Statistical Sampling for Substantive Testing

Statistical Sampling for Substantive Testing

MULTIPLE CHOICE:

1. In applying variables sampling, an auditor attempts to

a. Estimate a qualitative characteristic of interest. b. Determine various rates of occurrence for specified attributes. c. Discover at least one instance of a critical error. d. Predict a monetary population value within a range of precision.

ANSWER: D

2. The measure of variability of a statistical sample that serves as an estimate of the population variability is the

a. Basic precision. b. Range. c. Standard deviation. d. Interval.

ANSWER: C

3. An auditor selects a statistical sample from a large inventory of replacement parts. Which of the following audit objectives would be most appropriate if the sampling method used is probability proportional to size (PPS)?

a. The auditor plans to estimate the total dollar value of the inventory when a recorded book value does not exist.

b. The auditor plans to make a statement concerning the total dollar amount of error in the population from the recorded book value. c. The auditor wishes to accept or reject the hypothesis that the proportion of defective parts in the population is less than 5%. d. The auditor wishes to estimate the proportion of defective parts in the population.

ANSWER: B

4. Precision is defined as the range (plus or minus) within which the true answer most likely falls. Reliability, also referred to as the confidence level, is the likelihood that the sample range contains the true value. Of the following statements concerning precision and reliability, which one is not true?

a. Precision is set by the auditor and is a function of materiality and risk.

b. The narrower the range of precision, the smaller the sample size.

c. An increase in control risk, other parameters remaining unchanged, causes a narrowing of the precision range.

d. Overall audit risk is the complement of reliability.

ANSWER: B

5. In an application of mean per unit sampling, the following information has been obtained:

Reported book value $600,000 Point estimate (estimated total value) 591,000 Allowance for sampling risk (precision) +- 22,000 Tolerable error +- 45,000

The appropriate conclusion would be that the reported book value is

a. Acceptable only if the risk of incorrect rejection is at least twice the risk of incorrect acceptance. b. Acceptable. c. Not acceptable. d. Acceptable only if the risk of incorrect acceptance is at least twice the risk of incorrect rejection.

ANSWER: B

6. Sampling risk refers to the possibility that:

a. The auditor may use a less than optimal statistical method for the circumstances, e.g. difference estimation instead of ratio estimation. b. The auditor may fail to recognize an error that is included in the sample. c. Even though a sample is properly chosen, it may not be representative of the population. d. The confidence level and/or precision established by the auditor are not appropriate.

ANSWER: C

7. The primary reason for an auditor to use statistical sampling is to

a. Obtain a smaller sample than would be required by non-statistical sampling techniques. b. Obtain a sample more representative of the population than would be obtained by non-statistical sampling techniques. c. Allow the auditor to quantify, and therefore control, the risk of making an incorrect decision based on sample evidence. d. Meet requirements of Statements on Auditing Standards.

ANSWER: C

8. The variability of a population, as measured by the standard deviation, is the

a. Extent to which the individual values of the items in the population are spread about the mean. b. Degree of asymmetry of a distribution. c. Tendency of the means of large samples (at least 30 items) to be normally distributed. d. Measure of the closeness of a sample estimate to a corresponding population characteristic.

ANSWER: A

9. Management has asked the internal auditing staff to evaluate the efficiency of the accounts payable system, particularly whether it would be economically feasible to reduce the number of discounts lost through slow payment. The audit staff decides that it needs to determine the amount currently being lost and can do so by taking a sample from payments made in the last six months, examining each, and recording the amount, if any, of the discount lost. This information can be used to project the total amount of discounts lost for the period. This proposed plan is an example of

a. Attribute sampling. b. Acceptance sampling. c. Variables sampling. d. Discovery sampling.

ANSWER: C

10. During an audit involving testing of accounts receivable balances, an auditor decides to specify a precision interval of $100,000 instead of the $200,000 precision interval that was called for in the sampling plan contained in the audit program. Which of the following would be a result of the auditor's decision to narrow the precision interval?

a. An increase in the required sample size. b. A decrease in the required sample size. c. An increase in the population standard deviation. d. A decrease in the population standard deviation.

ANSWER: A

11. To determine the number of items to be selected in a sample for a particular substantive test of details, the auditor should consider all of the following except

a. Tolerable error. b. Expected error. c. Allowable risk of incorrect acceptance. d. Characteristics of the population.

ANSWER: D

12. Sample size

a. Increases with the use of higher confidence levels. b. Decreases with the use of higher confidence levels. c. Remains unchanged with changes in confidence levels. d. Increases with the use of lower confidence levels.

ANSWER: A

13. In sampling for variables, which of the following must be

known in order to estimate the appropriate sample size required to meet the auditor's needs in a given situation?

a. The total amount of the population. b. The desired standard deviation. c. The desired confidence level. d. The estimated rate of error in the population.

ANSWER: C

14. In a variables sampling application, which of the following will result when the confidence level is changed from 90% to 95%?

a. Standard error of the mean will not be affected. b. Non-sampling error will decrease. c. Sample size will increase. d. Point estimate of the arithmetic mean will increase.

ANSWER: C

15. An auditor wishes to estimate inventory shrinkage by weighing a sample of inventory items. From past experience, the auditor knows that a few specific items are subject to unusually large amounts of shrinkage. In using statistical sampling, the auditor's best course of action is to

a. Eliminate any of the items known to be subject to unusually large amounts of shrinkage. b. Increase the sample size to lessen the effect of the items subject to unusually large amounts of shrinkage. c. Stratify the inventory population so that items subject to unusually large amounts of shrinkage are reviewed separately. d. Continue to draw new samples until a sample is drawn which includes none of the items known to be subject to large amounts of shrinkage.

ANSWER: C

16. In conducting a substantive test of an account balance, an auditor hypothesizes that no material error exists. The risk that sample results will support the hypothesis when a material error actually does exist is the risk of

a. Incorrect rejection. b. Alpha error. c. Incorrect acceptance. d. Type I error.

ANSWER: C

17. Which of the following best illustrates the concept of sampling risk?

a. A randomly chosen sample may not be representative of the population as a whole on the characteristic of interest.

b. An auditor may select audit procedures that are not appropriate to achieve the specific objective. c. An auditor may fail to recognize errors in the documents examined for the chosen sample. d. The documents related to the chosen sample may not be available for inspection.

ANSWER: A

18. Using the following results from a variables sample, compute the achieved precision:

Population size = 10,000 Sample size = 144 Sample standard deviation = $24.00 Confidence level = 90% (Z = 1.65) Mean = $84.00

a. $5,702 b. $33,000 c. $20,000 d. $16,500

ANSWER: B

19. Which of the following sampling plans would be designed to estimate a numerical measurement of a population, such as a dollar value?

a. Discovery sampling. b. Numerical sampling. c. Sampling for variables. d. Sampling for attributes.

ANSWER: C

20. An auditor is using the mean-per-unit method of variables sampling to estimate the correct total value of a group of inventory items. Based on the sample, the auditor estimates with precision of +-4% and confidence of 90% that the correct total is $800,000. This means that:

a. There is a 4% chance that the actual correct total is less than $720,000 or more than $880,000. b. There is a 10% chance that the actual correct total is less than $768,000 or more than $832,000. c. The probability that the inventory is not significantly overstated is between 6% and 14%.

d. The inventory is not likely to be overstated by more than 4.4% ($35,200) nor understated by more than 3.6%($28,800).

ANSWER: B

21. Which of the following statements is correct concerning the auditor's use of statistical sampling?

a. An auditor needs to estimate the dollar amount of the standard deviation of the population to use classical variables sampling. b. An assumption of PPS sampling is that the underlying accounting population is normally distributed. c. A classical variables sample needs to be designed with special considerations to include negative balances in the sample. d. The selection of zero balances usually does not require special sample design considerations when using PPS sampling.

ANSWER: A

22. Which of the following statements concerning alpha and beta risk is true?

a. As alpha risk increases, beta risk decreases.

b. As inherent risk and/or control risk increase, beta risk should also increase.

c. As inherent risk and/or control risk increase, beta risk should decrease.

d. As a "rule of thumb," the auditor generally sets beta risk equal to or less than 10%, inasmuch as it is the basis for the audit opinion.

ANSWER: C

23. Statistical samples as compared to non-statistical samples permit the auditor to

a. Quantify and control sampling risk. b. Eliminate non-sampling errors of every type. c. Obtain smaller sample sizes in all cases. d. Use less complex formulas than those required to evaluate non-statistical samples.

ANSWER: A

24. Probability-proportional-to-size (PPS) sampling is less efficient if

a. Computerized account balances are being audited. b. Statistical inferences are to be made. c. The audit objective is oriented to understatements. d. The account contains a large number of transactions.

ANSWER: C

25. A number of factors influences the sample size for a substantive test of details of an account balance. All other factors being equal, which of the following would lead to a larger sample size?

a. Lowering of assessed level of control risk. b. Lowering of assessed inherent risk through use of analytical review procedures. c. Smaller expected frequency of errors. d. Smaller measure of tolerable error.

ANSWER: D

26. For variables sampling purposes, changes in certain parameters affect sample size positively while changes in others have a negative effect. In this regard, which of the following statements is true?

a. An increase in beta risk reduces sample size.

b. Population size affects sample size inversely, i.e., as population size increases, sample size decreases.

c. An increase in alpha risk increases sample size.

d. As materiality (M) increases, sample size increases.

ANSWER: A

27. A population that is physically separated into two or more groups based on the sample variation being less than that

for the entire population is called a

a. Systematic sample. b. Judgment sample. c. Simple random sample. d. Stratified sample.

ANSWER: D

28. The standard deviation of a sample will usually decrease with

a. A decrease in sample size. b. The use of stratification. c. An increase in desired precision. d. An increase in confidence level.

ANSWER: B

29. An auditor is primarily concerned with substantial overstatements of accounts receivable balances and expects few, if any, errors. In an effort to concentrate on the large dollar values, the auditor would logically employ

a. Ratio estimation. b. Probability proportional to size sampling c. Discovery sampling. d. Mean per unit sampling.

ANSWER: B

30. Consider the following sampling plan: An auditor took a sample of 30 receiving reports completed in the last 12 months. The receiving reports were filed along with their corresponding requisitions and purchase orders by the date on which the goods were moved from the receiving department to their storage area in the warehouse. Sampling from a five-place random digit table, the first two digits were used to determine the month (from 1 to 12) and the next two digits to determine the day of the month (from 1 to 31). The auditor discarded random numbers for which corresponding dates did not exist or on which the warehouse was closed. For each selected date the auditor then took as the sample item the second report in the file for that day. Can the auditor rely on the statistical inference of this test?

a. Yes, because the plan eliminated any possible deliberate or unconscious bias in the auditor's selection of the sample items. b. Yes, because a random number table was used to make the selection. c. No, because the auditor sampled without replacement. d. No, because all items do not have an equal chance of being selected.

ANSWER: D

31. In testing accounts receivable, an auditor sends out positive confirmation requests to 100 randomly selected customers. A customer returns the confirmation indicating that the balance is correct when, in fact, the balance is overstated. This is an example of

a. Tainting. b. Sampling error. c. Standard error. d. Non-sampling error.

ANSWER: D

32. An auditor wishes to use stratified sampling to estimate the book value of a large accounts receivable file. The auditor has a complete listing of the file in account number order. The listing is several pages long and includes the account balance of each customer. Several customers have very large account balances while most balances are small. Which of the following would be the most appropriate means of classifying members of this population into strata?

a. Account balance. b. Account number. c. Alphabetically by name. d. Numerically by pages of the listing.

ANSWER: A

33. Auditors who prefer statistical to non-statistical sampling believe that the principal advantage of statistical sampling flows from its unique ability to

a. Define the precision required to provide adequate satisfaction. b. Provide a mathematical measurement of risk.

c. Establish conclusive audit evidence with decreased audit effort. d. Promote a more legally defensible procedural approach.

ANSWER: B

34. In a probability-proportional-to-size sample with a sampling interval of $10,000, an auditor discovered that a selected account receivable with a recorded amount of $5,000 had an audit amount of $2,000. The projected error of this sample was

a. $3,000 b. $4,000 c. $6,000 d. $8,000

ANSWER: C

35. An auditor selects a preliminary sample of 100 items out of a population of 1,000 items. The sample statistics generate an arithmetic mean of $60, a standard deviation of $6, and a standard error of the mean of $.60. If the sample was adequate for the auditor's purposes and the auditor's desired precision was plus or minus $1,000, the minimum acceptable dollar value of the population would be

a. $61,000. b. $60,000. c. $59,000. d. $58,800.

ANSWER: C

36. During the audit of inventories an external auditor specified a precision of five percent instead of the four percent contained in the preliminary audit program. What would be the impact of the change in precision?

a. A decrease in population standard deviation. b. An increase in population standard deviation. c. A decrease in required sample size. d. An increase in required sample size.

ANSWER: C

37. The inventory of a subsidiary consists of 12,980 items valued at $19,625,000. The inventory data are not available in computer readable form, but are contained in 217 pages of printout with 60 lines per page. Each page has a page total. Based on the last audit, the auditor expects there to be a few errors of overstatement. The most efficient

sampling technique to test the reasonableness of the stated value is

a. Stratified mean-per-unit estimation. b. Probability proportional to size sampling (PPS) c. Difference estimation. d. Ratio estimation.

ANSWER: B

38. Statistical sampling would be appropriate to estimate the value of an auto dealer's 3,000 line-item inventory because statistical sampling is:

a. Reliable and objective. b. Thorough and complete. c. Thorough and accurate. d. Complete and precise.

ANSWER: A

39. In a variables estimation application involving a book value of $550,000, with a tolerable error of specified precision limits of plus or minus $50,000, the auditor has calculated a point estimate of the population value to be $490,000. At the 95% reliability level, sample results indicate an achieved precision of plus or minus $50,000. Based on these data, the auditor would

a. Conclude that the book value is acceptable at the 95% reliability level. b. Reject the book value and extend control testing. c. Conclude that the book value is acceptable at a level of reliability higher than 95%. d. Extend substantive testing.

ANSWER: D

40. What effect does an increase in the standard deviation have on the required sample size of mean-per-unit estimation and probability proportional to size sampling? Assume no change in any of the other characteristics of the population and no change in desired precision and confidence.

Mean-per-unit Estimation PPS

a. Decrease in sample size No change in sample size

b. No change in sample size Decrease in sample size c. Increase in sample size No change in sample size

d. No change in sample size Increase in sample size

ANSWER: C

41. Use of the difference estimation sampling technique to estimate dollar amounts is inappropriate when

a. The total book value is known and corresponds to the sum of all the individual book values. b. A book value for each sample item is unknown. c. There are some observed differences between audited values and book values. d. The audited values are nearly proportional to the book value.

ANSWER: B

42. Select the description which illustrates sampling risk.

a. Applying audit procedures which are inappropriate for the audit objectives. b. Failing to recognize errors or deviations in the documents examined. c. Arriving at incorrect statistical conclusions due to computational errors. d. Choosing a sample which has proportionately more errors than the population.

ANSWER: D

43. A bank auditor is interested in estimating the average account balance of its depositors based on a sample. This substantive test is an example of

a. Attribute sampling. b. Discovery sampling. c. Acceptance sampling. d. Variables sampling.

ANSWER: D

COMPLETION:

44. Of the two variations of classical variables sampling,

often results in smaller sample sizes

and is therefore more cost-effective under those

circumstances.

ANSWER: DIFFERENCE ESTIMATION

45. Detection risk, defined as the risk that material errors or fraud that are not prevented or detected by the client's system of internal control, will not be discovered by the auditor, is referred to as by statisticians.

ANSWER: BETA RISK

46. Rejecting a book value that is stated fairly is referred to as risk.

ANSWER: ALPHA

47. Standard deviation is defined as the degree of variation of individual values about the population mean. The more narrowly dispersed the values in the population, the (larger or smaller) the standard deviation.

ANSWER: SMALLER

48. In applying probability proportional to size sampling, the auditor rejects a book value as being materially overstated when upper error limit is (greater or less) than the tolerable error.

ANSWER: GREATER

49. For PPS sampling purposes, the allowance for sampling risk is the sum of basic precision plus the .

ANSWER: INCREMENTAL ALLOWANCE

50. The incremental allowance for precision need be calculated only when errors are found in logical units that are (larger or smaller) than the _______________

.

ANSWER: SMALLER, SAMPLING INTERVAL

51. As contrasted with classical variables sampling, probability proportional to size sampling has the advantage of automatically the population such that logical units equal to or greater than the _____________

_________ have a 100% chance of being included in the sample.

ANSWER: STRATIFYING, SAMPLING INTERVAL

52. Probability proportional to size sampling has its greatest applicability to situations in which the auditor suspects a few material errors of .

ANSWER: OVERSTATEMENT

53. In evaluating sampling results for mean per unit sampling purposes, achieved precision (A') must be equal to or ______ (greater or less) than desired precision (A). Otherwise, the actual beta risk is higher than specified in the sampling plan, and further sampling would be necessary to reduce the risk to an acceptable level.

ANSWER: LESS

MATCHING:

54. Match the following terms with their definitions:

A. Projected error

B. Achieved precision

C. Basic precision

D. Desired precision

E. Finite correction factor

F. Range of acceptability

H. Sampling risk

I. Standard deviation

J. Upper error limit

K. Reliability

L. Sampling interval

M. Precision

N. Logical sampling unit

____1. The risk that the auditor’s conclusions regarding a population are incorrect.

____2. A measure of population variability.

____3. The range within which the true answer most likely falls.

____4. The likelihood that the sample range contains the true

value.

____5. Required for mean per unit and difference estimation

sampling applications when the preliminary sample size

equals or exceeds 5% of the population size.

____6. The calculated range within which the true answer most

likely falls.

____7. The range of possible values considered acceptable to the

auditor.

____8. The item to which a randomly selected dollar attaches.

____9. The distance between two consecutive sample items.

____10. Auditor’s best estimate of overstatement error.

SOLUTION:

1. H

2. I

3. M

4. K

5. E

6. B

7. F

8. N

9. L

10. A

PROBLEM/ESSAY:

55. An auditor wishes to randomly select a sample of customers' accounts receivable for confirmation purposes. The population consists of 4,000 accounts with a book value of $6,000,000. The preliminary estimate of standard deviation (computer calculated from book value) is $25. The auditor considers $40,000 to be a material misstatement of accounts receivable. The auditor has assessed inherent risk at 60% and control risk at 70%, and has set alpha risk and overall audit risk at 10%.

a. Assuming the following Z values, calculate the sample size:

ALPHA/2

OR BETA Z VALUE

1.00% 2.33

2.00 2.06

2.50 1.96

5.00 1.65

10.00 1.29

20.0 .85

24.0 .71

b. An examination of the sample items produces a sample mean of $1,200 and a standard deviation of the sample items equal to $22. What is the estimated population value?

c. What is the achieved precision?

d. What is the minimum audit adjustment based on these

findings?

SOLUTION:

a. 35

b. $4,800,000

c. $24,539

d. Downward adjustment of $1,184,4539

56. An auditor suspects that the client's ending merchandise

inventory is materially overstated and wishes to apply

probability proportional to size sampling to estimate the extent of overstatement. The inventory consists of 10,000 line items on computer printouts with a total book value of $14,000,000. The auditor has set the risk of incorrect acceptance at 5%, the tolerable error at $180,000, and the anticipated error at $70,000.

a. Given the following reliability and expansion factors at a 5% risk of incorrect acceptance, calculate sample size:

OVERSTATEMENT

ERRORS RELIABILITY FACTOR EXPANSION FACTOR

0 3.00 1.60

1 4.75

2 6.30

3 7.76

b. What is the sampling interval?

c. In auditing the sample, the auditor discovered the following errors:

Item Book Value Audited Value

A $20,000 0

E $800,000 $400,000

S $220,000 0

Calculate the projected error.

d. Determine the allowance for sampling risk and the upper

error limit.

SOLUTION:

a. 618

b. $22,653

c. $642,653

d. Allowance for sampling risk = $84,949 ($67,959 + $16,990)

Upper error limit = $727,602 ($642,653 + $84,949)