CHAPTER 16BASIC AUDIT SAMPLING CONCEPTS

PSA 530 (Clarified) – “Audit Sampling”- establishes standards and provides guidance when the auditor has decided to use audit sampling in performing audit procedures- deals with the auditor’s use of statistical and non-statistical sampling when designing and selecting the audit sample, performing TOCs and test of details, and evaluating the results from the sample.

Audit sampling – the application of compliance or substantive procedure to less than 100% of the items within an account balance or class of transactions to enable the auditor to obtain and evaluate evidence of some characteristic of the balance or class and to form or assist in forming a conclusion concerning that characteristic.

NATURE AND PURPOSE OF AUDIT SAMPLING Sampling – a process whereby information is gained concerning the characteristic of population

(universe or field) of items by means of an examination of only a portion of the items comprising that population- considered an INFERENTIAL TECHNIQUE and it allows the auditor to draw conclusions about transactions or account balances without sustaining the time and cost of examining all available data.

WHY AUDITORS SAMPLEAuditors use audit sampling when

a) The nature and materiality of the balance or class does not demand a 100% auditb) A decision must be made about the balance or class; andc) The time and cost to audit 100% of the population would be too great.

TESTING PROCEDURES WHICH DO NOT INVOLVE SAMPLING1. Tests performed on 100% of the items within a population2. Inquiry and observation

a) Procedures that depend on segregation of duties or that otherwise provide no documentary evidence

b) Tracing 1 or a few transactions to obtain an understanding of an accounting system and its internal control.*Procedures performed to obtain an “understanding of the IC structure

sufficient to plan an audit” do NOT involve sampling.However, many TOCs used to assess control risk do involve sampling

3. Analytical procedures- analysis of significant ratios and trends including the resulting investigation of fluctuations and relationship that are inconsistent with other relevant information or deviate from predicted amounts.

SAMPLING vs NON-SAMPLING RISK Ultimate risk – combination of sampling and non-sampling risks.

- uncertainty that in every audit engagement a degree of risk is always present

SAMPLING RISK NON-SAMPLING RISK Risk that the auditor’s conclusion based

on a sample might be different from the conclusion they would reach if they examined every item in the entire population.

Probability that a material error will occur during the accounting process

Is a function of a client’s internal accounting system

The stronger the internal accounting control is, the lower the likelihood that a material error will occur

2 type of sampling risk1. The risk the auditor will conclude, in

case of a TOC, that control risk is lower than it actually is; or in case of a substantive test, that a material error does not exist when it fact it does.

This type of risk affects audit effectiveness and is more likely to lead to an inappropriate audit opinion.

2. The risk the auditor will conclude, in the case of a TOC, that control risk is higher that it actually is; or in case of a substantive test, that a material error exists when it fact it does not.

This type of risk affects audit efficiency as it would usually lead to additional work to establish that initial conclusions were incorrect.

- mathematical complements of these risks are termed CONFIDENCE LEVELS

Refers to the probability that a material error will not be discovered by the auditor in the performance of the substantive tests

Arises from factors that cause the auditor to reach an erroneous conclusion for any reason not related to the size of the sample

Can generally be reduced to low levels through implementation of appropriately designed quality control procedures by the CPA firm

If errors are material, the auditor cannot express an unqualified opinion unless the statements are corrected

NONSTATISTICAL AND STATISTICAL SAMPLINGNONSTATISTICAL SAMPLING STATISTICAL SAMPLING

Or Judgment sampling Procedures are designed and executed

on the basis of the auditor’s sound objective reasoning judgment

Although they can involve arithmetic concepts, subjective influence and probabilities reasoning, they are NOT formally derived from mathematical sampling proofs and theorems

Findings from judgment sampling must always be interpreted subjectively in the professional judgment of the auditor

3 common variations of nonstatistical samplinga) Haphazard sampling – consists of

sampling units without any conscious bias that is without any special reason for including or omitting items from the sample

- if used, it should not consist of items selected in a careless matter, the sample should be expected to be a representative of the population

b) Block selection – a block sample consists of all items in a selected time period, numerical sequence, or alphabetical sequence

- due to relatively large number of blocks needed to form a reasonable audit conclusion, block sampling CANNOT generally be relied upon to efficiently produce a representative sample and therefore is NOT an acceptable for statistical or nonstatiscal audit sampling

c) Judgmental Sampling – the auditor selects large or unusual items from the population and uses some other judgmental criterion for selection

- this method has a conscious bias and cannot be considered a representative selection method

Is a mathematically derived tool which provides the auditor with an OBJECTIVE basis for expressing conclusions about population characteristic based upon a sample of items from the population

Means any approach to sampling that has the following characteristics

1. Random selection of a sample2. Use of probability theory to evaluate

sample results, including measurement of sampling risk

Auditors can use statistical sampling when performing TOC for which documentary evidence exists that a procedure was performed

Examples of controls for which documentary evidence exists are

(1) those that are related to approving transactions such as sales order that bears the credit manager’s signature indicating approval of credit(2) keeping adequate documents and records(3) making independent checks on performance

SELECTING ITEMS FOR TESTING TO GATHER AUDIT EVIDENCE

When designing audit procedures, the auditor should determine appropriate means of selecting items for testing. The means available to the auditor are

a) Selecting all items (100% examination)

b) Selecting specific itemsc) Audit sampling

When auditors use statistical/ non statistical sampling methods, auditors must decide on:

STATISTICAL SAMPLING (cont.)

a) Selecting all items (100% examination)

b) Selecting specific itemsc) Which sampling techniques to use

- And they must draw conclusions from their sampling.

SELECTING ALL ITEMS 100% examination is unlikely in the

case of TOC; however, it is more common for substantive procedures

SELECTING SPECIFIC ITEMS the judgmental selection of specific

items is subject to non-sampling risk. Specific items selected may includea) High value or key itemsb) All items over a certain amountc) Items to obtain informationd) Items to test procedures

AUDIT SAMPLING The primary benefit of statistical

methods is the quantification of sampling risk

ATTRIBUTE AND VARIABLES SAMPLING TECHNIQUES- the auditor may use sampling procedures to measure both qualitative and quantitative characteristics of a population

ATTRIBUTE SAMPLING TECHNIQUE

Attribute estimation procedures – measure qualitative characteristics

The auditor uses attribute estimation procedures when he/ she is interested in measuring a qualitative feature, such as the presence/ absence of internal control procedure

When conducting TOCs, the auditors must consider both the risk of overreliance and the risk of underreliance

Risk of overreliance – risk that the sample supports reliance on IC when this reliance is unjustified

Risk of underreliance – risk that the sample does not support reliance on IC when the true compliance rate suggests that the auditor should rely on IC

- if this happens the result will be decreased efficiency in the form of increased substantive testing procedures

In compliance testing of an attribute, the auditor is interested in determining whether the population rate of occurrence exceeds a tolerable rate

Tolerable rate – maximum rate of exception that the auditor would be willing to accept in a population without altering his or her reliance on the attribute.

Once the auditor determines sample size using his/her desired reliability, tolerable rate of occurrence and expected rate of occurrence and selects items for examination, he/she classifies them as either possessing the attribute or not (deviations).

Based on the number of sample items containing deviations the sample rate of occurrence can be calculated

The auditor can determine the upper precision limit by considering thea) Sample sizeb) Reliabilityc) Number of observed deviations

Upper precision limit – is the rate of occurrence which statistically exceeds the actual population error rate

The auditor then compares the upper precision limit to tolerable error to determine whether he/she can rely on IC related to the attributes examined.

If the precision limit is LESS than the tolerable error, the sample results suggest that the auditor should rely on the IC examined.

VARIABLES SAMPLING TECHNIQUE

Variables estimation procedures – measure quantitative techniques

Variable estimation sampling – is used by the auditor during substantive testing This provides an estimate of a peso range within which the true audited value of the population

lies Example: the auditor uses variable estimation sampling to determine whether the book value of

an account balance is fairly stated

Variable estimation sampling exposes the auditor to 2 aspects of sampling risk1. Risk of incorrect rejection2. Risk of incorrect acceptance

Risk of incorrect rejection - or alpha risk - risk that the sample supports the conclusion that the recorded account balance is materially misstated when it is not materially misstated

Risk of incorrect acceptance - or beta risk - risk hat the sample supports the conclusion that the recorded account balance is not materially misstated - also referred to as detection risk

DESIGN OF THE SAMPLEThe factors that the auditor should consider in designing an audit sample are

1. Audit objectives2. Population and its characteristics3. Risk and assurance4. Tolerable error5. Expected error in the population6. Stratification

1. Audit objectives2. Population It is appropriate for the auditor to ensure that the population is

a) APPROPRIATE – to the objective of the sampling procedure, which will include consideration of the direction of testing

b) COMPLETE

The entire set of data from which the auditor desires to sample in order to reach a conclusion constitutes the population.

3. Risk and Assurance In planning the audit, the auditor uses professional judgment to assess the level of audit risk that is appropriate. This audit risk includes

a) Risk that material errors will occur (inherent risk)b) Risk that the client’s system of IC will not prevent/ correct such errors

(control risk)c) Risk that any remaining material errors will not be detected by the

auditor (detection risk)

The auditor is faced with sampling risk in both TOC and substantive procedures as follows

a) Tests of Controls1. Risk of underreliance – the risk that, although the sample result does

not support the auditor’s assessment of control risk, the actual compliance rate would support such an assessment

2. Risk of overreliance – the risk that, although the sample result supports the auditor’s assessment of control risk, the actual compliance rate would not support such an assessment

b) Substantive Procedures1. Risk of incorrect rejection – the risk that, although sample results

supports the conclusion that a recorded account balance/ class of transactions is materially misstated, in fact it is not materially misstated

2. Risk of incorrect acceptance – the risk that, although sample results supports the conclusion that a recorded account balance/ class of transactions is not materially misstated, in fact it is materially misstated.

Risk of underreliance & risk of incorrect rejection – affect AUDIT EFFICIENCY as they would ordinarily lead to additional work being performed by the auditor, or the entity, which would establish that the initial conclusions were incorrect.

Risk of overreliance & risk of incorrect acceptance – affect AUDIT EFFECTIVENESS and are more likely to lead to an erroneous opinion on the financial statements

4. Tolerable Error Maximum error in the population that the auditor would be willing to accept and still conclude that the result from the sample has achieved his audit objective.

There is an INVERSE RELATIONSHIP between the tolerable error and the

required sample size

In compliance proceduresThe tolerable error is the maximum rate of deviation from a prescribed control procedure that the auditor would be willing to accept without altering his planning reliance on the control being tested

In substantive proceduresThe tolerable error is the maximum monetary error in an account balance/ class of transactions that the auditor would be willing to accept to that he considers the results of all audit procedures he is able to conclude, with reasonable assurance, that the financial information is not materially in error.

5. Expected Error in the Population

The following factors should be considered in determining the expected error in population:

a) Error levels identified in previous auditsb) Changes in client proceduresc) Evidence available from his evaluation of the system of IC and from

results of analytical review procedures

6. Stratification Stratification is the process of dividing a population into subpopulation, that is, a group of sampling units which have similar characteristics (often monetary value).

Stratification enables the auditor to direct his efforts towards the items he considers would potentially contain the greater monetary error

Audit efficiency may be improved if the auditor stratifies a population by dividing it into discrete sub-populations which have an identifying characteristic

Objective of stratification is to reduce the variability of items within each stratum and therefore allow sample size to be reduced without a proportional increase in sampling risk

Value Weighted Selection It will often be efficient in substantive testing, particularly when testing for overstatements, to

identify the sampling unit as the individual monetary units that make up an account balance/ class of transactions.

This approach to defining the sampling unit ensures that audit effort is directed to the larger value items because they have greater chance of selection, and can result in smaller sample sizes.

This approach is ordinarily used in conjunction with the systematic method of sample selection and is most efficient when selecting from a computerized database

Selecting the Sample

Sample Selection MethodsMost commonly used selection methods for Statistical and Non-statistical sampling are as follows:

1. Random Sampling A simple random sample is a sample that is selected in such a way that every item in a

population has an equal chance of being selected Accomplished by using a printed table of random numbers/ computer software that

generates random numbers. To use this method, it is necessary to establish correspondence between the population

and the random numbers

2. Systematic Sampling In using this method, the auditor counts through the population and selects items on the

basis of a sampling interval which is determined by dividing the no. of physical items in the population by sample size

Interval may be based on certain number of items (every 25th voucher number) or on monetary totals (every 1,500 in cumulative value of the population)

One precaution that can be taken is to use several random starts

3. Stratified Random Sampling The auditor groups the population into subpopulation, or strata that are similar in

amount. Samples are then drawn from each stratum using 1 of the random selection methods

4. Sampling with Probability Proportional to Size Emphasizes larger peso items within an account balance The probability of an item being selected in this method is directly proportional to its

peso amount Each individual peso in the account balance has an equal chance of selection, but each

physical unit does not

Sample Size Sample size is affected by the level of sampling risk that the auditor is willing to accept The lower the risk the auditor is willing to accept, the greater the sample size will need to be

The size of a sample has a direct effect upon both the

1. Allowance for sampling risk2. Sampling risk

As the sample size decreases, both the sampling risk, and allowance for sampling risk increase

Sample size is also affected by 3 certain characteristics of the population being tested As the population increases in size, the sample size necessary to estimate the population with

specified sampling risk, and allowance for sampling risk increases In attributes sampling, sample size also increases as the expected population deviation rate

increases

Examples of Factors Influencing Sample Size for Tests of Control – CHAPTER 17 Step 6FACTOR EFFECT ON SAMPLE SIZE1. An increase in the extent to

which the auditor’s risk assessment takes into account relevant controls

Increase The more assurance the auditor intend to obtain from the operating effectiveness of control, the lower the auditor’s assessment of the ROMM will be, and the larger sample size will need to be.

2. An increase in the tolerable rate of deviation

Decrease

3. An increase in the expected rate of deviation of the population to be tested

Increase The higher the expected rate of deviation, the larger the sample size needs to be so that the auditor is in a position to make a reasonable estimate of the actual rate of deviation.

4. An increase in the auditor’s desired level of assurance that the tolerable rate of deviation is not exceeded by the actual rate of deviation in the population

Increase The greater the level of assurance that the auditor desires that the results of the sample are in fact indicative of the actual incidence of deviation in the population, the larger the sample size needs to be

5. An increase in the no. of sampling units in the population

Negligible effect For large populations, the actual size of the population has little, if any, effect on sample size.For small populations however, audit sampling may not be as efficient as alternative means of obtaining sufficient appropriate audit evidence

Examples of Factors Influencing Sample Size for Tests of Details – Chapter 18 Step 3FACTOR EFFECT ON SAMPLE SIZE1. An increase in the auditor’s

assessment of the ROMMIncrease The higher the auditor’s assessment of

ROMM, the larger the sample size needs to be.

2. An increase in the use of other substantive procedures directed at the same assertion

Decrease The more the auditor is relying on other substantive procedures to reduce to an acceptable level the detection risk regarding a particular population, the less assurance the auditor will require from sampling, and, therefore., the smaller the sample size can be

3. An increase in the auditor’s desired level of assurance that tolerable misstatement is not exceeded by actual misstatement in the population

Increase The greater the level of assurance that the auditor requires that the results of the sample are in fact indicative of the actual amount of misstatement in the population, the larger the sample size needs to be

4. An increase in tolerable misstatement

Decrease

5. An increase in the amount of misstatement the auditor expects to find in the population

Increase The greater the amount of misstatement the auditor expects to find in the population, the larger the sample size needs to be in order to make a reasonable estimate of the actual misstatement in the population.

6. Stratification of the population when appropriate

Decrease

7. The number of sampling units in the population

Negligible effect

Performing the Audit Procedure

Evaluation of Sample ResultsHaving carried out, on each sample item, those audit procedures that are appropriate to the particular audit objective, the auditor should:

1. Analyze any error detected in the sample2. Project the errors found in the sample to the population3. Assess the sampling risk

1. Analysis of Errors in the Sample When errors are identified, the auditor also needs to consider matters such as:

a) Direct effect of identified errors on the FS

b) The effectiveness of the accounting and IC systems and their effect on the audit approach when, for example, the errors results from management override of an internal control

2. Projection of Errors PSA 530 (Clarified) provides guidance in projecting errors as follows:

For substantive procedures, the auditor should project monetary errors found in the sample to the population, and should consider the effect of the projected error on the particular test objective and on other areas of the audit.

When an error has been established as an anomalous error, it may be excluded when projecting sample errors to the population

- projected errors plus anomalous errors for each stratum are then combined when considering the possible effect of errors on the total account balance or class of transactions.

For TOC, no explicit projection of errors is necessary since the sample error rate is also the projected rate of error for the population as a whole

When the population is divided into 2 or more subpopulations (stratification), the projection of errors is done separately for each subpopulation, and the results are added together

3. Assessing Sampling Risk The total of projected error plus anomalous error is the auditor’s best estimate of error in the population.

If the evaluation of sample results indicates that the preliminary assessment of the relevant characteristic of the population needs to be revised, the auditor may:

a) Request management to investigate identified errors and the potential for further errors, and to make any necessary adjustments; and/or

b) Modify planned audit proceduresc) Consider the effect on the audit report

ConclusionDetailed Audit Sampling Plan

AUDIT SAMPLING

Attributes Sampling in

Tests of Control

Statistical

Regular/ Classical Discovery Sequential/

Stop or Go

Nonstatistical

Variables Sampling in Substantive

Tests of Details

Statistical

Probability Proportional

to SizeClassical

Mean per unit Difference Estimation

Ratio Estimation Regression

Nonstatistical

Definition/ Description of Audit Sampling PlansAttributes Sampling Plan

This is used to test an entity’s rate of deviation from a prescribed control procedure Also called RATE OF OCCURRENCE It is an audit sampling in which auditor’s look for the presence/ absence of a control condition Example: an auditor might use it to plan to test controls for billing systems, disbursement

processing, inventory pricing, and depreciation among other things

Variables Sampling Plan This is used to test whether recorded account balances are fairly stated Example: an auditor might use it to test recorded peso amounts for receivables, inventory, fixed

asset additions among others

Statistical Sampling Plan A sampling technique in which an auditor uses the laws of probability to select and evaluate a

sample When using statistical sampling, auditors must select a random sample, which means every item

in the population must have an equal chance of being included in the sample

Nonstatistical Sampling Plan These plans rely exclusively on subjective judgment to determine sample size and to evaluate

sample resultsRegular/ Classical Attributes Sampling

This sampling plan enables the auditor to estimate the rate of occurrence of certain characteristics in the population

Example: the auditor might use this to plan to estimate the percentage of cash disbursements processed during the year that were not approved

Discovery Sampling This form of attributes sampling is designed to locate at least 1 deviation (exception) in the

population Often used in situations in which the auditors expect a very low rate of occurrence of some

critical deviations Purpose is to detect a least 1 deviation, with a predetermined risk of assessing control risk too

low, if the deviation rate in the population is greater than the specified tolerable deviation rate 1 important use of discovery sampling is to locate examples of suspected fraud Example: when the auditor attempts to locate a fraudulent cash disbursement

Sequential (Stop or Go) Sampling A sampling plan for which the sample is selected in several steps, with the need to perform each

step conditional on the results of the previous steps The results may either be so poor as to indicate that the control may not be relied upon, or so

good as to justify reliance at each step--Probability Proportional to Size (PPS)

Also referred to as PESO-UNIT SAMPLING This technique applies attributes sampling theory to develop an estimate of the total peso

amount of misstatement in a population Used as an alternative to classical variable sampling methods for performing substantive tests of

transactions/ balances Unlike classical variable sampling techniques that define the sampling unit as each transaction/

account balance in the population, PPS sampling defines the sampling unit as each INDIVIDUAL PESO making up the book value of the population.

A transaction/ account is selected for audit if a peso from that transaction/ account is selected from the population, therefore, each transaction/ account has a probability proportional to its size of being selected for inclusion in the sample.

Classical Variables Sampling Models These use normal distribution theory to evaluate selected characteristics of a population on the

basis of a sample of the items constituting the population These sampling applications provide the auditors with an estimate of a numerical quantity such

as the peso balances of an account This technique is primarily used by auditors to perform substantive tests

--

Mean-per-unit Estimation This is a classical variables sampling plan enabling the auditors to estimate the average peso

value (or other variable) of items in a population by determining the average value of items in a sample

Difference Estimation

Uses the difference between the audited (correct) values and book values of items in a sample to calculate the estimated total audited value of the population

Is used in lieu of ratio estimation when the differences are not nearly proportional to book values

Ratio Estimation Uses the ratio of audited (correct) values to book values of items in the sample to calculate the

estimated total audited value of the population Used in lieu of difference estimation when the differences are nearly proportional to book values

Regression Similar to difference and ratio approaches Has the effect of using both the average ratio and the average difference in calculating an

estimate of the total amount for the population

CHAPTER 17AUDIT SAMPLING FOR TESTS OF CONTROLS

STEPS IN THE APPLICATION OF SAMPLING IN TEST OF CONTROLSThe application of sampling in TOC auditing is a structured formal approach embodied in 10 steps

The 9-step framework helps auditors plan, perform, and evaluate TOC audit work

It also helps auditors accomplish a 10th step—careful documentation of work—by showing each of the 9 areas to be described in the working papers

1. Determine the control to be tested/ audit objective of the test2. Define the attributes and deviation conditions3. Define the population to be sampled and the sampling unit4. Specify the risk of assessing control risk too low (overreliance) and the tolerable deviation rate5. Estimate the expected population deviation rate6. Determine the sample size7. Select the sample8. Perform the TOC procedures on sample items9. Evaluate the sample results10. Document the sampling procedure

1. Determine the control to be tested/ audit objective of the test

TOC is performed to provide evidence about the design/ operating effectiveness of IC structure policies and procedures.It will also support the auditor’s planned assessed level of control risk.

2. Define the attributes and deviation conditions Professional judgment is applied by the auditor

Attributes – characteristics that provide evidence that an internal control procedure was actually performed

Deviation – occurs when a sample item does not have 1 or more of the identified attributes

3. Define the population to be sampled and the sampling unit

4. Specify the risk of assessing control risk too low (overreliance) and the tolerable deviation rate

The risk of assessing control risk too low—that is, the risk that the actual deviation rate EXCEEDS the tolerable deviation rate—is a critical risk in TOC because this risk impacts the effectiveness of the audit

Auditors specify the tolerable deviation rate based on1. Their planned assessed level of control risk2. Degree of assurance desired from the evidential

matter in the sample

PLANNED ASSESSED LEVEL OF CR TOLERABLE DEVIATION RATELOW 2% - 7%MODERATE 6% - 12%SLIGHTLY BELOW THE MAXIMUM 11% - 20%MAXIMUM omit test

5. Estimate the expected population deviation rate

This expected deviation rate is significant because it represents the rate that the auditors expect to discover in their sample from the population

6. Determine the sample size ** SUMMARY OF FACTORS INFLUENCING COMPLIANCE TEST SAMPLE SIZE** (table)

7. Select the sample Random samples may be selected using:a) Random number tables

b) Random number generatorsc) Systematic sampling

Random sample – is one in which every sampling unit in the population has an equal chance of being included in the sample.

Sampling with replacement – means that after an item has been selected for inclusion in the sample, it is placed back in the population so that it is subject to selection again. If an item is selected twice because the random number is drawn twice, the item must be included twice

Sampling without replacement – requires the selected items be included in the sample only once

Random number generators – are computer programs used to provide any length list of random numbers applicable to a given population

- a standard program in all generalized audit software packages.

Systematic sampling - involves selecting every 21st item in the population following 1 or more random starting points

8. Perform the TOC procedures on sample items **COMPLIANCE TEST RISK MATRIX

COMPLIANCE TEST RISK MATRIXRelevant IC is in fact

Compliance test indicates reliance on relevant IC should be:

Adequate for Planned Reliance

Inadequate for Planned Reliance

ACCEPTED Correct Decision Risk of overreliance

REJECTED Risk of underreliance

Correct Decision

9. Evaluate the sample results He/she may use the following steps:

1. Determine the sample deviation rate by using the formula:

SAMPLE DEVIATION RATE – no. of deviations observed Sample size

2. Determine the upper deviation rate and the allowance for sampling risk by using the following relationship:

UPPER DEVIATION RATE = SAMPLE DEVIATION RATE + ALLOWANCE FOR SAMPLING RISK*

*determined from the standard tables showing max deviation rates at specified risks of overreliance

3. Compare upper deviation rate and the tolerable rate of deviation and evaluate the effectiveness of a control accordingly

a. Under the statistical sampling plan If: Upper deviation rate < Tolerable deviation rate

It is implied that a control is effective and the results would support assessing control risk below the maximum or 100%

Conversely,

If: Upper deviation rate > Tolerable deviation rate

It would suggest that a control is NOT effective and the results would support assessing control risk at max level or 100%

b. In a nonstatistical sampling application, sampling deviation rate is compared with tolerable rate of expected population deviation rate

If: sampling deviation rate < tolerable rate/ expected population deviation rate

internal control is considered effective and would support assessing control risk below max

On the other hand, if:

sampling deviation rate > tolerable rate/ expected population deviation ratecontrol is NOT considered effective and therefore the auditor would assess control risk to be maximum

4. Consider qualitative info such as evidence of deliberate manipulation/ circumvention of IC

5. Reach an overall conclusion. The

auditor must relate the assessed control risk to detection risk for each FS assertion.

10. Document the sampling procedure

** SUMMARY OF FACTORS INFLUENCING COMPLIANCE TEST SAMPLE SIZE**

Conditions leading toFACTORS SMALLER SAMPLE SIZE LARGER SAMPLE SIZE

1. Planned reliance on internal control

- compliance tests are not performed when no reliance on IC is planned

Lower reliance on IC Higher reliance on IC

2. Allowable rate of deviation (tolerable error)

Higher acceptable rate of deviation for planned reliance on IC

Lower acceptable rate of deviation for planned reliance on IC

3. Likely rate of population deviation

- larger samples are necessary when deviations occur, in order to obtain more precise conclusions. However, high deviation rate normally warrant little, if any, reliance on IC and, therefore, compliance testing might be omitted.

Lower expected rate of deviation in population

Higher expected rate of deviation in population

4. Required confidence level

Decrease in required confidence level

Increase in required confidence level

5. Number of items in population

Virtually no effect on sample size unless population is small

NONSTATISTICAL ATTRIBUTES SAMPLING

CHAPTER 18AUDIT SAMPLING FOR SUBSTANTIVE TESTS

SUBSTANTIVE PROCEDURES Concerned with amounts and are of 2 types:

1. Analytical procedures2. Test of details of transactions and balances

The purpose of substantive procedures is to obtain audit evidence to detect material misstatements in the FS

When performing substantive tests of details, audit sampling, and other means of selecting items for testing and gathering audit evidence may be used to verify 1 or more assertions about a FS amount or to make an independent estimate of some amount

Statistical sampling techniques that auditors use in performing substantive tests require testing a hypothesis

Hypothesis test – an auditor estimates the account balance using sampling techniques, and in comparing the estimated amount to a FS amount, determines whether the difference between the estimated amount and the recorded amount allows the auditor to accept the recorded amount as fairly stated.

RISKS IN SUBSTANTIVE TESTS Nonsampling risk – risk that arises from human error; it cannot be quantitatively measured

- adequate supervision, proper review of audit working papers, and adherence to quality control standards can all reduce nonsampling risk

Sampling risk – can be measured quantitatively and when performing substantive tests, an auditor encounters 2 types of sampling risk:

1. Risk of incorrect rejection- sometimes called ALPHA RISK- risk that a sample supports the conclusion that a recorded balance is not materially

misstated when it is not. - this risk relates to EFFICIENCY because when an auditor concludes that an account

balance is misstated, the auditor and/or client generally perform additional procedures when in fact, it is not necessary

2. Risk of incorrect acceptance- sometimes called BETA RISK- risk that a sample supports the conclusion that a recorded account balance is not

materially misstated when the account is actually materially misstated.- this risk relates to EFFECTIVENESS because an auditor who accepts a client’s account

balance that is materially misstated may express an unqualified opinion on FS that do not warrant such opinion.

VARIABLE SAMPLING PLAN TECHNIQUES are:1. Probability-proportional-to-size sampling technique, and2. Classical variable sampling technique

1. PROBABILITY-PROPORTIONAL-TO-SIZE-SAMPLING (PPS)

PPS sampling is a sampling technique auditors use to estimate the maximum amount of overstatement of a recorded amount with measurable level of risk of making a decision error

Sometimes referred to as PESO-UNIT SAMPLING (PU) or CUMULATIVE-MONETARY-AMOUNT SAMPLING (CMA) uses attribute sampling technique

PPS sampling gets its name from the fact that the probability of an item’s being selected for inclusion in the sample is equal to its size in proportion to the size of the whole population

The auditor may use PPS sampling to conclude that an account balance is fairly stated when the maximum overstatement is less than the tolerable misstatement

ADVANTAGES DISADVANTAGES1) It increases the likelihood of including high

peso value items in the sample1. The assumption of PPS sampling that the

audited value of a sampling unit is neither less than 0 nor greater than the book value may not be consistent with the auditor’s objectives

2. PPS allows auditors to compute the sample size and evaluate the results by hand or by means of tables

2) Negative balance or 0 balances cannot be audited using PPS sampling

3. PPS sampling is easy to use 3) When misstatements are found using PPS sampling, the upper misstatement limit may be too high to be useful

4. PPS sampling enables the auditor to state conclusion in a peso amount

2. CLASSICAL VARIABLES SAMPLING

Use normal distribution theory to evaluate selected characteristics of a population on the basis of a sample of items constituting the population

Assumes a two-trailed approach which is appropriate since classical variables sampling models generally test for both overstatement and understatement

Central Limit Theorem – for large samples (greater than or equal to 30) the distribution of sample means tends to be normally distributed about its own mean which is equal to the true population mean, even if the population is not normally distributed

Standard deviation – measures the dispersion among the respective amounts of a particular characteristic, for all items in the population for which a sample estimate is developed

VARIATIONS OF CLASSICAL VARIABLES SAMPLING1. Mean-per-unit estimation is a classical variables sampling technique that

projects the sample average to the total population by multiplying the sample average by the number of items in the population.

a) Determine audit values for each sample item

b) Calculate the average audit amountc) Multiply this average audit amount times

the number of units in the population to obtain the estimated population value.

2. Difference estimation Is a classical variable sampling technique that uses the average difference between audited amounts and individual recorded amounts to estimate the total audited amount of a population and an allowance for sampling risk

a) Determine audit values for each sample item

b) Calculate the difference between the audit value and book value for each sample item

c) Calculate the average differenced) Determine the estimated population value

by multiplying the average difference by the total population units and adding or subtracting this value from the recorded book value

3. Ratio estimation Is a classical variables sampling technique that uses the ratio of audited amounts to recorded amounts in the sample to estimate the total peso amount of the population and an allowance for sampling risk

a) Determine audit values for each sample item

b) Calculate the ratio between the sum of sample audit values and sample book values

c) Determine the estimated population value by multiplying the total population book value times this ratio.

4. Regression approach Similar to the difference and ratio approaches

This approach has the effect of using both the average ratio and the average difference in calculating an estimate of the total amount for the population

5. Difference and ratio estimation Are used as alternative to mean-per-unit estimation.

The auditor should use these approaches when applicable because they require a smaller sample size (i.e., they are more efficient than mean-per-unit estimation)

a) One factor in the calculation of sample size for classical variables sampling models is the estimated standard deviation

If the sd of differences/ ratios is smaller than the sd of audit values, these 2 methods will produce a smaller sample size

I. Difference estimation will be used if the difference between sample audit values and book values are a relatively constant peso amount, regardless of account size

II. Ratio estimation will be used if the differences are a constant percentage of book values

b) In order to use either difference/ ratio estimation, the following constraints must be met:

(1) the individual book values must be known and must sum to the total of book value (2) there must be more than a few differences (20 is often suggested as a minimum) between audit and book values.

c) These 2 methods will usually be more efficient than mean-per-unit estimation when stratification of the population is not possible

6. Stratification Separates a population into relatively homogenous groups to reduce the sample size by minimizing the effect of variation of items (i.e., the standard deviation) in the populations.

a) Although stratification may be applied with any of the classical methods, it is

most frequently used with the mean-per-unit estimation method

b) Know that the primary objective of stratification is to decrease the effect of variance in the total population and thereby reduce sample size

STEPS IN VARIABLES SAMPLING FOR SUBSTANTIVE TESTS1. Determine the objective(s) of the test2. Define population and sampling unit3. Determine the initial sample size4. Using random sampling techniques to identify the actual items to audit5. Audit the selected items and identify misstatements6. Evaluate the sample results7. Document the sampling procedure

APPLICATIONS OF THE STEPS IN VARIABLES SAMPLING PLAN USING PPS TECHNIQUE FOR SUBSTANTIVE TESTS

1. Determine the objective(s) of the test For example, a sampling plan applied to substantive tests of details is designed either:

(1) to estimate an account balance that is not recorded within an entity’s accounts (called PESO-VALUE ESTIMATION)(2) to test the reasonableness of a recorded account balance (HYPOTHESIS TESTING)

Once an objective is stated the auditor must then identify the CHARACTERISTICS OF INTEREST.Ex: auditor’s objectives is to determine whether an account is fairly stated, the characteristics might be defined as MONETARY ERROR, that is, monetary difference between recorded and audited peso amount

2. Define population and sampling unit An audit population – consists of all the items constituting an account balance or class of transaction

Sampling unit – any of the individual elements constituting a population or in PPS sampling, the sampling is the individual peso

Logical sampling unit – each peso selected is associated with an account or transaction

3. Determine the initial sample size SAMPLE SIZE (n) =

Book Value (BV) x Reliability factor for risk of overstatement (RF) Tolerable - Anticipated x ExpansionMisstatement Misstatement factor for anticipated misstatement (TM) (AM) (EF)

BV- this is the recorded book value of the population being tested

RF- determined using the table from the “AICPA Accounting and Audit Guide: Audit Sampling,” 1983 p. 117 for the level of risk of incorrect acceptance and the equation to determine the factor for the risk the auditor is willing to acceptFORMULA:

RISK OF INCORRECT ACCEPTANCE (TD) =AUDIT RISK (AR)

IR x CR x Analytical Procedures Risk (AP)

TM – Tolerable misstatement is an auditor’s planned level of materiality for an account balance or class of transaction

AM – Anticipated/ Expected Misstatement is the auditor’s preliminary estimate of the amount of material misstatement contained in the population. Usually based on the auditor’s past experience with and knowledge of the client

EF – Expansion factor is used only when the auditor anticipates finding misstatements in the sample

Component Relationship to Sample Size

Book value DirectRisk of incorrect Inverse

acceptanceTolerable misstatement InverseAnticipated misstatement

Direct

Expansion factor for anticipated misstatement

Direct

4. Using random sampling techniques to identify the actual items to audit

(SELECT THE SAMPLE)

Systematic sampling with a sampling interval/ random sampling to select the items to audit may be employed when using PPS sampling

Systematic sampling – means that an auditor selects every nth item for inclusion in the population

SAMPLING INTERVAL (SI) =

Book value of the population (BV)Sample size (n)

5. Audit the selected items and identify misstatements

(AUDIT THE ITEMS IN THE PPS SAMPLE)SUBSTANTIVE TEST RISK MATRIX

SUBSTANTIVE TEST RISK MATRIXEvidence indicates relevant account balance should be

Relevant account balance is in fact:

Fairly stated Not fairly stated

Accepted Correct Decision

Risk of Incorrect Acceptance

Rejected Risk of Incorrect Rejection

Correct Decision

6. Evaluate the sample results(EVALUATE THE PPS SAMPLE RESULTS)

UPPER MISSTATEMENT LIMIT =Projected Misstatement

+ Basic Precision* + Incremental Allowance*(*ALLOWANCE FOR SAMPLING RISK)

Projected Misstatement – auditor’s best estimate of the amount of misstatement in the population

For logical sampling units less than the sampling interval:PROJECTED MISSTATEMENT FOR EACH AUDITED ITEM =*Book Value – Audited Value x Sampling interval Book Value*TAINTING PERCENTAGE

For logical sampling units greater than or equal to the sampling interval:PROJECTED MISSTATEMENT FOR EACH AUDITED ITEM =Book Value – Audited Value

The projected misstatements equals the sum of the individual misstatements

Allowance for sampling risk – in PPS sampling, this is the sum of basic precision and an incremental allowance for sampling risk

Basic precision – measure of the closeness of the estimate of projected misstatement to the population misstatement

BASIC PRECISION =Sampling interval x Reliability factor for risk of incorrect acceptance

Reliability factor for risk of incorrect acceptance – can be obtained from table on page 117 of 1983 AICPA Accounting and Audit Guide: Audit Sampling

Incremental allowance for sampling risk – is an allowance in PPS sampling to incorporate risk arising from not auditing the entire sampling interval. It is computed by:

a) Ranking in descending order the projected misstatements for logical sampling units less than the sampling interval

b) Multiplying the ranked projected misstatements by the incremental change in reliability factor, and

c) Subtracting the projected misstatement for the logical units that are smaller than the sampling

interval

7. Document the sampling procedure When the auditor finds, for example, that the upper misstatement limit exceeds tolerable error, recorded book value may be overstated. If this occurs, the auditor could:

a) Examine additional logical units from the population

b) Perform additional substantive tests directed toward the same audit objective, and

c) Following the above steps, have the client correct the errors found, reduce the upper misstatement limit accordingly and compare the revised upper misstatement limit with tolerable error

APPLICATION OF CLASSICAL VARIABLES SAMPLING PLAN USING CLASSICAL TECHNIQUES Classical variables sampling plans enable auditors to estimate a numerical quantity such as the

peso amount of an account balance This makes these techniques particularly useful for performing substantive tests

APPLICATION OF CLASSICAL VARIABLES SAMPLING PLAN USING CLASSICAL TECHNIQUES1. Mean-per-unit estimation This technique enables auditors to estimate the mean audited

value of the items in a population with specified sampling risk and allowance for sampling risk (precision), by determining the mean audited value of the items in a sample.

TOTAL AUDITED VALUE OF THE POPULATION =Audited value of the sample (sample mean) x no. of items in the population

PROJECTED MISSTATEMENT=Estimated total audited value – client’s book value

When using the mean-per-unit estimation, auditor generally stratify the population to reduce the variations within homogenous subgroups

SAMPLE SIZE (n) =Population size (N) – Est. population sd (Sxj) x Std normal deviate

for desired risk of incorrect rejection (Ur)Desired allowance for sampling risk (A)

DESIRED ALLOWANCE FOR SAMPLING RISK (A) =Tolerable misstatement (TM) x Ratio of desired allowance for

sampling risk to tolerable misstatement (R)

ESTIMATED POPULATION VALUE (X) =Population size (N) x Ave. value of items in sample (x)

ACHIEVED ALLOWANCE FOR SAMPLING RISK (A)=Population size (N) x Factor for risk of incorrect rejection (Ur) x Standard error of the mean (Sxj / square root of n)

ESTIMATED RANGE OF POPULATION (P)=Expected population value (X) +- Achieved allowance for sampling risk (A’)

2. Ratio estimation The auditors use a sample to estimate the ratio of the audited value of a population to its book value

The ratio is estimated by dividing the total audited value of a sample by the total book value of the sample items

An estimate of the correct population value is obtained by multiplying this estimated ratio by the total book value of the population

3. Difference estimation The auditors use a sample to estimate the average difference between the audited value and book value of items in a population.

The average difference is estimated by dividing the audited value and book value of a sample by the no. of items in the sample

The total difference between the book value of the population and its estimated correct value is determined by multiplying the estimated average difference by the no. of items in the population

4. Regression

CHOOSING BETWEEN RATIO ESTIMATION AND DIFFERENCE ESTIMATION

In using ratio/ difference estimation technique, the following are required:a) Each population item has a book valueb) An audited value may be determined for each sample item, andc) Difference between audited and book values (misstatements)

When these requirements are met, ratio/ difference estimation is often more efficient than mean-per-unit estimation.

When the size of misstatements is nearly proportional to book values of the items, the use of ratio estimation is appropriate because large accounts have large misstatements and smaller accounts have small misstatements.

The difference estimation technique is more appropriate when the size of misstatements is not approximately proportional to book value.

NONSTATISTICAL SAMPLING FOR SUBSTANTIVE TESTS Auditors using nonstatistical sampling techniques employ the following 2 methods to estimate

the total misstatement in the population:1. Ratio method – they divide the sum of the misstatements in the audited sample by the book

value of the sample, and they then multiply the result by the book value of the population.2. Average difference method – auditors divide the sum of the misstatements in the audited

sample by the no. of items in the sample, and they then multiply the result by the no. of items in the population.

FACTORS AFFECTING SAMPLE SIZE FOR SUBSTANTIVE TESTSFactor Change in Factor Effect on Required Sample Size

Auditors requirements:

Risk of incorrect rejection

Increase Decrease

Risk of incorrect acceptance

Increase Decrease

Tolerable misstatement Increase DecreasePopulation characteristics:

Population size Increase Increase (if population is small) Standard deviation (if

classical variables sampling is used)

Increase Increase

Expected misstatement (if probability-proportional-to-size sampling is used)

Increase Increase