1. Acceptance Sampling for Attributes Statistical Quality Control
SMU
EMIS 7364
NTU
TO-570-N
Acceptance Sampling for Attributes
Updated:
Statistical Quality Control
Dr. Jerrell T. Stracener, SAE Fellow

2. Decision Risk
Producer’s Risk – conforming lots are rejected
Consumer’s Risk – nonconforming lots are accepted.

3. Decision Risk True Situation
Lot meets Lot does not Decision requirements meet requirements
Accept Lot no error Type II error
Reject Lot Type I error no error

4. Lot-by-Lot Acceptance Sampling by Attributes
– Single Sampling

5. Single Sampling
N = lot size = the number of items in the lot from which the sample is to be drawn.
n = sample size = the number of items drawn at random from the lot
c = the maximum allowable number of defective items in the sample. More than c defectives in the sample will cause rejection of the lot.

6. Type A OC Function for Single Sampling Plan
Sampling Plan Specification
N = Lot Size
n = Sample Size
c = Acceptance number
D = true number of defectives in the lot
X = number of defective items in the random sample.

7. Type A OC Function for Single Sampling Plan
Probability Distribution of X
X ~ H(N, D, n)
Probability Mass Function of X

8. Type A OC Function for Single Sampling Plan

9. Example – Single Sampling Plan A
Determine and plot the OC Function for a single sampling plan specified by

10. Example Solution – Single Sampling Plan
OC(D) D

11. Concept
Suppose that the lot size N is large (theoretically
infinite). Under this condition, the distribution of the
number of defectives d in a random sample of n
items is binomial with parameters n and p, where p
is the fraction of defective items in the lot. An
equivalent way to conceptualize this is to draw lots
of N items at random from a theoretically infinite
process, and then to draw random samples of n from
these lots. Sampling from the lot in this manner is
the equivalent of sampling directly from the process.

12. Type B OC Function for Single Sampling Plan
Sampling Plan Specification
N = Lot Size
n = Sample Size
c = Acceptance number
N is infinite, or at least much larger than n
D = true number of defective items in the lot
p = proportion of the population that is defective, i.e.,
X = number of defective items in the random sample.

13. Type B OC Function for Single Sampling Plan
Probability Distribution of X
X ~ B(n, p)
Probability Mass Function of X

14. Acceptance Sampling
The probability of observing exactly x defective items is:
for x = 0, 1, . . ., n

15. OC Function
The probability of acceptance is the probability that
d is less than or equal to c, or:

16. Example – Single Sampling Plan B
Determine and plot the OC Function for a single sampling plan specified by
Compare the OC curve for N=500, n=98, c=2.

17. Example Solution – Single Sampling Plan
OC(p) p

18. Example
If the lot fraction defective is p = 0.02, n = 89 and
c = 2, then

19. Example
The OC curve is developed by evaluating PA(p) for
various values of p. The following table displays the
calculated value of several points on the curve.
The OC curve shows the discriminatory power of the
sampling plan. For example, in the sampling plan
n = 98, c = 2, if the lots are 2% defective, the
probability of acceptance is approximately This
means that if 100 lots from a process that
manufactures 2% defective product are submitted
to this sampling plan, we will expect, in the long run, to accept 74 of the lots and reject 26 of them.

20. Example
Probabilities of Acceptance for the Single-Sampling
plan n = 89, c = 2
Fraction Probability of
Defective, p Acceptance, Pa (p)

21. If N = 500, n =98 & c=2,
For comparison, select p= Then

22. Example -p 10 20 30 40 -D

23. Single Sample Test Plan Design
Probability Distribution of X
P0 = specified fraction defective
P1 = minimum acceptable fraction defective
a = producer’s risk
b = consumer’s risk

24. Test Procedure
To test
H0: p = p0
vs H1: p = p1
at the   100% level of significance,
Obtain a random sample of size n
Inspect the n items and determine the number, X, that are defective
Reject H0 if x > c, otherwise accept H0

25. Single Sampling Plan x
number
of defective
items
reject c .
accept 2 1 n
n = sample size
x = number of defective items
c = maximum number of defective items for acceptance

26. Operating Characteristic (OC) Function
Note that:
and

27. OC Curve 1
1 -   p0 p1 1 p

28. Test Plan Design
To determine a single sample test plan for testing H0: p = p0, specify values of p0, p1, , and  such that PA(p0) = 1 -  and PA(p1) = , then find the values of n and c that satisfy the following equations.
and

29. Lot-by-Lot Acceptance Sampling by Attributes
– Double Sampling

30. Double Sampling
A double-sampling plan is a procedure in which,
under certain circumstances, a second sample is
required before the lot can be sentenced. A double-
sampling plan is defined by four parameters:
n1 = sample size on the first sample
c1 = acceptance number of the first sample
n2 = sample size on the second sample
c2 = acceptance number for both samples

31. Single Sampling Plan number of defective items c1 2 1 Accept Reject n1
Accept
Reject n1 x
n1+n2
Continue .
c1+1 c2
. . .

32. Double Sampling - Advantages
The principal advantage of a double-sampling plan
with respect to single sampling is that it may reduce
the total amount of required inspection. Suppose that
the first sample taken under a double-sampling plan
is smaller than the sample that would be required
using a single-sampling plan that offers the consumer
the same protection. In all cases, then, in which a lot
is accepted or rejected on the first sample, the cost
of inspection will be lower for double sampling than it
would be for single sampling. It is also possible to
reject a lot without complete inspection of the second
sample (called curtailment of the second sample).

33. Double Sampling - Disadvantages
Double sampling has two potential disadvantages:
1. Unless curtailment is used on the second sample, under some circumstances double sampling may require more total inspection than would be required in a single-sampling plan that offers the same protection.
2. Double-sampling is administratively more complex, which may increase the opportunity for the occurrence of inspection errors. Furthermore, there may be problems in storing and handling raw materials or component parts for which one sample has been taken, but that are awaiting a second sample before a final lot dispositioning decision can be made.

34. Double Sampling - The OC Curve
The performance of a double-sampling plan can be
conveniently summarized by means of its operating-
characteristic (OC) curve. A double-sampling plan
has a primary OC curve that gives the probability of
acceptance as a function of lot of process quality. It
also has supplementary OC curves that show the
probability of acceptance as a function of lot
acceptance and rejection on the first sample.

35. Operation of the double-sampling plan with
n1 = 50, c1 = 1, n2 = 100, c2 = 3
Inspect a random sample of
n1 = 50 from the lot
d1 = number of observed defectives
Accept
the
lot
d1 > c1 = 3
Reject
the
lot
d1  c1 = 1
Inspect a random sample of
n2 = 100 from the lot
d2 = number of observed defectives
Accept
the
lot
Reject
the
lot
d1 + d2  c2 = 3
d1 + d2 > c2 = 3

36. Example
If denotes the probability of acceptance on the
combined samples, and and denote the
probability of acceptance on the first and second
samples, respectively, then
is just the probability that we will observe
d1  c1 = 1 defectives out of a random sample of
n1 = 50 items. Thus

37. Example
If p = 0.05 is the fraction defective in the incoming
lot, then
To obtain the probability of acceptance on the second
sample, we must list the number of ways the second
sample can be obtained. A second sample is drawn
only if there are two or three defectives on the first
sample - that is, if c1 < d1  c2.

38. Example
1. d1 = 2 and d2 = 0 or 1; that is, we find two defectives on the first sample and one or less defectives on the second sample. The probability of this is:
P(d1 = 2, d2  1) = P(d1 = 2) x P(d2  1)
= (0.261)(0.037)
= 0.009

39. Example
2. d1 = 3 and d2 = 0; that is, we find three defectives on the first sample and no defectives on the second sample. The probability of this is:
P(d1 = 3, d2  0) = P(d1 = 3) x P(d2 = 0)
= (0.220)(0.0059)
= 0.001

40. Example
Thus, the probability of acceptance on the second
sample is
The probability of acceptance of a lot that has fraction
defective p = 0.05 is therefore

41. Double Sampling - The OC Curve
OC Curves for the double-sampling plan with
n1 = 50, c1 = 1, n2 = 100, c2 = 3
Probability of acceptance on combined sample
Probability, P
Probability of rejection on first sample
Probability of acceptance on first sample
Lot fraction defective, p

42. Rectifying Inspection Programs

43. Rectifying Inspection Programs
Acceptance-sampling programs require corrective action when lots are rejected.
Generally takes the form of 100% inspection or screening of rejected lots, with all discovered defective items either removed for subsequent rework or return to the vendor, or replaced from a stock of known good items. Such sampling programs are called rectifying inspection programs.

44. Rectifying Inspection Programs (continued)
The inspection activity affects the final quality of the outgoing product. Suppose that the incoming lots to the inspection activity have fraction defective, po. Some of these lots will be accepted, and others will be rejected. The rejected lots will be screened, and their final fraction defective will be zero. However, accepted lots have fraction defective p0. Consequently, the outgoing lots from the inspection activity are a mixture of lots with fraction defective p0 and fraction defective zero, so the average fraction defective in the stream of outgoing lots is p1, which is less that p0, and serves to “correct” lot quality.

45. Rectifying Inspection
Rejected lots
Fraction defective 0
Incoming lots Fraction defective p0
Inspection activity
Outgoing lots Fraction defective p1
Fraction defective p0
Accepted lots

46. Rectifying Inspection Programs (continued)
Used in situations where the manufacturer wishes to know the average level of quality that is likely to result at a given stage of the manufacturing operations.
Used either at receiving inspection, in-process inspection of semi-finished products, or at final inspection of finished goods
The objective of in-plant usage is to give assurance regarding the average quality of material used in the next stage of the manufacturing operations.

47. Handling of Rejected Lots
The best approach is to return rejected lots to the vendor, and require it to perform the screening and rework activities.
Has the psychological effect of making the vendor responsible for poor quality
May exert pressure on the vendor to improve its manufacturing processes or to install better process controls.
Screening and rework take place at the consumer level because the components or raw materials are required in order to meet production schedules.

48. Average Outgoing Quality
Widely used for the evaluation of a rectifying sampling plan.
Is the quality in the lot that results from the application of rectifying inspection
Is the average value of lot quality that would be obtained over a long sequence of lots from a process with fraction defective p.

49. Average Outgoing Quality (AOQ)
The average fraction defective, called average outgoing quality is
Where the lot size is N and that all defectives are replaces with good units. Then in lots of size N, we have
N items in the lot that, after inspection, contain no defectives, because all discovered defectives are replaced
N – n items that, if the lots is rejected, contain no defectives
N – n items that, if accepted, contain (N-n)p defectives

50. Example
Suppose that N = 10,000, n = 89, and c = 2, and that the incoming lots are of quality p = Now at p = 0.01, we have Pa = , and the AOQ is
That is, the average outgoing quality is at 0.93% defective.

51. Example
Average outgoing quality will vary as the fraction defective of the incoming lots varies. The curve that plots average outgoing quality against incoming lot quality is called an AOQ curve.
AOQ
fraction defective, p

52. Average Total Inspection (ATI)
Another important measure relative to rectifying inspection is the total amount of inspection required by the sampling program. If the lots contain no defective items, no lots will be rejected, and the amount of inspection per lot will be the sample size n. If the items are all defective, every lot will be submitted to 100% inspection, and the amount of inspection per lot will be the lot size N. Of the lot quality is 0 < p < 1, the average amount of inspection per lot will vary between the sample size n and the lot size N.

53. Average Total Inspection (ATI)
Consider our previous example with N = 10,000, n = 89, and c = 2, and p = Then since Pa = , we have
Remember this is an average number of units inspected over many lots with fraction defective p = 0.01.

54. Determination of optimum quality level, p*
cost to achieve p
total cost
cost of inspection per lot
cost p* 1
incoming quality level ~p