Transcript Ch11a_s01_605

Acceptance Sampling
Outline
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Sampling
Some sampling plans
A single sampling plan
Some definitions
Operating characteristic curve
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Necessity of Sampling
• In most cases 100% inspection is too costly.
• In some cases 100% inspection may be impossible.
• If only the defective items are returned, repair or
replacement may be cheaper than improving quality.
But, if the entire lot is returned on the basis of sample
quality, then the producer has a much greater
motivation to improve quality.
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Some Sampling Plans
• Single sampling plans:
– Most popular and easiest to use
– Two numbers n and c
– If there are more than c defectives in a sample of
size n the lot is rejected; otherwise it is accepted
• Double sampling plans:
– A sample of size n1 is selected.
– If the number of defectives is less than or equal to
c1 than the lot is accepted.
– Else, another sample of size n2 is drawn.
– If the cumulative number of defectives in both
samples is more than c2 the lot is rejected;
otherwise it is accepted.
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Some Sampling Plans
• Sequential sampling
– An extension of the double sampling plan
– Items are sampled one at a time and the
cumulative number of defectives is recorded at
each stage of the process.
– Based on the value of the cumulative number of
defectives there are three possible decisions at
each stage:
• Reject the lot
• Accept the lot
• Continue sampling
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Some Definitions
• Acceptable quality level (AQL)
Acceptable fraction defective in a lot
• Lot tolerance percent defective (LTPD)
Maximum fraction defective accepted in a lot
• Producer’s risk, 
Type I error = P(reject a lot|probability(defective)=AQL)
• Consumer’s risk, 
Type II error = P(accept a lot| probability(defective)=LTPD)
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A Single Sampling Plan
Consider a single sampling plan with n = 10 and c = 2
• Compute the probability that a lot will be accepted
with a proportion of defectives, p = 0.10
• If a producer wants a lot with p = 0.10 to be accepted,
the sampling plan has a risk of _______________
• This is producer’s risk,  and AQL = 0.10
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A Single Sampling Plan
• Compute the probability that a lot will be accepted
with a proportion of defectives, p = 0.30
• If a consumer wants to reject a lot with p = 0.30, the
sampling plan has a risk of _____________
• This is consumer’s risk,  and LTPD = 0.30
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Approximation to Binomial Distribution
Under some circumstances, it may be desirable to
obtain  and  by an approximation of binomial
distribution
• Poisson distribution: When p is small and n is
moderately large (n>25 and np<5)
• Normal distribution: When n is very large, np(1-p)>5
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Operating Characteristic Curve
1.00
{
Probability of acceptance, Pa
1- =
0.05
0.80
OC curve for n and c
0.60
0.40
0.20
 = 0.10
{
0.02 0.04
AQL
0.06
0.08
0.10
0.12
Percent defective
0.14
0.16
LTPD
0.18
0.20
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Example: Samples of size 50 are drawn from lots 200
items and the lots are rejected if the number of defectives
in the sample exceeds 4. If the true proportion of
defectives in the lot is 10 percent, determine the
probability that a lot is accepted using
a. The Poisson approximation to the binomial
b. The normal approximation to the binomial
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Example: Samples of size 50 are drawn from lots 200
items and the lots are rejected if the number of defectives
in the sample exceeds 4. If the true proportion of
defectives in the lot is 10 percent, determine the
probability that a lot is accepted using
a. The Poisson approximation to the binomial
  np  50(0.10)  5
P { X  4 |   5}  0.4405
b. The normal approximation to the binomial
  np  50(0.10)  5
  np(1 - p )  50(0.1)( 0.90)  2.12
4.5 - 5 

P { X  4}  P  z 
  Pz  -0.2357
2.12 

 0.4052
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Operating Characteristic Curve
OC Curve by Poisson Approximation
n
c
AQL
LTPD
Proportion
Defective
(p)
0.005
0.01
0.015
0.02
0.025
0.03
100
4
0.02
0.08

0.052653017

0.0996324
np
0.5
1
1.5
2
2.5
3
Probability
of c or less
Defects
(Pa)
0.999827884
0.996340153
0.981424064
0.947346983
0.891178019
0.815263245
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0.
00
5
0.
02
5
0.
04
5
0.
06
5
0.
08
5
0.
10
5
0.
12
5
0.
14
5
0.
16
5
0.
18
5
Probability of acceptance
Operating Characteristics Curve
1.2
1
0.8
0.6
0.4
0.2
0
Proportion of Defectives
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