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**Auditing & Assurance Services, 6e**

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**David Letterman, American comedian and television host**

Module G Variables Sampling USA Today has come out with a new survey – apparently, three out of every four people make up 75 percent of the population. David Letterman, American comedian and television host

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Module G Objectives Define variables sampling and understand when it is used in the audit. Understand the basic process underlying monetary unit sampling (MUS) and when to use it. Identify the factors affecting the size of an MUS sample and calculate the sample size for an MUS application. Evaluate the sample results for an MUS by calculating the projected misstatement, incremental allowance for sampling risk, and basic allowance for sampling risk. Understand the basic process underlying classical variables sampling and the use of classical variables sampling in the audit. Understand the use of nonstatistical sampling for variables sampling.

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Variables Sampling Variables sampling is used to estimate the amount (or value) of a population Substantive procedures Estimate account balance or misstatement Compare estimated account balance or misstatement to recorded amount or tolerable misstatement Approaches Monetary unit sampling (MUS) Classical variables sampling

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**Major Topics Monetary Unit Sampling (MUS) Classical Variables Sampling**

Basics of MUS Determining Sample Size Selecting and Measuring Sample Items Evaluating Sample Results Classical Variables Sampling Nonstatistical Sampling

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**Monetary Unit Sampling (MUS)**

Defines the sampling unit as an individual dollar (or other monetary unit) in an account balance Auditor will select individual dollars (or monetary units) for examination Auditor will verify the entire “logical unit” containing the selected dollar (or monetary unit) Accounts receivable: Customer account Inventory: Inventory item

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**Advantages of MUS Results in more efficient (smaller) sample sizes**

Selects transactions or components reflecting larger dollar amounts Effective in identifying overstatement errors Asset and revenue accounts Generally simpler to use than classical variables sampling

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Disadvantages of MUS Provides a conservative (higher) estimate of misstatement Not effective for understatement or omission errors Liabilities and expenses Expanding sample is difficult if initial conclusion is to reject the account balance Requires special consideration for accounts with zero or negative balances

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**Major Topics Monetary Unit Sampling (MUS) Classical Variables Sampling**

Basics of MUS Determining Sample Size Selecting and Measuring Sample Items Evaluating Sample Results Classical Variables Sampling Nonstatistical Sampling

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**Effect of Factors on Sample Size**

How Determined Sampling risk (risk of incorrect acceptance) Inverse Using the audit risk model and based on prior assessments of audit risk, risk of material misstatement, and analytical procedures risk Tolerable misstatement Based on recorded account balance and relationship between the recorded account balance and important financial statement subtotals Expected misstatement Direct Based on prior experience with the client (for recurring audits) or a pilot sample (for initial audits) Population size Based on the recorded balance in the account balance or class of transactions

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**Summary: Sampling Risks Under Variables Sampling**

Decision Based on Population Account is not misstated (AM ≤ TM) Account is misstated (AM > TM) (ULM ≤ TM) Correct decision Risk of incorrect acceptance (ULM > TM) Risk of incorrect rejection Decision Based on Sample AM = Actual misstatement TM = Tolerable misstatement ULM = Upper limit on misstatements

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**Using MUS Tables See Exhibit G.2 for Sample Size Table Inputs**

Risk of incorrect acceptance Expected misstatement Tolerable misstatement Population size

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**Example Parameters Calculations Risk of incorrect acceptance = 5%**

Expected misstatement = $100,000 Tolerable misstatement = $500,000 Population size = $1,000,000 Calculations Ratio of expected to tolerable misstatement: $100,000 ÷ $500,000 = 0.20 Tolerable misstatement as a percentage of population: $500,000 ÷ $1,000,000 = 50%

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**Tolerable Misstatement as a Percentage of Population**

Step 3: Select column for TM as % of population = 50% Step 1: Select entries for risk of incorrect acceptance = 5% Risk of incorrect acceptance Ratio of Expected to Tolerable Misstatement Tolerable Misstatement as a Percentage of Population 50% 30% 10% 8% 5% - 6 10 30 38 0.10 8 13 37 46 0.20 16 47 58 0.30 12 20 60 75 0.40 17 27 81 102 Step 4: Read sample size at junction of row and column Step 2: Select row for ratio of EM to TM = 0.20

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**Major Topics Monetary Unit Sampling (MUS) Classical Variables Sampling**

Basics of MUS Determining Sample Size Selecting and Measuring Sample Items Evaluating Sample Results Classical Variables Sampling Nonstatistical Sampling

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**MUS: Selecting Sample Items**

Use systematic random sampling Calculate sampling interval as: Population size ÷ Sample size Process Identify random start Skip number of items equal to sampling interval Select item (dollar in account) and examine entire logical unit containing that item (customer account) May select same logical unit multiple times

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**MUS: Measuring Sample Items**

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**Major Topics Monetary Unit Sampling (MUS) Classical Variables Sampling**

Basics of MUS Determining Sample Size Selecting and Measuring Sample Items Evaluating Sample Results Classical Variables Sampling Nonstatistical Sampling

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**MUS: Evaluating Sample Results**

Determine the upper limit on misstatements, which has a (1 – Risk of incorrect acceptance) of equaling or exceeding the true amount of misstatement Components: Projected misstatement Incremental allowance for sampling risk Basic allowance for sampling risk

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**Projected Misstatement**

Assumes entire sampling interval contains same percentage of misstatement as the logical unit examined by auditors Calculated for each misstatement as: Sampling interval x Tainting % Do not project misstatements if the logical unit > sampling interval

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**Incremental Allowance for Sampling Risk**

Adjusts the projected misstatement to control exposure to risk of incorrect acceptance Allows for the possibility that the remainder of the sampling interval might be misstated by a higher percentage than the logical unit Procedure: Rank all projected misstatements in descending order Determine incremental confidence factor for each misstatement Multiply projected misstatement by (incremental confidence factor – 1)

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**Basic Allowance for Sampling Risk**

Provides a measure of the misstatement that might exist in sampling intervals in which a misstatement was not detected Calculated as: Sampling interval x Confidence factor

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**MUS: Evaluating Sample Results**

Projected Misstatement $ XX,XXX Incremental allowance for sampling risk XX,XXX Basic allowance for sampling risk Upper limit on misstatements $ XXX,XXX 1 2 3

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**Upper Limit on Misstatements**

If ULM = $50,000 and risk of incorrect acceptance = 5% $0 $50,000 95% probability (1 – risk of incorrect acceptance) 5% probability (risk of incorrect acceptance)

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**MUS: Making the Decision**

Upper Limit on Misstatement Tolerable Misstatement Account balance is not misstated ≤ Upper Limit on Misstatement Tolerable Misstatement Account balance is misstated >

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**Decisions under MUS Account balance is not misstated**

Suggest correction of identified misstatements Investigate cause of misstatements Account balance is misstated Increase sample size to attempt and reduce upper limit on misstatements Recommend adjustment to reduce misstatement below tolerable misstatement

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**Major Topics Monetary Unit Sampling (MUS) Classical Variables Sampling**

Basics of MUS Determining Sample Size Selecting and Measuring Sample Items Evaluating Sample Results Classical Variables Sampling Nonstatistical Sampling

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**Classical Variables Sampling**

Uses normal distribution theory and the central limit theorem to provide an estimated range of Recorded account balance or class of transactions Misstatement in an account balance or class of transactions Basic methodology Determine estimated range of account balance or misstatement Evaluate using tolerable misstatement

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**Additional Considerations in Classical Variables Sampling**

Consider the following additional factors in determining sample size Risk of incorrect rejection Population variability To reduce population variability, auditors may choose to stratify the population

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**Example Assume Recorded balance = $300,000**

Tolerable misstatement = $10,000 Estimated balance = $292,500 Precision = $2,275 Risk of incorrect acceptance = 10% Risk of incorrect rejection = 15%

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**90% probability of including true recorded balance**

Example (continued) Estimate ± Precision $292,500 ± $2,275 = $290,225 to $294,775 $290,225 $294,775 $300,000 90% probability of including true recorded balance Difference between recorded balance and far end of interval < Tolerable misstatement

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**Classical Variables Sampling Approaches**

Mean-per-unit: Assumes each item in population (component of account) has similar balance Estimates recorded balance by multiplying number of components by average audited value Difference estimation: Assumes each item in population (component of account) has similar difference between recorded and audited value Estimates the amount of misstatement by multiplying number of components by average misstatement Estimates recorded balance using estimated misstatement Ratio estimation: Assumes a constant percentage misstatement in population Estimates recorded balance by multiplying recorded balance by ratio of audited value to recorded balance

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**Sampling Methods MUS Classical Variables Sampling**

Overstatement errors are greatest concern Both overstatement and understatement errors are of concern Standard deviation difficult to estimate Standard deviation can be estimated Smaller number of misstatements anticipated Larger number of misstatements anticipated Population has high degree of variability and large dollar components exist Population is homogenous (in terms of dollar balances) and large dollar components do not exist

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**Major Topics Monetary Unit Sampling (MUS) Classical Variables Sampling**

Basics of MUS Determining Sample Size Selecting and Measuring Sample Items Evaluating Sample Results Classical Variables Sampling Nonstatistical Sampling

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**Nonstatistical Sampling**

Permissible under GAAS Does not permit auditors to control exposure to sampling risk Major differences in: Determining sample size Selecting sample items Evaluating sample results

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