misstatement limit



Misstatements DetectedIn the sample of 93 items the following misstatements

were found:

€€3,284 3,284 ÷ €21,893 = 15%÷ €21,893 = 15%Because the Axa balance of €32,549 is greater than the interval of €26,882, no sampling risk is added. Since all the euros in the large accounts are audited, there is no

sampling risk associated with large accounts.

Customer Book Value Audit Value Difference Tainting Factor

Good Hospital 21,893€ 18,609€ 3,284€ 15%Marva Medical Supply 6,705 4,023 2,682 40%Axa Corp. 32,549 30,049 2,500 NA

Learn Heart Centres 15,000 - 15,000 100%



Compute the Upper Misstatement LimitWe compute the upper misstatement limit by calculating basic precision and ranking the detected misstatements

based on the size of the tainting factor from the largest to the smallest.

(0.15 (0.15 × €26,882 × €26,882 × 1.4 = €5,645)× 1.4 = €5,645)

Customer

Tainting Factor

Sample Interval

Projected Misstatement

95% Upper Limit

Upper Misstatement

Basic Precision 1.00 26,882€ NA 3.0 80,646€ Learn Heart Centres 1.00 26,882 26,882 1.7 (4.7 - 3.0) 45,700 Marva Medical 0.40 26,882 10,753 1.5 (6.2 - 4.7) 16,130 Good Hospital 0.15 26,882 4,032 1.4 (7.6 - 6.2) 5,645

Add misstatments greaterthat the sampling interval:

Axa Corp. NA 26,882 NA 2,500 Upper Misstatement Limit 150,621€



Evaluation6. Calculate the projected misstatement and the upper limit on misstatement.7. Draw final conclusions.

We compare the tolerable misstatement to the upper misstatement limit. If the upper misstatement limit is less than or equal to the tolerable misstatement, we

conclude that the balance is not materially misstated.

In our example, the final decision is whether the accounts receivable balance

is materially misstated or not.


Steps in MUS SamplingIn our example the upper misstatement limit of €150,621 is

greater than the tolerable misstatement of €125,000, so the auditor concludes that the accounts receivable balance

is materially misstated.

When faced with this situation, the auditor may:

1. Increase the sample size.

2. Perform other substantive procedures.

3. Request the client adjust the accounts receivable balance.

4. If the client refuses to adjust the account balance, the auditor would consider issuing a qualified or adverse opinion.


Risk When Evaluating Account Balances

Auditor's Decision Basedon Sample Evidence Not Materially Misstated Materially Misstated

Supports the fairness of the account balance Correct decision Risk of incorrect

acceptance (Type II) Does not support the

fairness of the account balance

Risk of incorrect rejection (Type I)

Correct Decision

True State of Financial Statement Account


Effect of Understatement Misstatements

MUS is not particularly effective at detecting understatements. An understated account is less likely to be selected than an overstated account.

The most likely error will be reduced by €2,688The most likely error will be reduced by €2,688((– 0.10 × €26,882)– 0.10 × €26,882)

Customer Book Value

Audit Value Difference

Tainting Factor

Wayne County Medical 2,000€ 2,200€ (200)€ -10%


Non-Statistical Sampling for Tests of Account Balances

The sampling unit for non-statistical sampling is normally a customer account, an individual

transaction, or a line item on a transaction. When using non-statistical sampling, the following items

must be considered:• Identifying individually significant items.Identifying individually significant items.

• Determining the sample size.Determining the sample size.

• Selecting sample items.Selecting sample items.

• Calculating the sample results.Calculating the sample results.


Identifying Individually Significant Items

The items to be tested individually are items that may contain potential misstatements that

individually exceed the tolerable misstatement. These items are tested 100% because the

auditor is not willing to accept any sampling risk.


Determining the Sample Size and Selecting the Sample

SampleSize =

Sampling Population book valueTolerable – Expected misstatement × Confidence

factor

Auditing standards require that the sample items be selected in such a way that the sample can be expected to represent the population.


Calculating the Sample Results

One way of projecting the sampling results to the population is to apply the misstatement ratio in the

sample to the population. This approach is known as ratio projection.

If the population total is €200,000, the projected misstatement would be €20,000 (€200,000 × 10%)

Assume the Assume the auditor finds auditor finds €1,500 in €1,500 in misstatements in misstatements in a sample of a sample of €15,000. The €15,000. The misstatement misstatement ratio is 10%.ratio is 10%.


Calculating the Sample Results

A second method is the difference projection. This method projects the average misstatement of each

item in the sample to all items in the population.

The projected misstatement would be €30,000 (€3 × 10,000).

Assume Assume misstatements in a misstatements in a sample of 100 items sample of 100 items total €300 (for an total €300 (for an average average misstatement of €3), misstatement of €3), and the population and the population contains 10,000 contains 10,000 items.items.


Non-Statistical Sampling Example

The auditor’s of Calabro Wireless Service have decided to use non-statistical sampling to examine the accounts receivable balance. Calabro has a total of 11,800 (15 + 250 + 11,535) accounts with a balance of €3,717,900.

The auditor’s stratify the accounts as follows:



The auditor decides . . .•Based on the results of the tests of controls, the risk of material misstatement is assessed as low.

•The tolerable misstatement is €55,000, and the expected misstatement is €15,000.

•The desired level of confidence is moderate based on the other audit evidence already gathered.

•All customer account balances greater than €25,000 are to be audited.



× Confidence factor

SampleSize =

Sampling population book valueTolerable - Expected misstatement

SampleSize =

€3,167,900€40,000 × 1.2 = 95 (rounded)

€€3,717,900 3,717,900 – €550,000– €550,000

€€55,000 55,000 – €15,000– €15,000



The auditor sent positive confirmations to each of the 110 (95 + 15) accounts selected. Either the

confirmations were returned or alternative procedures were successfully used. Four customers indicated that

their accounts were overstated and the auditors determined that the misstatements were the result of unintentional error by client personnel. Here are the

results of the audit testing:

Amount ofBook Value Audit Value Over-

Stratum Book Value of Sample of Sample Statement>€25,000 550,000€ 550,000€ 549,500€ 500€ >€3,000 850,500 425,000 423,000 2,000 <€3,000 2,317,400 92,000 91,750 250



As a result of the audit procedures, the following projected misstatement was prepared:

The total projected misstatement of €10,800 is less than the expected misstatement of €15,000, so the

auditors may conclude that there is a low risk that the true misstatement exceeds the tolerable

misstatement.

Amount of ProjectedStratum Misstatement Misstatement

>€25,000 500€ 500€ >€3,000 2,000 4,002 <€3,000 250 6,298

Total projected misstatement 10,800€ €250 ÷ 92,000 × €2,317,400

Ratio of Misstatementin Stratum Tested

100%€2,000 ÷ 425,000 × €850,500


Why Did Statistical Sampling Fall Out Of Favour?

1.Firms found that some auditors were over relying on statistical sampling techniques to the exclusion of good judgement.

2.There appears to be poor linkage between the applied audit setting and traditional statistical sampling applications.


Classical Variables Sampling

Classical variables sampling uses normal distribution theory to evaluate the characteristics of a population based on sample data. Auditors most commonly use classical variables sampling to estimate the size of

misstatement.

Sampling distributions are formed by plotting the Sampling distributions are formed by plotting the projected misstatements yielded by an infinite projected misstatements yielded by an infinite

number of audit samples of the same size taken number of audit samples of the same size taken from the same underlying population.from the same underlying population.



A sampling distribution is useful because it

allows us to estimate the probability of

observing any single sample result.

In classical variables sampling, the sample

mean is the best estimate of the

population mean.



Advantages1. When the auditor expects a relatively large

number of differences between book and audited values, this method will normally result in smaller sample size than MUS.

2. The techniques are effective for both overstatements and understatements.

3. The selection of zero balances generally does not require special sample design considerations.



Disadvantages1. Does not work well when little or no misstatement is

expected in the population.

2. To determine sample size, the auditor must estimate the standard deviation of the audit differences.

3. If few misstatements are detected in the sample data, the true variance tends to be underestimated, and the resulting projection of the misstatements and the related confidence limits are not likely to be reliable.


Applying Classical Variables Sampling

Defining the Sampling UnitThe sampling unit can be a customer account, an individual transaction, or a line item. In auditing accounts receivable, the auditor can define the

sampling unit to be a customer’s account balance or an individual sales invoice included in the

account balance.



Determining the Sample Size

whereCC = Confidence coefficientSD = Estimated standard deviation of audit differences.

SampleSize =

Population size (in sampling units) × CC × SDTolerable misstatement – Estimated

misstatement

2



The Confidence Coefficient (CC) is associated with the desired level of confidence. The desired level of

confidence is the complement of the risk that the auditor will mistakenly accept a population as fairly stated when the true population misstatement is greater than tolerable

misstatement. Desired Level of

Confidence CC Value95% 1.96 90% 1.65 80% 1.28 70% 1.04



The year-end balance for accounts receivable contains 5,500 accounts with a book value of

€5,500,000. The tolerable misstatement for accounts receivable is set at €50,000. The expected

misstatement has been judged to be €20,000. The desired confidence is 95%. Based on work

completed last year, the auditor estimates the standard deviation at €31.

Let’s calculate sample size.SampleSize

5,500 × 1.96 × €31€50,000 – €20,000

2= = 125



Calculating the Sample ResultsThe sample selection usually relies on random-selection techniques. Upon completion, 30 of the customer accounts selected contained misstatements that totalled €330.20. Our first calculation is the mean misstatement in an individual account which is calculated as follows:

Meanmisstatementper sampling

item

=Total audit difference

Sample size

€330.20125

=€2.65



The mean misstatement must be projected to the population

€14,575 = 5,500 × €2.65

Population size × Mean misstatementper sampling item

Projectedpopulation

misstatement=

(in sampling units)



The formula for the standard deviation is . . .

SD =

Total squaredaudit differences –

Mean differenceper sampling item2

SampleSize ×

Sample size – 1

= €36,018.32 – (125 × 2.652)125 – 1

= €16.83



Confidencebound

Populationsize CC

SD

Sample size× ×=

= 5,500 × 1.96 × €16.83125√

Confidenceinterval

Projectedmisstatement

Confidencebound±=

= €14,575 ± €16,228

€16,228



If both limits are within the bounds of tolerable misstatement, the evidence supports the conclusion

that the account is not materially misstated.

(€50,000) €50,000

Lowerlimit

(€1,653)

Projectedmisstatement

€14,575

Upperlimit

€30,803

€0

Tolerable Misstatement


End of Chapter 9