ČSN ISO 28596 - Přejímací postupy pro přejímku srovnáváním - Dvouúrovňové přejímky pro audity a kontrolu s využitím předběžné informace

3.1.23 tolerance proportion

largest value p0 of the proportion nonconforming such that the target population is considered as acceptable

####

Symbols and abbreviated terms

sample sizes in stage i

one stage sample size with same OC as two stage sampling plan

number of misstated items (nonconforming items) found in ni

D

confidence interval for the proportion of misstatements (nonconforming items)

p

proportion of misstatements (nonconforming units)

lower limit of D

upper limit of D

p0

tolerance proportion

I.cp

integrated actual coverage

nominal confidence level

a, b

shape parameters of the beta distribution

acceptance number in stage i

rejection number in stage i

c.type I

conditional probability of erroneous acceptance

c.type II

conditional probability of erroneous rejection

integrated probability of entering the second stage

I.ASN

integrated average sample number

N

lot size

OC

operating characteristic function

Pa

probability of acceptance (OC function at a specified value p)

Selecting and operating a two-stage sampling plan under prior information

General

Table 1 to Table 5 in Clause 7 provide two-stage sampling plans

The aim of the application of a two-stage sampling plan is two-fold:

a) enable a decision on whether or not the actual proportion nonconforming p exceeds the tolerance proportion p0. In statistical terminology, the decision problem can be considered as a test of the hypothesis

b) provide a confidence interval for the actual proportion nonconforming

The design of the sampling plans assures that the probabilities of both decision errors 1) erroneous rejection of H, and 2) erroneous acceptance of H are bounded.

Selecting a sampling plan

Sampling plans can be obtained from Table 1 to Table 5 in Clause 7. The cell entries Table 1 to Table 5 display:

a) upper left: n1 sample size in stage 1;

b) upper right:

c) lower left: n2 sample size in stage 2;

d) lower right:

The sampling plans are indexed in p0 (tolerance proportion),

The nominal confidence level

The level of prior information shall be specified on an ordinal scale named Trust, by choosing among the values {low, mid, high}. The Trust level low shall be used if no prior experience or bad prior experience with populations submitted for inspection exists. The Trust level high shall be used if there is strong evidence of good performance. The Trust level mid shall be used if there is weak evidence of good performance or strong evidence of in-between performance.

See Annex H for further technical background on the prior information model and the Trust scale.

Sampling and decision procedure

The decision by a two-stage sampling plan

Stage 1:

Draw a random sample of size n1, determine the number x1 of nonconforming units among the n1 sampled units. Decide according to the subsequent cases a), b), and c):

a)

b)

c)

Stage 2:

If, in stage 1, the case c) occurs and enforces entering stage 2, proceed as follows:

Draw a second random sample of size n2, determine the number x2 of nonconforming units among the n2 sampled units. Decide according to the subsequent cases a) and b):

a)

b)

Estimation of the actual proportion nonconforming

The sample proportion nonconforming is

p̂ is an unbiased estimator of the actual proportion nonconforming p in the population. The sampling uncertainty inherent in the estimator p is expressed by a confidence interval. A two-sided confidence interval D =

Application paradigms: lot inspection and financial auditing

Details of two standard application paradigms for the two-stage decision procedure are described below:

— for the inspection of lots of discrete product items, see 6.1

— financial auditing, with two targets: for testing for the compliance of an internal control system (test of controls), and test of details in the course of substantive procedures, see 6.2

Lot inspection

Sampling

Samples shall be drawn from the lot by simple random sampling. When the lot consists of sub-lots or strata, identified by some rational criterion, representative sampling shall be used in such a way that the number of items sampled is proportional to the number of items in the sub-lot or stratum.

Acceptance of loss

All items in the sample shall be inspected and the nonconforming items shall be counted.

Acceptability of a lot shall be determined by the use of the obtained sampling plans. If the number of nonconforming items found in the sample is equal to or less than the acceptance number

Disposition of non-accepted lots

The disposition of lots not accepted shall be agreed in advance by all interested parties.

Lots with one or more nonconforming units

If a lot has been accepted, the right is reserved not to accept any item found nonconforming during the acceptance sampling inspection that led to lot acceptance.

Resubmitted lots

A lot that has been inspected but not accepted shall only be resubmitted for re-inspection if

a) the purchaser is satisfied that all misstated items (nonconforming items) have been removed or replaced by conforming items, and

b) all interested parties agree.

The responsible authority shall determine the method of re-inspection to be applied.

Financial auditing

Purposes in the risk-oriented auditing process

The relevant purposes in the risk-oriented auditing process are:

1) test of controls, i.e. tests of compliance in the evaluation of the internal control system (ICS);

2) test of details for selected purposes in course of substantive procedures.

In any case, the auditor notifies the result of the sampling procedure and the subsequent decision in the audit documentation.

Target population, proportion nonconforming and tolerance proportion p0

In the framework of the evaluation of the ICS, the target population is a totality of internal control events over a specified time frame. The proportion nonconforming is the rate of control events which deviate within a specified time frame from the prescribed internal control procedures. The tolerance proportion p0 is the rate of deviation from prescribed internal control procedures considered as tolerable for the purposes of financial auditing within a specified time frame.

In the framework of a test of details, the target population is a totality of statements in a specified account balance or class of transactions. The proportion nonconforming is the rate of misstatements in the target population of statements. The tolerance proportion p0 is the rate of misstatements considered as tolerable for the purposes of financial auditing.

Acceptance and rejection in the case of a test of compliance of the ICS

Both acceptance and rejection affect the auditor’s assessment of the control risk. In the case of acceptance, the auditor rather tends to choose a lower value of the control risk. As a consequence, the amount of auditing efforts in course of subsequent substantive procedures decreases. In the case of rejection, the auditor rather tends to choose a higher value of the control risk. As a consequence, the amount of auditing efforts in course of subsequent substantive procedures increases.

Acceptance and rejection in the case of a test of details

Both acceptance and rejection affect the auditor’s judgment on the existence of material misstatement in the targeted audit population. However, the final conclusion of the auditor is affected by various additional factors, in particular, further test of details, analytical procedures, qualitative assessment of the type of nonconformities.

Examples

Example 1: Lot inspection

A consumer buys a set of screws and can tolerate 3 % of failures. Suppose, the consumer’s confidence in having a low

If the concerned parties are also interested in the operating indicators, Table I.2 provides this additional information: c.type I

Example 2: Auditing of an internal control system (purchase process)

An auditor inspects the purchase process of a medium-size retailer of office equipment to evaluate the effectiveness of the respective part of the relevant ICS. In the case subject to auditing, there are a large number of purchases per year with a high quantity of different suppliers. In a first step, the auditor evaluates the appropriateness of the process design. As a result of an interview and observation, the purchase process consists of the following stages: needs assessment, purchase order, incoming goods, invoice receipt and verification, payment processing, adjustment of general ledger. In these stages, numerous different controls have been identified, which shall ensure that the purchase process operates appropriately. After having assessed the appropriateness of the process design, the auditor determines the kind of controls of each stage, which are subject to further investigation. For example, the auditor selects in the stage of “incoming goods“ the control, whether the goods delivered correspond to the goods ordered in quantity and quality. Therefore, the auditor prompts the retailer to prove that the responsible staff has duly signed the delivery notes of all incoming goods. The signature should indicate that the quantity and quality of each incoming good have been checked (e. g. information from purchase order) and were considered as appropriate. The auditing target is the proportion p of missing or unsigned or inappropriately signed delivery notes. The auditor assumes 5 % (

In view of the large number of incoming goods, the auditor proceeds by sampling inspection. Calculations in the framework of the risk-oriented auditing process impose for the internal control system (ICS) auditing step a confidence level of

The auditor takes a random sample of size n1=32 of goods incoming events from the ERP system. The inspection of the 32 goods incoming events reveals that all corresponding delivery were duly signed, i.e. the number of nonconforming units in the sample is

Example 3: Auditing of an integral control system (sales process)

An auditor inspects the sales process of a medium-size retailer of steel products to evaluate the effectiveness of the respective part of the relevant internal control system (ICS). There are a large number of sales per year with a high quantity of different customers. In a first step, the auditor evaluates the appropriateness of the sales process design. As a result of an interview and the auditor’s own observations, the sales process consists of the following stages: submission of tenders, order acceptance, goods outgoing, invoicing, payment processing, post entries to general ledger. In these stages, numerous different controls have been identified, which shall ensure that the sales process operates appropriately. After having assessed the appropriateness of the process design, the auditor determines the kind of controls of each stage, which are subject to further investigation. For example, the auditor considers in the stage of “invoicing“ the control of whether the realisation principle has been observed appropriately. Therefore, the auditor asks the retailer to prove that, with respect to all single sales, the realisation of the turnover has been recorded in the correct period. The auditor would accept 3 % of incorrectly recorded turnovers as tolerable.

In view of the large number of outgoing invoices per year, the auditor proceeds by sampling inspection. For each sampled invoice, the auditor investigates whether the turnover was realised correctly. Calculations in the framework of the risk-oriented auditing process impose for the internal control system (ICS) auditing step a confidence level of

Example 4: Auditing test of details (accounts receivable)

An auditor inspects the accounts receivables of a medium-size retailer of office equipment with respect to accuracy of statements at balance sheet date. The auditor imposes a tolerance of

In view of the large number of accounts receivable, the auditor proceeds by sampling inspection. Calculations in the framework of the risk-oriented auditing process impose for the test of details auditing step a confidence level of

Example 5: Auditing test of details (raw materials)

An auditor inspects the raw materials inventory of a medium-size retailer of steel products with respect to accurate value assessment at balance sheet date. The auditor imposes a tolerance of

In view of the large variety of raw materials, the auditor proceeds by sampling inspection. The auditor imposes for the test of details a confidence level of

Sampling plans

Sampling plans with acceptance/rejection numbers for varying nominal confidence levels, stages arranged row-wise are given in Tables 1 to 5.

Table 1 — Sampling plans under nominal confidence level γ = 0,70

Trust

p0 in proportion

0,01

0,02

0,03

0,04

0,05

0,06

Low

181      (0; 4)

797      (9; 10)

91       (0; 4)

449     (10; 11)

60       (0; 4)

393    (13; 14)

45      (0; 4)

299    (13; 14)

36      (0; 4)

260    (14; 15)

30      (0; 4)

217    (14; 15)

Medium

148      (0; 4)

599      (7; 8)

74       (0; 4)

299     (7; 8)

49      (0; 4)

200    (7; 8)

37       (0; 4)

150      (7; 8)

30       (0; 4)

120      (7; 8)

25      (0; 4)

100    (7; 8)

High

120      (0; 7)

557      (6; 7)

60       (0; 6)

278     (6; 7)

40      (0; 6)

147    (5; 6)

30       (0; 6)

126      (6; 7)

24        (0; 5)

103      (5; 6)

20      (0; 5)

82      (5; 6)

Trust

p0 in proportion

0,07

0,08

0,09

0,1

0,15

0,2

Low

26      (0; 4)

185    (14; 15)

22       (0; 4)

150    (13; 14)

20        (0; 4)

121    (12; 13)

18       (0; 4)

109    (12; 13)

12      (0; 4)

58      (10; 11)

9        (0; 4)

44      (10; 11)

Medium

22      (0; 4)

85      (7; 8)

19      (0; 4)

75      (7; 8)

17      (0; 4)

66      (7; 8)

15      (0; 4)

50      (6; 7)

10      (0; 4)

41      (7; 8)

8        (0; 4)

29      (7; 8)

High

17      (0; 5)

64      (5; 6)

15      (0; 5)

59      (5; 6)

13      (0; 5)

56      (5; 6)

12      (0; 5)

38      (4; 5)

8        (0; 4)

34      (5; 6)

6        (0; 4)

23      (5; 6)

Table 2 — Sampling plans under nominal confidence level γ = 0,80

Trust

p0 in proportion

0,02

0,03

0,04

0,05

0,06

0,07

Low

110      (0; 5)

600      (13; 14)

74        (0; 5)

460      (15; 16)

55         (0; 5)

343      (15; 16)

44       (0; 5)

275     (15; 16)

37       (0; 5)

229      (15; 16)

31      (0; 5)

197   (15; 16)

Medium

94        (0; 5)

392     (9; 10)

63        (0; 5)

228      (8; 9)

47        (0; 5)

169      (8; 9)

38       (0; 5)

135     (8; 9)

32        (0; 5)

112      (8; 9)

27      (0; 5)

97      (8; 9)

High

80        (0; 7)

333      (7; 8)

53        (0; 8)

214      (7; 8)

40         (0; 7)

151      (7; 8)

32        (0; 6)

113      (6; 7)

27         (0; 6)

94         (6; 7)

23      (0; 6)

81      (6; 7)

Trust

p0 in proportion

0,08

0,09

0,1

0,15

0,2

Low

27       (0; 5)

160    (14; 15)

24      (0; 5)

142    (14; 15)

22      (0; 5)

128   (14; 15)

15      (0; 5)

76      (13; 14)

11      (0; 5)

48      (11; 12)

Medium

24      (0; 5)

85      (8; 9)

21      (0; 5)

76      (8; 9)

19      (0; 5)

57      (7; 8)

13      (0; 5)

52      (9; 10)

10      (0; 5)

33      (8; 9)

High

20      (0; 6)

71      (6; 7)

18      (0; 6)

64      (6; 7)

16      (0; 6)

52      (6; 7)

10      (0; 6)

37      (6; 7)

8        (0; 5)

26      (6; 7)

Table 3 — Sampling plans under nominal confidence level γ = 0,90

Trust

p0 in proportion

0,02

0,03

0,04

0,05

0,06

0,07

Low

150      (0; 7)

710      (16; 17)

100       (0; 7)

547      (18; 19)

75        (0; 7)

407      (18; 19)

60        (0; 7)

306      (17; 18)

50        (0; 7)

255      (17; 18)

43        (0; 7)

219      (17; 18)

Medium

131      (0; 7)

463      (11; 12)

88        (0; 7)

309      (11; 12)

66        (0; 7)

232      (11; 12)

52        (0; 7)

185      (11; 12)

44        (0; 7)

154      (11; 12)

38        (0; 7)

132      (11; 12)

High

114      (0; 11)

440      (10; 11)

76        (0; 9)

275      (9; 10)

57        (0; 8)

211      (9; 10)

45        (0; 9)

163      (9; 10)

38        (0; 8)

139      (9; 10)

32        (0; 8)

119      (9; 10)

Trust

p0 in proportion

0,08

0,09

0,1

0,15

0,2

Low

37        (0; 7)

192      (17; 18)

33       (0; 7)

171    (17; 18)

30       (0; 7)

141    (16; 17)

17       (0; 6)

83       (14; 15)

13       (0; 6)

51      (12; 13)

Medium

33        (0; 7)

116      (11; 12)

29       (0; 7)

92       (10; 11)

26      (0; 7)

72      (9; 10)

16       (0; 7)

62      (11; 12)

12       (0; 6)

42      (10; 11)

High

28         (0; 8)

104      (9; 10)

25       (0; 7)

81       (8; 9)

22      (0; 8)

63      (7; 8)

15       (0; 7)

53      (9; 10)

11       (0; 6)

37      (9; 10)

Table 4 — Sampling plans under confidence value γ = 0,99

Trust

p0 in proportion

0,02

0,03

0,04

0,05

0,06

0,07

Low

188      (0; 9)

802      (18; 19)

126      (0; 9)

631      (21; 22)

94      (0; 9)

472    (21; 22)

76        (0; 9)

357      (20; 21)

63        (0; 9)

298      (20; 21)

54        (0; 9)

240      (19; 20)

Medium

169      (0; 9)

553      (13; 14)

112      (0; 9)

363      (13; 14)

84      (0; 9)

271    (13; 14)

67        (0; 9)

216      (13; 14)

56         (0; 8)

201      (14; 15)

48        (0; 8)

184      (15; 16)

High

149     (0; 11)

500      (11; 12)

99        (0; 10)

332      (11; 12)

74      (0; 10)

253    (11; 12)

59         (0; 10)

199      (11; 12)

49         (0; 10)

177      (12; 13)

42        (0; 10)

166      (13; 14)

Trust

p0 in proportion

0,08

0,09

0,1

0,15

0,2

Low

47        (0; 9)

208      (19; 20)

38        (0; 8)

189      (19; 20)

34        (0; 8)

171      (19; 20)

23      (0; 8)

99      (17; 18)

17      (0; 8)

64      (15; 16)

Medium

42        (0; 8)

165      (15; 16)

35        (0; 8)

144      (15; 16)

32        (0; 8)

130      (15; 16)

21      (0; 8)

72      (13; 14)

16      (0; 8)

49      (12; 13)

High

36        (0; 11)

148      (13; 14)

32        (0; 10)

132      (13; 14)

29         (0; 10)

117      (13; 14)

19      (0; 9)

58      (10; 11)

14      (0; 8)

44      (10; 11)

Table 5 — Sampling plans under nominal confidence level γ = 0,99

Trust

p0 in proportion

0,03

0,04

0,05

0,06

0,07

Low

177      (0; 13)

806      (27; 28)

133      (0; 13)

603      (27; 28)

107     (0; 13)

481      (27; 28)

89        (0; 13)

402      (27; 28)

76       (0; 13)

343     (27; 28)

Medium

165      (0; 13)

578      (20; 21)

123      (0; 13)

408      (19; 20)

99         (0; 13)

302      (18; 19)

82        (0; 13)

283      (20; 21)

70        (0; 12)

263      (21; 22)

High

152      (0; 17)

540      (18; 19)

113       (0; 15)

374      (17; 18)

90         (0; 15)

284      (16; 17)

75         (0; 17)

268      (18; 19)

64        (0; 16)

240      (19; 20)

Trust

p0 in proportion

0,08

0,09

0,1

0,15

0,2

Low

62       (0; 12)

305      (27; 28)

55        (0; 12)

259      (26; 27)

50        (0; 12)

222      (25; 26)

33        (0; 12)

126      (22; 23)

23      (0; 11)

91      (21; 22)

Medium

59       (0; 12)

231      (21; 22)

52        (0; 12)

192      (20; 21)

47         (0; 12)

152      (18; 19)

31        (0; 12)

94        (17; 18)

22      (0; 11)

56      (14; 15)

High

56        (0; 16)

209      (19; 20)

49        (0; 16)

178      (18; 19)

44         (0; 15)

129      (15; 16)

29        (0; 15)

86        (15; 16)

21      (0; 13)

48      (12; 13)

(informative) Confidence intervals

The methodology of the decision procedure in 4.3 is based on shortest two-sided confidence intervals under prior information for an unknown proportion. See Reference [5] for technical details on shortest confidence intervals under prior information. The hypothesis test compares the confidence intervals against the desired tolerance p0.

The decision procedure in stages 1 and 2 proceeds as follows, see Figure A.1 for an illustration.

In stage 1, after drawing n1 items and observing x1 nonconforming units, a confidence interval for p is calculated. The decision in stage 1 proceeds according to the following scenarios: