Overview of Sampling

1. Overview of Sampling There are three kinds of lies: lies, damned lies, and statistics – Benjamin Disraeli ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

2. Audit Sampling • The application of an audit procedure to less than 100 percent of the items within an account balance or class of transactions for the purpose of evaluating some characteristic of the balance or class ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

3. How Audit Sampling Differs from Sampling in Other Professions • Generally evaluates whether amount is misstated rather than determine value. • Distribution is skewed. • Other evidence is gathered ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

4. When is Sampling Used? • Study and evaluation of internal control • Selecting control procedures to verify compliance • Attribute sampling • Substantive procedures • Selecting components or transactions of account balances for verification • Variables sampling • Dual Purpose tests ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

5. Sampling Risk • Risk that the decision made based on the sample differs from the decision that would have been made by examining the population • Cause is a nonrepresentative sample • Controlled by: • Determining an appropriate sample size • Ensuring that all items have an equal opportunity of selection • Mathematically evaluating sample results ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

6. Sampling Approaches • Statistical sampling methods use the laws of probability to: • Select sample items • Evaluate sample results • Statistical sampling methods control the auditor’s exposure to sampling risk • Nonstatistical sampling violates one or both of the above criteria • Both statistical sampling and nonstatistical sampling can be used in a GAAS audit ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

7. Major Steps in Sampling • Determine sample plan and method of selection • Determine the objective • Define characteristic of interest • Define the population • Determine sample size • Select the sample • Measure sample items • Evaluate sample results Planning Performing Evaluating ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

8. Factors Affecting Sample Size ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

9. Precision and Reliability • Precision (Allowance for sampling risk) • Closeness of sample estimate to true population value • Reliability (Confidence) • Likelihood of achieving a given level of precision ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

10. Example of Evaluating Sample Results • Precision Interval Estimate Estimate Estimate - Precision + Precision ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

11. Example of Evaluating Sample Results (Continued) • A 90% probability exists that the true average age is between 33 and 53 years (sample estimate  precision) • A 10% probability exists that the true average age is less than 33 years or greater than 53 years ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

12. Example of Evaluating Sample Results (Continued) • Precision Interval 33 yrs 43 yrs 53 yrs Estimate Estimate Estimate - Precision + Precision ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

13. Example of Evaluating Sample Results (Continued) • Precision Interval 33 yrs 43 yrs 53 yrs Estimate Estimate Estimate - Precision + Precision 90% ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

14. Attribute Sampling There are five kinds of lies: lies, damned lies, statistics, politicians quoting statistics, and novelists quoting politicians on statistics – Stephen K. Tagg ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

15. Attribute Sampling • Used to estimate the extent to which a characteristic exists within a population • Used in the auditor’s study of internal control • Goal: Estimate the rate at which the client’s internal control is failing to function effectively and compare to an allowable level (tolerable deviation rate) ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

16. Decisions in Attribute Sampling Sample Tolerable Rely on Deviation  Deviation controls as Rate Rate planned Sample Tolerable Reduce Deviation  Deviation planned reliance Rate Rate on controls ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

17. Risks Associated with Attribute Sampling ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

18. Summary of Sampling Risks • Effectiveness Losses • Risk of assessing control risk too low (overreliance) • Risk of incorrect acceptance • Efficiency Losses • Risk of assessing control risk too high (underreliance) • Risk of incorrect rejection ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

19. Major Steps in Attribute Sampling • Determine the objective • Define deviation conditions • Define the population • Determine sample size • Select the sample • Measure sample items • Evaluate sample results Planning Performing Evaluating ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

20. Major Steps in Attribute Sampling: Planning • Determine the objective of sampling • Identify key controls that the auditor intends to rely upon • Define deviation conditions • Instance in which control is not functioning as intended • Define the population • Should reflect all potential applications of the control during the period being examined ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

21. Major Steps in Attribute Sampling: Performing • Determine sample size • Sampling risk (Risk of assessing control risk too low) • Expected deviation rate • Tolerable deviation rate • Population size (not applicable in most instances) • Select sample items • Measure sample items ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

22. Factors Affecting Sample Size ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

23. How to Determine Sample Size? • Select AICPA Sample Size table corresponding to desired risk of assessing control risk too low • Identify row related to appropriate expected deviation rate (EDR) • Identify column related to appropriate tolerable deviation rate (TDR) • Determine sample size at junction of row for EDR and column for TDR ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

24. Sample Size Example • Parameters • Risk of assessing control risk too low = 5% • Expected deviation rate = 2% • Tolerable deviation rate = 7% ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

25. Sample Size Example (Continued) • From AICPA Table (5% risk) Tolerable Deviation Rate EDR 2% 3% 4% 5% 6% 7% 1.00% * * 156 93 78 66 2.00% * * * 181 127 3.00% * * * * 195 129 88 ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

26. Major Steps in Attribute Sampling: Performing • Determine sample size • Select sample items • For statistical sampling, use unrestricted random selection or systematic random selection • Haphazard selection or block selection are not appropriate for statistical sampling • Measure sample items ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

27. Major Steps in Attribute Sampling: Performing • Determine sample size • Select sample items • Measure sample items • Perform appropriate test of controls • Calculate sample deviation rate = No. deviations  sample size ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

28. Major Steps in Attribute Sampling: Evaluating • Evaluate sample results • Problem with sample deviation rate is that it may result from a nonrepresentative sample • Need to “adjust” sample deviation rate to control for the risk of assessing control risk too low • Calculate a Computed Upper Limit (CUL) ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

29. Evaluating Exceptions • Voided Documents • Unused or inapplicable documents • Misstatement in estimating sequence • Consider qualitative effect of deviations ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

30. Computed Upper Limit (CUL) • There is a (1 minus risk of assessing control risk too low) probability that the true population deviation rate is less than or equal to the CUL • There is a (risk of assessing control risk too low) probability that the true population deviation rate exceeds the CUL • Example: CUL = 6%, Risk of assessing control risk too low = 5% 95% probability5% probability 0% 6% ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

31. Components of the CUL CUL XXX Sample Deviation Rate (XXX) Allowance for Sampling Risk XXX or Sample Deviation Rate XXX Allowance for Sampling Risk XXX CUL XXX ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

32. How to Determine the CUL • Select AICPA Sample Evaluation Table corresponding to desired risk of assessing control risk too low • Identify row related to appropriate sample size • If cannot locate exactly, round down to next lowest sample size • Identify column related to number of deviations noted • Determine CUL at junction of row for sample size and column for deviations ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

33. CUL Example • Parameters • Sample size = 50 • Risk of assessing control risk too low = 5% • No. of deviations = 3 • Sample deviation rate 3  50 = 6% ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

34. CUL Example (Continued) • From AICPA Table (5% risk) No. of deviations found n 0 1 2 3 4 5 40 7.3 11.4 15.0 18.3 * * 50 5.9 9.2 12.1 17.4 19.9 60 4.9 7.7 10.2 12.5 14.7 16.8 14.8 ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

35. CUL Example (Continued) CUL = 14.8% Sample Deviation Rate = 6% Allowance for Sampling Risk = 8.8% ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

36. Making the Decision • If CUL > Tolerable Deviation Rate: • Conclude that internal control is not functioning effectively • Options • Increase sample size in hopes of supporting planned level of control risk • Increase level of control risk, leading to more effective substantive procedures (lower detection risk) ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

37. Making the Decision • If CUL  Tolerable Deviation Rate • Conclude that the internal control is functioning effectively • Options • Maintain planned level of control risk, leading to planned effectiveness of substantive tests • Consider a further reduction in control risk, leading to less effective substantive procedures (higher detection risk) ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

38. Sequential (Stop-or-Go Sampling) • Select initial (smaller) sample and consider results • Decision • Rely on control; discontinue sampling • Cannot rely on control • Select additional items; make decision • Discontinue sampling • Advantage is that evidence may support reliance on control with a relatively small sample size • Disadvantage is that auditor may continually extend the sample, creating inefficiencies ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

39. Discovery Sampling • Used when deviations from control are expected to be infrequent but very critical • Allows the auditor to • Determine the necessary sample size to find at least one example of a deviation if such deviations exist • Determine the probability that the rate of occurrence of a deviation is less than a specific (low) level ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

40. Non Statistical Test • If sample deviation rate exceeds expected rate then reject. ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

41. Variables Sampling USA Today has come out with a new survey-apparently three out of every four people make up 75% of the population – David Letterman ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

42. Variables Sampling • Used to estimate the amount (or value) of a population • Substantive procedures • Estimate the account balance or misstatement • Compare account balance or misstatement to recorded amount or tolerable error • Types of variables sampling approaches • Probability proportional to size (PPS) sampling • Classical variables sampling • Non Statistical Sampling ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

43. Decisions in Variables Sampling Sample Tolerable Account Estimate of  Error balance is Error (Materiality) not misstated Sample Tolerable Account Estimate of  Error balance is Error (Materiality) misstated ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

44. Risks Associated with Variables Sampling ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

45. Nonstatistical Sampling • Does not control the auditor’s exposure to sampling risk • Permitted under generally accepted auditing standards • Differences • Does not consider sampling risk in determining sample size or evaluating sample results • May use a nonprobabilistic selection technique (block or haphazard selection) ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

46. Basic Procedure for Nonstatistical Sampling • Select sample • Does not explicitly consider sampling risk in determining sample size • May use block or haphazard selection methods • Measure sample items • Evaluate sample results • Does not consider sampling risk in projected results to population • Compare determined misstatement to tolerable error ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

47. Nonstatistical Sampling for Substantive Tests of Details • Identify individually significant items • Determine sample size • Consider variation within the population ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

48. Factors Influencing Sample Size • Assessment of inherent risk • Assessment of control risk • Assessment of risk from other substantive tests • Tolerable misstatement • Expected errors and variance of population • Number of items in population ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

49. Sample Size Formula • Population's recorded amount X assurance factor Tolerable Misstatement ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.

50. Example • Recorded amount $190,000 • Examine 100 percent—12 items totaling $70,000 • Tolerable misstatement $4,000 • Inherent risk and control risk “slightly below the maximum” • Risk that other substantive procedures will fail to detect material misstatement—moderate • Sample Size= 60 ($120,000x2/$4,000) ©2005 by the McGraw-Hill Companies, Inc. All rights reserved.