The risk of incorrect acceptance and the likelihood of assessing control risk too low relate to the?
 Effectiveness of the audit.
 Efficiency of the audit.
 Preliminary estimates of materiality levels.
 Allowable risk of tolerable error.
Effectiveness of the audit.
The risk of incorrect acceptance and the likelihood of assessing control risk too low both relate to the effectiveness of the audit. The risk of incorrect acceptance is the risk that the auditor will accept a balance as fairly stated when it really isn't. The risk of assessing control risk too low is the risk that the auditor will conclude that the controls are effective when they are not effective.
In either case, the auditor has made an erroneous judgment that could result in the failure to detect a material misstatement and the issuance of an incorrect opinion.
An auditor may decide to increase the risk of incorrect rejection (i.e. wants to do more work than necessary) when?
 Increased reliability from the sample is desired.
 Many differences (audit value minus recorded value) are expected.
 Initial sample results do not support the planned level of control risk.
 The cost and effort of selecting additional sample items are low.
The cost and effort of selecting additional sample items are low.
The risk of incorrect rejection relates to the efficiency of the audit. If the auditor incorrectly believes that a material misstatement exists, the auditor will perform extra auditing procedures to try to find the material misstatement. After performing the extra work, the auditor will then reach the correct conclusion that the balance is fairly stated.
When the auditor decides to increase the risk of incorrect rejection, the auditor is increasing the risk of audit inefficiency, i.e., of doing too much work. This decision might seem appropriate if the cost and effort of selecting additional sample items are low.
In attribute sampling, a 25% change in which of the following factors will have the smallest effect on the size of the sample?
 Tolerable rate of deviation.
 Number of items in the population.
 Degree of assurance desired.
 Planned assessed level of control risk.
Number of items in the population
Population size is not considered in determining the sample size for an attributes sampling application. The tables that are used to determine the sample size for attributes sampling are based upon an assumption that the population is very, very large. Hence, an increase in the population size would have a negligible effect.
As the population size increases, the sample size should increase proportionately, but only for very small populations. There is virtually no effect on sample size for larger populations.
PPS Sampling Approach when account balance selected is:
 Greater than the sampling interval
 Less than the sampling interval
Remember that PPS sampling enables the auditor to directly control for the risk of incorrect acceptance by requiring the auditor to specify the desired level of that risk.
In a probabilityproportionaltosize application, the projected error of the sample is the amount of the difference between the book value and the audit value when the amount of the account examined is greater than the sampling interval.
When the recorded balance of the account involved is less than the sampling interval, the auditor must determine the "tainting" percentage and apply that percentage to the sampling interval.
Examples
 In a probabilityproportionaltosize sample with a sampling interval of $5,000, an auditor discovered that a selected account receivable with a recorded amount of $10,000 had an audit amount of $8,000. What is projected error?
 $2,000. As the selected account receivable was $10,000 and the sampling interval was $5,000, the projected error was $2,000 (the actual difference between the recorded amount and the audit value).
 In a PPS sampling application, the sampling interval was $6,000. The auditor discovered that a selected account receivable having a recorded amount of $5,000 had an audit amount of $1,000. What was the projected error associated with this sample?
 $4,800 In this case the tainting percentage = [($5,000 − $1,000)/$5,000] = 80%. Accordingly, the projected misstatement is $6,000 × 80% = $4,800.
Define Attribute Sampling and its intended use
Attribute Sampling is a statistical approach used with Tests of Controls (i.e. Internal Control).
It requires the use of a probabilistic sample selection method (random or systematic sampling). Attribute sampling allows the auditor to estimate the proportion of population items containing a specified characteristic. The characteristic auditors are concerned with for tests of controls is deviations from internal controls.
Sampling Risk vs. NonSampling Risk

Sampling risk results from performing an audit procedure on less than one hundred percent of the population. It represents the risk that the audit sample is not representative of the population. In other words, that the auditor's evaluation of a population based on an audit sample is different from what it would be if the entire population was tested.

Nonsampling risk results from human error. It represents the risk that the selected audit procedure is not appropriate for the intended purpose or the evidence from an audit procedure is misinterpreted. Nonsampling risk exists regardless of the number of items selected from a population for testing.
Sampling Risk
 Test of Controls
 Substantive Testing
There are two aspects to sampling risk when performing tests of controls:
 The risk of assessing control risk too low represents the risk that an audit sample supports the conclusion that the design and operation of an internal control is effective when in fact it is not.
 The risk of assessing control risk too high represents the risk that an audit sample supports the conclusion that the design and operation of an internal control is not effective when in fact it is effective.
Sampling Risk
 Test of Controls
 Substantive Testing
Similarly there are two aspects to sampling risk when performing substantive tests:
 The risk of incorrect acceptance represents the risk that an audit sample supports the conclusion that a material misstatement does not exist when in fact a material misstatement does exist. This risk is similar to the risk of assessing control risk too low.
 The risk of incorrect rejection represents the risk that an audit sample supports the conclusion that a material misstatement exists when in fact a material misstatement does not exist. This risk is similar to the risk of assessing control risk too high.
Sample size for attribute sampling can be determined by reference to attribute sampling tables. These sample determination tables require the auditor to establish three factors:
 also what is relation of factors to sample size?

Risk of assessing control risk too low represents the risk that the auditor concludes that the design and operation of an internal control is effective when in fact it is not. The level used for this risk is based on the auditor's desired control risk assessment. The lower the desired control risk assessment the lower the needed risk of assessing control risk too low. This risk is inversely related to sample size.

Expected Population Deviation Rate represents the auditor's best estimate of the population deviation rate. This rate is normally based on prior experience with the client. This rate is directly related to sample size.

Tolerable Deviation Rate represents the highest deviation rate the auditor could accept and still conclude that the design and operation of an internal control is effective. This rate is based on the tolerable misstatement relative to the number and dollar size of traansactions included in the population. Tolerable misstatement represents the maximum misstatement that could occur before the population would be considered materially misstated. The lower the required tolerable misstatement relative to the number and dollar size of transactions the lower the needed tolerable deviation rate. This rate is inversely related to sample size.
Probabilityproportionatetosize Sampling (Monetaryunit Sampling)
 PPS is used with substantive tests
PPS allows the auditor to estimate the total misstatement of a population. It is different from other sampling approaches used by auditors in that each dollar in the population is treated as a separate sampling unit instead of each customer, invoice, check, vendor, etc.
Also, PPS sampling requires special design considerations when there are negative balances, while classical methods do not.
PPS Sample Size Factors
 what is relationship of factors to sample size?
Sample size for a PPS application can be determined by reference to a PPS sampling table. The factors considered when determining sample size for substantive tests are:
 Risk of incorrect acceptance represents the risk that the auditor concludes that a material misstatement does not exist when in fact a material misstatement does exist. The level used for this risk is based on the auditor's planned detection risk and other planned substantive tests. A higher risk of incorrect acceptance is used with a higher planned detection risk and/or other planned substantive tests. This risk is inversely related to sample size.
 Expected Number of Misstated Dollars represents the auditor's best estimate of the number of sample dollars that will be misstated. This estimate is based on prior experience with the client and is normally expected to be zero when PPS is used. This estimate is directly related to sample size.
 Tolerable Misstatement represents the highest misstatement that could occur before the population would be considered materially misstated. This amount has an inverse relationship with sample size.
 Reported $ Balance represents the total recorded dollar balance in the population.
NonStatistical Sampling
Nonstatistical sampling can be used with tests of controls or substantive tests. Nonstatistical sampling does not require the use of a probabilistic selection method.
 The main advantage of nonstatistical sampling is that it is less complex and less time consuming than statistical sampling.
 The main disadvantage is that sampling theory cannot be used to quantify sampling risk.
NonStatistical Sampling Factors
 what is relationship of factors to sample size?
 remember same as attribute factors
Sample size for nonstatistical sampling is left entirely to the auditor’s professional judgement. The factors considered when determining sample size for tests of controls using a nonstatistical approach are the same as those considered for attributes sampling. The factors considered when determining sample size for substantive tests using a nonstatistical approach are:
 Risk of incorrect acceptance represents the risk that the auditor concludes that a material misstatement does not exist when in fact a material misstatement does exist. The level used for this risk is based on the auditor's planned detection risk and other planned substantive tests. A higher risk of incorrect acceptance is used with a higher planned detection risk and/or other planned substantive tests. This risk is inversely related to sample size.
 Risk of incorrect rejection represents the risk that the auditor concludes that a material misstatement exists when in fact a material misstatement does not exist. The level used for this risk is based on the cost and difficulty of obtaining additional evidence. A lower risk of incorrect acceptance is used when more costly or difficult evidence will be required if expanded testing is needed. This risk is inversely related to sample size.
 Expected Misstatement represents the auditor's best estimate of the population misstatement. This estimate is normally based on prior experience with the client. This estimate is directly related to sample size.
 Tolerable Misstatement represents the highest misstatement that could occur before the population would be considered materially misstated. This amount has an inverse relationship with sample size.
Attributes Sampling Steps
Note that AS is used exclusively with tests of controls
Attributes sampling (used in tests of controls) reaches a conclusion in terms of a rate of occurrence.
 Determine the objectives of the test
 Define the deviation conditions (this is different from VS!!!)
 Define the population
 Define the period covered by the test
 Define the sampling unit
 Consider the completeness of the population
Variables Sampling Steps
 Types of Variable Sampling
 Steps
Note that VS is used exclusively with substantive testing of details
Variables sampling (used in substantive testing) reaches a conclusion in dollar amounts (or possibly in units).
Types
 Probabilityproportionaltosize (PPS) sampling
 Classical variables sampling techniques
Steps
 Determine the objectives of the test
 Define the population
 Define the sampling unit
 Consider the completeness of the population
 Identify individually significant items
 Select an audit sampling technique
Systematic Sampling
Every nth (population size/sample size) item is selected after a random start. When a random starting point is used, this method provides every sampling unit in the population an equal chance of being selected. If the population is arranged randomly, systematic selection is essentially the same as random number selection.
 Advantage: population items do not have to be numbered.
 Disadvantage: the population may be systematically ordered, this can be overcome by using multiple random starts.
Difference and Ratio Estimation Sampling
Alternatives to meanperunit estimation (i.e. more efficient than MPU)
 Difference estimation will be used if the differences between sample audit values and book values are a relatively constant dollar amount, regardless of account size.
 Ratio estimation will be used if the differences are a constant percentage of book values.
In order to use either difference or ratio estimation, the following constraints must be met:
 The individual book values must be known and must sum to the total book value.
 There must be more than a few differences (20 is often suggested as a minimum) between audit and book values.
The ratio estimation sampling technique is based upon comparing the ratio of the book value to the audited value of the sampled items. This method cannot be used when book values for the sample items are unknown.
Audit Sampling Definition
Audit sampling is the application of an audit procedure to less than 100% of the items within an account balance or class of transactions for the purpose of evaluating some characteristic of the balance or class.
PPS vs. Variables Sampling
Advantages of PPS sampling  most common in actual practice
 Generally easier to use.
 Size of sample not based on variation of audited amounts.
 Automatically results in a stratified sample.
 Individually significant items are automatically identified.
 Usually results in a smaller sample size if no misstatements are expected.
 Can be easily designed and sample selection can begin before the complete population is available.
Advantages of classical variables sampling
 May result in a smaller sample size if there are many differences between audited and book values.
 Easier to expand sample size if that becomes necessary.
 Selection of zero balances does not require special sample design considerations.
 Inclusion of negative balances does not require special sample design considerations.
Which of the following is the primary objective of probabilityproportionaltosample (PPS) size?
To identify overstatement errors
 PPS sampling is most effective in detecting overstatements, since the likelihood of an item's selection increases with the recorded magnitude of the item.
An auditor examining inventory may appropriately apply sampling for attributes in order to estimate the?
 Average price of inventory items.
 Percentage of slowmoving inventory items.
 Dollar value of inventory.
 Physical quantity of inventory items.
Percentage of slowmoving inventory items.
Remember that attribute sampling is used to determine a rate of occurrence as a test of controls.
Attribute sampling expresses a conclusion about the population in terms of a rate of occurrence. Accordingly, determining the percentage of slowmoving inventory items would be an appropriate attribute sampling application.
What should the auditor do when the actual deviation rate plus the allowance for sampling rate exceed the tolerable rate?
When the deviation rate plus the allowance for sampling risk exceeds the tolerable rate, the assessed level of control risk may increase.
There are many kinds of statistical estimates that an auditor may find useful, but basically every accounting estimate is either of a quantity or of an error rate. The statistical terms that roughly correspond to "quantities" and "error rate," respectively, are:
 Attributes and variables.
 Variables and attributes.
 Constants and attributes.
 Constants and variables.
Variables and attributes.
 Variables sampling is measurement of a continuous variable (i.e., a variable such as number of items in inventory or amount of accounts receivable). The variable being estimated can take on a continuum of values (e.g., zero to infinity). Variable sampling is based upon the normal distribution.
 Attribute sampling is based on the binomial distribution and sample items must be binary in nature (i.e., right or wrong, correct or incorrect, etc.) Attribute sampling is most applicable to tests of compliance and variables sampling is most applicable to substantive tests.
In estimation sampling for attributes, which one of the following must be known in order to appraise the results of the auditor’s sample?
 Estimated dollar value of the population.
 Standard deviation of the values in the population.
 Actual occurrence rate of the attribute in the population.
 Sample size.
Sample size. The sample size is needed in order to evaluate the results of the attribute sampling since it is used to calculate the deviation rate for the sample.
When using a statistical sampling plan, the auditor would probably require a smaller sample if the:
 Population increases.
 Desired sampling risk decreases.
 Desired sampling risk increases.
 Expected error occurrence rate increases.
Desired sampling risk increases.
Increased sampling risk allows a larger margin of error and suggests a smaller sample size. i.e. The auditor can use a smaller sample size if the auditor is willing to accept a higher level of risk.
In PPS sampling, sample size is calculated as:
("Reliability factor" × Book value)
Tolerable misstatement (net)
In PPS sampling, sample size is determined by the following: (1) a "reliability factor" from applicable AICPA tables; (2) the population book value; and (3) the tolerable misstatement, net of any expected misstatements.
True or False:
For accounts having book values less than the sample interval, the auditor calculates "projected misstatement" by applying the "tainting percentage" to the sample interval from which that account was selected.
TRUE
In planning a statistical sample for a test of controls, an auditor increased the expected population deviation rate from the prior year’s rate because of the results of the prior year’s tests of controls and the overall control environment. The auditor most likely would then increase the planned
 Tolerable rate.
 Allowance for sampling risk.
 Risk of assessing control risk too low.
 Sample size.
Sample size.
An increased population deviation rate will require an increase in sample size when the planned assessed level of control risk is to remain the same.
Expected Population Deviation Rate (EPDR)
 if EPDR > Tolerable = ?
 Relationship btw. EPDR and Sample Size?
Expected population deviation rate (expected rate of occurrence)—An estimate of the deviation rate in the entire population.
 If the expected population deviation rate exceeds the tolerable rate, tests (tests of controls/attributes sampling) will not be performed.
 Although the risk of assessing control risk too high is often not explicitly controlled when determining attributes sample size, it can be controlled to some extent by specifying a conservative (larger) expected deviation rate.
 There is a direct relationship (e.g., as one increases, the other also increases) between the expected deviation rate and sample size.
The expected population deviation rate is determined by?
 Typically determined by using last year’s deviation rate adjusted judgmentally for current year changes in the control procedure, or by determining the deviation rate in a small preliminary sample.
 Used only to determine sample size and not to evaluate sample results, so the estimate need not be exact.
Because a deviation from a control procedure does not necessarily result in a misstatement (e.g., an unapproved invoice may still represent a valid business expenditure), the rate of misstatements is generally?
Lower than the expected population deviation rate.
What is a dual purpose test?
 The size of a sample designed for dualpurpose testing should be?
Dualpurpose tests are those in which a single sample is used to test a control and to serve as a substantive test of a recorded balance or class of transactions. When a dualpurpose test is used, auditors select the sample size as the higher of that required for the two purposes. For example, if the test of control test required 35 items, and the substantive test required 40, both tests would be performed using the 40 items.
 The larger of the samples that would otherwise have been designed for the two separate purposes.
When to use Attributes Sampling vs. Variables Sampling

Attributes sampling is used to estimate the error rate for internal control activities. Attributes sampling is generally used when there is a trail of documentary evidence.

Variables sampling is used when the auditor wishes to estimate dollar or quantity errors, not when estimating error rates. i.e. substantive testing.
Remember that tests of controls are designed to provide reasonable assurance that internal control is operating effectively. For example, attributes sampling might test controls for voucher processing, billing systems, payroll systems, inventory pricing, fixedasset additions, and depreciation computations.
The major reason that the difference and ratio estimation methods would be expected to produce audit efficiency is that the
 Number of members of the populations of differences or ratios is smaller than the number of members of the population of book values.
 Beta risk may be completely ignored.
 Calculations required in using difference or ratios estimation are less arduous and fewer than those required when using direct estimation.
 Variability of the populations of differences or ratios is less than that of the populations of book values or audited values.
Variability of the populations of differences or ratios is less than that of the populations of book values or audited values.
Difference and ratio estimation methods measure the difference between audit and book values or the ratio of audit to book values. As these differences should not be great, the population of these differences will have little variance. In statistical sampling the less variation in a population, the smaller the required sample to provide an estimate of the population. In other words, difference and ratio estimation methods are more efficient because the differences between audit and book values are expected to vary less than the actual items in the population.
What is Discovery Sampling?
A special case of sampling for attributes. In discovery sampling there should be no opportunity to observe more than one occurrence, because it is designed to sample for serious or critical errors. Once a critical or serious error is discovered, the sampling plan will probably be abandoned and a more comprehensive examination undertaken.
In assessing sampling risk, the risk of incorrect rejection and the risk of assessing control risk too high relate to the
 Efficiency of the audit.
 Effectiveness of the audit.
 Selection of the sample.
 Audit quality controls.
Efficiency of the audit. These two errors generally result in an auditor performing unnecessary additional procedures.
Which of the following ordinarily would be the greatest consideration in planning an effective sample for a test of controls?
 Preliminary judgments about materiality levels.
 The auditor’s allowable risk of assessing control risk is too high.
 The level of detection for the account.
 The auditor’s allowable risk of assessing control risk is too low.
The auditor’s allowable risk of assessing control risk is too low. Auditors use attributes sampling for tests of controls, and to use attributes sampling, auditors need the
 allowable risk of assessing control risk too low,
 the expected population deviation rate,
 and the tolerable rate.
Which of the following would be a consideration in planning a sample for a test of subsequent cash receipts?
 Preliminary judgments about materiality levels.
 The amount of bad debt writeoffs in the prior year.
 The size of the intercompany receivable balance.
 The auditor’s allowable risk of assessing control risk is too low.
Preliminary judgments about materiality levels. The auditor must determine which differences are material and which are not so to determine the size and nature of the sample.
Remember that subsequent cash receipts is often used as a substantive procedure and the allowable risk of assessing control risk is too low relates to internal control!
Which of the following statistical selection techniques is least desirable for use by an auditor?
 Systematic selection.
 Stratified selection.
 Block selection.
 Sequential selection.
Block selection.
Sample items should be selected in such a way that the sample can be expected to be representative of the population allowing all items in the population to have an equal opportunity of being selected. A block sample consisting of sequential items may preclude items from meeting the above criteria.
If certain forms are not consecutively numbered
 Selection of a random sample probably is not possible.
 Systematic sampling may be appropriate.
 Stratified sampling should be used.
 Random number tables cannot be used.
Systematic sampling may be appropriate. Systematic sampling is a procedure where a random start is obtained and then every nth item is selected. For example, a sample of forty from a population of a thousand would require selecting every 25th item after obtaining a random start between items 1 through 25. Therefore, although the forms are not consecutively numbered, the use of systematic sampling may be appropriate.
If the size of the sample to be used in a particular test of attributes has not been determined by utilizing statistical concepts, but the sample has been chosen in accordance with random selection procedures
 No inferences can be drawn from the sample.
 The auditor has committed a nonsampling error.
 The auditor may or may not achieve the desired risk of assessing control risk too low.
 The auditor will have to evaluate the results by reference to the principles of discovery sampling.
The auditor may or may not achieve the desired risk of assessing control risk too low.
Although the use of random sampling makes it possible to calculate statistical conclusions, the wrong size sample will affect the achieved risk of assessing control risk too low. For example, a sample which is too small will lead to a higher level of assessing control risk too low (more sampling risk) than desired.
If an auditor, planning to use statistical sampling, is concerned with the number of a client’s sales invoices that contain mathematical errors, the auditor would most likely utilize
 Random sampling with replacement.
 Sampling for attributes.
 Sampling for variables.
 Stratified random sampling.
Sampling for attributes. The auditor is concerned with the existence of an error or the nonexistence of an error; therefore, attribute sampling is appropriate.
ATTRIBUTES SAMPLING SUMMARY OF RELATIONSHIPS TO SAMPLE SIZE
Increases in (Effects on Sample Size)
 Risk of assessing control risk too low (Decrease)
 Tolerable rate (Decrease)
 Expected population deviation rate (Increase)
 Population (Increase (slightly for large samples))