Solution manual auditing and services 2e by louwers MODE

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 statisti

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 representthat 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

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 sampling

2 Define the characteristic of interest

3 Define the population

4 Determine the sample size

5 Select sample items

6 Measure sample items

7 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

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 notreduce 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

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

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

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

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

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

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

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

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 samplesize 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.

E.53 Sampling and Nonsampling Risk

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

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 areunrelated 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 youruniversity) 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

b To ensure a representative sample, you should consider the following events or

characteristics that may result in nonrepresentative levels of attendance (students may listothers):

• “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 ± Precision

Precision Interval = 600 ± 30 = 570 to 630 (daily)Precision Interval = 17,100 to 18,900 (monthly)

2 Precision Interval = Sample Estimate ± Precision

Precision Interval = 680 ± 150 = 530 to 830 (daily)Precision Interval = 15,900 to 24,900 (monthly)

3 Precision Interval = Sample Estimate ± Precision

Precision Interval = 490 ± 35 = 455 to 525 (daily)Precision Interval = 13,650 to 15,750 (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)

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 reflectthe 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 Theprimary 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)

• 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

E.58 Sample Evaluation

A Precision Interval = Sample Estimate ± Precision

Precision Interval = 56 ± 20 = 36 to 76

B Sampling Risk = 1 - Reliability

Sampling 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 - Reliability

0.10 = 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)

F Sampling Risk = 1 - Reliability

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)