module e

Module E - Overview of Sampling

MODULE E

Overview of Sampling

LEARNING OBJECTIVES

Review Checkpoints

Exercises, Problems, and Simulations

1. Understand the basic principles of sampling, including the differences between statistical and nonstatistical sampling and sampling and nonsampling risk.

1, 2, 3, 4, 5, 6 51, 52, 53, 54, 55 (parts a – c), 56, 63 (parts a – c), 66, 67, 71 (parts a – e)

2. Understand the basic steps and procedures used in conducting a sampling plan.

7, 8, 9, 10, 11, 12, 13

55 (part d), 57, 58, 59, 60, 61, 62, 63 (parts d – g), 64, 65, 68, 70

3. Identify the two situations in which sampling is used in an audit examination.

14, 15, 16, 17, 18, 19, 20, 21

71

4. Understand how the basic steps and procedures used in a sampling plan apply to an audit examination.

22, 23, 24 69, 72, 73, 74

SOLUTIONS FOR REVIEW CHECKPOINTS

E.1 Sampling can be used by the auditor during the study and evaluation of a client’s internal control and the substantive procedures.

E.2 Sampling risk is the possibility that the decision made based on the sample differs from the decision that would have been made if the entire population had been examined, a sampling error. Sampling error arises when the sample drawn from the population does not appropriately represent that population.

E.3 Nonsampling risk represents the probability that an incorrect conclusion is reached because of reasons unrelated to the nature of the sample, a nonsampling error. Nonsampling error arises because of errors in judgment or execution of the sampling plan.

E.4 Sampling risk is controlled by the auditor by (1) determining an appropriate sample size and (2) evaluating sample results to consider the possibility that the sample does not appropriately represent the population.

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E.5 Statistical sampling plans apply the laws of probability to select sample items for examination and evaluate sample results. Statistical sampling plans differ from nonstatistical sampling plans in terms of the methods used to determine the appropriate sample size and evaluate the sample results. In a statistical sampling plan, these methods control exposure to sampling risk, whereas they do not do so in a nonstatistical sampling plan.

E.6 Either statistical sampling or nonstatistical sampling can be used under generally accepted auditing standards. Nonstatistical sampling should not be used solely to reduce sample size.

E.7 1. Determine the objective of sampling2. Define the characteristic of interest3. Define the population4. Determine the sample size5. Select sample items6. Measure sample items7. Evaluate the sample results

E.8 It is important to carefully define the population of interest, since the results of the entire sampling application will be based on the population from which the sample is drawn.

E.9 Sampling risk has an inverse relationship with sample size; that is, as a lower level of sampling risk is necessary, the individual needs to select a larger sample (and vice versa).

E.10 Four methods commonly used to select sample items are (1) unrestricted random selection, (2) systematic random selection, (3) haphazard selection, and (4) block selection.

When using unrestricted random selection, a series of random numbers is identified and the random numbers are matched to numbered items in the corresponding population.

When using systematic random selection, a random starting point is selected within the population. A fixed number of items are bypassed and the corresponding item in the population is selected. This process is continued until a number of items equal to the appropriate sample size is selected.

Haphazard selection identifies sample items in a nonsystematic manner, with no deliberate effort to match random numbers to sample items.

Block selection identifies a series of contiguous (adjacent) units for selection.E.11 Unrestricted random selection or systematic random selection are used with statistical sampling

because these methods (1) provide a reasonable likelihood of selecting a representative sample, (2) allow the probability of selecting sample items to be determined, and (3) allow the sample selection process to be replicated.

E.12 The precision (or allowance for sampling risk) is the numeric distance from the estimated population value in which the true (but unknown) population value may lie with a given probability.

Reliability (or confidence) is the likelihood of achieving a given level of precision.

The precision interval is a range around the sample estimate that has a likelihood equal to reliability (or 100 percent minus the sampling risk) of including the true population value.

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E.13 The following are the basic steps in evaluating sample results:

1. Select and measure sample items to determine the sample estimate.2. Based on the acceptable sampling risk, determine the reliability and related precision.3. Form the precision interval by adding and subtracting the precision from the sample

estimate.4. Determine whether the hypothesized (or acceptable) value falls within the precision

interval.

E.14 Attribute sampling is used to determine the extent to which a particular characteristic (or attribute) exists within a population. In an audit examination, attribute sampling is used in the study and evaluation of internal control and subsequent assessment of control risk.

E.15 The tolerable deviation rate is the maximum rate of deviations from a control that an auditor will permit without reducing the planned reliance on internal control. The auditor will compare an “adjusted” sample deviation rate (upper limit deviation rate) to the tolerable deviation rate to determine the extent to which the auditor can rely on internal control.

E.16 The risk of assessing control risk too high (risk of underreliance) occurs when the auditor’s sample indicates that the control is not functioning effectively when, in fact, it is doing so. When this risk occurs, the auditor’s upper limit deviation rate exceeds the tolerable deviation rate. However, unknown to the auditor, the true population deviation rate is less than the tolerable deviation rate.

The risk of assessing control risk too low (risk of overreliance) occurs when the auditor’s sample indicates that the control is functioning effectively when, in fact, it is not. When this risk occurs, the auditor’s upper limit deviation rate is less than the tolerable deviation rate. However, unknown to the auditor, the true population deviation rate exceeds the tolerable deviation rate.

The assessing control risk too high results in an efficiency loss for the auditor, since more extensive substantive procedures are performed than is necessary. The assessing control risk too low exposes the auditor to an effectiveness loss, since the auditor’s substantive procedures will not reduce audit risk to the acceptable level.

E.17 The risk of assessing control risk too low is of greater concern to the auditor, since it may eventually result in the auditor issuing an unqualified opinion on financial statements that are materially misstated.

E.18 Variables sampling is used to examine a population when the auditor wants to estimate the amount (or value) of some characteristic of that population. Variables sampling is used by the auditor when performing substantive procedures to evaluate the fairness of an account balance or class of transactions.

E.19 Tolerable error is the amount of misstatement that the auditor is willing to allow in an account balance or class of transactions without concluding that it is materially misstated. The auditor will compare an “adjusted” sample misstatement (upper error limit) to the tolerable error to determine whether the account balance is materially misstated.

E.20 The two sampling risks associated with variables sampling are the risk of incorrect acceptance and the risk of incorrect rejection. The risk of incorrect acceptance is the likelihood that the sample results indicate the account balance is fairly stated when, in fact, it is materially misstated. The risk of incorrect rejection is the likelihood that the sample results indicate the account balance is materially misstated when, in fact, it is fairly stated.

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Incorrect acceptance exposes the auditor to an effectiveness loss, because the auditor will make an incorrect conclusion and issue an inappropriate opinion on the financial statements.

Incorrect rejection exposes the auditor to an efficiency loss, because additional transactions or components will be examined by the auditor prior to proposing an adjustment to the client’s account balance.

E.21 The auditor is more concerned with the risk of incorrect acceptance because it may result in issuing an unqualified opinion on financial statements that are materially misstated.

E.22 The objective of attribute sampling is to assess the operating effectiveness of a key control. The objective of variables sampling is to estimate the amount of misstatement in an account balance or class of transactions.

E.23 The factors that affect the sample size in an attribute sampling application (as well as their relationship to sample size) are shown below:

Population size (direct relationship) Expected deviation rate (direct relationship) Tolerable deviation rate (inverse relationship) Sampling risk (inverse relationship)

E.24 The factors that affect the sample size in a variables sampling application (as well as their relationship to sample size) are shown below:

Population size (direct relationship) Expected error (direct relationship) Tolerable error (inverse relationship) Sampling risk (inverse relationship) Population variability (direct relationship)

SOLUTIONS FOR MULTIPLE-CHOICE QUESTIONS

E.25 a. Incorrect When sampling, the auditor performs procedures on less than 100 percent of the items in a balance.

b. Correct When sampling, the auditor performs procedures on less than 100 percent of the items in a balance to form a conclusion about the entire balance.

c. Incorrect Becoming familiar with an accounting system is not an application of audit sampling.

d. Incorrect Analytical procedures are not an application of audit sampling.

E.26 a. Incorrect Statistical sampling is characterized by statistical calculation of the results.

b. Incorrect Statistical sampling is characterized by representative sample selection.c. Correct Statistical sampling is characterized by both representative sample

selection and statistical calculation of the results.d. Incorrect Statistical sampling is characterized by both representative sample

selection and statistical calculation of the results.

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E.27 a. Incorrect Sampling is typically most appropriate for populations consisting of a large number of items.

b. Correct Sampling is appropriate when the need for precise information about the population is less important.

c. Incorrect More critical decisions would typically increase the need to examine the entire population.

d. Incorrect Sampling is not appropriate when the costs of an incorrect decision are extremely high.

E.28 a. Incorrect Audit risk is the risk that the auditor issues an unqualified opinion on financial statements that are materially misstated.

b. Incorrect There is no term known as examination risk.c. Correct This response represents the correct definition of sampling risk.d. Incorrect Nonsampling risk is related to errors in judgment and execution and not

to the nature of the sample.

E.29 NOTE TO INSTRUCTOR: Since this question asks students to identify the statement that will not assist in controlling the auditor’s exposure to sampling risk, the response labeled “correct” will not assist in controlling the auditor’s exposure to sampling risk and the those labeled “incorrect” will assist in controlling the auditor’s exposure to sampling risk.

a. Incorrect This method assists in controlling the auditor’s exposure to sampling risk.

b. Correct Performing the appropriate audit procedure is related to nonsampling risk, not sampling risk.

c. Incorrect This method assists in controlling the auditor’s exposure to sampling risk.

d. Incorrect This method assists in controlling the auditor’s exposure to sampling risk.

E.30 a. Incorrect Only statistical sampling methods measure the auditor’s exposure to sampling risk.

b. Incorrect Generally accepted auditing standards permit the use of either statistical sampling or nonstatistical sampling methods.

c. Incorrect Samples can be selected either randomly or judgmentally under nonstatistical sampling methods.

d. Correct Nonstatistical sampling is typically less complex than statistical sampling.

E.31 a. Incorrect Block selection identifies a series of contiguous units for examination.b. Incorrect Haphazard selection is characterized by the auditor selecting sample

items in a nonsystematic fashion.c. Incorrect Systematic random selection uses a random starting point and then

bypasses a fixed number of items in selecting sample items.d. Correct Unrestricted random selection uses a series of random numbers to

identify sample items.

E.32 a. Incorrect See the response to (c) below.b. Incorrect See the response to (c) below.c. Correct The random starting point (10) is the first item selected. The sampling

interval would be calculated as 5 (100 20 = 5). Adding the sampling interval to the random starting point would result in item 15 being selected (10 + 5 = 15), followed by item 20 (15 + 5 = 20).

d. Incorrect See the response to (c) above.

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E.33 a. Incorrect Systematic random selection provides a relatively high likelihood of yielding a representative sample.

b. Incorrect There is no difference in the sample size provided by systematic random selection and other selection methods.

c. Correct Because systematic random selection bypasses items between selection, a limitation may occur if the population is not randomly ordered.

d. Incorrect Systematic random selection can be used with statistical sampling plans.

E.34 a. Incorrect Block selection cannot be used with statistical sampling applications.b. Correct Of these two methods, only unrestricted random selection can be used

with statistical sampling applications.c. Incorrect Unrestricted random selection can be used with statistical sampling

plans; block selection cannot be used with statistical sampling applications.

d. Incorrect Unrestricted random selection can be used with statistical sampling applications.

E.35 a. Incorrect Block selection cannot be used with statistical sampling applications.b. Incorrect Neither block selection nor haphazard selection can be used with

statistical sampling applications.c. Incorrect Haphazard selection cannot be used with statistical sampling

applications.d. Correct Unrestricted random selection and systematic random selection can be

used with statistical sampling applications.

E.36 a. Incorrect The confidence is the likelihood that the precision interval contains the true (but unknown) population value.

b. Incorrect The mean is the average of the observations in the sample.c. Correct The precision represents a range around the sample estimate that

has a certain likelihood (equal to reliability) of including the true population value.

d. Incorrect The precision interval is the sample estimate plus and minus the precision.

E.37 a. Correct The confidence is the likelihood that the precision interval contains the true (but unknown) population value.

b. Incorrect The mean is the average of the observations in the sample.c. Incorrect The precision represents the closeness of a sample estimate to the true

population value.d. Incorrect The likelihood that the interval contains the true population value is

confidence, or one minus sampling risk.

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E.38 NOTE TO INSTRUCTOR: Since this question asks students to identify the statement that is not true, the response labeled “correct” is not true and those labeled “incorrect” are true.

a. Incorrect Because there is a 90 percent (1 minus sampling risk) probability that the interval contains the true population value, there is a 10 percent probability that the true population value lies outside of this interval.

b. Correct There is a 10 percent probability (sampling risk) that the true population value is less than 60 or greater than 70.

c. Incorrect Reliability equals one minus sampling risk, or 90 percent (1 minus 10 percent = 90 percent).

d. Incorrect Since the precision interval is symmetrical around the sample estimate, the sample estimate is the average of the precision interval, or 65 [(60 + 70) 2 = 65)]. The precision can be determined as the distance between either bound of the precision interval and the sample estimate, or 5 (65 – 60 = 5).

E.39 a. Incorrect Sampling risk can occur in either statistical or nonstatistical sampling applications.

b. Incorrect This response is the definition of nonsampling risk.c. Incorrect This response is the definition of inherent risk.d. Correct This response is the definition of sampling risk.

E.40 a. Correct Both the risk of incorrect acceptance and risk of assessing control risk too low relate to the effectiveness of an audit.

b. Incorrect The risk of incorrect rejection and the risk of assessing control risk too high relate to the efficiency of the audit.

c. Incorrect Only the risk of assessing control risk too low is related to control risk assessments.

d. Incorrect Only the risk of incorrect acceptance is related to evidence about assertions in financial statements.

E.41 a. Correct Attribute sampling is most frequently used during the auditor’s study of internal control.

b. Incorrect Control sampling is not a type of sampling.c. Incorrect Probability proportional to size sampling is a form of variables

sampling, which is used in the auditor’s substantive procedures.d. Incorrect Variables sampling is used in the auditor’s substantive procedures.

E.42 a. Incorrect While attribute sampling occurs as part of the use of the audit risk model, it is most closely associated with the assessment of control risk.

b. Correct Attribute sampling is most closely associated with the assessment of control risk.

c. Incorrect While attribute sampling will allow the auditor to determine the acceptable level of detection risk, it is most closely associated with the assessment of control risk.

d. Incorrect Attribute sampling is unrelated to inherent risk.

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E.43 a. Incorrect If the upper limit deviation rate exceeds the tolerable deviation rate, auditors will reduce their reliance on controls. These responses are reversed.

b. Correct If the upper limit deviation rate exceeds the tolerable deviation rate, the auditor will reduce their reliance on controls.

c. Incorrect The auditor’s evaluation of controls is not related to the expected deviation rate.

d. Incorrect The auditor’s evaluation of controls is not related to the expected deviation rate.

E.44 a. Incorrect Statistical sampling methods do not necessarily provide the auditor with greater assurance.

b. Correct Statistical sampling methods allow the auditor to quantitatively measure the exposure to sampling risk.

c. Incorrect Either statistical or nonstatistical sampling methods can convert samples into dual-purpose tests for substantive procedures.

d. Incorrect Judgments are required to assess various factors that affect sample size under both statistical and nonstatistical sampling.

E.45 a. Incorrect See the response to (d) below.b. Incorrect See the response to (d) below.c. Incorrect See the response to (d) below.d. Correct The auditor would compare an estimation of the deviation rate to the

tolerable deviation rate when using sampling in the study of internal control.

E.46 NOTE TO INSTRUCTOR: Since this question asks students to identify the item that would not expose an individual to nonsampling risk, the response labeled “correct” would not expose an individual to nonsampling risk and the responses labeled “incorrect” would expose the individual to nonsampling risk.

a. Incorrect Measuring the characteristic in an inappropriate manner would result in exposure to nonsampling risk.

b. Correct If items are selected that are not representative of the population, this represents sampling risk, not nonsampling risk.

c. Incorrect A mistake in measurement (whether intentional or unintentional) would result in exposure to nonsampling risk.

d. Incorrect Because (b) would not result in exposure to nonsampling risk, this choice would not be appropriate.

E.47 NOTE TO INSTRUCTOR: Since this question asks students to identify the statement that is not true with respect to nonstatistical sampling, the response labeled “correct” is not true and those labeled “incorrect” are true.

a. Correct Either statistical or nonstatistical sampling can be used in an audit conducted in accordance with generally accepted auditing standards.

b. Incorrect Nonstatistical sampling considers various factors in determining sample size.

c. Incorrect Nonstatistical sampling does provide an estimate of the characteristic of interest.

d. Incorrect Nonstatistical sampling requires the use of judgment in establishing factors that will be used to determine sample size, among other areas.

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E.48 a. Incorrect Because a 90 percent probability exists that the average weight is between 110 and 130 pounds, the likelihood that the average weight is greater than 130 pounds is less than 10 percent, since the average weight could be less than 110 pounds.

b. Incorrect Because a 90 percent probability exists that the average weight is between 110 and 130 pounds, the likelihood that the average weight is less than 110 pounds is less than 10 percent, since the average weight could be greater than 130 pounds.

c. Incorrect The likelihood that the average weight is less than 110 pounds or greater than 130 pounds is one minus reliability, or 10 percent (not 90 percent).

d. Correct A reliability (90 percent) probability exists that the average weight is in the interval bounded by the sample estimate ± precision (in this case, 120 pounds ± 10 pounds, or 110 pounds to 130 pounds).

E.49 a. Correct This step would be performed last (see responses below).b. Incorrect The desired level of reliability must be determined prior to selecting the

sample, which precedes examining sample items. c. Incorrect Determining the objective of the sampling application is the first step in

the sampling application.d. Incorrect Determining the appropriate sample size would occur prior to

examining sample items.

E.50 a. Incorrect In both cases, the correct decision with respect to the client’s internal control or account balances should ultimately be made by the auditor.

b. Correct Both risks may result in the auditor performing more extensive substantive procedures than necessary to control audit risk to acceptable levels.

c. Incorrect Only the risk of incorrect rejection is related to preliminary estimates of materiality.

d. Incorrect Only the risk of incorrect rejection is related to tolerable error.

SOLUTIONS FOR EXERCISES, PROBLEMS, AND SIMULATIONS

E.51 Sampling Risk

a. Sampling risk is the risk that the decision made based on the sample is different from the decision that would have been made if the entire population were examined.

The two types of sampling risk are if you conclude that (1) the temperature will be above 50 degrees when it will be below 50 degrees and (2) the temperature will be below 50 degrees when it will be above 50 degrees.

b. The costs of committing sampling risk if you conclude that the temperature will be above 50 degrees is that you will not pack heavy clothing and will be uncomfortable or forced to purchase heavy clothing during your trip.

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The cost of committing sampling risk if you conclude that the temperature will be below 50 degrees is that you will unnecessarily pack heavy clothing.

c. If you use statistical sampling, your sampling plan would consider the acceptable level of sampling risk in determining the appropriate sample size and evaluating sample results. Nonstatistical sampling methods would not consider sampling risk in determining sample size or evaluating sample results.

d. The primary advantage of statistical sampling is that it allows you to explicitly control your exposure to sampling risk. The primary disadvantage of using statistical sampling is that it is typically more complex and time-consuming that nonstatistical sampling.

E.52 Sampling and Nonsampling Risk

a. Nonsampling risk (misplaced distance markers would result in the failure to accurately record yardage).

b. Sampling risk (golfers playing in a club championship would typically be of a higher skill level than those not playing in a club championship).

c. Nonsampling risk (because a 3-wood will not hit the ball as long as a driver, your conclusion may be affected by the club used and not the golf ball).

d. Sampling risk (a sample of golfers that includes only females will not be representative of the population).

e. Nonsampling risk (the conclusions may be affected by the fact that the comparison golf balls are older, not inferior in quality to the Wilson golf balls).

f. Nonsampling risk (failure to accurately record yardage may influence your conclusions).

NOTE TO INSTRUCTOR: Some students might classify items (c) and (e) as neither sampling risk nor nonsampling risk, since they do not apparently relate to an error made in recording results. However, one could contend that the failure to ensure that the proper golf club or a new golf ball is being used as a form of error. The important point to raise is that these issues do not relate to the representativeness of the sample, while items (b) and (d) do relate to the representativeness of the sample.


1. This situation is characteristic of sampling risk if the passengers seated in the rows you selected had either higher or lower income than that of the average passenger on the airplane. This could occur if one of the rows you randomly selected was in the first class cabin or business class cabin (in this case, your sample would typically provide a higher average income than the average passenger on the plane). In addition to this example, the possibility exists that individuals of relatively high and low incomes could be seated in different rows throughout the airplane.

2. This situation is characteristic of nonsampling risk because of the error made by including swimmers from events other than the freestyle race. This error is not related to the representativeness of the sample of swimmers, but including events other than that of interest in the sample.

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3. This situation is characteristic of sampling risk, because various characteristics of honors students (intelligence, extent of preparation, diligence in completing the examination, etc.) may not be representative of the population of students. Interestingly, it is not clear whether the sample of honors students will complete the examination more quickly (because of superior intelligence and examination preparation) or more slowly (because of more diligence in completing the examination) than non-honors students.

4. This situation is characteristic of sampling risk, because the sample of students attending schools in small, college towns are likely to differ from the overall population of sixth grade students. Specifically, these students are likely to be more aware of college education in general; in addition, their parents are more likely to have received a college education, as they are employed by the university. As a result, it is likely that including these students in your sample would provide a relatively high sample estimate of the percentage of students who plan on attending college.

E.53 Sampling and Nonsampling Risk (Continued)

5. This situation is characteristic of nonsampling risk. Assuming you selected a representative sample, your sampling procedure might yield an inappropriate conclusion because of mistakes in converting various currencies into U.S. dollars. These mistakes are unrelated to the representativeness of your sample.

E.54 Basic Sampling

a. Sampling is the process of making a statement about a population of interest based on examining only a subset (or sample) of that population.

b. The primary advantage of sampling is efficiency; you could make your conclusion in a fraction of the time (by examining only a subset of all students enrolled at your university) than would be necessary if the entire population were examined.

The primary disadvantage of sampling is related to effectiveness. The decision reached based on the sample might differ from the decision that would be made if the entire population were examined.

c. Sampling is more likely to be used if (1) the need for exact information is less important (for example, it is less important to know if you are taller than the average student at your university) or (2) if the number of students enrolled at your university is larger.

E.55 Basic Sampling

a. Some possible methods of estimating the number of patrons on the evenings you visit include:

Counting the patrons at one show time for one screen and assuming that all other show times and screens are similar in terms of attendance.

Counting the patrons at all show times for one screen and assuming that all other screens are similar in terms of attendance.

Counting the patrons at all screens for one show time and assuming that the other show times are similar in terms of attendance.

Counting all patrons attending on a given evening for all show times and screens.

Count patrons at a random sample of shows taken from the 780 (21 x 20 + 9x40) shows.

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b. To ensure a representative sample, you should consider the following events or characteristics that may result in nonrepresentative levels of attendance (students may list others):

“Blockbuster” or other popular movies that may have an unusually large audience.

The existence of other local events (such as a high school graduation or sporting event) that may result in unusually low attendance at the theater.

Poor weather, which has an indeterminate effect on attendance (some patrons may prefer to attend movies during poor weather; others may find it too uncomfortable to leave their homes).

Good weather, which may result in unusually low attendance if patrons preferred other types of activities to attending movies.

The time of the show (matinee versus evening versus late), as overall attendance may differ by time.

The day of the week (popular television shows airing on certain weeknights may influence attendance).

Weekend versus weekdays (weekends would typically have more attendance).

E.55 Basic Sampling (Continued)

c. The two types of sampling risk are (1) concluding that the average attendance exceeds 15,000 patrons per month when it is less than 15,000 patrons per month and (2) concluding that the average attendance is less than 15,000 patrons per month when it exceeds 15,000 patrons per month.

Of these two risks, most people would conclude that the first of these is greater. If you believe that the average attendance will support the theater and it will not, you will lose your investment and have an unsuccessful business. If you believe that the average attendance will not support the theater and it will, you will have missed a potentially profitable opportunity.

d. In all of these cases, you will need to convert the daily estimates to monthly estimates by multiplying by 30 days.

1. Precision Interval = Sample Estimate PrecisionPrecision Interval = 600 30 = 570 to 630 (daily)Precision Interval = 17,100 to 18,900 (monthly)



In the first two cases, you would conclude that the movie theater would be profitable, since the lower bound of the precision interval exceeds the criterion level of attendance (15,000 patrons). In the third case, you would be unable to reach this conclusion, since the lower bound of the precision interval (13,650) is less than the criterion level of attendance (15,000 patrons).

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E.56 Basic Sampling

a. Sampling is the process of making a statement about a population of interest based on examining only a subset (or sample) of that population.

The primary advantage of sampling is the time savings of examining only a subset of flights instead of all flights by competing airlines. The primary disadvantage of sampling is that, if a nonrepresentative sample of flights is selected, the results may not truly reflect the population of interest and Northeast Airlines may reach an incorrect conclusion.

b. To identify the population of flights, you would first need to clearly define the airlines against which Northeast wishes to compete. In this case, it appears that there are four other airlines. Once these airlines have been defined, you could obtain a flight schedule or other information that contained a listing of all flights into the proposed “hub” airline. This list would represent the population of flights from which you would select your sample.

E.56 Basic Sampling (Continued)

c. Sampling risk is the possibility that the decision made based on the sample differs from the decision that would have been made if the entire population were examined. The primary cause of sampling risk is the selection of a sample that is not representative of the population from which it is drawn.

Nonsampling risk is the probability that an incorrect conclusion is reached because of reasons unrelated to the nature of the sample. The primary cause of nonsampling risk is a mistake in evaluating sample items or interpreting sample results.

Sampling risk could occur if the sample of flights you select differs from the population of flights into the proposed airport (see (d) for a more complete discussion of why this might occur). Nonsampling risk could occur if some error was made in classifying a flight (an on-time flight was erroneously classified as late, or vice versa) or an error was made in compiling the sample results or calculating the sample estimate.

d. While not comprehensive, some characteristics that might influence the on-time arrival of flights (and, therefore, exposure to sampling risk) are:

Airline (because of relative operating advantages and disadvantages, some airlines will have a greater percentage of on-time arrivals than others).

Origin of flight (flights from busy airports are more likely to be delayed; flights from areas with more severe weather are more likely to be delayed).

Time of day of flight (flights with peak departure and arrival times are more likely to be delayed).

Length of flight (longer flights may have more opportunity to compensate for ground delays and be more likely to arrive on time).

Day of week (flights during peak travel days are more likely to be delayed).

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Capacity of flight (more crowded flights are more likely to be delayed as a result of longer passenger boarding and seating times).

e. After estimating the on-time arrival rate for the airlines currently serving this airport, you would form a precision interval based on the desired level of reliability. The on-time arrival rates included in this interval could be compared to Northeast’s rates (82 percent) to see if Northeast could be competitive at this airport.

E.57 Sample Evaluation

a. Sample Estimate: The estimate of the true population value based on the sample drawn to represent that population.

Precision: The precision is the numeric distance from the estimated population value in which the true (but unknown) population value may lie with a given probability.

Reliability: The probability the true (but unknown) population value lies within the precision interval.

b. The preferred candidate could have between 42 percent and 54 percent of the vote (48 percent 6 percent).

c. A 99 percent likelihood exists that the preferred candidate’s share of the vote is between 42 percent and 54 percent.

d. Since only two candidates are seeking office, it is uncertain as to whether the preferred candidate will be able to attract 50 percent of the vote, since the share could be as low as 42 percent.

e. In this case, the preferred candidate’s vote could range from 47 percent to 49 percent (48 percent 1 percent). While this suggests that the candidate has at least 47 percent of the vote, it also means that the candidate almost certainly cannot expect a majority of the vote.


A. Precision Interval = Sample Estimate PrecisionPrecision Interval = 56 20 = 36 to 76

B. Sampling Risk = 1 - ReliabilitySampling Risk = 1 - 0.95 = 0.05 or 5%

C. The precision can be determined by taking the difference between either bound of the precision interval and the sample estimate (for example, 85 – 80 = 5).

D. Sampling risk = 1 - Reliability0.10 = 1 – ReliabilityReliability = 0.90 or 90%

E. The sample estimate can be determined by taking the difference between either bound of the precision interval and the precision (for example, 131 – 10 = 121).

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F. Sampling Risk = 1 - ReliabilitySampling Risk = 1 – 0.98 = 0.02 or 2%


a. Sample 1:

Precision Interval = Sample Estimate PrecisionPrecision Interval = 26 5 = 21 to 31

Sample 2:


Sample 3:


b. Sample 1: There is a 95 percent likelihood that the average age of an NFL fan is between 21 and 31 years.

Sample 2: There is a 95 percent likelihood that the average age of an NFL fan is between 31 and 37 years.

Sample 3: There is a 95 percent likelihood that the average ago of an NFL fan is between 32 and 48 years.

c. For sample 1, since the upper bound of the precision interval (31 years) is less than 35 years, the NFL could reliably conclude that the average age of its fan base is less than 35 years. For samples 2 and 3, because the upper bound of the precision interval (37 years and 48 years, respectively) exceeds 35 years, the NFL could not reliably conclude that the average age of its fan base is less than 35 years at the specified confidence.

d. If sampling risk is increased, the reliability (or confidence) decreases, since reliability equals one minus sampling risk. As a result, the precision interval would tighten and the NFL would be more likely to conclude that its fan base has an average age of less than 35 years (assuming that the sample estimate is an accurate representation of the true population average).

On the other hand, if sampling risk is decreased, the precision interval would widen and the NFL would be less likely to conclude that its fan base has an average age of less than 35 years (assuming that the sample estimate is an accurate representation of the true population average).

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a. Precision is the numeric distance from the estimated population value in which the true (but unknown) population value may lie with a given probability. Reliability is the probability the true (but unknown) population value lies within the precision interval.

These terms are related in that a level of precision is associated with a given level of reliability. As a higher (lower) level of reliability is desired, a wider (tighter) precision interval results.

b. Precision Interval = Sample Estimate PrecisionPrecision Interval = 2.5 0.7 = 1.8 to 3.2

c. A lower sampling risk will correspond with a higher reliability. Because the precision interval will need to have a larger likelihood of including the true population value (reliability), the level of precision will increase, resulting in a wider precision interval.

d. 1. Precision Interval = Sample Estimate PrecisionPrecision Interval = 2.5 0.7 = 1.8 to 3.2

Because the lower bound of the precision interval is greater than 1.5, Gloria would conclude that the average number of children per household exceeds 1.5 children.

2. Precision Interval = Sample Estimate PrecisionPrecision Interval = 2.5 1.4 = 1.1 to 3.9

Because the lower bound of the precision interval is less than 1.5, Gloria would not be able to conclude that the average number of children per household exceeds 1.5 children.

3. Precision Interval = Sample Estimate PrecisionPrecision Interval = 2.5 1.8 = 0.7 to 4.3

Because the lower bound of the precision interval is less than 1.5, Gloria would not be able to conclude that the average number of children per household exceeds 1.5 children.

e. The differences noted in (d) above are the direct result of different levels of reliability. If a given interval needs to have a higher likelihood of including the true (but unknown) population value (higher reliability or confidence and lower sampling risk), that interval needs to be larger (wider). This relationship can be seen examining the differences in the three scenarios noted above.

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a. Based on the sample results, Alice can conclude that there is a 95 percent probability that her level of support is between 48 percent and 58 percent (53 percent ± 5 percent). Therefore, while a reasonable probability exists that she might win in the election, it is less than the desired 95 percent level and Alice could not have 95 percent confidence of receiving a majority of the ballots cast. This is because the precision interval includes levels below 50 percent.

b. 1. In this case, the level of support would be between 45 percent and 61 percent (53 percent ± 8 percent). Since the precision interval includes levels below 50 percent, Alice could not have 99 percent confidence of receiving a majority of the ballots cast.

2. In this case, the level of support would be between 51 percent and 55 percent (53 percent ± 2 percent). Since the precision interval does not include levels below 50 percent, Alice could have 90 percent confidence of receiving a majority of the ballots cast.

c. As one wishes to be more confident that the precision interval includes the true (but unknown) population value (i.e., higher level of reliability), the precision interval needs to be wider (i.e., larger precision). In this particular case, as Alice desires higher levels of confidence regarding her ability to receive a majority of the ballots cast, the lower bound of the precision interval will be lower.

d. Summarized below is a brief description of how some of the issues in your notes of the sampling process affect Alice Evans’ ability to rely on the sample evidence (numbers correspond to numbers from your sampling notes). Many of these issues relate to whether the individual(s) surveyed were representative of the population of voters in Alice’s district (sampling risk). There are no definitive “right” or “wrong” answers; the main objective is to have students begin to understand how various facets of a sampling plan may introduce sampling and nonsampling risk into the evaluation process.

1. Alice’s ability to rely on the results could be affected if (1) the four neighborhoods chosen were not representative of the population of eight neighborhoods comprising Alice’s district and (2) individuals who responded to the door-to-door inquiry were not representative of those who did not respond.

2. Alice’s ability to rely on the results could be affected if the individuals who were home during the hours in which the neighborhood was canvassed were not representative of voters in Alice’s district.

3. While this question does not introduce sampling risk, respondents may view it differently (and respond differently) than a more direct question about whether they would vote for Alice Evans.

E.61 Sample Evaluation (part d, Continued)

4. The use of telephone surveys in one neighborhood may result in a nonrepresentative sample, particularly if individuals screened their phone calls or otherwise chose not to participate (although they could also do so for a door-to-door survey).

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5. Cases where a household indicated support for Alice while displaying signs for her rival may reflect an undecided voter or a desire to minimize the amount of time spent with the pollster. In any case, these instances should be interpreted carefully.

6. Clinton’s misunderstanding will obviously have an effect on the results, because truly undecided voters may provide responses that do not truly reflect their feelings. Similar to (5), these responses should be interpreted carefully.

7. Bush’s mistake may result in nonregistered voters’ responses being included in the sample results. Since these individuals will not vote, they are not a part of the population of voters and their inclusion in the sample will misrepresent Alice’s level of support (this misrepresentation could be in either a favorable or unfavorable direction).


a. The sample estimate would be $1,750,000 (50,000 x $35 = $1,750,000).

b. Since the sample estimate ($1,750,000) exceeds the targeted amount to be raised ($1,500,000), it appears that the fundraising campaign could be successful.

c. The primary limitation of relying only on the sample estimate is that the sample of citizens may not be representative of the population of the College Bryan area. More specifically, these individuals may be willing to provide a greater level of financial support to the fundraising campaign than citizens who were not included in the sample. This would result in Marts believing that the campaign would be successful when, in fact, it would not be successful.

d. Sampling risk is the possibility that the decision made based on the sample differs from the decision that would have been made if the entire population were examined.

Factors that could result in Marts’ exposure to sampling risk are characteristics of sample items that result in the sample not being representative of the population from which it is drawn. Since this example represents an individual’s willingness to participate in a fundraising campaign to assist in building a new recreation center, the following should be considered:

Household income (those with higher incomes could be more likely to support the campaign).

Age and composition of household, particularly the number of children (those with children could be more likely to support the campaign).

Past support of various fundraising efforts (those who have supported efforts in the past could be more likely to support the campaign).

Proximity of address to proposed recreation center (those living closer to the facility could be more likely to support the campaign).

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E.62 Sample Evaluation (Continued)

e. Sample Estimate ± Precision = Precision Interval

1. $1,750,000 ± $100,000 = $1,650,000 to $1,850,000

2. $1,750,000 ± $200,000 = $1,550,000 to $1,950,000

3. $1,750,000 ± $300,000 = $1,450,000 to $2,050,000

f. Based on the precision intervals calculated in (e), it appears that the fundraising campaign is likely to be successful. In cases (1) and (2), the lower bound of the precision interval exceeds the fundraising target of $1,500,000. In the one case where the lower bound is less than the fundraising target, it is very close to this target ($1,450,000). Assuming that these precisions corresponded to a reliability of 99 percent, the likelihood that Marts will exceed its fundraising target and have a successful fundraising campaign is quite high.

g. As the reliability decreases, the precision will also decrease. In this instance, since the sample estimate exceeds the fundraising target, a decrease in the precision will result in the lower bound of the precision interval being higher. As a result, Marts would be more likely to conclude that its fundraising campaign would be successful.

E.63 Basic Sampling: Comprehensive

a. The primary advantage of sampling is efficiency; Reagan could make his conclusion in a fraction of the time (by examining only a subset of all households in Anytown, USA) than would be necessary if the entire population were examined.

The primary disadvantage of sampling is related to effectiveness. Reagan’s decision based on the sample might differ from the decision that he would make by examining the entire population.

b. The advantage of statistical sampling is that it allows Reagan to explicitly control his exposure to sampling risk. The primary disadvantage of statistical sampling is that it is typically more costly and time-consuming that nonstatistical sampling.

c. 1. Sampling risk is the possibility that the decision made based on the sample differs from the decision that would have been made if the entire population were examined.

Nonsampling risk is the risk that an incorrect conclusion is reached because of reasons unrelated to the nature of the sample.

2. Reagan can control his exposure to sampling risk by mathematically determining the appropriate sample size and mathematically evaluating sample results. He can control his exposure to nonsampling risk primarily by exercising care during the sampling and evaluation process.

3. An example of sampling risk would be selecting a nonrepresentative sample. This may occur if Reagan limited his selection of households to those in either unusually high or low income neighborhoods.

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Examples of nonsampling risk would be failing to properly record household income levels or making a mathematical error in tabulating the sample results.

E.63 Basic Sampling: Comprehensive (Continued)

d. Reagan would identify 100 random numbers (either using a random number table or computer program) and match those random numbers to items in the population. Once these items were identified, Reagan would then determine the income for these households.

e. Selecting all of the houses on a small number of streets may not provide a representative sample, since income levels would probably be fairly similar on these streets. Essentially, Reagan would be sampling a small number of items (streets) and would risk selecting streets that did not represent the population of Anytown, USA with respect to average levels of household income. This is an example of block selection.

f. The precision interval would be $36,000 to $42,000 ($39,000 $3,000 = $36,000 to $42,000).

Based on this information, you would tell Reagan that there is a 90 percent (1 minus sampling risk of 10 percent) probability that the true average household income is between $36,000 and $42,000. As a result, Reagan could reliably conclude that the average household income exceeds $35,000, since the lower bound of the precision interval ($36,000) exceeds this criterion level.

g. The precision interval would be $32,000 to $52,000 ($42,000 $10,000 = $32,000 to $52,000).

Based on this information, you would tell Reagan that there is a 90 percent (1 minus sampling risk of 10 percent) probability that the true average household income is between $32,000 and $52,000. While the average household income could be as high as $52,000, it could also be as low as $32,000. As a result, Reagan could not reliably conclude that the average household income exceeds $35,000.

E.64 Sample Selection

a. Sampling risk is the possibility that the decision made based on the sample differs from the decision that would have been made if the entire population were examined. Arthur can control sampling risk by (1) mathematically determining the appropriate sample size and (2) mathematically evaluating the sample results.

b. The population would be defined as all registered voters residing in Hoops County. These voters can be defined using the appropriate zip codes in the electronic data file maintained in the State Commissioner’s office.

c. Because the population is maintained in an electronic format, Arthur can use a computer program with an unrestricted random selection method or a systematic random selection method. Prior to doing so, he needs to identify the appropriate zip code(s) for voters registered in Hoops County.

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When using an unrestricted random selection method, Arthur would identify a series of random numbers and match those numbers to voters in the data file. This can be done using voter registration numbers or simply renumbering the voters in the data file beginning with the number 1. If he chooses a systematic random selection method, Arthur would identify a random starting point in the population and then select every nth

voter in the data file.

Alternatively, Arthur could use haphazard selection methods by nonsystematically selecting voters from a printed copy of the electronic data file (for example, by picking some number of pages and sampling the identified voters).

Finally, Arthur could use a block selection method to select a series of contiguous items (a series of contiguous voter registration numbers, addresses, names, etc.).

E.64 Sample Selection (Continued)

d. One of the limitations of using haphazard selection and block selection is that they are less likely to provide a representative sample compared to unrestricted random selection and systematic random selection. With respect to the use of any of these methods, care should be taken to ensure that Arthur selects a sample of voters representing different genders, ethnicities, income levels, family sizes, and other factors that may influence their propensity to support or oppose the funding for a sports stadium. One way to do this would be to sort the electronic data file alphabetically by last name.

E.65 Sample Selection Methods

a. To ensure a representative sample, you should consider factors that may influence customer satisfaction. These factors include type of service, length of service, and name of carrier.

b. Four major methods used to select samples are:

Unrestricted random selection (or random selection) involves numbering items in the population and selecting items for examination based on random numbers picked from a random number table or generated by a computer program.

Systematic random selection (or systematic selection) involves randomly selecting a starting point in a population and bypassing a fixed number of items between selections.

Haphazard selection involves selecting sample items in a nonsystematic manner.

Block selection involves selecting sample items by choosing a series of contiguous (or adjacent) items.

c. With an appropriate sample size of 150 items, you would select the sample as follows using the above methods:

Unrestricted random selection: Number the customers in the database (from 1 to 10,500), identify 150 random numbers between 1 and 10,500, and select the corresponding customer record.

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Systematic random selection: Select a random starting point within the population and then select every 70th customer record thereafter (10,500 customers ÷ sample size of 150 = 70 items).

Haphazard selection: Select 150 customer records in a nonsystematic manner (for example, selecting the first 150 even numbered customers in the database).

Block selection: Select 15 customers on ten streets.E.65 Sample Selection Methods

d. This portion illustrates the impact of a randomly-ordered population when using various selection methods. In this case, the fact that the population is arranged based on length of service is problematic, since one could assume that longer-term subscribers have a higher level of satisfaction than more recent subscribers. In most instances, the use of unrestricted random selection and haphazard selection are not influenced by the ordering of the population.

1. Because this population is not randomly arranged and cannot be sorted, the use of systematic random selection may be problematic (since the fixed number of items bypassed will be similar in terms of length of service). In addition, since contiguous items selected under a block selection method will also be similar in terms of length of service, this method may result in a nonrepresentative sample being selected (for example, all customers could be either long-term subscribers or recent subscribers).

2. Despite the fact that the population is maintained electronically, the issues noted in (a) above still exist.

3. Because the population can be sorted, you would sort the population to ensure it is in random order with respect to factors influencing customer satisfaction. Using customer name or telephone number would appear to be most appropriate, since neither factor appears to be related to satisfaction.

e. 1. You would not be able to determine whether your sample includes subscribers residing throughout the subscription area.

2. While the classification of length of service into these three categories will allow you to ascertain that your sample includes a variety of subscription lengths, it will not allow you to determine whether your sample includes a representative number of customers that have subscribed for relatively long periods of time (for example, five years or longer).

3. This feature might result in long-term subscribers being misclassified as recent subscribers, making it more difficult for you to determine that your sample includes a representative number of both types of subscribers.

MODE-22



a. This characteristic may affect sampling risk, since Arthur needs to consider the time of day during which he conducts his sample to ensure that he can include these households. Because many dual-income career couples may not have children (or may have a smaller number of children), the failure to include these households may overstate the average number of family members.

b. This characteristic may affect sampling risk, because Arthur would need to select homes from each of the different price ranges for his sample. It is uncertain whether more expensive homes would be occupied by larger families or smaller families, but the types of families choosing these homes may differ from those in smaller, newer homes.

c. This characteristic may affect sampling risk, because homes near the park would be expected to have a larger number of family members. While Arthur would need to include some of these homes in his sample, if a disproportionate number of these homes were included in his sample, the sample estimate may overstate the average number of family members.

d. This characteristic may affect sampling risk, because homes near the wooded area would be expected to have a smaller number of family members. While Arthur would need to include some of these homes in his sample, if a disproportionate number of these homes were included in his sample, the sample estimate may understate the average number of family members.

e. This characteristic may affect nonsampling risk, because visitors may be mistakenly recorded as family members.

NOTE TO INSTRUCTOR: (a) – (d) above may also be subject to nonsampling risk if errors in recording observations (number of family members) or calculations were made. However, the characteristics described in (a) – (d) do not by themselves provide exposure to nonsampling risk.


a. 1. Sampling risk would be affected if individuals who work out in the morning are more highly-motivated and disciplined than those who work out during other times of the day. Presumably, these higher levels of motivation and discipline would result in a greater weight loss.

Nonsampling risk would not be affected by this characteristic of the methodology.

2. Sampling risk would be affected if males are more or less likely than females to experience a weight loss as a result of the workout regimen or dietary restrictions.


3. Sampling risk would be affected if the lifestyles of these types of individuals provide them with a greater opportunity to exercise or observe a more restricted diet, resulting in a greater weight loss.

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4. Sampling risk would be affected, since individuals who frequently walk to their offices are more likely to be in good physical condition and some of the observed weight loss could be the result from this additional physical activity.


5. Nonsampling risk would be affected if participants did not provide you with accurate self-reported weights.

Sampling risk would not be affected by this characteristic of the methodology.

6. Nonsampling risk would be affected if participants did not completely observe the workout regimen or dietary restrictions.

Sampling risk would not be affected by this characteristic of the methodology.

b. Some improvements to the sampling methodology are as follows:

1. If a population of individuals between the ages of 30 to 35 from within the community can be identified and obtained, select participants from this listing as opposed to members of Health Busters. If such a listing is not available, select members attending Health Busters at various times throughout the day. This might overcome some of the issues with respect to the composition of your current sample (mostly males, single or married with no children, individuals working out at Health Busters early in the morning, etc.) and provide you with a more representative sample of individuals aged 30 to 35 years. As a result, this improvement will reduce your exposure to sampling risk.

2. Either observe workouts and dietary restrictions in some manner or have participants provide you with a log of their workouts and diet during the two-week period. While using a log introduces issues with respect to self-reporting of information, it is not feasible for you to observe all workouts and meals and is an improvement compared to the current methodology. This will reduce your exposure to nonsampling risk.

E.67 Sampling and Nonsampling Risk (part b, Continued)

3. Physically weigh the individuals before and following the two-week workout regimen and dietary restriction. This will overcome issues related to self-reporting of information and reduce your exposure to nonsampling risk.

E.68 Sample Selection Methods

a. Precautions that should be considered to enhance the likelihood of a representative sample include ensuring that the sample includes observations from different groups of students, as noted below (student responses may include other considerations as well):

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Students taught by different professors. Students having different abilities, as measured by grade point average and other

characteristics. Students enrolled in classes at different times of the day (morning, afternoon,

and evening classes). Students enrolled in different types of business courses (accounting, finance,

management, and marketing). Students having different personal characteristics (gender, weight, ethnicity).

Selecting sample items from the different groups as defined above will allow you to ensure that any observed results are because of the intended treatment of interest (coffee) and not other factors. For example, if you provide coffee to students in Professor Wynn’s class and withhold coffee from students in Professor Selleck’s class, you couldn’t determine if the difference was because of the coffee or the teaching methods used by Professors Wynn and Selleck.

b. (1) Identify 100 random numbers using a random number table or computer program and “match” the student from the listing that corresponds to the random number selected.

(2) Select a random starting point (say the third student), which is the first observation. Calculate the sampling interval by dividing the population size (2,200 students) by the appropriate sample size (100 students), obtaining a sampling interval of 22 students (2,200 100 = 22). Bypass 22 students to select the 25th student as the second observation and continue to proceed through the listing of students until a total of 100 have been selected.

(3) Select 100 students in some sort of a nonsystematic yet nonbiased way. For example, you could pick the first, third, fifth, seventh, and ninth students on each of the first 20 pages of the listing, for a total of 100 students (5 students x 20 pages = 100 students).

(4) Select one page at random and use that page (which contains 100 names) as your sample.

E.68 Sample Selection Methods (Continued)

c. Unrestricted random selection and systematic random selection are advantageous in that they are more likely to provide a representative sample and that another individual can replicate the sample given the criteria of the selection. Both of these methods have a disadvantage in that they are more time consuming (and, therefore, costly) to perform. An additional disadvantage of systematic random selection is that this method bypasses a fixed number of items between selections. This feature could result in a nonrepresentative sample being selected if the population is not arranged in a random order.

Haphazard and block selection are advantageous in that they are less time consuming (and, therefore, costly) to perform. However, a primary disadvantage of these methods is that they are less likely to provide a representative sample. In addition, it is typically more difficult for another individual to replicate the sample given the criteria of the search.

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E.69 Factors Affecting Sample SizeAttribute

SamplingVariablesSampling

Population Size D D

Expected deviation rate/Expected error D D

Tolerable deviation rate/Tolerable error I I

Sampling Risk I I

Population Variability U D

E.70 General Sampling

a. Alex’s decision to use sampling would be based on the need for exact information and the size of the population being considered. In this instance, he would be more likely to use sampling if the need for exact information is less important. For example, if the store would be successful if the average number of children per household is exactly 1.5, he would be more likely to examine the entire population than if less exact information is required (such as the average number of children exceeding 1.3 per household). Also, Alex would be more likely to use sampling if the number of households within a one-mile radius was larger, since examining the entire population would be less viable in this situation.

b. The major advantage of using sampling in this situation is efficiency; that is, Alex can attempt to draw a conclusion based on examining only a subset of the entire population. The primary disadvantage of using sampling is related to effectiveness. Because Alex’s decision will be based on only a subset of the items in the population, there is a chance that sampling will not provide him with the correct answer to his question.

c. Sampling risk is the risk that the decision made based on the sample is different from the decision that would have been made if the entire population had been examined.

One outcome that may reflect sampling risk is if Alex’s sample results indicate that the average number of children per household is less than 1.3 when, in fact, it is greater than 1.3. In this situation, Alex would decide not to open the ice cream and candy shop at that location. The loss to Alex would be the annual income that is foregone from a successful business.

A second outcome that may reflect sampling risk is if Alex’s sample results indicate that the average number of children per household is more than 1.3 when, in fact, it is less than 1.3. In this situation, Alex would decide to open the ice cream and candy shop at that location, but it would be unsuccessful. The loss to Alex would be his initial investment in the unsuccessful business.

There is no one “correct” answer as to which risk is of more concern to Alex. For the first outcome, Alex would fail to realize income related to a successful business. In the second outcome, Alex would lose the $150,000 related to his initial investment. The student’s answer would depend upon their perceptions of the relative effect(s) of the $150,000 initial investment and the $50,000 annual income on Alex’s net worth. If Alex could easily withstand the loss of $150,000, the first outcome may be of greater concern; if not, the second outcome would be of greater concern.

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d. Alex would be far greater concerned with the first outcome, since the annual income foregone ($50,000 per year) is far more substantial than the initial investment that could potentially be lost from an unsuccessful venture ($10,000).

e. Nonsampling risk is the probability that an incorrect conclusion is reached because of reasons unrelated to the nature of the sample. Potential nonsampling risks that may be encountered by Alex are incorrectly recording the number of children in sample household(s), mistaking playmates for children residing in sample household(s), or making a mathematical error in determining the average number of children per household.

E.71 Audit Sampling: Types of Audit Samples

a. Attribute sampling is a form of sampling used to determine the extent to which some characteristic (attribute) exists within a population of interest; used by auditors during tests of controls.

Variables sampling is a form of sampling used to examine a population to estimate the amount or value of some characteristic of that population; used by auditors during their substantive procedures.

A dual-purpose test is an audit procedure that can be used as both a test of controls and a substantive test.

b. 1. This is not an audit sample because the results will not be projected to the population. The procedure is only used to gain an understanding of the internal controls in operation.

2. This is an example of attribute sampling. You are not looking to estimate a dollar amount, but the rate of error occurrence in internal control.

3. This is an example of a dual-purpose test. You are testing for internal control errors, and you will use the sample to project a dollar amount of an account balance.

4. This is an example of variables sampling. You are using the sample to estimate the monetary amount of an account balance.

5. This is not an audit sample as you are examining the entire population.

6. This is an attributes sample because you are not looking for monetary errors.

7. Even though you will not inquire of all personnel, this is not sampling as considered by SAS 39.

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E.72 Audit Simulation: Various Sampling Concepts

a. This statement is not correct Both statistical and nonstatistical sampling require the auditor to use professional judgment in planning, performing, and evaluating a sample and in relating the audit evidence produced by the sample to other audit evidence when forming a conclusion about the related account balance of class of transactions (AU 350.03).

b. This statement is not correct Either statistical or nonstatistical sampling methods are permissible under generally accepted auditing standards (AU 350.04). However, it is true that statistical sampling is more time-consuming than nonstatistical sampling.

c. This stateement is not correct Nonsampling risk may result in an error in evaluating sample results, regardless of the representativeness of the sample. Nonsampling risk may result in the failure to reach an appropriate conclusion with respect to the sample (AU 350.11).

d. This statement is correct A lower sampling risk will result in a larger number of items being selected; that is, sample size varies inversely with sampling risk (AU 350.10).

e. This statement is not correct Sample items should be selected in such a way that the sample is representative of the population and all items in the population have an equal opportunity to be selected (AU 350.39). As a result, despite the difficulty in examining transactions with Wimbledon, these transactions cannot be automatically excluded from the items subject to selection.

f. This statement is not correct If the purpose of the auditor’s application of a procedure to less than 100 percent of the population is something other than evaluating a trait of the entire account balance or class of transactions, the application is not considered sampling (AU 9350.02). A walk-through that is conducted to understand the nature of transactions is an example of such an exclusion.

E.73 Audit Simulation: General Sampling

TO: Mason & Jarr, CPAsFROM: Consultant-AdvisorSUBJECT: Application of audit sampling standards (SAS 39, AU 350, 9350)

At your request, I have reviewed the audit work in the case files you provided. Herein are my conclusions about proper application of the audit sampling standards in SAS 39 (AU 350).

a. Work to Understand the Accounting System The sample of three purchase orders and subsequent tracing the cash disbursement documents and procedures is not considered “audit sampling,” and SAS 39 does not apply. The work was properly done for the purpose of obtaining a preliminary understanding of internal control, not for making a judgment about the effectiveness of control procedures. Audit sampling standards apply to samples taken for the purpose of reaching a conclusion about a whole population of data—in this case the cash disbursements controls—and not to work done to obtain a general understanding of a client’s internal control. (AU 9350.02)

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b. Inventory Count Accuracy Test The sample of inventory items for recounting is a sampling application covered by SAS 39. Ms. Jarr took the sample for the purpose of making an overall judgment of the accuracy of the counting procedure. However, the sample did not meet SAS 39 requirements because it does not appear to have been representative (AU 350.24). Only the largest-quantity items were chosen, and the others were ignored. These items were probably the most likely to be miscounted.

In addition to the above concern, Ms. Jarr did not appropriately consider sampling risk in projecting the results to the population (AU 350.26). Simply stated, the rate of miscount (8 percent) assumed that the entire population was misstated to the same extent as the sample (16 200 = 8 percent). This is not appropriate.

c. Short-Term Debt Outstanding The audit of all of the outstanding commercial paper notes is not a sampling application. Audit sampling is the application of audit procedures to less than 100 percent of the items in a balance (AU 350.01). Ms. Jarr’s work examined all items comprising the short-term commercial paper amount on the balance sheet.

d. Management Representations Audit sampling is not involved in the procedure of obtaining management representations. While it may initially appear that Ms. Jarr “sampled” client officers to obtain management representations, this evidence is considered to represent the entire body of evidence under examination.

NOTE TO INSTRUCTOR: The following professional guidance is more closely related to management representations than audit sampling.

Management representations are ordinarily obtained from those members of management with overall responsibility for financial and operating matters, such as the chief executive officer, chief financial officer, and others with equivalent positions whom the auditor believes are responsible for and knowledgeable about, directly or through others in the organization, the matters covered by the representations. Such members of management normally include the chief executive officer and chief financial officer or others with equivalent positions in the entity. (AU 333.09)

E.73 Audit Simulation: General Sampling (Continued)

e. Ms. Jarr divided the Repairs and Maintenance account into two “populations.” Under SAS 39, this is acceptable and, in fact, the recorded or book value of the items is specifically mentioned as an example of a characteristic that may be used to subdivide a population (AU 350.22)

While the process of subdividing the population appears to have been appropriate, Ms. Jarr’s selection of sample items within these groups was not. She audited all the items in one population (the $278,000 expense entries each over $5,000), and this was not a sampling application. The second population ($75,000 for which each entry was less than $5,000) also was not a sampling application; Ms. Jarr applied analytical procedures with respect to these entries (AU 9350.02).

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Instructors should feel free to observe that the “analytical comparison” is very casual, and the $75,000 of smaller entries is actually not audited very well. The purpose of auditing the larger entries was to reach a conclusion about the entire account, even though very little audit effort was given to the population of smaller entries.

E.74 Kaplan CPA Exam Simulation: Sampling Terminology

C Sampling Risk The chance that auditor's conclusion will be wrong because only a portion of the population was examined.

Sampling risk is the chance that auditor's conclusion will be wrong because only a portion of the population was examined.

D Judgment sampling Estimates the amount of sampling risk that the auditor faces purely by human guess.

Judgment sampling estimates the amount of sampling risk that the auditor faces purely by human guess.

K Tolerable misstatement Measures the sufficiency of the evidential matter obtained.

Statistical sampling measures the sufficiency of the evidential matter obtained.

F Attributes sampling Estimates a percentage and is often used in tests of controls.

Attributes sampling estimates a rate of occurrence and is often used in the tests of controls.

I Variables sampling Estimates a total and is often used in substantive tests.

Variable sampling estimates a total and is often used in substantive tests.

E Tolerable deviation rate The maximum error rate that the auditor is willing to accept without modifying the planned assessed level of control risk.

Tolerable deviation rate is the maximum error rate that the auditor is willing to accept without modifying the planned assessed level of control risk.

H Tolerable misstatement The size of the largest misstatement in the account being examined that (when combined with misstatements in all other accounts) would still not cause the financial statements to be materially misstated.

Tolerable misstatement is the size of the largest misstatement in the account being examined that (when combined with misstatements in all other accounts) would still not cause the financial statements to be materially misstated.

G Non-sampling risk The chance that auditor's conclusion will be wrong for reasons that would happen even if every item had been tested. Includes human errors such as the failure to recognize a misstatement and the misinterpretation of results.

Non-sampling risk is the chance that the auditor's conclusion will be wrong for reasons that would happen even if every item had been tested. Includes human error such as the failure to recognize a misstatement and the misinterpretation of results.

MODE-30

module e

Documents

objective of sampling

sampling module e overview

nonstatistical sampling

basic principles of

entire sampling application

sample results

e module e overview

lower level of sampling