Statistical Sampling

FE/JA

Statistical Sampling

1

A.

B.

C. Attribute Sampling

1. Tests of controls

2. Compliance testing

D. Variable Sampling

1. Estimate dollar values

2. Substantive testing

E. Guiding Concepts

1. Normal Curve: Assume population has properties of normal curve

• Central Limit Theorem: Arithmatic means of all possible samples, ifplotted, would form a bell-shaped curve.

2. Randomness: Must use unrestricted random sample

3. Variability

a. Causes uncertainty

b. Leads to larger sample size

c. Measured by standard deviation

Independently answer multiple choice question numbers 1-3

USE OF SAMPLING

Third Standard of Field Work: “Sufficient competent evidential matter is to be obtained through inspection, observation, inquiries, and confirmations to afford a reasonable basis for an opinion regarding the financial statements under audit.”

Audit Risk: The culmination of all risks taken in auditing. Statistical sampling allows the auditor to quantify risk.

1.

2.

3.

Inherent Risk: The risk that material misstatements are occurring in the financial statements.

Control Risk: The risk that internal controls do not detect material misstatements in the financial statements.

Detection Risk: The risk that the will fail to detect materialmisstatements in the financial statements.

auditor

--- Internal control

Bell-shaped curve

2

Statistical Sampling

VARIABLE SAMPLING

A. Mean-Per-Unit Projection

Example 1: Mean-Per-Unit Point Estimate

Population size: 2,000 items Sample size: 100 Audit value of sample: $25,000 Standard deviation: $100 Standard error of the mean: $10

Step 1 : Audit value $25,000 Step 2: Mean-per-unit $ 250Sample size / 100 Units x 2,000Mean-per-unit $ 250 Point estimate $500,000

B. Tolerable Misstatement & Precision

Exhibit 1: 68% Confidence/Reliability

1 standarddeviation

1 standarddeviation

Point estimate

Bell-shaped curve

Statistical Sampling

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Exhibit 2: 95% Confidence/Reliability

1. Precision

Example 2: Precision @ 68% Confidence/Reliability

Standard deviation 1Standard error of the mean x$ 10Subtotal $ 10Population x 2,000Achieved precision ±$20,000

Example 3: Precision @ 95% Confidence/Reliability

Standard deviation 1.96Standard error of the mean x$ 10Subtotal $ 19.6Population x 2,000Achieved precision ±$39,200

2. Tolerable misstatement

Notes:

1.96 standarddeviations

1.96 standarddeviations

Upper precision limitLower precision limit

4

Statistical Sampling

SAMPLING RISK

A. Sampling Risk

1. Probability that the sample is wrong

2. At 95% confidence, 5% sampling risk

3. At 99% confidence, 1% sampling risk

B. Non-Sampling Risk

1. Auditor uses wrong audit procedures

2. Auditor improperly evaluates evidence

Notes:

Statistical Sampling

5

A. Divide population into smaller populations

Example 4: Stratified Sample

Population of 11,200 equipment transactions

• 11,000 transactions for $500 or less• 200 transactions for $1,000,000 or more

Define as two populations with relatively little variability

B. Increases audit efficiency

1. Reduce variability

2. Smaller sample sizes

Notes:

STRATIFIED SAMPLING

Independently answer multiple choice question numbers 4-6

Variable sampling

6

Statistical Sampling

POINT ESTIMATES

A. Ratio Estimation

Example 5

Audited value of sample $ 8,000 Book value of sample / 10,000 Ratio 80%

Book value of population $100,000 Ratio 80% Point estimate $ 80,000

B. Difference Estimation

Example 6

Book value of sample $10,000 Audited value of sample $ 8,000 Difference $ 2,000 / 100 items = $20 (negative)

Population 1,000 items x ($20) = ($20,000) adjustment

Book value of population: $100,000 - $20,000 = $80,000 point estimate

Notes:

IIndependently answer multiple choice question numbers 7-10

VARIABLE SAMPLING

A. Sampling Risk

1. Alpha Risk

a. Financial statements are correct

b. Sample says incorrect

c. Modify opinion unnecessarily

d. Risk of incorrect rejection

2. Beta Risk

a. Financial statements are incorrect

b. Sample says correct

c. Fail to modify opinion

d. Risk of incorrect acceptance

B. Sample Size

Exhibit 3

variability small => Smaller sample Directvariability large => Larger sample Relationship

Tolerable misstatement small => Larger sample InverseTolerable misstatement large => Smaller sample Relationship

Statistical Sampling

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C. Nature

1. Evaluate value of something

2. Substantive tests

Notes:

1.96 standarddeviations

1.96 standarddeviations

8

Statistical Sampling

ATTRIBUTES SAMPLING

A. Nature

1. Seeking errors

2. Compliance (tests of controls)

B. Confidence/Reliability

Exhibit 4

Notes: Point estimate

Upper precision limit

Lower precision limit

C. Sampling Risk

1. Alpha Risk

a. Internal control is effective

b. Sample says material weakness exists

c. Assess control risk too high

d. Risk of underreliance

e. Result: Too much substantive testing

2. Beta Risk

a. Internal control not effective

b. Sample says no material weakness exists

c. Assess control risk too low

d. Risk of overreliance

e. Result: Too little substantive testing

D. Sample Size

1. Direct relationship between expected deviation rate and sample size

a. Highly expected deviation rate => sample size

b. Low expected deviation rate => sample size

2. Inverse relationship between tolerable error rate and sample size

a. High tolerable error rate => sample size

b. Low tolerable error rate => sample size

Exhibit 5

High expected deviation rate = Larger sample DirectLow expected deviation rate = Smaller sample relationship

High tolerable rate = Smaller sample InverseLow tolerable rate = Larger sample relationship

Statistical Sampling

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Larger

Smaller

SmallerLarger

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Statistical Sampling

RISK SUMMARY

A. Alpha Risk

1. Variable Sampling: Risk of incorrect rejection

2. Attribute Sampling: Risk of underreliance

B. Beta Risk

1. Variable Sampling: Risk of incorrect acceptance

2. Attribute Sampling: Risk of overreliance

Notes:

Memorize

Alpha, Beta

DISCOVERY SAMPLING

A. Type of Attribute Sampling

B. Use With Critical Items

1. Predetermined probability of discovering one deviation

2. Expected deviation rate very low

Notes:

Statistical Sampling

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Independently answer multiple choice question numbers 11-18.

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Statistical Sampling

PPS SAMPLING

A. Variable Sampling Method

1. Probability-proportional-to-size (dollar-unit) sampling

2. Uses attribute sampling tables to assist in sampling variables

Answer multiple choice question number 19

Example 7

Refer to Question #19

Tolerable misstatement $ 24,000 Reliability factor / 3.00 Sampling interval $ 8,000

Book value of population $240,000 Sampling interval /$ 8,000 Sample size 30

Answer multiple choice question number 20-21

B. Advantages of PPS Sampling

1. Easier to use than other methods

2. Automatic stratification

3. Sample size not affected by variability

4. If the expected error rate is small, then a very small sample size is adequate.

5. Sampling can begin before entire population is available

Notes:

SAMPLING PLAN STEPS

A. Define Population

B. Choose Method

1. Mean-per-unit estimation

2. Difference estimation

3. Ratio estimation

4. Stratified sampling

5. PPS

C. Determine Sample Size

1. Variable Sampling

a. Variability

b. Tolerable misstatement

c. Expected misstatement

2. Attribute Sampling

a. Tolerable deviation (error) rate

b. Expected deviation (error) rate

D. Determine Acceptable Risk Level

E. Select Unrestricted Random Sample

F. Audit Unrestricted Random Sample

G. Evaluate Results

H. Document Procedures

Notes:

Statistical Sampling

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IIndependently answer problem number 2 (simulation)

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Statistical Sampling

Problem 1 (42 to 53 minutes)

1. An auditor is testing internal controlprocedures that are evidenced on an entity’s vouchers by matching random numbers with voucher numbers. If a random number matches the number of a voided voucher, that voucher ordinarily should be replaced by another voucher in the random sample if the vouchera. Constitutes a deviationb. Has been properly voidedc. Cannot be locatedd. Represents an immaterial dollar

amount (0253)

2. When performing a test of controlswith respect to control over cash receipts, an auditor may use a systematic sampling technique with a start at any randomly selected item. The biggest disadvantage of this type of sampling is that the items in the populationa.

b.

c.

d.

3. For which of the following audit testswould an auditor most likely use attribute sampling ?a.

b.

c.

d.

4. In performing tests of controls overauthorization of cash disbursements, which of the following statistical sampling methods would be most appropriate?a. Variablesb. Stratifiedc. Ratiod. Attributes (3921)

5. Which of the following samplingmethods would be used to estimate a numerical measurement of a population, such as a dollar value?a. Discovery samplingb. Numerical samplingc. Sampling for attributesd. Sampling for variables (2782)

6. Which of the following courses ofaction would an auditor most likely follow in planning a sample of cash disburse-ments if the auditor is aware of several unusually large cash disbursements?a.

b.

c.

d.

Must be systematically replaced in the population after samplingMay systematically occur more than once in the sampleMust be recorded in a systematic pattern before the sample can be drawnMay occur in a systematic pattern, thus destroying the sample random-ness (0267)

Making an independent estimate of the amount of a LIFO inventoryExamining invoices in support of the valuation of fixed asset additionsSelecting accounts receivable for confirmation of account balancesInspecting employee time cards for proper approval by supervisors

(4682)

Increase the sample size to reduce the effect of the unusually large disbursementsContinue to draw new samples until all the unusually large disburse-ments appear in the sampleSet the tolerable rate of deviation at a lower level than originally plannedStratify the cash disbursements population so that the unusually large disbursements are selected

(2973)

MULTIPLE CHOICE QUESTIONS

Properly voided

Variable Sampling

Attribute Sampling

Destroy

Statistical Sampling

15

7. Using statistical sampling to assist inverifying the year-end accounts payable balance, an auditor has accumulated the following data:

Using the ratio estimation technique, the auditor’s estimate of year-end accounts payable balance would bea. $6,150,000b. $6,000,000c. $5,125,000d. $5,050,000 (7496)

8. Stratified mean per unit (MPU)sampling is a statistical technique that may be more efficient than unstratified MPU because it usuallya.

b.

c.

d.

9. When planning a sample for asubstantive test of details, an auditor should consider tolerable misstatement for the sample. This consideration shoulda.

b.

c.

d.

10. The use of the ratio estimationsampling technique is most effective whena.

b.

c.

d.

11. Given random selection, the samesample size, and the same precision requirement for the testing of two unequal populations, the risk of assessing control risk too low on the smaller population isa.

b.

c.

d.

May be applied to populations where many monetary errors are expected to occurProduces an estimate having a desired level of precision with a smaller sample sizeIncreases the variability among items in a stratum by grouping sampling units with similar charac-teristicsYields a weighted sum of the strata standard deviations that is greater than the standard deviation of the population (0247)

Be related to the auditor’s business riskNot be adjusted for qualitative factorsBe related to preliminary judgments about materiality levelsNot be changed during the audit process (0255)

The calculated audit amounts are approximately proportional to the client’s book amounts.A relatively small number of differ-ences exist in the population.Estimating populations whose records consist of quantities, but not book valuesLarge overstatement differences and large understatement differ-ences exist in the population.

(5669)

The same as assessing control risk too low for the larger populationHigher than assessing control risk too low for the larger populationLower than assessing control risk too low for the larger populationIndeterminate relative to assessing control risk too low for the larger population (7498)

BalanceNumber of Book determinedaccounts balance by the auditor

Population 4,100 $5,000,000 ?Sample 200 $ 250,000 $300,000

Tolerable misstatement = Materiality

300,000 / 250,000x 5,000,000 = 6,000,000

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Statistical Sampling

12. Which of the following statisticalsampling methods is most useful to audi-tors when performing tests of controls ?a. Ratio estimationb. Variable samplingc. Difference estimationd. Discovery sampling (0268)

13. In which of the following cases wouldthe auditor be most likely to conclude that all of the items in an account under consideration should be examined rather than tested on a sample basis ?

14. While performing a test of detailsduring an audit, an auditor determined that the sample results supported the conclusion that the recorded account balance was materially misstated. It was, in fact, not materially misstated. This situation illustrates the risk ofa. Assessing control risk too highb. Assessing control risk too lowc. Incorrect rejectiond. Incorrect acceptance (4708)

15. Which of the following factors isgenerally not considered in determining the sample size for a test of controls ?a. b.c.

d.

16. The sample size of a test of controlsvaries inversely with

17. As a result of tests of controls, anauditor assessed control risk too low and decreased substantive testing. This assessment occurred because the true deviation rate in the population wasa.

b.

c.

d.

18. In performing testing of controls,the auditor will normally find thata.

b.

c.

d.

Population sizeTolerable rateRisk of assessing control risk too lowExpected population deviation rate

(0262)

More than the risk of assessing control risk too low based on the auditor’s sampleMore than the deviation rate in the auditor’s sampleLess than the risk of assessing control risk too low based on the auditor’s sampleLess than the deviation rate in the auditor’s sample (3907)

The level of risk is directly propor-tionate to the rate of misstatement.The rate of deviations in the sampleexceeds the rate of misstatement inthe accounting records.The rate of misstatement in the sampleexceeds the rate of deviations.All unexamined items result inmisstatements in the accountingrecords. (9911)

The measure Misstatementof tolerable frequency is

misstatement is expected to bea. Large Lowb. Small Highc. Large Highd. Small Low

(9911)

Expected population Tolerabledeviation rate rate

a. Yes Yesb. No Noc. Yes Nod. No Yes

(4709)DirectInverse

Attribute

Statistical Sampling

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19. Hill has decided to use probability-proportional-to-size (PPS) sampling, sometimes called dollar-unit sampling, in the audit of a client’s accounts receivable balances. Hill plans to use the following PPS sampling table:

20. Which of the following statements iscorrect concerning probability-proportional-to-size (PPS) sampling, also known as dollar-unit sampling ?a.

b.

c.

d.

21. In statistical sampling methods usedin substantive testing, an auditor most likely would stratify a population into meaningful groups ifa.

b.

c.

d.

The sampling distribution should approximate the normal distribution.Overstated units have a lower probability of sample selection than units that are understated.The auditor controls the risk of incorrect acceptance by specifying that risk level for the sampling plan.The sampling interval is calculated by dividing the number of physical units in the population by the sample size. (0243)

Probability proportional to size (PPS) sampling is used.The population has highly variable recorded amounts.The auditor’s estimated tolerable misstatement is extremely small.The standard deviation of recorded amounts is relatively small. (5668)

TABLEReliability Factors for Errors of overstatement

Number ofoverstatement Risk of incorrect acceptancemisstatements 1% 5% 10% 15% 20%

0 4.61 3.00 2.31 1.90 1.611 6.64 4.75 3.89 3.38 3.002 8.41 6.30 5.33 4.72 4.283 10.05 7.76 6.69 6.02 5.524 11.61 9.16 8.00 7.27 6.73

Additional Information

Tolerable misstatement (net of effect of expected misstatement) $24,000Risk of incorrect acceptance 20%Number of misstatements allowed 1Recorded amount of accounts receivable $240,000Number of accounts 360

What sample size should Hill use ?a. 120b. 108c. 60d. 30 (7497)

Stratify

24,000 / 3.00 = 8,000240,000 / 8,000 =30

PPS Sampling

TABLEReliability Factors for Errors of overstatement

Number ofoverstatement Risk of incorrect acceptance

misstatements 1% 5% 10% 15% 20%0 4.61 3.00 2.31 1.90 1.611 6.64 4.75 3.89 3.38 3.002 8.41 6.30 5.33 4.72 4.283 10.05 7.76 6.69 6.02 5.524 11.61 9.16 8.00 7.27 6.73

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Statistical Sampling

SIMULATION

Scenario Response #1 Response #2 Response #3 Response #4

Scenario Response #1 Response #2 Response #3 Response #4

Edwards has decided to use probability-proportional-to-size (PPS) sampling, sometimes called dollar-unit sampling, in the audit of a client’s accounts receivable balance. Few, if any, misstatements of account balance overstatement are expected. Edwards plans to use the following PPS sampling table.

Research Question: What authoritative reference provides guidance to an auditor on choosing between statistical and non-statistical sampling ?

Paragraph Reference Answer:

Problem 2 (15 to 25 minutes)

Statistical Sampling

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Scenario Response #1 Response #2 Response #3 Response #4

Identify the advantages of using PPS sampling over classical variables sampling.

P.12

Advantages of PPS Sampling

1.Easier to use than other methods

2.Automatic stratification

3.Sample size not affected by variability

4.If the expected error rate is small, then a very small sample size is adequate.

5.Sampling can begin before entire population is available

Calculate the total projected misstatement if the following three misstatements werediscovered in a PPS sample.

Recorded amount Audit amount Sampling interval 1st misstatement $ 400 $ 320 $1,000 2nd misstatement 500 0 1,000 3rd misstatement 3,000 2,500 1,000

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Statistical Sampling

Scenario Response #1 Response #2 Response #3 Response #4

Scenario Response #1 Response #2 Response #3 Response #4

Calculate the sampling interval and the sample size Edwards should use given thefollowing information.

Tolerable misstatement $ 15,000Risk of incorrect acceptance 5%Number of misstatements allowed 0Recorded amount of accounts receivable $300,000

Sample interval = Tolerable misstatement / Reliability factor15,000 / 3 = 5,000

Book value / Sampling interval = Sample size300,000 / 5,000 = 60

(400 - 320) / 400 x 1,000 = 200200 + 1,000 + 500 = 1,700(1st) (2nd) (3rd)

Statistical Sampling Ver. FE/JA

1. Ans. BTest に選ばれた Voucher を Replace できるのは？ Choice A 間違い、Deviation（誤りや Error）であれ

ば監査調書に Documentation しなくてはいけません。めどくさいなと思って、違うのに変えてはいけ

ません。Choice B も同様。Test に選ばれた Voucher が発見できなのであれば、内部統制の問題点とし

て指摘します。Choice D、金額が小さくても 選ばれたのであればちゃんと Test します。

2. Ans. DSystematic sampling＝例：5 飛びで Test していこう。これでは Random sampling と違い、Sampleが equal chance で選ばれていない。Destroy randomness.

3. Ans. DAttribute sampling=内部統制の Sampling＝金額の Test には使用しない。Choice D 以外は金額の

Sampling＝Variable sampling。

4. Ans. D問題文に Tests of control＝attributes sampling。ほかの Choice は Variable sampling。

5. Ans. DSampling methods for dollar value＝金額の Sampling＝Variable sampling。

6. Ans. D母集団の中に金額の大きくかけ離れた Sample が存在するのであれば ＝Stratify(分ける)母集団。

7. Ans. B300000÷250000＝1.2 ５Million×1.2＝６Million。なぜ監査の Value のほうが多いのか？クライア

ントは A/P（負債）を過少申告しています。

8. Ans. BStratify すると母集団が割れて、金額の大きいのと小さいので分けて Sampling します。これにより、

精度が向上＋Sampling の数も少なくて済みます。

9. Ans. CTolerable misstatement＝許容できるミス＝Materiality。

10. Ans. ARatio estimation＝Proportional to book amounts。

11. Ans. C二つの母集団、ひとつは 1000 件、もうひとつは 100 件。Sample をそれぞれから 10 件選んだ場合、

少ない母集団は 全体の 10%を Test したことになり、大きい母集団は１％（1000 件のうち 10 件）。

よって、CR を Too Low にしてしまう Risk は Small population のほうが Lower。

12. Ans. D

Tests of control=Attribute sampling=Discovery sampling (Attribute sampling の一つ)。

13. Ans. B

Sampling しない、全件 Test するのは？ 許容できるミスは少ない＋ミスが頻繁に起きている。

14. Ans. C

Sample をテストした結果、重大な誤りがあると結論。しかし、実は誤りではなく正常だった＝Incorrect

rejection

15. Ans. A

Sample size に影響を及ぼさないのは＝Population size。

16. Ans. D

Test of control において Sample size に Inverse 関係があるのは？

Tolerable rate（TR）。 TR is small = More sample。TR is large= Less sample。

17. Ans.：B

Test した結果、CR を下げ過ぎてしまいました＝信頼しすぎ＝Population 全体の Deviation Rate＞

Sample deviation rate＝実際は Error やミスが多く存在している。

18. Ans. B

内部統制の Test で Error やミスが発見＝必ずしも財務諸表の Misstatement ではありませ。例：Sample

で選ばれた Voucher にサインが無かった＝Deviation。取引は正常に行われていて A/P－Ledger にも記

入。これであれば、帳簿は問題ありません。Rate of deviation in the sample > Rate of misstatement in the

accounting records.

19. Ans. D

24000÷3.00＝$8000 Interval。 240000÷8000＝30 件。

20. Ans. C

PPS sampling＝The auditor controls both risk of incorrect acceptance and number of misstatements

allowed。

21. Ans. B

Stratify is used for = The population has highly variable amounts（変動の大きい金額）。

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Simulation Solution

Solution 2

Response #1: Research Element (1 point)

Paragraph Reference Answer: AU 350.46

Guidance: AU 350.46 states, "Statistical sam-pling helps the auditor (a) to design an efficient sample, (b) to measure the sufficiency of the evidential matter obtained, and (c) to evaluate the sample results. By using statistical theory, the auditor can quantify sampling risk to assist himself in limiting it to a level he considers acceptable. However, statistical sampling

involves additional costs of training auditors, designing individual samples to meet the statistical requirements, and selecting the items to be examined. Because either nonstatistical or statistical sampling can provide sufficient evidential matter, the auditor chooses between them after considering their relative cost and effectiveness in the circumstances."

Response #2: PPS Sampling (3 points)

The advantages of PPS sampling over classical

variables sampling are as follows:

PPS sampling is generally easier touse than classical variables sampling.

Size of a PPS sample is hot based onthe estimated variation of auditedamounts.

PPS sampling automatically results in astratified sample.

Individually significant items are

automatically identified. If no misstatements are expected, PPS

sampling will usually result in a smallersample size than classical variablessampling.

A PPS sample can be easily designedand sample selection can begin beforethe complete population is available.

Response #3: PPS Sampling (3 points)

Sampling Tolerable Misstatement

Interval ＝ Reliability Factor for

Misstatement of

Overstatement

＝$15000÷3.00＝$5000

Sample Size ＝Recorded Amount

Sample Interval

$300000÷$5000＝60

Response #4: PPS Sampling (3 points)

Recorded amount

Audit amount Tainting

Sampling Interval

Projected misstatement

1st misstatement $ 400 $ 320 20% $1,000 $ 200

2nd misstatement 500 0 100% 1,000 1,000

3rd misstatement 3,000 2,500 * 1,000 500

Total projected misstatement $1,700

*The recorded amount is greater than thesampling interval; therefore, the projectedmisstatement equals the actualmisstatement.