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Sampling and small populations Ljubljana, 12-13 October 2009

Sampling and small populations Ljubljana, 12-13 October 2009. Henk W. H. Geilen RE RA CISA Senior auditmanager Dutch Audit Authority. Outline. Introduction Why Sampling What is Sampling “Small populations” Conclusion. Why sampling ?. A true Story A long time ago……….

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Sampling and small populations Ljubljana, 12-13 October 2009

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  1. Sampling and small populations Ljubljana, 12-13 October 2009 Henk W. H. Geilen RE RA CISA Senior auditmanager Dutch Audit Authority Ljubljana, 12-13 October 2009

  2. Outline • Introduction • Why Sampling • What is Sampling • “Small populations” • Conclusion Ljubljana, 12-13 October 2009

  3. Why sampling ? • A true Story • A long time ago………. Ljubljana, 12-13 October 2009

  4. Ljubljana, 12-13 October 2009

  5. What’s the problem ? • Situation A (no errors) • What to do? • Stop? • Seen Enough? Ljubljana, 12-13 October 2009

  6. What is Sampling (2) • Situation B (errors found) • What to do? • Take more items? • Correcting only the errors found? • …… Ljubljana, 12-13 October 2009

  7. So….. • You never know whether you have seen enough items • So did you do “enough” work? • “Enough” means: • Not to few • Not to much Ljubljana, 12-13 October 2009

  8. Mathematics ? • Total number of balls • Number of red balls • Chance : Red /total • Number of draws • If you put the balls back after drawing • Multiply chances for each draw Ljubljana, 12-13 October 2009

  9. Mathematics/Formula • Red/(Red + White) * Red/( Red + White)… • Number of draws: n • Red / (Red +White) = Red / Total = p • Chance: ß • ß = pn Ljubljana, 12-13 October 2009

  10. So What ? • Audit • We know “total” • We don’t know “error” p • We don’t want to know “Chance” ß • ….. Ljubljana, 12-13 October 2009

  11. Audit • The “object” is good • We mean the “object" is good ENOUGH • So there can be a little error • Let’s call it MATERIALITY • Hé isn’t that p Ljubljana, 12-13 October 2009

  12. Chance • 100% assurance • Is this possible ? • Is this necessary ? • The auditor defines his risk • This means: the chance that he accepts the population while the error is higher than acceptable Ljubljana, 12-13 October 2009

  13. Where are we ? • We have a formula • ß = pn • Let’s say ß = 0,05 • So the risk is 5% or the Probability = 95 % • Say the materiality = 1 % (0,01) • Then we can calculate n = the number of samples • 0,05 = (1 - 0,01)n • N = 300 Ljubljana, 12-13 October 2009

  14. This means • If you want to accept 1 percent errors (max) • You sample 300 times • You find zero errors • The risk is (less than) 5 % • or • If you take a sample of 300 items and find 0 errors you know 95 % sure that the maximum error in the population is 1 percent. Ljubljana, 12-13 October 2009

  15. Playing with n • Different materiality: • f.e. 2 percent n = 300/ 2 = 150 • f.e. 0,5 percent n = 300 / 0,5 =600 • Different risk: • 10 percent : n * p = 231 • 15 percent : n * p = 190 Ljubljana, 12-13 October 2009

  16. Based on 0 errors • Risk 5 percent • 1 error : n * p = 475 • 2 errors : n * p = 630 • 3 errors : n * p = 776 • 4 errors : n * p = 916 • 5 errors : n * p = 1052 Ljubljana, 12-13 October 2009

  17. Risk • ß (Beta) Risk : The risk that you accept the population on the basis that the error is less or equal than the materiality • α (Alpha) Risk : The risk that you don’t accept the population while in fact there is no error Ljubljana, 12-13 October 2009

  18. What is important • Random / A-select • Meaning : each element has an equal change of being sampled • The sampled set has the same characteristics as the population Ljubljana, 12-13 October 2009

  19. First Conclusions Sampling : • A method to calculate how much work to do Parameters : • Materiality • Confidence level • Not size of the population Ljubljana, 12-13 October 2009

  20. Back to small populations • The questions are/stay: • How many items (operations) to audit • What items (operations) to audit • How to audit an item (operation) • Non statistical sample Ljubljana, 12-13 October 2009

  21. How can we do it (how much)? • Step 1: Audit all “big” items (big means > materiality) • Step 2: Calculate sample size for other items based on results system audit • This defines the confidence level Ljubljana, 12-13 October 2009

  22. In detail Total population (all operations) MinusBig operations Is Population to Sample MultiplyConfidence Level Is “tempresult” Devide(Population to sample) multiply Materiality Is Sample Size Ljubljana, 12-13 October 2009

  23. Ljubljana, 12-13 October 2009

  24. Example 141.596.219 (393 operations) Minus15.312.795 ( 4 operations) Is126.283.424 Multiply 1,21 (70 %) Is152.802.943 Divide2.525.668 (0.02 *126.283.424 ) Is 60 Sample Size (plus 4) Ljubljana, 12-13 October 2009

  25. Which items (1) • Random • Verify • Has sample same characteristics as population • Average.. • MA, Region etc • Looks like risk model Commission Ljubljana, 12-13 October 2009

  26. What items (2) • Not the same characteristics = • Draw again • After a few draws …. • Expand sample Ljubljana, 12-13 October 2009

  27. After the audit Total error fractions Divide sample size Is average error fraction Multiply population to sample Is projected error Plus error in big operations Is Total error Ljubljana, 12-13 October 2009

  28. Evaluation • Calculate materiality • Compare with total error • Example • 0,02 * 141.596.219 = 2.831.924 • Vs 5.096.546 Ljubljana, 12-13 October 2009

  29. What can you influence ? 141.596.219 (393 operations) Minus15.312.795 ( 4 operations) Is126.283.424 Multiply1,21 (70 %) Is152.802.943 Divide2.525.668(0.02 *126.283.424 ) Is 60 Sample Size (plus 4) Ljubljana, 12-13 October 2009

  30. Reminder • What was the goal? • Sample? • Audit? • Audit Opinion? Ljubljana, 12-13 October 2009

  31. Conclusions • No miracle solutions (Mathematic not Magic) • basic principles equal to statistic sampling • definition of the audit objective • definition of the population • definition of the characteristics to test • define confidence level, materiality • random selection Ljubljana, 12-13 October 2009

  32. More conclusions • non-statistical sample size rather large to support audit judgment • consider increasing tests of controls to determine confidence level for sampling • difficult to determine precision of error projection • representativeness of the sample needs to be evaluated Ljubljana, 12-13 October 2009

  33. Thank you for your attention! Henk Geilen h.w.h.geilen@rad.nl +31 6 3100 7794 Ljubljana, 12-13 October 2009

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