• APPLIED STATISTICS


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    • Abstract: APPLIED STATISTICSPrinciples of acceptance samplingRenu Kaul and S. Roy ChowdhuryReaderDepartment of StatisticsLady Shri Ram College for WomenLajpat Nagar, New Delhi4-Jan-2007 (Revised 20-Nov-2007)CONTENTS

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APPLIED STATISTICS
Principles of acceptance sampling
Renu Kaul and S. Roy Chowdhury
Reader
Department of Statistics
Lady Shri Ram College for Women
Lajpat Nagar, New Delhi
4-Jan-2007 (Revised 20-Nov-2007)
CONTENTS
Problem of lot acceptance
Acceptance Quality Level (AQL)
Lot Tolerance Percent Defective (LTPD)
Average Outgoing Quality Limit (AOQL)
Average Amount of Total Inspection (ATI)
Average Sample Number (ASN)
Producer’s risk (Pp)
Consumer’s risk (Pc)
Operating characteristic curve (OC)
Acceptance – Rejection Sampling Plans
Rectifying Sampling Inspection Plans
Keywords
Acceptance Quality Level (AQL); Lot Tolerance Percent Defective (LTPD); Consumer’s risk; Producer’s risk;
OC curve; ASN curve; ATI curve; AOQ curve; Rectifying inspection plans; Single sampling; Double sampling;
Sequential sampling
Problem of lot acceptance
Another important aspect of statistical quality control is lot/ product control. However, from
various practical and economic considerations it is not feasible to inspect each and every lot
fully. The usual practice is to draw a random sample from the lot and take a decision about
the lot disposition on the basis of sample fraction defective i.e. take recourse to sampling
inspection. When a customer is offered a lot there are following three choices before him for
accepting a lot:
a) without any inspection accept the lot
b) after 100% inspection accept the lot
c) select a fraction of the lot as sample and use an appropriate sampling plan
Sampling inspection has a number of advantages as:
a) it reduces time
b) it is applicable to destructive sampling
c) it reduces the inspection error considerably
d) it is less expensive
e) it reduces damage as there is lesser handling of the product.
When inspection is for the purpose of acceptance or rejection of the product based on certain
quality standards the procedure employed is called Acceptance Sampling. Accepted sampling
plans may be classified by attributes and variables. Here we shall talk of acceptance sampling
plans for attributes first. It means that items will be judged as defective/ bad or non-defective/
good. Further, a sampling plan may be of either type viz. the acceptance-rejection type or
acceptance-rectification type. In an acceptance-rejection sampling inspection plan lots are
either accepted or rejected on the basis of the sample. In an acceptance-rectification sampling
inspection plan if we do not accept on the basis of the sample, we take recourse to 100%
inspection and in either case replace all defectives by non-defectives.
Before embarking upon a detailed discussion of the sampling inspection plans we shall first
discuss a few concepts, which are quite important in their discussion:
Acceptance Quality Level (AQL)
This is the quality level of the supplier’s process that the consumer would consider to be
acceptable. Let p1 be the fraction defective in a lot, which is fairly of good quality. Then
P[rejecting a lot of quality p1 ] = 0.05
P[accepting a lot of quality p1 ] = 0.95
p1 is known as the acceptance quality level.
Lot Tolerance Percent Defective (LTPD)
It is the maximum fraction defective ( p t ) in the lot that the consumer will tolerate. 100 p t is
called lot tolerance percent defective. The probability of accepting lots with fraction defective
p t or greater is very small.
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Average Outgoing Quality Limit (AOQL)
Let p be the fraction defective in the lot before inspection, also called the incoming quality.
Then the expected number of fraction defectives remaining in the lot after the application of
the sampling inspection plan is known as average outgoing quality. The maximum value of
the AOQ, the maximum being taken with respect to p is called average outgoing quality limit.
Average Amount of Total Inspection (ATI)
The expected value of the sample size required for coming to a decision in an acceptance-
rectification sampling inspection plan calling for 100% inspection of the rejected lots is
called average amount of total inspection. It is a function of the incoming quality. The curve
obtained on plotting ATI against p is called the ATI Curve.
Average Sample Number (ASN)
The expected value of sample size required for coming to a decision whether to accept or
reject a lot in an acceptance- rejection sampling inspection plan is called ASN. The curve
obtained on plotting ASN against p is called the ASN curve.
Producer’s risk (Pp)
A producer can be an individual, a firm or a department that produces goods and supplies
them to another individual, firm or department. As our decisions are based on sampling
inspection plans he is always under the constant risk that sometimes certain lots of
satisfactory quality be rejected. Further let the producer’s average fraction defective be p i.e.
a quality standard, which he has been able to maintain over a long period of time. Then
probability of rejecting a lot of quality p is called producer’s risk and is denoted by Pp .
Pp = P[rejecting a lot of quality p ] = α (1)
Consumer’s risk (Pc)
By consumer here we will mean the person, firm or department that receives the articles from
the producer. Just like producer, consumer is also faced with the risk of accepting a lot of
unsatisfactory quality on the basis of sampling inspection. Thus probability of accepting a lot
of quality pt is called consumer’s risk and is denoted by Pc .
Pc = P[accepting a lot of quality p t ] = β (2)
Operating characteristic curve (OC)
The curve obtained on plotting the acceptance probability L(p) against p, the lot fraction
defective is called OC curve. The discriminatory power of a sampling plan is revealed by its
OC curve. The greater the slope of the OC. curve the greater the discriminatory power. By
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increasing the sample size we can increase the discriminatory power of the sampling plan. Of
course, the acceptance number c should be kept proportional to n. The ideal OC curve looks
like
However, this ideal OC curve can never be attained in reality.
There are basically two types of sampling plans:
1) Acceptance-rejection sampling plans and
2) Acceptance-rectification sampling plans or Rectifying sampling plans
1. Acceptance – Rejection Sampling Plans
In these plans a decision to accept or reject a lot is taken on the basis of the samples drawn
from it.
Single Sampling Plan
In this sampling plan, the decision to accept or reject the lot is based on a single sample.
Draw a random sample of size n from the lot consisting of N units. If the number of
defectives (d) in the sample is less than or equal to c accept the lot, otherwise reject it.
Decision Flow Chart for a Single Sampling Plan
Consider a random sample of
size n for inspection
Compare number of defectives (d) with
the acceptance number (c)
YES NO
dc 2 (acceptance number for both the samples combined) reject the
lot. However, if c1 +1 ≤ d1 ≤ c 2 , take another sample of size n 2 . Let d 2 be the number of
defectives observed in the second sample. If d1 +d 2 ≤ c 2 , accept the lot. If d1 +d 2 > c 2 , reject
the lot.
Decision Flow Chart for Double Sampling Plan
Consider first sample of size n1
for inspection
Compare number of defectives (d1) with the
acceptance numbers (c1) and (c2)
YES NO
d1 < c1
Accept the Lot No Yes
d1 >c2
c1+1 r m (22)
and continue sampling by taking one more observation if
am < dm


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