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American Journal of Sustainable Cities and Society
Available online on http://www.rspublication.com/ajscs/ajsas.html
Issue 3, Vol. 1 January 2014
ISSN 2319 – 7277
Weighted Binomial for Single Sampling Plan
M.Latha#1, K.Abdul Samathu#2
#1.Dr.M.Latha,Head and Associate Professor, Department of Statistics, Government Arts
College, Udumalpet - 642126, 9566324923.
#2.K.Abdul Samathu, Research Scholar, Department of Statistics, Government Arts College,
Udumalpet-642126, 9976758575
ABSTRACT
In this paper , single sampling plan (SSP) with weighted binomial distribution
(WBD) is obtained. Operating characteristic (OC) curve for this SSP is constructed and
analyzed. The effect of increasing sample size, increasing weight of the distribution and
increasing both of them simultaneously is observed. comparison is made with SSP with
binomial distribution.
Keywords
Weighted binomial distribution, single sampling plan, operating characteristics curve.
Corresponding Author : K.Abdul Samathu
INTRODUCTION
Acceptance sampling plan is the process of evaluating a portion of the product in a lot
for the purpose of accepting or rejecting the entire lot. In case of conventional sampling
plans, right from Hameker. considerable amount of work has been carried out with regard to
selection of sampling plan.Dodge (1959) has studied sampling inspection based on the SSP.
Dodge and Romig (1959) have studied on sampling inspections tables.
Weighted Distribution :
Traditional environmetric theory and practice have been occupied largely with
randomization and replication. But in environmental and ecological work, observations also
fall in the non experimental, non- replicated, and nonrandom categories. The problems of
model specification and data interpretation then acquire special importance and great concern.
The theory of weighted distributions provides a unifying approach for these problems.
Weighted distributions take into account the method of ascertainment, by adjusting the
probabilities of actual occurrence of events to arrive at a specification of the probabilities of
those events as observed and recorded. Failure to make such adjustments can lead to incorrect
conclusions. The concept of weighted distributions can be traced to the study of the effect of
methods of ascertainment upon estimation of frequencies by Fisher(1934) . In extending the
basic ideas of Fisher (1965) , Rao (1985) saw the need for a unifying concept and identified
various sampling situations that can be modeled by what he called weighted distributions.
Rao (1965) introduced the concept of weighted distribution when the samples are
recorded without a sampling frame that enables random samples to be drawn. The weight
function that usually appears in the scientific and statistical literature is ω(X) =X, which
provide the size – biased version of the random variable. The size – biased version of order k,
which corresponds to the weight ω(X) =Xk, for k any real positive number has also been
widely used. Joan Del Castillo and Pérez- Casany(1998) applied the weighted Poisson
distribution that results from the modification of the Poisson distribution with the weight
ω(X) =Xk can also considered as a mixture of the size – biased version of the Poisson
distribution. They fit the weighted Poisson distribution for over dispersion (aggregation) and
under dispersion (repulsion) situation. Patil and Rao (1973). and Ratnaparki (1986) have
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American Journal of Sustainable Cities and Society
Available online on http://www.rspublication.com/ajscs/ajsas.html
Issue 3, Vol. 1 January 2014
ISSN 2319 – 7277
proved that given a random variable X, the weighted version Xk is stochastically greater or
smaller than the original random variable X according as the weight function ω(X) is
monotone increasing or decreasing to X.
Weighted Binomial Distribution :
The probability mass function of weighted Binomial distribution is given by:
𝑝 𝑋, 𝑛, 𝑝, 𝑘 =
𝑋 𝑘 𝑃 𝑋;𝑛,𝑝
𝑋 𝑘 𝑃 𝑋;𝑛,𝑝
𝑋 = 0,1,2, …
𝑝 𝑋, 𝑛, 𝑝, 1 =
=
𝑛−1
𝑥−1
(1)
𝑥
𝑛
𝑥
𝑝 𝑥 (1 − 𝑝)𝑛−𝑥
𝑤ℎ𝑒𝑛 𝑘 = 1,
𝑛𝑝
𝑝 𝑥−1 (1 − 𝑝)𝑛−𝑥 𝑥 = 1,2, …
Sample
(2)
Probability Value (P)
c
0.01
0.05
0.25
0.5
10
1
0.009135
0.03151
0.01877
0.00097
2
0.00083
0.01492
0.05631
0.00878
3
0.000033
0.00314
0.03515
0.00092
1
0.00868
0.02438
0.00445
0.00003
0.00122
0.01796
0.02078
0.00042
0.00008
0.00614
0.04503
0.00277
0.00826
0.01886
0.00105
0.0000009
0.00158
0.01886
0.00669
0.000018
0.00014
0.00893
0.02008
0.00016
0.00785
0.01459
0.00025
2.98023×10-8
0.00190
0.01844
0.002006
7.15256×10-7
3
0.00022
0.01116
0.000769
8.22544×10-6
1
0.00747
0.01129
0.000059
9.31323×10-10
2
0.00218
0.01724
0.00057
2.70084×10-8
3
0.00030
0.01270
0.00268
3.78117×10-7
15
2
3
20
1
2
3
25
1
2
30
Probability of Acceptance pa(p)
N
Table – 1
Tables are constructed for different values of (n,c) Figure 1-3 give. the OC curves for c = 1,2
and 3 and n = 10,15,20,25.30,35,40 and 45
Single sampling plan with weighted binomial distribution
Various single sampling plan studies involve lot size and defective ratio as important
factors that can have some bearing on the main study. Patil (1973) has shown how a
weighted distribution arises as a result of size –biased sampling. Consider the data in table 1
relating to lot and samples.
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Issue 3, Vol. 1 January 2014
ISSN 2319 – 7277
In some cases there may be situations where it is not necessary to include the case of
zero defectives (d=0) It will be helpful to the producer of not considering the case of d=0 for
the calculation of probability of acceptance In such situation the weighted binomial
distribution (WBD) may be useful for the calculation of probability of acceptance.
This sampling plan may be used for large lots having defective items The operating
procedure is as follows :
For each lot, select a sample of n units and test each unit for conformance to the specified
requirements.
Category I :
The lot having non defective items are combined as first category and they are send to
the next stage of the process directly.
Category II :
The lots having defective items are considered for the calculation of risks
(i) Select a random sample of size of n from a lot of size N
(ii) if 1≤ d ≤ c accept the lot ,replacing defective pieces found in the sample by non-defective
ones.
(iii). If d > c inspect the entire lot and replace all the defective pieces by standard ones.
Operating Procedure of Weighted Binomial Single sampling plan (Tree)
Inspection
If d=0
Accept the lot
If d > 0
1≤ d ≤ c
Accept the lot
d>c
continue to the inspection
Operating Characteristic Curve :
The OC curve is constructed for category II. When weight k=1 Operating
Characteristic curve table is given below :
𝑐
𝑃𝑎 (𝑝) =
𝑥=1
𝑛 − 1 𝑥−1
𝑝
1−𝑝
𝑥−1
𝑛−𝑥
Comparison with single sampling plan with Binomial Distribution :
𝑛
Pa(SSP Binomial ) = 𝑐𝑥=1 𝑥 𝑝 𝑥 𝑞 𝑛−𝑥
and
𝑐
𝑃𝑎 (𝑆𝑆𝑃 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑏𝑖𝑛𝑜𝑚𝑖𝑎𝑙) =
𝑥=1
𝑛 − 1 𝑥−1
𝑝
1−𝑝
𝑥−1
𝑛−𝑥
It is observed that the plan based on weighted binomial distribution is more favorable
to the producer when the lots having minimum number of defective items. Moreover the lots
with zero defective have been accepted as a separate category.
When n=20 and c=3, for percent defective p=0.01, the probability of acceptance is
0.00096 in case of binomial distribution and 0.000144 when weighted binomial distribution is
considered.
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American Journal of Sustainable Cities and Society
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Issue 3, Vol. 1 January 2014
ISSN 2319 – 7277
when n=15,c=2 , for percent defective p=0.5 , the probability of acceptance is
0.003204 in case of binomial distribution and 0.000427 when weighted binomial distribution
is considered.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure-1
C=1 and n=10,15,20,25,30,35,40 and 45
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure-2
C=2 and n=10,15,20,25,30,35,40 and 45
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure-3
C=3 and n=10,15,20,25,30,35,40 and 45
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Issue 3, Vol. 1 January 2014
ISSN 2319 – 7277
Comparison of three OC functions based on inflection points:
For 1< c ≤ n-2 Probability of acceptance Pa has one and only point of inflection at
p=p* (say) for 0< Pa <1, The OC curve is strictly convex p>p*and concave for p<p*. The
point inflection for Binomial, weighted binomial and Poisson distribution are c/n-1, c-1/n-2
and c/n respectively.
When n=10, c=2 the inflection point of the OC curve based on Binomial Distribution
is 0.222 (p*B) ,In case of Weighted Binomial Distribution the inflection points is 0.125
(p*WB) and when the Poisson Distribution is considered ,the inflection point is 0.2(p*G)
p*WB< p*B< p*G
CONCLUSION
It is observed that for any particular single sampling plan the probability of
acceptance based on weighted binomial distribution is less than that of binomial distribution
which implies that the consumer has more benefit on the usage of weighted binomial
distribution . Also, on comparing the inflection points of weighted binomial distribution ,
Binomial distribution and Poisson distribution , weighted binomial distribution has minimum
point of inflection for a given single sampling plan.
REFERENCES
1.Rao.C.R (1965) Weighted distribution arising out of methods of ascertainment , in a
celebration of statistics, A.C.Alkinson & S.E.Fienberg, eds, Springer-Verlag. New york.
2.Patil G.P & Rao C.R (1973) Weighted distributions and size biased sampling with
applications to wildlife populations and human families.
3.Patil G.P (2002) Weighted distributions, John wiley & sons Ltd, Chichester.
4.Dodge H.F and Romig H.G (1959) Sampling inspection tables, John wiley & sons,Inc,
New york.
5. Suresh .K.K and Latha.M Selection of Bayesian single sampling plan with gamma prior
distribution (2002)
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