CHAPTER 4 ACCEPTANCE SAMPLING PLANS - …shodhganga.inflibnet.ac.in/bitstream/10603/8131/10/10_chapter 4.pdf · 103 CHAPTER – 4 ACCEPTANCE SAMPLING PLANS 4.1 Introduction Acceptance

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CHAPTER – 4

ACCEPTANCE SAMPLING PLANS

4.1 Introduction

Acceptance sampling is concerned with inspection and

decision making regarding products, one of the oldest aspects of quality

assurance. In the 1930‟s and 1940‟s, acceptance sampling was one of the

major components of the field of statistical quality control and was used

primarily for incoming or receiving inspection.

A typical application of acceptance sampling is as follows. A

company receives a shipment of product from a vendor. This product is

often a component or raw material used in the company‟s manufacturing

process. A sample is taken from the lot and some quality characteristic of

the units in the sample is inspected. On the basis of the information in this

sample, a decision is made regarding lot disposition. Usually, this

decision is either to accept or to reject the lot. Sometimes we refer to this

decision as lot sentencing. Accepted lots are put into production; rejected

lots may be returned to the vendor or may be subjected to some other lot

disposition action. While it is customary to think of acceptance sampling

as a receiving inspection activity, there are other uses of sampling

methods. For example, frequently a manufacturer will sample and inspect

its own product at various stages of production. Lots that are accepted are

sent forward for further processing, while rejected lots may be reworked

or scrapped.

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It is the purpose of acceptance sampling to sentence lots, not to

estimate the lot quality. Most acceptance sampling plans are not designed

for estimation purposes.

Acceptance sampling plans do not provide any direct form of

quality control. Acceptance sampling simply accepts / rejects lots. Even if

all lots are of the same quality, sampling plan will accept some lots and

reject others, the accepted lots being no better than the rejected ones.

Process controls are used to control and systematically improve quality,

but acceptance sampling is not. The most effective use of acceptance

sampling is not to “inspect quality into the product” but rather as an audit

tool to ensure that the output of a process conforms to requirements.

We generally use 100% inspection in situations where the

component is extremely critical and passing any defectives would result

in an unacceptably high failure cost at subsequent stages.

Acceptance sampling is most likely to be useful in the following

situations:

1. When testing is destructive.

2. When the cost of 100% inspection is extremely high.

3. When 100% inspection is not technologically feasible or

would require so much calendar time that production

scheduling would be seriously impacted.

4. When there are many items to be inspected and the

inspection error rate is sufficiently high that 100% inspection

might cause a higher percentage of defective units to be

passed than would occur with the use of a sampling plan.

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5. When the vendor has an excellent quality history and some

reduction in inspection from 100% is desired, but the

vendor‟s process capability is sufficiently low as to make no

inspection an unsatisfactory alternative.

There are risks of accepting “bad” lots and rejecting “good” lots.

Less information is usually generated about the product or about the

process that manufactured the product.

Acceptance sampling is a “middle ground” between the extremes

of 100% inspection and no inspection. If often provides a methodology

for moving between these extremes as sufficient information is obtained

on the control of the manufacturing process that produces the product.

While there is no direct control of quality in the application of an

acceptance sampling plan to an isolated lot, when that plan is applied to a

stream of lots from a vendor, it becomes a means of providing protection

for both the producer of the lot and the consumer. It also provides for an

accumulation of quality history regarding the process that produces the

lot, and it may provide feedback that is useful in process control, such as

determining when process controls at the vendor‟s plant are not adequate.

Finally, it may place economic or psychological pressure on the vendor to

improve the production process.

Acceptance sampling plans can be classified as variables and

attributes. Variables, of course are quality characteristics that are

measured in numerical scale. Attributes are quality characteristics that are

expressed on a “go, no-go” basis. The primary disadvantage of variables

sampling plans is that the distribution of the quality characteristic must be

known. In most of the standard variable acceptance sampling plans

assume that the distribution of the quality characteristic is normal. If the

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distribution of the quality characteristic is not normal, and a plan based on

the normal assumption is employed. Serious departures from the

advertised risks of accepting or rejecting lots of given quality may be

experienced. Another disadvantage of variable sampling plan is that a

separate sampling plan must be employed for each quality characteristic

that is being inspected. For example, if an item is inspected with respect

to four quality characteristics, it is necessary to have four separate

variables inspection sampling plans. If this same product were being

inspected under attribute sampling, one attribute sampling plan could be

employed. Finally it is possible that the use of variable sampling plans

will lead to rejection of a lot even though the actual sample inspected

does not contain any defective items.

There are two general types of variable sampling procedures, (i)

plans that control the lot or process fraction defective (or non

conforming), and (ii) plans that control a lot or process parameter.

Consider a variable sampling plan to control the lot or process fraction

non-conforming. Since the quality characteristic is a variable, there will

exist either a lower specification limit (LSL), an upper specification limit

(USL) or both, that define the acceptable values of this parameter.

The fraction defective in the lot is a function of the lot or process

parameters. Variable sampling plans can also be used to give assurance

regarding the average quality of a material instead of the fraction

defective. The general approach employed in this type of variable

sampling is statistical hypothesis testing.

In scaled densities a null hypothesis about scale parameter such as

“the scale parameter is greater than or equal to a specified value‟‟ is

equivalent to saying that the “average life of a product governed by the

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given scaled density exceeds a specified value”. Acceptance of this

hypothesis by a test procedure means that the sample life times used for

testing indicate that the lot from which the sample is drawn is a good lot.

Similarly rejection of the hypothesis implies that the lot is a bad lot. Also

instead of mean an equally competent population measure of central

tendency is median especially for skewed models (Balakrishnan et al.

(2007)). Accordingly, acceptance of a null hypothesis about population

median „the median life of a product governed by a scaled density

exceeds a specified value‟ may also indicate that the lot from which the

sample is drawn is a good lot. As an extension, because median is the

fiftieth percentile, a similar test procedure for testing a null hypothesis

can be considered for any general percentile of the model also. This is

parallel between the testing of hypothesis in scaled densities and

sampling plans is the basis of this chapter.

Acceptance sampling plans in statistical quality control concern

with accepting or rejecting a submitted lot of a large size of products on

the basis of the quality of products inspected in a sample taken from the

lot. If the quality of the product that is inspected is the life time of the

product that is put for testing, after the completion of sampling

inspection what we have is a sample of life times of the sampled

products. If a decision to accept or reject the lot subject to the risks

associated with the two types of errors (rejecting a good lot/ accepting a

bad lot) is possible, such a procedure may be termed as „Acceptance

sampling based on life tests‟ or „Reliability test plans‟. Such a procedure

obviously requires the specification of the probability model governing

the life of the products. Exponential distribution-the CFR model is the

central distribution in reliability studies. Epstein (1954) developed

reliability test plans for exponential distribution. Gupta and Groll (1961)

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constructed sampling plans similar to those of Epstein (1954) based on

Gamma distribution. Sampling plans similar to those of Gupta and Groll

(1961) are developed by Kantam and Rosaiah (1998) for half-logistic

distribution and Kantam et al. (2001) for log-logistic distribution,

Rosaiah and Kantam (2005) for the inverse Rayleigh distribution and

Srinivasa Rao et al. (2009b) for Marshall-Olkin extended Lomax

distribution. Sampling plans in a new approach for log-logistic

distribution are suggested by Kantam et al. (2006a) and Srinivasa Rao

et al. (2009a) for Marshall-Olkin extended Lomax distribution

All these authors considered the design of acceptance sampling

plans based on the population mean under a truncated life test. Whereas

Lio et al. (2010) have considered acceptance sampling plans from

truncated life tests based on the Birnbaum-Saunders distribution for

percentiles and they proposed that the acceptance sampling plans based

on mean may not satisfy the requirement of engineering on the specific

percentile of strength or breaking stress. When the quality of a specified

low percentile is concerned, the acceptance sampling plans based on the

population mean could pass a lot which has the low percentile below the

required standard of consumers. Furthermore, a small decrease in the

mean with a simultaneous small increase in the variance can result in a

significant downward shift in small percentiles of interest. This means

that a lot of products could be accepted due to a small decrease in the

mean life after inspection. But the material strengths of products are

deteriorated significantly and may not meet the consumer‟s expectation.

Therefore, engineers pay more attention to the percentiles of lifetimes

than the mean life in life testing applications. Moreover, most of the

employed life distributions are not symmetric. In viewing Marshall and

Olkin (2007), the mean life may not be adequate to describe the central

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tendency of the distribution. This reduces the feasibility of acceptance

sampling plans if they are developed based on the mean life of products.

Actually, percentiles provide more information regarding a life

distribution than the mean life does. When the life distribution is

symmetric, the 50th

percentile or the median is equivalent to the mean

life. Hence, developing acceptance sampling plans based on percentiles of

a life distribution can be treated as a generalization of developing

acceptance sampling plans based on the mean life of items. Balakrishnan

et al. (2007) proposed the acceptance sampling plans could be used for

the quantiles and derived the formulae whereas Lio et al. (2010)

developed for the acceptance sampling plans for any other percentiles of

the Birnbaum-Saunders (BS) model. They have developed the acceptance

sampling plans for percentile by replace the scale parameter by the

100qth percentile in the BS distribution function. Srinivasa Rao and

Kantam (2010) are developed acceptance sampling plans from truncated

life tests based on the log-logistic distribution for percentiles.

In this chapter, we develop acceptance sampling plans based on

average (mean) for the inverse Rayleigh distribution under a truncated

life test, along with operating characteristic and relevant tables, example

based on real life data sets are provided in Section 4.2. The proposed

sampling plans based on percentiles, along with operating characteristic

function and examples based on real life data sets are provided for the

illustration in Section 4.3. A comparative study of the two approaches is

given in Section 4.4. The discussion and some conclusions are made in

Section 4.5.

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4.2 Acceptance sampling plans - Averages

We assume that the life time of a product follows an inverse

Rayleigh distribution with scale parameter . A common practice in life

testing is to terminate a life test by a pre-determined time t and note the

number of failures (assuming that a failure is well-defined). One of the

objectives of these experiments is to set a lower confidence limit on the

average life. It is then desired to establish a specified average life with a

given probability of at least P*. The decision to accept the specified

average life occurs if and only if the number of observed failures at the

end of the fixed time t does not exceed a given number c- called the

acceptance number. The test may get terminated before the time t is

reached when the number of failures exceeds c in which case, the

decision is to reject the lot. For such a truncated life test and the

associated decision rule, we are interested in obtaining the smallest

sample sizes necessary to achieve the objective.

A sampling plan consists of

(i) The number of units n on test,

(ii) The acceptance number c,

(iii) The maximum test duration t and

(iv) The ratio t/o, where o is the specified average life

The consumer‟s risk, i.e., the probability of accepting a bad lot (the

one for which the true average life is below the specified life o) not to

exceed 1-P*, so that P

* is a minimum confidence level with which a lot of

true average life below o is rejected, by the sampling plan. For a fixed

P* our sampling plan is characterized by (n, c, t/o). Here we consider

sufficiently large lots so that the binomial distribution can be applied. The

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problem is, for given values of P* (0<P

*<1), o and c the smallest

positive integer n is to be determined such that

0

(1 ) 1c

i n i

i

np p P

i

(4.2.1)

where p=F(t;o) is given by (4.2.1 ) indicates the failure probabilities

before time t, which depends only on the ratio t/, it is sufficient to

specify this ratio for designing the experiment.

If the number of observed failures before t is less than or equal to c,

from inequality (4.2.1) we obtain

F(t ; ) F(t ; o) o (4.2.2)

The minimum values of n satisfying the inequality (4.2.1) are

obtained and given in Table (4.2.1) for P*=0.75, 0.90, 0.95, 0.99 ;

t/o=1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 3.0, 3.5, 4.0 and c = 0, 1, 2, …,

10.

If p=F(t; ) is small and n is large (as is true in some cases of our

present work), the binomial probability may be approximated by Poisson

probability with parameter =np so that the left hand side of (4.2.1) can

be written as

1!0

ic eP

ii

(4.2.3)

where =nF(t;). The minimum values of n satisfying (4.2.3) are

obtained for the same combination of P*, t/o and c values as those used

for (4.2.1). The results are given in Table 4.2.2

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Operating characteristic function

In the problem of choosing an acceptance sampling plan, interest

centers on how it will perform in actual practice not merely for one value

of a specified lot quality parameter (specified average in the present case)

but for all possible values that might be encountered. In other words, the

probabilities of a lot being accepted should have clear relation with

changes in lot quality parameter. In the present situation any downward

shift in the average life of the product should reveal a significant fall in

the probability of acceptance of the lot. This is best shown by

considering the graph of acceptance probability of the sampling plan

against the lot quality parameter. Such a graph is called operating

characteristic (O.C) curve. There are two distinct types of operating

characteristic curves called type A and type B according as sampling is

from a finite universe or infinite universe. Our sampling plans will have

type B operating characteristic curves.

The operating characteristic function of the sampling plan (n, c,

t/o) gives the probability L(p) of accepting the lot with :

L(p)=0

(1 )c

i n i

i

np p

i

(4.2.4)

where p=F(t;) is considered as a function of , i.e., the lot quality

parameter. It can be seen that operating characteristic is an increasing

function of . For given P*, t/o , the choice of c and n will be made on

the basis of operating characteristics. Values of operating characteristics

as a function of /o for a few sampling plans are given in Table 4.2.3

when the plan is based on binomial probability. Similar values are given

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in Table 4.2.2 for a sampling plan based on Poisson probabilities are not

much different from that of Table 4.2.1, and hence, O.C values for these

are not presented.

Scope of population average relative to a producer’s risk

In life test data required for sampling plans of quality control, we

know that lots having the average life above a specification are termed as

good lots. It may be noted that the quality parameter considered here is

population average life of the product a case of “the more, the better”.

That is, all lots with products having average life beyond the specified life

(σ>σ0) are regarded as good lots. On the application of sampling plan if a

good lot is not accepted sometimes, it results in a wrong decision also

called an error in classical testing of statistical hypothesis. The probability

of rejecting a good lot on the application of an acceptance sampling is

called producer‟s risk. It may be recalled that the sampling plans

presented by us in section 4.2 are constructed for a given consumer‟s

risk - probability of accepting a bad lot. Therefore, in this section we

attempt to investigate how far the specified and the unknown population

average fluctuate without disturbing producer‟s risk. In other words, let

the producer‟s risk be 0.05. Let 0 be the ratio of specified average to

unknown average. We attempt to find the range of values of 0 which

will ensure a producers risk not beyond 0.05 if a specified sampling plan

is adopted. We know that the distribution function of the population

distribution can be written as a function of 0 .

It should be noted that the probability p may be obtained as a

function of 0 , as

p=F(t/)=F[(t/o)(o/)]. (4.2.5)

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where p=F (t;).

The value 0 is the smallest number for which the following

inequality

0

(1 ) 0.95c

i n i

i

np p

i

(4.2.6)

For a given sampling plan (n, c, t/o) on the basis of binomial

probabilities and specified confidence level P*, the minimum values of

0 satisfying the inequality (4.2.6) are given in Table 4.2.4. Similar

values for sampling plan constructed using Poisson probabilities are not

much different from binomial probabilities and hence, these are not

presented.

Illustration of the Tables and Example for Test plan

Assume that the life distribution is an inverse Rayleigh distribution

and the experimenter is interested in showing that the true unknown

average life is at least 1000 hours. Let the consumer‟s risk be set to

1-P*=0.25. It is desired to stop the experiment at t=1000 hours. Then for

an acceptance number c=2, the required n is the entry in Table 4.2.1

corresponding to the values of 1-P*=0.25, t/o=1.0 and c=2. This number

is n=10. Thus n=10 units have to be put on test. If during 1000 hours, no

more than 2 failures out of 10 units are observed, then the experimenter

can assert with a confidence level of P*=0.75 that the average life is at

least 1000 hours. If the Poisson approximation to binomial probability is

used, the value of n=11 is obtained for the same situation from Table

4.2.2.

For the sampling plan (n=10, c=2, t/o=1.0) under inverse Rayleigh

model, the operating characteristic values from Table 4.2.3 are

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/o : 2 4 6 8 10 12

L(p) : 0.999 1.000 1.000 1.000 1.000 1.000

This shows that, if the true mean life is twice the specified mean

life ( 0 =2) the producer‟s risk is approximately 0.001.

From Table 4.2.4, we can get the value of 0 for various choices

of c, t/o in order that the producer‟s risk may not exceed 0.05. Thus, in

the above example we obtain the values of 0 = 1.81. That is, the

product should have an average life of 1.81 times the specified average

life 1000 hours in order that under the above acceptance sampling plan

(10, 2, 1.0) the product be accepted with probability of at least 0.95. The

actual average life necessary to transship 95 percent of the lots is

provided by Table 4.2.4.

Example: Consider the following ordered failure times of the release of

a software given in terms of hours from the starting of the execution of

the software denoting the times at which the failure of the software is

experienced (Wood, 1996). This data can be regarded as an ordered

sample of size 10 with observations (Xi , i=1, 2, 3, . . . , 10) : 519, 968,

1430, 1893, 2490, 3058, 3625, 4422, 5218, 5823.

Let the specified average life be 1000 hrs and the testing time be

1250 hrs, this leads to ratio of t/o=1.25 with corresponding n and c as

10, 2 from Table 4.2.1 for P*=0.95. Therefore, the sampling plan for the

above sample data is (n=10, c=2, t/o=1.25). Based on the observations,

we have to decide whether to accept the product or reject it. We accept

the product only, if the number of failures before 1250 hrs is less than or

equal to 2. However, the confidence level is assured by the sampling

plan only if the given life times follow an inverse Rayleigh distribution,

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we have compared with the sample quantiles and the corresponding

population quantiles and found a satisfactory agreement .Thus the

adoption of the decision rule of the sampling plan seems to be justified.

We see that in the sample of 10 failures there are 2 failures at 519 and

968 hrs before 1250 hrs, therefore we accept the product.

4.3 Acceptance sampling plans - Percentiles

Assume that the lifetime of a product follows a inverse Rayleigh

distribution with scale parameter . For 0 1q , the 100qth percentile (or

the qth quantile) is given by

1/2

lnqt q

(4.3.1)

The qt is increases as q increases. Let 1/ 2

ln q

. Then, equation

(4.3.1) implies that

qt . (4.3.2)

To develop acceptance sampling plans for the inverse Rayleigh

percentiles, the scale parameter in the inverse Rayleigh cdf is replaced

by equation (4.3.2) and the inverse Rayleigh cdf is rewritten as

2

( ) /( ) ; 0qt t

F t e t

.

Lettingqt t , F(t) can be rewritten emphasizing its dependence on

as

21( ; ) ; 0F t e t

.

Taking partial derivative with respect to , we have

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21

3

( ; ) 2; 0

F te t

.

A common practice in life testing is to terminate the life test by a

pre-determined time t, the probability of rejecting a bad lot be at least P ,

and the maximum number of allowable bad items to accept the lot be c.

The acceptance sampling plan for percentiles under a truncated life test is

to set up the minimum sample size n for the given acceptance number c

such that the consumer‟s risk, the probability of accepting a bad lot, does

not exceed 1-P . A bad lot means that the true 100qth

percentile, qt , is

below the specified percentile, 0

qt . Thus, the probability P is a confidence

level in the sense that the chance of rejecting a bad lot with 0

q qt t is at

least equal to P . Therefore, for a given P , the proposed acceptance

sampling plan can be characterized by the triplet 0( , , )qn c t t .

Minimum sample size

For a fixed P , our sampling plan is characterized by 0( , , )qn c t t .

Here we consider sufficiently large sized lots so that the binomial

distribution can be applied. The problem is to determine for given values

of P (0 <P <1), 0

qt and c, the smallest positive integer n required to

assert that 0

q qt t must satisfy

0 0

0

1 1c

n in i

i

i

p p P

, (4.3.3)

where 0( ; )p F t is the probability of a failure during the time t given a

specified 100qth percentile of lifetime 0

qt and depends only on 0

0 qt t ,

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since ( ; ) 0, ( ; )F t F t is a non-decreasing function of .

Accordingly, we have

0 0( , ) ( , )F t F t ,

or equivalently,

00( , ) ( , ) q qF t F t t t .

The smallest sample size n satisfying the inequality (4.3.3) can be

obtained for any given q, 0

qt t , P . Whereas, the smallest sample size n

calculation in Section 4.2 only needs input values for 0t and P .

Hence, the proposed process to find the smallest sample size in this case

is the same as the procedure provided by in Section 4.2 or the inverse

Rayleigh model except in place of 0t replace by 0

qt t at q. To save

space, only the results of small sample sizes for q=0.10, 0.25. 0.50, 0.75,

0.90; 0

qt t =0.7, 0.9, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5; P =0.75, 0.90, 0.95, 0.99;

c = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are reported in Tables 4.3.1- 4.3.5. These

tables reveal that as percentile values increase the smallest sample size

require are decreases for all combinations of parameters of, 0

qt t , P

and c.

If 0( ; )p F t is small and n is large the binomial probability may

be approximated by Poisson probability with parameter λ = np so that the

left side of (4.3.3) can be written as

0

1 P!

ic

i

ei

, (4.3.4)

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where λ = n. 0( ; )F t . The minimum values of n satisfying (4.3.4)

are obtained for the same combination of q, 0

qt t and P values as those

used for (4.3.3). The results are reported in Tables 4.3.6 to 4.3.10.These

tables reveal that as percentile values increase the smallest sample size

require are decreases for all combinations of parameters of 0

qt t , P

and c. Also as compared with binomial approach, the Poisson approach

gives the smaller sample size.

Operating characteristic of the sampling plan 0( , , )qn c t t

The operating characteristic (OC) function of the sampling plan

0( , , )qn c t t is the probability of accepting a lot. It is given as

0

( ) 1c

n in i

i

i

L p p p

, (4.3.5)

where ( ; )p F t . It should be noticed that ( ; )F t can be represented as

a function of qt t . Therefore,

0

1( )

q q

tp F

t d where 0

q q qd t t . Using

equation (4.3.5), the OC values and OC curves can be obtained for any

sampling plan 0( , , )qn c t t . To save space, we present Tables 4.3.11–

4.3.15 to show the OC values for the sampling plan 0

0.1( , 5, )n c t t .

Figures 4.3.1 – 4.3.5 shows the OC curves for the sampling plan

0

0.1( , , )n c t t with P =0.90 for 0 0.8 , for q=0.10, 0.25. 0.50, 0.75, 0.90,

when c= 0,1,2,3,4,5,6,7,8,9,10. These OC curves show that as percentile

increases the OC values are decreases for all given combinations.

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Figure 4.3.1: OC curves for c = 0,1,2,3,4,5,6,7,8,9,10, respectively under

P =0.90, 0 0.8 based on the 10th percentile, 0.1d d , of inverse

Rayleigh distribution.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

L(P

)

d

Operating Characterstic Curve




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

L(p

)

d

Operating Characterstic Curve

121







0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

L(P

)

d

Operating Characteristic Curve

122




0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

L(P

)

d

Operating Characteristic Curve

Producer’s Risk

The producer‟s risk is defined as the probability of rejecting the lot

when 0

q qt t . For a given value of the producer‟s risk, say , we are

interested in knowing the value of qd to ensure the producer‟s risk is less

than or equal to if a sampling plan 0( , , )qn c t t is developed at a

specified confidence level p . Thus, one needs to find the smallest value

qd according to equation (4.3.5) as

0

1 1c

n in i

i

i

p p

, (4.3.6)

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where 0

1( )

q q

tp F

t d , 0

q q qd t t . To save space, based on sampling plans

0( , , )qn c t t the minimum ratios of 0.1 0.25 0.5 0.75 0.9, , , andd d d d d for the

acceptability of a lot at the producer‟s risk of =0.05 are presented in

Tables 4.3.16 to 4.3.20.

Illustrative Examples

In this section, we consider two examples with real data sets are

given to illustrate the proposed acceptance sampling plans. The first data

set is of the data given arisen in tests on endurance of deep groove ball

bearings (Lawless, 1982, p.228). The data are the number of million

revolutions before failure for each of the 23 ball bearings in life test and

they are: 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96,

54.12, 55.56, 67.80, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12,

105.84, 127.92, 128.04 and 173.40. The second data set is obtained from

Proschan (1963) and represents times between successive failures of air

conditioning (AC) equipment in a Boeing 720 airplane and they are as

follows: 12, 21, 26, 27, 29, 29, 48, 57, 59, 70, 74, 153, 326, 386, 502.

As the confidence level is assured by this acceptance sampling plan

only if the lifetimes are from the inverse Rayleigh distribution. Then, we

should check, if it is reasonable to admit that the given sample comes

from the inverse Rayleigh distribution by the goodness of fit test and

model selection criteria. The first data set was used by Sultan (2007) to

demonstrate the goodness of fit for generalized exponential distribution

and Gupta and Kundu (2001) fitted for extended exponential distribution.

However, the acceptance sampling plans under the truncated life test

based on the inverse Rayleigh distribution for percentiles has not yet been

developed. We fit the inverse Rayleigh distribution to the two data sets

124

separately. We used the Kolmogorov-Smirnov (K-S) tests for each data

set to the fit the inverse Rayleigh model. It is observed that for Data Sets I

and II, the K-S distances are 0.12091 and 0.21378 with the corresponding

p values are 0.85028 and 0.43879 respectively. For data sets I and II, the

chi-square values are 0.3052 and 2.6383 respectively. Therefore, it is

clear that inverse Rayleigh model fits quite well to both the data sets.

Example 1

Assume that the lifetime distribution is inverse Rayleigh

distribution and that the experimenter is interested to establish the true

unknown 10th

percentile lifetime for the ball bearings to be at least 20

million revolutions with confidence p=0.90 and the life test would be

ended at 30 million revolutions, which should have led to the ratio 0

0.1t t =

1.50. Thus, for an acceptance number c =5 and the confidence level

p=0.90, the required sample size n found from Table 4.3.1 should be at

least 23. Therefore, in this case, the acceptance sampling plan from

truncated life tests for the inverse Rayleigh distribution 10th percentile

should be 0( , , )qn c t t = (23, 5, 1.5). Based on the ball bearings data, the

experimenter must have decided whether to accept or reject the lot. The

lot should be accepted only if the number of items of which lifetimes are

less than or equal to the scheduled test lifetime, 30 million revolutions,

was at most 5 among the first 23 observations. Since there were 2 items

with a failure time less than or equal to 30 million revolutions in the

given sample of n =23 observations, the experimenter would accept the

lot, assuming the 10th percentile lifetime 0.1t of at least 20 million

revolutions with a confidence level of P=0.90. The OC values for the

acceptance sampling plan 0( , , )qn c t t = (23, 5, 1.5) and confidence level

125

P=0.90 under inverse Rayleigh distribution from Table 4.3.11 is as

follows:

0

0.1 0.1t t

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

OC 0.0884 0.6461 0.9723 0.9995 1.0000 1.0000 1.0000 1.0000

This shows that if the true 10th percentile is equal to the required 10

th

percentile ( 0

0.1 0.1t t = 1.00) the producer‟s risk is approximately 0.9116

(=1- 0.0884). The producer‟s risk is almost equal to zero when the true

10th percentile is greater than or equal to 2.00 times the specified 10

th

percentile.

From Table 4.3.16, the experimenter could get the values of 0.1d

for different choices of c and 0

0.1t t in order to assert that the producer‟s

risk was less than 0.05. In this example, the value of 0.1d should be

1.4550 for c = 5, 0

0.1t t =1.0 and P=0.90. This means the product can have

a 10th percentile life of 1.4550 times the required 10

th percentile lifetime

in order that under the above acceptance sampling plan the product is

accepted with probability of at least 0.95.

Alternatively, assume that products have a inverse Rayleigh

distribution and consumers wish to reject a bad lot with probability of

P=0.75. What should the true 10th

percentile life of products be so that

the producer‟s risk is 0.05 if the acceptance sampling plan is based on an

acceptance number c =5 and 0

0.1t t =0.8? From Table 4.3.16, we can find

that the entry for P=0.75, c = 5, and 0

0.1t t =0.8 is 0.1d = 1.1680. Thus, the

manufacturer‟s product should have a 10th percentile life at least 1.1680

126

times the specified 10th percentile life in order for the products to be

accepted with probability 0.75 under the above acceptance sampling plan.

Table 4.3.1 indicates that the number of products required to be tested is

n = 270 so that the sampling plan is 0

0.1( , , )n c t t = (270, 5, 0.8).

Example 2

Suppose an experimenter would like to establish the true unknown

10th percentile lifetime for the data set regarding the failure of air

conditioning (AC) equipment in a Boeing 720 airplane mentioned above

to be at least 10 and the life test would be ended at 20, which should have

led to the ratio 0

0.1t t = 2.0. The goodness of fit test for these 15

observations were verified and showed that inverse Rayleigh model as a

reasonable goodness of fit for these 15 observations. Thus, with c = 5 and

P=0.90, the experimenter should find from Table 4.3.1 the sample size n

must be at least 15 and the sampling plan to be 0

0.1( , , )n c t t = (15, 5, 2.00).

Since there is a one item with a failure time less than 20 in the given

sample of n = 15 observations, the experimenter would accept the lot,

assuming the 10th

percentile lifetime 0.1t of at least 10 with a confidence

level of P=0.90.

4.4. Comparative study of the two approaches

The sampling plans based on the inverse Rayleigh population mean

developed in Section 4.2 and based on percentile developed in Section

4.3. It shows that the minimum sample sizes are larger than those

reported in Tables 4.3.1 and 4.3.2 of this article for the 10th and 25

th

percentile for both binomial and Poisson approximation whereas

minimum sample sizes are smaller than those reported in Tables 4.3.1 and

4.3.2 of this article for the 50th , 75

th and 90

th percentile for both

127

binomial and Poisson approximation. Here, 0

0 qt t for the sampling

plans based on qth

percentile is replaced by 0 0t with 0 as a specific

population mean for the acceptance plans based on the inverse Rayleigh

population mean. Therefore, the acceptance sampling plans based on the

inverse Rayleigh population mean could have less chance to report a

failure than the acceptance sampling plans based on percentile. The

acceptance sampling plans based on population mean could accept the lot

of bad quality of the qth percentiles. The minimum sample sizes are

reported in Table 4.3.1 of this thesis for the 10th percentiles are compared

with the minimum sample sizes are reported in Table 1 of Lio et al.

(2010). It shows that the minimum sample sizes using inverse Rayleigh

population are smaller than those reported in Table 1 of Lio et.al. (2010)

for Birnbaum-Saunders population for the 10th

percentile when

0 1.0 . Whereas, the minimum sample sizes using inverse Rayleigh

population are larger than those reported in Table 1 of Lio et al. (2010)

for Birnbaum-Saunders population for the 10th percentile when 0 1.0 .

4.5. Conclusions

This chapter has derived the acceptance sampling plans based on

the inverse Rayleigh mean and percentiles when the life test is truncated

at a pre-fixed time. The procedure is provided to construct the proposed

sampling plans for the mean and percentiles of the inverse Rayleigh

distribution. To ensure that the life quality of products exceeds a specified

one in terms of the life percentile, the acceptance sampling plans based on

percentiles should be used. Some useful tables are provided and applied

to establish acceptance sampling plans for two examples.

128

Table 4.2.1: Minimum sample sizes necessary to assert the average life to exceed

a given value-o with probability P* and the corresponding

acceptance number-c under inverse Rayleigh distribution using

Binomial probabilities

P* C

t/o

1.0 1.25 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00

0.75 0 4 2 2 2 1 1 1 1 1 1

1 7 5 4 3 3 3 3 2 2 2

2 10 7 5 5 4 4 4 4 3 3

3 13 9 7 6 6 5 5 5 5 4

4 16 11 9 8 7 7 6 6 6 6

5 19 13 11 9 9 8 8 7 7 7

6 22 15 12 11 10 9 9 8 8 8

7 25 17 14 12 11 11 10 9 9 9

8 28 19 16 14 13 12 11 11 10 10

9 31 21 17 15 14 13 13 12 11 11

10 34 23 19 17 15 14 14 13 12 12

0.90 0 6 4 3 2 2 2 2 2 1 1

1 10 6 5 4 4 3 3 3 3 3

2 13 9 7 6 5 5 5 4 4 4

3 17 11 9 7 7 6 6 5 5 5

4 20 13 11 9 8 8 7 7 6 6

5 23 16 12 11 10 9 9 8 7 7

6 27 18 14 12 11 10 10 9 9 8

7 30 20 16 14 13 12 11 10 10 9

8 33 22 18 15 14 13 12 12 11 11

9 36 24 20 17 15 14 14 13 12 12

10 39 27 21 18 17 16 15 14 13 13

0.95 0 7 4 3 3 2 2 2 2 2 2

1 11 7 6 5 4 4 4 3 3 3

2 15 10 8 7 6 5 5 5 4 4

3 19 13 10 8 7 7 6 6 6 5

4 23 15 12 10 9 8 8 7 7 6

5 26 17 14 12 10 10 9 8 8 8

6 30 20 15 13 12 11 10 10 9 9

7 33 22 17 15 13 12 12 11 10 10

8 36 24 19 16 15 14 13 12 11 11

9 40 26 21 18 16 15 14 13 13 12

10 43 29 23 20 18 17 16 15 14 13

0.99 0 11 7 5 4 4 3 3 3 2 2

1 16 10 8 6 5 5 5 4 4 4

2 20 13 10 8 7 7 6 5 5 5

3 24 16 12 10 9 8 8 7 6 6

4 28 18 14 12 11 10 9 8 8 7

5 32 21 16 14 12 11 10 9 9 8

6 35 23 18 15 14 13 12 11 10 10

7 39 26 20 17 15 14 13 12 11 11

8 43 28 22 19 17 15 15 13 13 12

9 46 31 24 20 18 17 16 15 14 13

10 50 33 26 22 20 18 17 16 15 14

129

Table 4.2.2 :Minimum sample sizes necessary to assert the average life to exceed

a given value-o with probability P* and the corresponding

acceptance number-c under inverse Rayleigh distribution using

Poisson approximation

P* c

t/o

1.0 1.25 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00

0.75 0 4 3 3 2 2 2 2 2 2 2

1 7 5 4 4 3 3 3 3 3 3

2 11 8 6 6 5 5 5 5 5 5

3 14 10 8 8 7 7 6 6 6 6

4 18 12 10 9 9 8 8 7 7 7

5 21 15 12 11 10 10 9 9 9 8

6 24 17 14 12 11 11 11 10 10 10

7 27 19 16 14 13 12 12 11 11 11

8 30 21 17 15 14 14 13 13 12 12

9 33 23 19 17 16 15 14 14 13 13

10 36 25 21 19 17 16 16 15 15 14

0.90 0 7 5 4 4 3 3 3 3 3 3

1 10 7 6 5 5 5 5 5 4 4

2 15 10 9 8 7 7 7 6 6 6

3 19 13 11 10 9 9 8 8 8 8

4 22 16 13 12 11 10 10 9 9 9

5 26 18 15 13 12 12 11 11 11 10

6 29 20 17 15 14 13 13 12 12 12

7 32 23 19 17 16 15 14 14 13 13

8 36 25 21 19 17 16 16 15 15 14

9 39 27 23 20 19 18 17 16 16 16

10 42 30 25 22 20 19 19 18 17 17

0.95 0 9 6 5 5 4 4 4 4 4 4

1 13 9 8 7 6 6 6 6 5 5

2 17 12 10 9 9 8 8 7 7 7

3 22 15 13 11 10 10 10 9 9 9

4 25 18 15 13 12 12 11 11 10 10

5 29 20 17 15 14 13 13 12 12 12

6 33 23 19 17 16 15 14 14 13 13

7 36 25 21 19 17 17 16 15 15 14

8 40 28 23 21 19 18 17 17 16 16

9 43 30 25 22 21 20 19 18 18 17

10 47 33 27 24 22 21 20 19 19 19

0.99 0 13 9 8 7 6 6 6 6 5 5

1 18 13 11 9 9 8 8 8 8 7

2 23 16 14 12 11 11 10 10 10 9

3 28 20 16 14 13 13 12 12 11 11

4 32 23 19 17 15 15 14 13 13 13

5 36 25 21 19 17 16 16 15 15 14

6 40 28 23 21 19 18 18 17 16 16

7 44 31 25 23 21 20 19 18 18 18

8 48 34 28 25 23 22 21 20 19 19

9 52 36 30 27 25 23 23 21 21 20

10 55 39 32 28 26 25 24 23 22 22

130

Table 4.2.3: Operating characteristic values of the sampling plan (n, c, t/o) for

a given P* under inverse Rayleigh distribution

P* c n t/o

/o

2 4 6 8 10 12

0.75 2 10 1.00 0.999 1.000 1.000 1.000 1.000 1.000

7 1.25 0.987 1.000 1.000 1.000 1.000 1.000

5 1.50 0.963 1.000 1.000 1.000 1.000 1.000

5 1.75 0.873 1.000 1.000 1.000 1.000 1.000

4 2.00 0.856 1.000 1.000 1.000 1.000 1.000

4 2.25 0.753 1.000 1.000 1.000 1.000 1.000

4 2.50 0.645 0.998 1.000 1.000 1.000 1.000

4 3.00 0.453 0.983 1.000 1.000 1.000 1.000

3 3.50 0.625 0.980 1.000 1.000 1.000 1.000

3 4.00 0.528 0.950 0.999 1.000 1.000 1.000

0.90 2 13 1.00 0.998 1.000 1.000 1.000 1.000 1.000

9 1.25 0.973 1.000 1.000 1.000 1.000 1.000

7 1.50 0.901 1.000 1.000 1.000 1.000 1.000

6 1.75 0.796 1.000 1.000 1.000 1.000 1.000

5 2.00 0.736 1.000 1.000 1.000 1.000 1.000

5 2.25 0.586 0.999 1.000 1.000 1.000 1.000

5 2.50 0.449 0.996 1.000 1.000 1.000 1.000

4 3.00 0.453 0.983 1.000 1.000 1.000 1.000

4 3.50 0.311 0.937 0.999 1.000 1.000 1.000

4 4.00 0.214 0.856 0.996 1.000 1.000 1.000

0.95 2 15 1.00 0.998 1.000 1.000 1.000 1.000 1.000

10 1.25 0.963 1.000 1.000 1.000 1.000 1.000

8 1.50 0.861 1.000 1.000 1.000 1.000 1.000

7 1.75 0.712 1.000 1.000 1.000 1.000 1.000

6 2.00 0.611 1.000 1.000 1.000 1.000 1.000

5 2.25 0.586 0.999 1.000 1.000 1.000 1.000

5 2.50 0.449 0.996 1.000 1.000 1.000 1.000

5 3.00 0.249 0.963 1.000 1.000 1.000 1.000

4 3.50 0.311 0.937 0.999 1.000 1.000 1.000

4 4.00 0.214 0.856 0.996 1.000 1.000 1.000

0.99 2 20 1.00 0.994 1.000 1.000 1.000 1.000 1.000

13 1.25 0.926 1.000 1.000 1.000 1.000 1.000

10 1.50 0.769 1.000 1.000 1.000 1.000 1.000

8 1.75 0.626 1.000 1.000 1.000 1.000 1.000

7 2.00 0.491 1.000 1.000 1.000 1.000 1.000

7 2.25 0.309 0.998 1.000 1.000 1.000 1.000

6 2.50 0.294 0.992 1.000 1.000 1.000 1.000

5 3.00 0.249 0.963 1.000 1.000 1.000 1.000

5 3.50 0.136 0.873 0.999 1.000 1.000 1.000

5 4.00 0.075 0.736 0.990 1.000 1.000 1.000

131

Table 4.2.4: Minimum ratio of true mean life to specified mean life for the

acceptability of a lot with producer’s risk of 0.05 under inverse

Rayleigh distribution

P* c

t/o

1.0 1.25 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00

0.75 0 2.45 2.92 3.48 3.48 3.93 4.51 5.31 6.43 7.49 8.16

1 2.01 2.32 2.52 2.83 3.23 3.62 3.77 4.30 5.01 5.63

2 1.81 1.97 2.27 2.38 2.66 3.02 3.62 3.62 4.11 4.51

3 1.68 1.85 2.01 2.32 2.38 2.66 3.12 3.77 3.48 3.93

4 1.59 1.78 1.97 2.11 2.38 2.38 2.83 3.35 3.77 4.30

5 1.54 1.74 1.81 2.11 2.16 2.45 2.59 3.02 3.48 3.93

6 1.49 1.62 1.81 1.97 2.01 2.27 2.45 2.83 3.23 3.62

7 1.46 1.59 1.71 1.85 2.06 2.11 2.27 2.66 3.02 3.48

8 1.44 1.59 1.71 1.85 1.97 2.01 2.45 2.52 2.92 3.23

9 1.42 1.51 1.65 1.78 1.89 2.11 2.32 2.38 2.74 3.12

10 1.40 1.51 1.65 1.71 1.81 2.01 2.21 2.27 2.59 2.92

0.90 0 2.66 3.12 3.48 3.93 4.51 5.01 6.01 6.43 7.49 8.16

1 2.11 2.45 2.74 3.12 3.23 3.62 4.30 5.01 6.01 6.43

2 1.93 2.16 2.45 2.59 2.92 3.23 3.62 4.30 4.75 5.63

3 1.78 2.01 2.16 2.45 2.59 2.92 3.12 3.77 4.30 4.75

4 1.68 1.93 2.06 2.27 2.52 2.66 3.12 3.35 3.77 4.30

5 1.65 1.81 2.01 2.21 2.32 2.59 2.92 3.02 3.48 3.93

6 1.59 1.74 1.89 2.06 2.21 2.45 2.74 3.23 3.23 3.62

7 1.56 1.71 1.89 2.06 2.21 2.32 2.59 3.02 3.02 3.48

8 1.51 1.68 1.78 1.97 2.11 2.21 2.66 2.83 3.23 3.62

9 1.49 1.65 1.78 1.89 2.01 2.21 2.52 2.66 3.12 3.48

10 1.49 1.59 1.71 1.89 2.01 2.16 2.38 2.59 2.92 3.35

0.95 0 2.66 3.12 3.62 3.93 4.51 5.01 6.01 6.92 8.16 8.97

1 2.16 2.52 2.83 3.12 3.48 3.93 4.30 5.01 6.01 6.43

2 1.97 2.27 2.52 2.74 2.92 3.23 3.93 4.30 4.75 5.63

3 1.85 2.11 2.27 2.45 2.83 2.92 3.48 4.11 4.30 4.75

4 1.78 1.97 2.16 2.38 2.52 2.83 3.12 3.77 3.77 4.30

5 1.68 1.89 2.11 2.21 2.52 2.59 2.92 3.35 3.93 4.51

6 1.65 1.81 1.97 2.16 2.32 2.45 2.92 3.23 3.62 4.11

7 1.62 1.74 1.93 2.06 2.21 2.45 2.74 3.02 3.48 3.93

8 1.56 1.71 1.85 2.06 2.21 2.32 2.66 2.83 3.23 3.62

9 1.54 1.68 1.85 1.97 2.11 2.21 2.52 2.92 3.12 3.48

10 1.54 1.68 1.81 1.97 2.11 2.27 2.59 2.83 2.92 3.35

0.99 0 2.83 3.23 3.77 4.30 4.75 5.31 6.43 6.92 8.16 8.97

1 2.32 2.66 2.92 3.23 3.62 4.11 4.75 5.63 6.43 6.92

2 2.11 2.38 2.66 2.92 3.23 3.48 3.93 4.75 5.31 6.01

3 1.97 2.21 2.45 2.74 2.92 3.23 3.77 4.11 4.75 5.31

4 1.85 2.06 2.32 2.59 2.83 3.02 3.35 3.93 4.30 4.75

5 1.78 2.01 2.21 2.38 2.59 2.74 3.12 3.62 3.93 4.51

6 1.74 1.93 2.11 2.32 2.52 2.74 3.12 3.48 3.93 4.51

7 1.71 1.89 2.06 2.21 2.45 2.59 2.92 3.23 3.77 4.11

8 1.65 1.81 2.01 2.21 2.32 2.59 2.83 3.23 3.48 3.93

9 1.65 1.78 1.93 2.11 2.27 2.45 2.83 3.12 3.35 3.77

10 1.59 1.78 1.93 2.06 2.21 2.38 2.74 3.02 3.23 3.62

132

Table 4.3.1: Minimum sample sizes necessary to assert the 10th

percentile to

exceed given values, 0

0.1t , with probability Pand the corresponding

acceptance number, c, for the inverse Rayleigh distribution using the

binomial approximation.

P C

0

0.1/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 50 24 14 4 2 2 1 1

1 98 46 27 7 4 3 3 3

2 143 67 39 10 6 5 4 4

3 186 87 51 14 8 7 6 5

4 228 107 62 17 10 8 7 7

5 270 127 73 20 12 10 9 8

6 312 146 85 23 14 11 10 9

7 353 165 96 26 16 13 11 10

8 394 184 107 29 18 14 13 12

9 434 203 118 32 20 16 14 13

10 474 222 129 35 22 17 15 14

0.90 0 83 39 22 6 3 2 2 2

1 141 66 38 10 6 4 4 3

2 193 90 52 14 8 6 5 5

3 243 113 65 17 10 8 7 6

4 290 136 78 21 12 10 8 8

5 337 158 91 23 15 11 10 9

6 383 179 104 27 17 13 11 10

7 428 200 116 31 19 15 13 12

8 473 221 128 34 21 16 14 13

9 517 242 140 37 23 18 16 14

10 560 262 152 40 25 19 17 15

0.95 0 108 50 29 7 4 3 3 2

1 172 80 46 12 7 5 4 4

2 228 106 61 16 9 7 6 5

3 281 131 76 20 12 9 8 7

4 332 155 89 23 14 11 9 8

5 382 178 103 27 16 12 11 10

6 430 201 116 30 18 14 12 11

7 478 223 129 34 20 16 14 12

8 524 245 142 37 22 17 15 14

9 571 267 154 41 25 19 16 15

10 616 288 167 44 27 21 18 16

0.99 0 166 77 44 11 6 4 4 3

1 240 112 64 16 9 7 6 5

2 304 142 81 20 12 9 7 6

3 364 169 97 25 14 11 9 8

4 420 196 113 29 17 13 11 10

5 475 221 127 33 19 15 12 11

6 528 246 142 36 21 16 14 12

7 580 271 156 40 24 18 15 14

8 631 294 170 44 26 20 17 15

9 681 318 183 47 28 22 18 17

10 731 341 197 51 30 23 20 18

133


percentile to

exceed a given values, 0

0.25t , with probability Pand the

corresponding acceptance number, c, for the inverse Rayleigh

distribution using the binomial approximation.

P c

0

0.25/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 12 7 5 2 2 1 1 1

1 23 15 10 5 3 3 3 2

2 34 21 15 7 5 4 4 4

3 44 28 20 9 6 6 5 5

4 54 34 24 11 8 7 6 6

5 64 40 29 13 10 8 8 7

6 74 47 33 15 11 10 9 8

7 84 53 38 17 13 11 10 9

8 93 59 42 19 14 12 11 11

9 103 65 47 21 16 13 12 12

10 113 71 51 23 17 15 14 13

0.90 0 19 12 9 3 2 2 2 2

1 33 21 15 6 4 4 3 3

2 45 28 20 8 6 5 5 4

3 57 36 25 11 8 6 6 5

4 68 43 30 13 9 8 7 7

5 79 50 35 15 11 9 8 8

6 90 56 40 17 13 11 10 9

7 101 63 45 20 14 12 11 10

8 111 70 50 22 16 13 12 12

9 122 76 55 24 17 15 14 13

10 132 83 59 26 19 16 15 14

0.95 0 25 16 11 4 3 2 2 2

1 40 25 18 7 5 4 4 3

2 53 33 23 10 7 6 5 5

3 66 41 29 12 9 7 6 6

4 78 49 34 14 10 9 8 7

5 89 56 40 17 12 10 9 8

6 101 63 45 19 14 11 10 10

7 112 70 50 21 15 13 12 11

8 123 77 55 24 17 14 13 12

9 134 84 60 26 19 16 14 13

10 145 91 65 28 20 17 16 15

0.99 0 38 24 17 6 4 3 3 3

1 56 34 24 10 6 5 5 4

2 71 44 31 12 8 7 6 6

3 85 52 37 15 10 9 8 7

4 98 61 43 18 12 10 9 8

5 111 69 49 20 14 12 10 10

6 123 77 54 23 16 13 12 11

7 135 84 60 25 18 15 13 12

8 148 92 65 27 19 16 14 13

9 159 99 70 30 21 18 16 15

10 171 107 76 32 23 19 17 16

134


percentile to

exceed a given values,0




P c

0

0.5/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 4 3 2 2 1 1 1 1

1 8 6 5 3 3 2 2 2

2 11 9 7 5 4 4 3 3

3 14 11 10 6 5 5 5 4

4 18 14 12 8 6 6 6 5

5 21 17 14 9 8 7 7 7

6 24 19 16 11 9 8 8 8

7 28 22 18 12 10 9 9 9

8 31 24 21 13 11 11 10 10

9 34 27 23 15 13 12 11 11

10 37 29 25 16 14 13 12 12

0.90 0 6 5 4 2 2 2 1 1

1 10 8 7 4 3 3 3 3

2 14 11 9 6 5 4 4 4

3 18 14 12 7 6 5 5 5

4 22 17 14 9 7 7 6 6

5 26 20 17 10 9 8 7 7

6 29 23 19 12 10 9 9 8

7 33 26 21 14 11 10 10 9

8 36 28 24 15 13 12 11 11

9 40 31 26 17 14 13 12 12

10 43 34 28 18 15 14 13 13

0.95 0 8 6 5 3 2 2 2 2

1 13 10 8 5 4 3 3 3

2 17 13 11 6 5 5 4 4

3 21 16 13 8 7 6 5 5

4 25 19 16 10 8 7 7 6

5 29 22 18 11 9 8 8 8

6 32 25 21 13 11 10 9 9

7 36 28 23 15 12 11 10 10

8 40 31 26 16 13 12 11 11

9 43 34 28 18 15 13 13 12

10 47 37 30 19 16 15 14 13

0.99 0 12 9 7 4 3 3 2 2

1 17 13 11 6 5 4 4 3

2 22 17 14 8 6 5 5 5

3 26 20 17 10 8 7 6 6

4 31 24 19 12 9 8 8 7

5 35 27 22 13 11 9 9 8

6 39 30 25 15 12 11 10 10

7 43 33 27 17 14 12 11 11

8 47 36 30 18 15 13 12 12

9 51 39 33 20 16 15 14 13

10 55 42 35 22 18 16 15 14

135


percentile to




distribution using the binomial approximation.

P c

0

0.75/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 2 2 1 1 1 1 1 1

1 4 3 3 2 2 2 2 2

2 6 5 5 4 3 3 3 3

3 7 7 6 5 5 4 4 4

4 9 8 7 6 6 5 5 5

5 11 10 9 7 7 6 6 6

6 12 11 10 8 8 8 7 7

7 14 13 12 10 9 9 8 8

8 16 14 13 11 10 10 9 9

9 17 16 15 12 11 11 11 10

10 19 17 16 13 12 12 12 11

0.90 0 3 2 2 2 1 1 1 1

1 5 4 4 3 3 2 2 2

2 7 6 6 4 4 4 3 3

3 9 8 7 6 5 5 5 4

4 11 9 9 7 6 6 6 6

5 12 11 10 8 7 7 7 7

6 14 13 12 9 9 8 8 8

7 16 14 13 11 10 9 9 9

8 18 16 15 12 11 10 10 10

9 20 18 16 13 12 11 11 11

10 21 19 18 14 13 12 12 12

0.95 0 3 3 3 2 2 1 1 1

1 6 5 5 3 3 3 3 2

2 8 7 6 5 4 4 4 4

3 10 9 8 6 5 5 5 5

4 12 10 9 7 7 6 6 6

5 14 12 11 9 8 7 7 7

6 16 14 13 10 9 8 8 8

7 17 15 14 11 10 10 9 9

8 19 17 16 12 11 11 10 10

9 21 19 17 14 12 12 11 11

10 23 20 19 15 14 13 13 12

0.99 0 5 4 4 3 2 2 2 2

1 8 7 6 4 4 3 3 3

2 10 9 8 6 5 5 4 4

3 12 11 9 7 6 6 5 5

4 14 12 11 8 7 7 7 6

5 16 14 13 10 9 8 8 8

6 18 16 15 11 10 9 9 9

7 20 18 16 12 11 10 10 10

8 22 20 18 14 12 12 11 11

9 24 21 19 15 13 13 12 12

10 26 23 21 16 15 14 13 13

136


percentile to





P c

0

0.9/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 1 1 1 1 1 1 1 1

1 3 2 2 2 2 2 2 2

2 4 4 4 3 3 3 3 3

3 5 5 5 4 4 4 4 4

4 6 6 6 5 5 5 5 5

5 8 7 7 6 6 6 6 6

6 9 9 8 8 7 7 7 7

7 10 10 9 9 8 8 8 8

8 11 11 11 10 9 9 9 9

9 13 12 12 11 10 10 10 10

10 14 13 13 12 12 11 11 11

0.90 0 2 2 1 1 1 1 1 1

1 3 3 3 2 2 2 2 2

2 5 4 4 4 3 3 3 3

3 6 6 5 5 5 4 4 4

4 7 7 7 6 6 5 5 5

5 9 8 8 7 7 6 6 6

6 10 9 9 8 8 8 7 7

7 11 11 10 9 9 9 8 8

8 12 12 11 10 10 10 9 9

9 14 13 13 11 11 11 11 10

10 15 14 14 13 12 12 12 11

0.95 0 2 2 2 1 1 1 1 1

1 4 3 3 3 3 2 2 2

2 5 5 5 4 4 3 3 3

3 6 6 6 5 5 5 4 4

4 8 7 7 6 6 6 6 5

5 9 9 8 7 7 7 7 7

6 11 10 10 8 8 8 8 8

7 12 11 11 10 9 9 9 9

8 13 12 12 11 10 10 10 10

9 14 14 13 12 11 11 11 11

10 16 15 14 13 12 12 12 12

0.99 0 3 3 2 2 2 2 2 1

1 5 4 4 3 3 3 3 3

2 6 6 5 5 4 4 4 4

3 8 7 7 6 5 5 5 5

4 9 9 8 7 6 6 6 6

5 11 10 9 8 8 7 7 7

6 12 11 11 9 9 8 8 8

7 13 13 12 10 10 9 9 9

8 15 14 13 12 11 11 10 10

9 16 15 14 13 12 12 11 11

10 17 16 16 14 13 13 12 12

137


percentile to




Poisson approximation.

P c

0

0.1/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 51 24 14 4 3 3 2 2

1 82 39 23 7 4 4 3 3

2 138 65 38 11 7 6 5 5

3 185 87 51 15 10 8 7 7

4 229 108 63 18 12 10 9 8

5 271 128 75 21 14 11 10 9

6 313 147 86 24 16 13 12 11

7 354 167 97 27 18 14 13 12

8 395 186 109 31 20 16 14 14

9 436 205 120 34 22 18 16 15

10 476 224 131 37 24 19 17 16

0.90 0 85 40 24 7 5 4 3 3

1 132 62 36 10 7 6 5 5

2 192 91 53 15 10 8 7 7

3 244 115 67 19 12 10 9 9

4 292 138 80 23 15 12 11 10

5 339 160 93 26 17 14 12 12

6 385 181 106 30 19 16 14 13

7 430 203 118 33 21 18 16 15

8 475 224 130 37 24 19 17 16

9 519 244 143 40 26 21 19 18

10 563 265 155 43 28 23 20 19

0.95 0 110 52 30 9 6 5 4 4

1 165 78 46 13 9 7 6 6

2 229 108 63 18 12 10 9 8

3 283 133 78 22 14 12 10 10

4 335 158 92 26 17 14 12 12

5 384 181 106 30 19 16 14 13

6 433 204 119 33 22 18 16 15

7 481 226 132 37 24 20 17 16

8 528 248 145 41 26 21 19 18

9 574 270 158 44 28 23 21 19

10 620 292 170 48 31 25 22 21

0.99 0 169 80 47 13 9 7 6 6

1 237 112 65 19 12 10 9 8

2 306 144 84 24 15 13 11 11

3 367 173 101 28 18 15 13 13

4 424 200 117 33 21 17 15 15

5 479 225 132 37 24 19 17 16

6 533 251 146 41 26 22 19 18

7 585 275 160 45 29 24 21 20

8 636 299 175 49 31 26 23 22

9 686 323 188 53 34 28 25 23

10 736 346 202 57 36 30 27 25

138


percentile to




distribution using the Poisson approximation.

P c

0

0.25/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 13 8 6 3 2 2 2 2

1 20 13 9 5 4 3 3 3

2 33 21 16 7 6 5 5 5

3 45 29 21 10 8 7 6 6

4 55 35 26 12 9 8 8 8

5 65 42 30 14 11 10 9 9

6 75 48 35 16 13 11 10 10

7 85 54 39 18 14 13 12 11

8 95 60 44 21 16 14 13 13

9 104 66 48 23 17 15 14 14

10 114 73 53 25 19 17 16 15

0.90 0 21 13 10 5 4 3 3 3

1 32 20 15 7 6 5 5 5

2 46 30 21 10 8 7 7 6

3 59 37 27 13 10 9 8 8

4 70 45 32 15 12 10 10 9

5 81 52 38 18 14 12 11 11

6 92 59 43 20 15 14 13 12

7 103 66 48 22 17 15 14 14

8 114 72 52 25 19 17 16 15

9 124 79 57 27 21 18 17 16

10 135 86 62 29 22 20 18 18

0.95 0 27 17 12 6 5 4 4 4

1 40 25 19 9 7 6 6 6

2 55 35 25 12 9 8 8 7

3 68 43 31 15 11 10 10 9

4 80 51 37 17 13 12 11 11

5 92 59 43 20 15 14 13 12

6 104 66 48 22 17 15 14 14

7 115 73 53 25 19 17 16 15

8 126 80 58 27 21 19 17 17

9 138 87 63 30 23 20 19 18

10 148 94 68 32 24 22 20 19

0.99 0 41 26 19 9 7 6 6 6

1 57 36 26 12 10 9 8 8

2 74 47 34 16 12 11 10 10

3 88 56 41 19 15 13 12 12

4 102 65 47 22 17 15 14 13

5 115 73 53 25 19 17 16 15

6 128 81 59 27 21 19 17 17

7 140 89 64 30 23 20 19 18

8 152 97 70 33 25 22 21 20

9 164 105 76 35 27 24 22 22

10 176 112 81 38 29 26 24 23

139


percentile to




distribution using the Poisson approximation.

P c

0

0.5/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 5 4 3 2 2 2 2 2

1 7 6 5 4 3 3 3 3

2 12 9 8 6 5 5 5 4

3 15 12 11 7 7 6 6 6

4 19 15 13 9 8 7 7 7

5 22 18 15 11 9 9 9 8

6 26 21 18 12 11 10 10 10

7 29 23 20 14 12 11 11 11

8 32 26 22 15 13 13 12 12

9 36 29 24 17 15 14 13 13

10 39 31 27 18 16 15 15 14

0.90 0 7 6 5 4 3 3 3 3

1 11 9 8 5 5 5 4 4

2 16 13 11 8 7 6 6 6

3 20 16 14 10 8 8 8 8

4 24 19 16 11 10 9 9 9

5 28 22 19 13 12 11 11 10

6 32 25 22 15 13 12 12 12

7 35 28 24 17 14 14 13 13

8 39 31 26 18 16 15 15 14

9 42 34 29 20 17 16 16 16

10 46 37 31 21 19 18 17 17

0.95 0 9 8 6 5 4 4 4 4

1 14 11 10 7 6 6 5 5

2 19 15 13 9 8 7 7 7

3 23 19 16 11 10 9 9 9

4 28 22 19 13 11 11 10 10

5 32 25 22 15 13 12 12 12

6 35 28 24 17 15 14 13 13

7 39 31 27 18 16 15 15 14

8 43 34 29 20 18 17 16 16

9 47 37 32 22 19 18 17 17

10 51 40 34 24 21 19 19 18

0.99 0 14 11 10 7 6 6 5 5

1 20 16 13 9 8 8 7 7

2 25 20 17 12 10 10 10 9

3 30 24 21 14 12 12 11 11

4 35 28 24 16 14 13 13 13

5 39 31 27 18 16 15 15 14

6 44 35 30 20 18 17 16 16

7 48 38 32 22 20 18 18 17

8 52 41 35 24 21 20 19 19

9 56 45 38 26 23 21 21 20

10 60 48 41 28 24 23 22 22

140


percentile to exceed a

given values, 0

0.75t , with probability Pand the corresponding acceptance

number, c, for the inverse Rayleigh distribution using the Poisson

approximation.

P c

0

0.75/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 3 2 2 2 2 2 2 2

1 4 4 3 3 3 3 3 3

2 6 6 6 5 5 4 4 4

3 8 8 7 6 6 6 6 6

4 10 9 9 8 7 7 7 7

5 12 11 10 9 8 8 8 8

6 14 13 12 10 10 9 9 9

7 16 14 13 12 11 11 10 10

8 17 16 15 13 12 12 12 12

9 19 17 16 14 13 13 13 13

10 21 19 18 15 14 14 14 14

0.90 0 4 4 4 3 3 3 3 3

1 6 6 5 5 4 4 4 4

2 9 8 7 6 6 6 6 6

3 11 10 9 8 8 7 7 7

4 13 12 11 10 9 9 9 9

5 15 14 13 11 10 10 10 10

6 17 16 15 12 12 12 11 11

7 19 17 16 14 13 13 13 13

8 21 19 18 15 14 14 14 14

9 23 21 19 17 16 15 15 15

10 25 22 21 18 17 17 16 16

0.95 0 5 5 4 4 4 4 4 4

1 8 7 7 6 5 5 5 5

2 10 9 9 8 7 7 7 7

3 13 12 11 9 9 9 8 8

4 15 14 13 11 10 10 10 10

5 17 15 15 12 12 12 11 11

6 19 17 16 14 13 13 13 13

7 21 19 18 15 15 14 14 14

8 23 21 20 17 16 16 15 15

9 25 23 21 18 17 17 17 17

10 27 25 23 20 19 18 18 18

0.99 0 8 7 7 6 5 5 5 5

1 11 10 9 8 7 7 7 7

2 14 12 12 10 10 9 9 9

3 16 15 14 12 11 11 11 11

4 19 17 16 14 13 13 12 12

5 21 19 18 15 15 14 14 14

6 23 21 20 17 16 16 16 15

7 26 23 22 19 18 17 17 17

8 28 25 24 20 19 19 18 18

9 30 27 26 22 21 20 20 20

10 32 29 27 23 22 22 21 21

141


percentile to exceed a

given values, 0

0.9t , with probability Pand the corresponding acceptance

number, c, for the inverse Rayleigh distribution using the Poisson

approximation.

P c

0

0.9/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 2 2 2 2 2 2 2 2

1 3 3 3 3 3 3 3 3

2 5 5 5 4 4 4 4 4

3 6 6 6 6 6 6 6 6

4 8 8 7 7 7 7 7 7

5 9 9 9 8 8 8 8 8

6 11 10 10 9 9 9 9 9

7 12 12 11 11 10 10 10 10

8 13 13 13 12 12 11 11 11

9 15 14 14 13 13 13 13 13

10 16 15 15 14 14 14 14 14

0.90 0 3 3 3 3 3 3 3 3

1 5 5 4 4 4 4 4 4

2 7 6 6 6 6 6 6 6

3 8 8 8 7 7 7 7 7

4 10 10 9 9 9 9 9 9

5 11 11 11 10 10 10 10 10

6 13 12 12 12 11 11 11 11

7 14 14 14 13 13 12 12 12

8 16 15 15 14 14 14 14 14

9 17 17 16 15 15 15 15 15

10 19 18 18 17 16 16 16 16

0.95 0 4 4 4 4 4 4 4 4

1 6 6 6 5 5 5 5 5

2 8 8 7 7 7 7 7 7

3 10 9 9 9 8 8 8 8

4 11 11 11 10 10 10 10 10

5 13 12 12 12 11 11 11 11

6 14 14 14 13 13 13 12 12

7 16 15 15 14 14 14 14 14

8 18 17 17 16 15 15 15 15

9 19 18 18 17 17 16 16 16

10 20 20 19 18 18 18 18 18

0.99 0 6 6 6 5 5 5 5 5

1 8 8 8 7 7 7 7 7

2 10 10 10 9 9 9 9 9

3 12 12 12 11 11 11 11 11

4 14 14 13 13 12 12 12 12

5 16 15 15 14 14 14 14 14

6 18 17 17 16 15 15 15 15

7 19 19 18 17 17 17 17 17

8 21 20 20 19 18 18 18 18

9 23 22 21 20 20 20 20 19

10 24 23 23 22 21 21 21 21

142

Table 4.3.11: Operating characteristic values of the sampling plan

0

0.1( , 5, / )n c t t for a given Punder inverse Rayleigh distribution.

P n

0

0.1/t t

0

0.1 0.1/t t

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

0.75 270 0.8 0.2495 0.9995 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

127 0.9 0.2444 0.9959 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

73 1.0 0.2499 0.9849 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

20 1.5 0.2188 0.7973 0.9887 0.9998 1.0000 1.0000 1.0000 1.0000

12 2.0 0.2329 0.6471 0.9192 0.9908 0.9995 1.0000 1.0000 1.0000

10 2.5 0.1646 0.4635 0.7664 0.9330 0.9872 0.9984 0.9999 1.0000

9 3.0 0.1234 0.3403 0.6097 0.8235 0.9393 0.9840 0.9968 0.9995

8 3.5 0.1438 0.3327 0.5599 0.7581 0.8897 0.9582 0.9868 0.9965

0.90 337 0.8 0.0993 0.9984 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

158 0.9 0.0969 0.9885 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

91 1.0 0.0976 0.9608 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

23 1.5 0.0884 0.6461 0.9723 0.9995 1.0000 1.0000 1.0000 1.0000

15 2.0 0.0637 0.3823 0.7943 0.9688 0.9978 0.9999 1.0000 1.0000

11 2.5 0.0878 0.3355 0.6673 0.8920 0.9773 0.9969 0.9997 1.0000

10 3.0 0.0529 0.2054 0.4635 0.7238 0.8936 0.9693 0.9933 0.9989

9 3.5 0.0525 0.1742 0.3783 0.6097 0.7989 0.9153 0.9708 0.9917

0.95 382 0.8 0.0493 0.9971 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

178 0.9 0.0494 0.9804 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

103 1.0 0.0479 0.9358 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

27 1.5 0.0412 0.5277 0.9529 0.9991 1.0000 1.0000 1.0000 1.0000

16 2.0 0.0393 0.3089 0.7427 0.9571 0.9967 0.9999 1.0000 1.0000

12 2.5 0.0444 0.2329 0.5650 0.8412 0.9633 0.9945 0.9995 1.0000

11 3.0 0.0211 0.1164 0.3355 0.6155 0.8350 0.9478 0.9877 0.9978

10 3.5 0.0174 0.0835 0.2374 0.4635 0.6909 0.8560 0.9458 0.9834

0.99 475 0.8 0.0099 0.9917 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

221 0.9 0.0099 0.9518 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

127 1.0 0.0100 0.8633 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

33 1.5 0.0076 0.3184 0.8939 0.9973 1.0000 1.0000 1.0000 1.0000

19 2.0 0.0081 0.1486 0.5757 0.9077 0.9914 0.9996 1.0000 1.0000

15 2.5 0.0045 0.0637 0.2964 0.6494 0.8937 0.9805 0.9978 0.9998

12 3.0 0.0080 0.0627 0.2329 0.5077 0.7662 0.9192 0.9796 0.9962

11 3.5 0.0053 0.0373 0.1402 0.3355 0.5768 0.7832 0.9113 0.9709

143


0


P n

0

0.25/t t

0

0.25 0.25/t t

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

0.75 64 0.8 0.2435 0.9787 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

40 0.9 0.2466 0.9452 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000

29 1.0 0.2317 0.8927 0.9985 1.0000 1.0000 1.0000 1.0000 1.0000

13 1.5 0.1987 0.6274 0.9198 0.9921 0.9996 1.0000 1.0000 1.0000

10 2.0 0.1383 0.4130 0.7201 0.9097 0.9803 0.9970 0.9997 1.0000

8 2.5 0.2007 0.4301 0.6696 0.8461 0.9425 0.9827 0.9958 0.9992

8 3.0 0.0934 0.2345 0.4301 0.6327 0.7965 0.9033 0.9605 0.9861

7 3.5 0.1673 0.3173 0.4903 0.6553 0.7894 0.8837 0.9420 0.9738

0.90 79 0.8 0.0982 0.9482 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

50 0.9 0.0911 0.8717 0.9993 1.0000 1.0000 1.0000 1.0000 1.0000

35 1.0 0.0976 0.7932 0.9960 1.0000 1.0000 1.0000 1.0000 1.0000

15 1.5 0.0890 0.4605 0.8516 0.9824 0.9990 1.0000 1.0000 1.0000

11 2.0 0.0704 0.2880 0.6111 0.8580 0.9657 0.9944 0.9994 1.0000

9 2.5 0.0842 0.2543 0.4995 0.7322 0.8869 0.9623 0.9900 0.9979

8 3.0 0.0934 0.2345 0.4301 0.6327 0.7965 0.9033 0.9605 0.9861

8 3.5 0.0454 0.1264 0.2600 0.4301 0.6053 0.7553 0.8645 0.9329

0.95 89 0.8 0.0496 0.9179 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

56 0.9 0.0464 0.8126 0.9988 1.0000 1.0000 1.0000 1.0000 1.0000

40 1.0 0.0433 0.6931 0.9922 1.0000 1.0000 1.0000 1.0000 1.0000

17 1.5 0.0363 0.3165 0.7653 0.9667 0.9979 0.9999 1.0000 1.0000

12 2.0 0.0340 0.1922 0.5028 0.7960 0.9456 0.9905 0.9989 0.9999

10 2.5 0.0321 0.1383 0.3492 0.6058 0.8134 0.9313 0.9803 0.9955

9 3.0 0.0287 0.1051 0.2543 0.4570 0.6614 0.8207 0.9195 0.9693

8 3.5 0.0454 0.1264 0.2600 0.4301 0.6053 0.7553 0.8645 0.9329

0.99 111 0.8 0.0094 0.8242 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000

69 0.9 0.0092 0.6599 0.9965 1.0000 1.0000 1.0000 1.0000 1.0000

49 1.0 0.0085 0.5027 0.9796 0.9999 1.0000 1.0000 1.0000 1.0000

20 1.5 0.0083 0.1629 0.6172 0.9299 0.9949 0.9998 1.0000 1.0000

14 2.0 0.0070 0.0768 0.3140 0.6526 0.8883 0.9773 0.9970 0.9997

12 2.5 0.0038 0.0340 0.1460 0.3685 0.6335 0.8379 0.9456 0.9860

10 3.0 0.0080 0.0429 0.1383 0.3083 0.5215 0.7201 0.8625 0.9433

10 3.5 0.0022 0.0137 0.0521 0.1383 0.2802 0.4595 0.6402 0.7891

144


0


P n

0

0.5/t t

0

0.5 0.5/t t

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

0.75 21 0.8 0.2330 0.8244 0.9923 0.9999 1.0000 1.0000 1.0000 1.0000

17 0.9 0.2002 0.7255 0.9719 0.9991 1.0000 1.0000 1.0000 1.0000

14 1.0 0.2120 0.6746 0.9455 0.9963 0.9999 1.0000 1.0000 1.0000

9 1.5 0.1951 0.4721 0.7461 0.9121 0.9780 0.9960 0.9995 0.9999

8 2.0 0.1210 0.2899 0.5059 0.7088 0.8555 0.9396 0.9787 0.9936

7 2.5 0.1622 0.3089 0.4798 0.6445 0.7801 0.8768 0.9375 0.9713

7 3.0 0.0899 0.1838 0.3089 0.4509 0.5920 0.7169 0.8167 0.8892

7 3.5 0.0528 0.1130 0.2000 0.3089 0.4303 0.5528 0.6661 0.7630

0.90 26 0.8 0.0814 0.6580 0.9777 0.9997 1.0000 1.0000 1.0000 1.0000

20 0.9 0.0850 0.5659 0.9400 0.9976 1.0000 1.0000 1.0000 1.0000

17 1.0 0.0717 0.4593 0.8722 0.9888 0.9996 1.0000 1.0000 1.0000

10 1.5 0.0971 0.3226 0.6230 0.8511 0.9585 0.9918 0.9988 0.9999

9 2.0 0.0412 0.1426 0.3242 0.5468 0.7461 0.8818 0.9543 0.9852

8 2.5 0.0432 0.1210 0.2506 0.4176 0.5918 0.7433 0.8555 0.9271

7 3.0 0.0899 0.1838 0.3089 0.4509 0.5920 0.7169 0.8167 0.8892

7 3.5 0.0528 0.1130 0.2000 0.3089 0.4303 0.5528 0.6661 0.7630

0.95 29 0.8 0.0399 0.5510 0.9632 0.9995 1.0000 1.0000 1.0000 1.0000

22 0.9 0.0452 0.4622 0.9103 0.9960 0.9999 1.0000 1.0000 1.0000

18 1.0 0.0481 0.3939 0.8408 0.9848 0.9994 1.0000 1.0000 1.0000

11 1.5 0.0451 0.2084 0.5000 0.7766 0.9309 0.9851 0.9977 0.9997

9 2.0 0.0412 0.1426 0.3242 0.5468 0.7461 0.8818 0.9543 0.9852

8 2.5 0.0432 0.1210 0.2506 0.4176 0.5918 0.7433 0.8555 0.9271

8 3.0 0.0172 0.0526 0.1210 0.2258 0.3595 0.5059 0.6460 0.7650

8 3.5 0.0076 0.0245 0.0602 0.1210 0.2087 0.3192 0.4428 0.5677

0.99 35 0.8 0.0084 0.3554 0.9193 0.9985 1.0000 1.0000 1.0000 1.0000

27 0.9 0.0079 0.2496 0.8086 0.9884 0.9998 1.0000 1.0000 1.0000

22 1.0 0.0085 0.1918 0.6904 0.9597 0.9981 1.0000 1.0000 1.0000

13 1.5 0.0083 0.0760 0.2905 0.6059 0.8514 0.9623 0.9935 0.9992

11 2.0 0.0036 0.0268 0.1070 0.2732 0.5000 0.7172 0.8699 0.9513

9 2.5 0.0099 0.0412 0.1157 0.2433 0.4120 0.5904 0.7461 0.8602

9 3.0 0.0028 0.0130 0.0412 0.0996 0.1951 0.3242 0.4721 0.6187

8 3.5 0.0076 0.0245 0.0602 0.1210 0.2087 0.3192 0.4428 0.5677

145

Table 4.3.14: Operating characteristic values of the sampling plan0

0.75( , 5, / )n c t t

for a given Punder inverse Rayleigh distribution.

P n

0

0.75/t t

0

0.75 0.75/t t

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

0.75 11 0.8 0.1699 0.5124 0.8271 0.9641 0.9955 0.9997 1.0000 1.0000

10 0.9 0.1485 0.4330 0.7390 0.9196 0.9834 0.9976 0.9998 1.0000

9 1.0 0.1657 0.4215 0.6978 0.8838 0.9671 0.9931 0.9989 0.9999

7 1.5 0.2013 0.3707 0.5551 0.7191 0.8418 0.9206 0.9644 0.9857

7 2.0 0.0800 0.1655 0.2818 0.4169 0.5551 0.6814 0.7862 0.8654

6 2.5 0.2413 0.3505 0.4628 0.5708 0.6687 0.7529 0.8220 0.8761

6 3.0 0.1745 0.2589 0.3505 0.4442 0.5357 0.6213 0.6984 0.7655

6 3.5 0.1314 0.1976 0.2717 0.3505 0.4309 0.5100 0.5855 0.6555

0.90 12 0.8 0.0997 0.3997 0.7561 0.9432 0.9923 0.9994 1.0000 1.0000

11 0.9 0.0770 0.3065 0.6338 0.8723 0.9708 0.9955 0.9995 1.0000

10 1.0 0.0781 0.2755 0.5641 0.8088 0.9395 0.9861 0.9976 0.9997

8 1.5 0.0608 0.1630 0.3215 0.5087 0.6854 0.8225 0.9118 0.9614

7 2.0 0.0800 0.1655 0.2818 0.4169 0.5551 0.6814 0.7862 0.8654

7 2.5 0.0365 0.0800 0.1457 0.2322 0.3344 0.4448 0.5551 0.6577

7 3.0 0.0187 0.0423 0.0800 0.1332 0.2013 0.2818 0.3707 0.4634

7 3.5 0.0105 0.0242 0.0468 0.0800 0.1247 0.1804 0.2460 0.3191

0.95 14 0.8 0.0306 0.2222 0.5987 0.8840 0.9815 0.9983 0.9999 1.0000

12 0.9 0.0379 0.2078 0.5277 0.8149 0.9533 0.9923 0.9992 0.9999

11 1.0 0.0343 0.1699 0.4367 0.7209 0.9017 0.9754 0.9955 0.9994

9 1.5 0.0159 0.0627 0.1657 0.3269 0.5183 0.6978 0.8346 0.9211

8 2.0 0.0144 0.0446 0.1042 0.1980 0.3215 0.4614 0.6006 0.7240

7 2.5 0.0365 0.0800 0.1457 0.2322 0.3344 0.4448 0.5551 0.6577

7 3.0 0.0187 0.0423 0.0800 0.1332 0.2013 0.2818 0.3707 0.4634

7 3.5 0.0105 0.0242 0.0468 0.0800 0.1247 0.1804 0.2460 0.3191

0.99 16 0.8 0.0083 0.1119 0.4438 0.8039 0.9632 0.9962 0.9998 1.0000

14 0.9 0.0081 0.0859 0.3381 0.6792 0.9026 0.9815 0.9977 0.9998

13 1.0 0.0056 0.0562 0.2345 0.5318 0.7990 0.9405 0.9877 0.9982

10 1.5 0.0037 0.0219 0.0781 0.1944 0.3678 0.5641 0.7390 0.8649

9 2.0 0.0022 0.0104 0.0335 0.0826 0.1657 0.2823 0.4215 0.5659

8 2.5 0.0043 0.0144 0.0365 0.0764 0.1377 0.2206 0.3215 0.4329

8 3.0 0.0015 0.0054 0.0144 0.0317 0.0608 0.1042 0.1630 0.2364

8 3.5 0.0006 0.0023 0.0063 0.0144 0.0286 0.0511 0.0838 0.1275

146


0


P n

0

0.9/t t

0

0.9 0.9/t t

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75

0.75 8 0.8 0.1082 0.2648 0.4724 0.6762 0.8311 0.9252 0.9718 0.9909

7 0.9 0.2065 0.3786 0.5643 0.7278 0.8486 0.9251 0.9670 0.9870

7 1.0 0.1497 0.2884 0.4535 0.6170 0.7556 0.8582 0.9251 0.9640

6 1.5 0.2449 0.3553 0.4686 0.5770 0.6750 0.7589 0.8273 0.8805

6 2.0 0.1462 0.2188 0.2992 0.3837 0.4686 0.5507 0.6276 0.6974

6 2.5 0.0962 0.1462 0.2035 0.2664 0.3327 0.4007 0.4686 0.5346

6 3.0 0.0678 0.1039 0.1462 0.1935 0.2449 0.2992 0.3553 0.4121

6 3.5 0.0503 0.0775 0.1096 0.1462 0.1865 0.2299 0.2757 0.3231

0.90 9 0.8 0.0353 0.1252 0.2925 0.5073 0.7102 0.8570 0.9408 0.9794

8 0.9 0.0633 0.1688 0.3308 0.5200 0.6963 0.8311 0.9175 0.9645

8 1.0 0.0381 0.1082 0.2279 0.3865 0.5575 0.7119 0.8311 0.9108

7 1.5 0.0377 0.0824 0.1497 0.2380 0.3419 0.4535 0.5643 0.6668

7 2.0 0.0130 0.0298 0.0573 0.0971 0.1497 0.2142 0.2884 0.3694

6 2.5 0.0962 0.1462 0.2035 0.2664 0.3327 0.4007 0.4686 0.5346

6 3.0 0.0678 0.1039 0.1462 0.1935 0.2449 0.2992 0.3553 0.4121

6 3.5 0.0503 0.0775 0.1096 0.1462 0.1865 0.2299 0.2757 0.3231

0.95 9 0.8 0.0353 0.1252 0.2925 0.5073 0.7102 0.8570 0.9408 0.9794

9 0.9 0.0168 0.0659 0.1728 0.3380 0.5315 0.7102 0.8440 0.9269

8 1.0 0.0381 0.1082 0.2279 0.3865 0.5575 0.7119 0.8311 0.9108

7 1.5 0.0377 0.0824 0.1497 0.2380 0.3419 0.4535 0.5643 0.6668

7 2.0 0.0130 0.0298 0.0573 0.0971 0.1497 0.2142 0.2884 0.3694

7 2.5 0.0055 0.0130 0.0257 0.0449 0.0717 0.1066 0.1497 0.2004

7 3.0 0.0027 0.0065 0.0130 0.0231 0.0377 0.0573 0.0824 0.1132

7 3.5 0.0015 0.0036 0.0072 0.0130 0.0214 0.0330 0.0482 0.0674

0.99 11 0.8 0.0028 0.0216 0.0895 0.2377 0.4524 0.6722 0.8385 0.9346

10 0.9 0.0040 0.0233 0.0826 0.2035 0.3812 0.5790 0.7520 0.8740

9 1.0 0.0083 0.0353 0.1010 0.2171 0.3759 0.5507 0.7102 0.8330

8 1.5 0.0045 0.0150 0.0381 0.0794 0.1427 0.2279 0.3308 0.4437

8 2.0 0.0009 0.0032 0.0086 0.0194 0.0381 0.0671 0.1082 0.1621

7 2.5 0.0055 0.0130 0.0257 0.0449 0.0717 0.1066 0.1497 0.2004

7 3.0 0.0027 0.0065 0.0130 0.0231 0.0377 0.0573 0.0824 0.1132

7 3.5 0.0015 0.0036 0.0072 0.0130 0.0214 0.0330 0.0482 0.0674

147

Table 4.3.16: Minimum ratio of true 0.1d for the acceptability of a lot for the

inverse Rayleigh distribution and producer’s risk of 0.05.

P c

0

0.1/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 2.5950 2.5950 2.5950 2.5950 3.1589 3.4219 3.9924 4.5626

1 1.3084 1.3695 1.6935 2.0121 2.3321 2.7988 3.2723 3.7286

2 1.2439 1.2935 1.5439 1.8080 2.1299 2.3381 2.7233 3.1153

3 1.2076 1.2525 1.4883 1.6935 2.0121 2.2624 2.3935 2.7315

4 1.1855 1.2222 1.4298 1.6181 1.8372 2.0530 2.3935 2.7397

5 1.1688 1.1996 1.3883 1.5625 1.8008 2.0437 2.2075 2.5265

6 1.1539 1.1871 1.3572 1.5232 1.6903 1.9106 2.0623 2.3563

7 1.1438 1.1733 1.3314 1.4908 1.6779 1.8044 1.9433 2.2183

8 1.1352 1.1628 1.3122 1.4643 1.5982 1.8262 1.9988 2.2852

9 1.1282 1.1539 1.2953 1.4434 1.5982 1.7455 1.9066 2.1810

10 1.1226 1.1467 1.2807 1.4253 1.5361 1.6717 1.8262 2.0907

0.90 0 2.5950 2.5950 2.5950 2.6614 3.1589 3.7908 4.4225 5.0542

1 1.3552 1.4253 1.7973 2.1968 2.5195 3.0221 3.2723 3.7286

2 1.2862 1.3412 1.6534 1.9558 2.2568 2.5549 2.9824 3.4072

3 1.2456 1.2935 1.5545 1.8188 2.1151 2.4190 2.6364 3.0120

4 1.2206 1.2629 1.5056 1.7256 2.0210 2.2022 2.5694 2.9438

5 1.2012 1.2405 1.4550 1.6966 1.8829 2.1603 2.3810 2.7233

6 1.1855 1.2238 1.4164 1.6445 1.8409 2.0300 2.2292 2.5478

7 1.1733 1.2092 1.3990 1.6010 1.8080 2.0165 2.2401 2.5549

8 1.1643 1.1965 1.3736 1.5652 1.7289 1.9186 2.1299 2.4319

9 1.1554 1.1871 1.3512 1.5361 1.7126 1.9186 2.0346 2.3261

10 1.1481 1.1779 1.3333 1.5106 1.6474 1.8447 1.9516 2.2292

0.95 0 2.5950 2.5950 2.5950 2.7531 3.3269 3.9922 4.4225 5.0542

1 1.3799 1.4526 1.8484 2.2568 2.6441 3.0221 3.5261 4.0258

2 1.3084 1.3674 1.6935 2.0121 2.3624 2.7152 2.9824 3.4072

3 1.2665 1.3198 1.6067 1.9106 2.2022 2.5407 2.8161 3.2258

4 1.2389 1.2862 1.5361 1.8080 2.0956 2.3261 2.5694 2.9438

5 1.2189 1.2612 1.4957 1.7355 1.9558 2.2624 2.5195 2.8785

6 1.2028 1.2422 1.4550 1.6779 1.9026 2.1249 2.3685 2.7071

7 1.1902 1.2271 1.4320 1.6297 1.8636 2.1004 2.2401 2.5549

8 1.1794 1.2157 1.4055 1.5926 1.7832 2.0032 2.2401 2.5621

9 1.1703 1.2044 1.3904 1.5870 1.7590 1.9186 2.1450 2.4516

10 1.1628 1.1949 1.3695 1.5571 1.7422 1.9146 2.0576 2.3502

0.99 0 2.5950 2.5950 2.5950 2.8775 3.4413 4.1296 4.6575 5.3229

1 1.4231 1.5031 1.9268 2.3563 2.8249 3.2960 3.7133 4.2319

2 1.3472 1.4120 1.7590 2.1400 2.5126 2.8337 3.1696 3.6245

3 1.3028 1.3592 1.6779 1.9857 2.3321 2.6441 2.9630 3.3818

4 1.2718 1.3256 1.6124 1.9066 2.2129 2.5126 2.8337 3.2373

5 1.2491 1.2972 1.5625 1.8262 2.1249 2.3441 2.6364 3.0120

6 1.2321 1.2789 1.5181 1.7624 2.0076 2.2852 2.4783 2.8337

7 1.2189 1.2612 1.4908 1.7322 1.9600 2.1706 2.4516 2.7988

8 1.2060 1.2473 1.4667 1.6903 1.9186 2.1400 2.3381 2.6674

9 1.1965 1.2355 1.4388 1.6534 1.8829 2.0530 2.3202 2.6518

10 1.1886 1.2255 1.4231 1.6210 1.8188 2.0346 2.2346 2.5549

148



P c

0

0.25/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 1.6955 1.8186 2.4433 3.2574 3.6763 4.4113 5.1454 5.8810

1 1.4762 1.5439 2.0437 2.4062 3.0120 3.6101 3.6390 4.1736

2 1.3716 1.4388 1.8262 2.1968 2.5056 3.0120 3.5125 4.0258

3 1.3217 1.3820 1.6998 1.9433 2.4254 2.6441 3.0836 3.5261

4 1.2844 1.3314 1.6181 1.8947 2.2022 2.3810 2.7816 3.1807

5 1.2560 1.3065 1.5598 1.8560 2.0346 2.4384 2.5549 2.9155

6 1.2405 1.2789 1.5156 1.7422 2.0530 2.2795 2.3747 2.7152

7 1.2238 1.2647 1.4786 1.7322 1.9391 2.1450 2.2292 2.5478

8 1.2092 1.2456 1.4503 1.6474 1.8409 2.0346 2.3747 2.7152

9 1.1980 1.2372 1.4276 1.6474 1.7556 1.9350 2.2624 2.5840

10 1.1886 1.2238 1.4055 1.5843 1.7973 2.0210 2.1603 2.4716

0.90 0 1.7857 1.9314 2.5725 3.2574 4.0715 4.8854 5.7003 6.5176

1 1.5413 1.6385 2.1200 2.5913 3.2489 3.6101 4.2123 4.8146

2 1.4343 1.5131 1.8907 2.3321 2.7397 3.2960 3.5125 4.0258

3 1.3778 1.4411 1.8044 2.1810 2.4254 2.9155 3.0836 3.5261

4 1.3373 1.3926 1.7094 1.9988 2.3685 2.6441 3.0836 3.5261

5 1.3084 1.3592 1.6415 1.9433 2.1915 2.4384 2.8514 3.2605

6 1.2825 1.3333 1.5870 1.8986 2.1810 2.4649 2.6596 3.0423 7 1.2647 1.3141 1.5760 1.8008 2.0623 2.3261 2.5056 2.8604

8 1.2508 1.2972 1.5387 1.7832 1.9600 2.2075 2.5767 2.9438

9 1.2372 1.2844 1.5081 1.7094 1.9728 2.2457 2.4582 2.8074

10 1.2271 1.2682 1.4835 1.6998 1.8947 2.1552 2.3563 2.6911

0.95 0 1.8322 1.9682 2.6614 3.4308 4.0715 4.8854 5.7003 6.5176

1 1.5760 1.6810 2.1810 2.7315 3.2489 3.8880 4.2123 4.8146

2 1.4667 1.5466 1.9900 2.4384 2.9155 3.2960 3.8388 4.3956

3 1.4077 1.4786 1.8484 2.2681 2.5988 2.9155 3.4072 3.8880

4 1.3674 1.4276 1.7489 2.0859 2.4988 2.8425 3.0836 3.5261

5 1.3353 1.3968 1.7062 2.0165 2.3202 2.6288 2.8514 3.2605

6 1.3103 1.3674 1.6504 1.9643 2.1810 2.4649 2.8785 3.2841

7 1.2898 1.3432 1.6038 1.8674 2.1654 2.4716 2.7152 3.1046

8 1.2736 1.3256 1.5898 1.8372 2.0623 2.3502 2.5767 2.9438

9 1.2612 1.3084 1.5545 1.8152 2.0576 2.2457 2.4582 2.8074

10 1.2491 1.2953 1.5258 1.7489 1.9814 2.2738 2.5195 2.8785

0.99 0 1.8956 2.0467 2.7820 3.5486 4.2885 5.1454 6.0053 6.8620

1 1.6327 1.7422 2.3202 2.8337 3.4072 4.0984 4.5496 5.2002

2 1.5258 1.6152 2.0717 2.5195 3.0423 3.4990 4.0800 4.6664

3 1.4550 1.5387 1.9474 2.3441 2.8337 3.2723 3.6390 4.1545

4 1.4142 1.4859 1.8713 2.2237 2.6062 3.0021 3.3201 3.7908

5 1.3799 1.4480 1.7902 2.1349 2.5195 2.7902 3.2489 3.7133

6 1.3552 1.4164 1.7489 2.0717 2.3747 2.7397 3.0525 3.4855

7 1.3314 1.3926 1.6966 2.0210 2.3321 2.5988 2.8877 3.2960

8 1.3160 1.3716 1.6534 1.9350 2.2292 2.4716 2.7480 3.1368

9 1.2990 1.3512 1.6356 1.9026 2.2075 2.4716 2.7647 3.1586

10 1.2880 1.3392 1.6010 1.8751 2.1249 2.3747 2.6518 3.0321

149



P c

0

0.50/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 2.1833 2.3034 3.4555 4.1590 5.1983 6.2402 7.2796 8.3188

1 1.8008 1.9268 2.5549 3.4072 3.6832 4.4170 5.1706 5.9102

2 1.6504 1.7224 2.3261 2.8425 3.5537 3.6101 4.2123 4.8146

3 1.5309 1.6565 2.0623 2.4919 3.1153 3.7439 3.6390 4.1736

4 1.4835 1.5706 2.0076 2.2457 2.8074 3.3693 3.2605 3.7286

5 1.4480 1.5106 1.8598 2.3026 2.5767 3.0941 3.6101 4.1356

6 1.3990 1.4643 1.8484 2.1501 2.3998 2.8785 3.3568 3.8388

7 1.3820 1.4276 1.7489 2.0210 2.2512 2.6991 3.1586 3.6101

8 1.3492 1.4253 1.6625 1.9146 2.3935 2.5549 2.9824 3.4072

9 1.3373 1.3990 1.6748 1.9857 2.2852 2.4319 2.8337 3.2373

10 1.3122 1.3757 1.6067 1.9026 2.1863 2.3202 2.7071 3.0941

0.90 0 2.3151 2.5093 3.4555 4.6081 5.7580 6.2402 7.2796 8.3188

1 1.8947 2.0576 2.7563 3.4072 4.2517 5.1125 5.9488 6.8446

2 1.7256 1.8335 2.4716 3.1046 3.5537 4.2517 4.9727 5.6883

3 1.6268 1.7422 2.2022 2.7480 3.1153 3.7439 4.3535 4.9727

4 1.5652 1.6474 2.1200 2.4919 3.1153 3.3693 3.9386 4.5045

5 1.5181 1.6095 1.9685 2.4851 2.8785 3.0941 3.6101 4.1356

6 1.4835 1.5545 1.9391 2.3261 2.6831 3.2258 3.3568 3.8388

7 1.4573 1.5106 1.9106 2.1915 2.5265 3.0321 3.1586 3.6101

8 1.4186 1.4981 1.8225 2.2183 2.6062 2.8785 3.3568 3.8388

9 1.4011 1.4667 1.8116 2.1200 2.4851 2.7397 3.1918 3.6536

10 1.3862 1.4388 1.7455 2.0346 2.3810 2.6212 3.0628 3.4990

0.95 0 2.3602 2.5725 3.6381 4.6081 5.7580 6.9089 8.0606 9.2123

1 1.9685 2.1053 2.8969 3.6684 4.2517 5.1125 5.9488 6.8446

2 1.7832 1.9146 2.4716 3.1046 3.8880 4.2517 4.9727 5.6883

3 1.6779 1.7762 2.3143 2.9343 3.4329 3.7439 4.3535 4.9727

4 1.6067 1.7094 2.2129 2.6831 3.1153 3.7439 3.9386 4.5045

5 1.5571 1.6385 2.0623 2.4851 2.8785 3.4590 4.0258 4.6189

6 1.5181 1.6038 2.0165 2.4649 2.9061 3.2258 3.7594 4.2918

7 1.4883 1.5571 1.9771 2.3321 2.7397 3.0321 3.5398 4.0437

8 1.4620 1.5387 1.8907 2.2183 2.6062 2.8785 3.3568 3.8388

9 1.4434 1.5056 1.8713 2.2346 2.4851 2.9824 3.1918 3.6536

10 1.4253 1.4762 1.8044 2.1450 2.5407 2.8604 3.0628 3.4990

0.99 0 2.4585 2.6645 3.7646 4.8515 6.0654 6.9089 8.0606 9.2123

1 2.0437 2.2183 3.0021 3.8551 4.5956 5.5157 5.9488 6.8446

2 1.8751 2.0121 2.6752 3.2960 3.8880 4.6664 5.4496 6.2344

3 1.7590 1.8907 2.4851 3.0836 3.6684 4.1169 4.8146 5.5157

4 1.6966 1.7867 2.3563 2.8249 3.3445 4.0258 4.3745 5.0000

5 1.6356 1.7289 2.2075 2.7480 3.1046 3.7286 4.0258 4.6189

6 1.5926 1.6872 2.1450 2.5840 3.0836 3.4855 4.0617 4.6425

7 1.5545 1.6356 2.0956 2.5478 2.9155 3.2960 3.8388 4.3956

8 1.5258 1.6095 2.0032 2.4319 2.7732 3.1260 3.6536 4.1736

9 1.5006 1.5870 1.9728 2.3321 2.7902 3.1807 3.4855 3.9730

10 1.4786 1.5571 1.9474 2.3261 2.6831 3.0525 3.3322 3.8066

150



P c

0

0.75/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 3.2182 3.2274 4.8412 6.4574 8.0678 9.6834 11.2956 12.9166

1 2.3747 2.6441 3.4329 4.5725 5.7241 6.8966 7.9872 9.1996

2 2.1654 2.4062 3.3080 3.7286 4.6664 5.6180 6.5488 7.4627

3 2.0530 2.1299 2.8969 3.8715 4.0437 4.8662 5.6529 6.4558

4 1.8713 1.9350 2.6137 3.4855 3.6101 4.3328 5.0556 5.7971

5 1.8335 1.9268 2.3998 3.2031 3.2960 3.9557 4.6189 5.2910

6 1.7224 1.8044 2.2346 2.9824 3.7286 3.6684 4.2717 4.8924

7 1.7094 1.8080 2.3563 2.7988 3.4990 3.4329 4.0080 4.5725

8 1.6297 1.7191 2.2292 2.6441 3.3080 3.2373 3.7750 4.3122

9 1.6268 1.7322 2.1249 2.5126 3.1476 3.7750 3.5817 4.0984

10 1.5652 1.6625 2.0346 2.3998 3.0021 3.6101 3.4072 3.9047

0.90 0 3.2182 3.5751 5.3639 6.4574 8.0678 9.6834 11.2956 12.9166

1 2.5621 2.8514 3.9557 5.2910 5.7241 6.8966 7.9872 9.1996

2 2.3026 2.5549 3.3080 4.4170 5.5157 5.6180 6.5488 7.4627

3 2.1552 2.2795 3.2031 3.8715 4.8403 5.7971 5.6529 6.4558

4 1.9728 2.1915 2.9061 3.4855 4.3535 5.2301 6.1087 7.0028

5 1.9186 2.0392 2.6831 3.2031 4.0080 4.8146 5.6180 6.4103

6 1.8751 2.0076 2.4988 3.3322 3.7286 4.4823 5.2301 5.9488

7 1.7797 1.8986 2.5549 3.1368 3.4990 4.1929 4.8924 5.6180

8 1.7590 1.8868 2.4254 2.9727 3.3080 3.9730 4.6425 5.2910

9 1.7422 1.8080 2.3143 2.8337 3.1476 3.7750 4.3956 5.0277

10 1.6779 1.8044 2.2183 2.7152 3.0021 3.6101 4.2123 4.8146

0.95 0 3.3886 3.7662 5.3639 7.1536 8.0678 9.6834 11.2956 12.9166

1 2.6911 2.9922 3.9557 5.2910 6.5963 7.9177 7.9872 9.1996

2 2.4062 2.5549 3.6101 4.4170 5.5157 6.6445 7.7160 8.8417

3 2.2401 2.3935 3.2031 3.8715 4.8403 5.7971 6.7935 7.7821

4 2.0576 2.1915 2.9061 3.8715 4.3535 5.2301 6.1087 7.0028

5 1.9900 2.1299 2.8877 3.5817 4.0080 4.8146 5.6180 6.4103

6 1.9391 2.0859 2.7071 3.3322 3.7286 4.4823 5.2301 5.9488

7 1.8447 1.9771 2.5549 3.1368 3.9216 4.1929 4.8924 5.6180

8 1.8152 1.9558 2.4254 2.9727 3.7286 3.9730 4.6425 5.2910

9 1.7937 1.8751 2.4649 2.8337 3.5398 3.7750 4.3956 5.0277

10 1.7289 1.8674 2.3685 2.9533 3.3944 4.0800 4.2123 4.8146

0.99 0 3.5062 3.8953 5.6472 7.1536 8.9405 10.7354 12.5250 14.3205

1 2.8785 3.1046 4.2717 5.6883 6.5963 7.9177 9.2937 10.5932

2 2.5621 2.7647 3.8388 4.8146 6.0277 6.6445 7.7160 8.8417

3 2.3747 2.4919 3.4200 4.2717 5.3533 5.7971 6.7935 7.7821

4 2.1968 2.3685 3.1153 3.8715 4.8403 5.8343 6.1087 7.0028

5 2.1101 2.2852 3.0628 3.8551 4.4603 5.3533 6.2775 7.1685

6 2.0483 2.2183 2.8694 3.6101 4.1736 5.0000 5.8343 6.6934

7 1.9944 2.1101 2.7152 3.4072 3.9216 4.7148 5.5157 6.2775

8 1.9516 2.0717 2.7152 3.2373 4.0437 4.4603 5.2002 5.9488

9 1.8790 1.9900 2.5988 3.0836 3.8551 4.2517 4.9727 5.6883

10 1.8522 1.9685 2.4988 3.1586 3.6982 4.0800 4.7642 5.4496

151



P c

0

0.90/t t

0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5

0.75 0 4.8003 5.3325 8.0038 10.6724 13.3333 16.0102 18.6741 21.3493

1 3.3944 3.7750 5.6883 7.5873 9.4877 11.3895 13.2100 15.1976

2 3.2723 3.6390 4.6189 6.1920 7.7160 9.2937 10.8460 12.3153

3 2.8785 3.1918 4.0080 5.3533 6.6934 8.0580 9.3897 10.7181

4 2.5913 2.8785 3.5817 4.7893 5.9880 7.1685 8.3542 9.5877

5 2.3810 2.6441 3.2723 4.3745 5.4496 6.5488 7.6511 8.7566

6 2.4783 2.4582 3.6982 4.0437 5.0556 6.0680 7.0572 8.1301

7 2.3321 2.3084 3.4722 3.7750 4.7148 5.6883 6.6445 7.5873

8 2.2129 2.4582 3.2723 3.5676 4.4603 5.3533 6.2344 7.1124

9 2.1053 2.3441 3.1153 3.3818 4.2319 5.0839 5.9102 6.7935

10 2.0165 2.2401 2.9727 3.9730 4.0258 4.8403 5.6529 6.4558

0.90 0 5.3169 5.3325 8.0038 10.6724 13.3333 16.0102 18.6741 21.3493

1 3.9216 4.3745 5.6883 7.5873 9.4877 11.3895 13.2100 15.1976

2 3.2723 3.6390 5.4825 6.1920 7.7160 9.2937 10.8460 12.3153

3 3.1696 3.1918 4.7893 6.4103 6.6934 8.0580 9.3897 10.7181

4 2.8785 3.2031 4.3328 5.7604 5.9880 7.1685 8.3542 9.5877

5 2.6596 2.9533 3.9730 5.2910 5.4496 6.5488 7.6511 8.7566

6 2.4783 2.7563 3.6982 4.9188 6.1501 6.0680 7.0572 8.1301

7 2.5336 2.5913 3.4722 4.6189 5.7971 5.6883 6.6445 7.5873

8 2.4062 2.4582 3.2723 4.3745 5.4825 5.3533 6.2344 7.1124

9 2.2910 2.5478 3.1153 4.1545 5.2002 6.2344 5.9102 6.7935

10 2.1968 2.4450 3.3693 3.9730 4.9727 5.9488 5.6529 6.4558

0.95 0 5.3169 5.9077 8.0038 10.6724 13.3333 16.0102 18.6741 21.3493

1 3.9216 4.3745 6.5488 8.7566 9.4877 11.3895 13.2100 15.1976

2 3.5817 3.9904 5.4825 7.2833 7.7160 9.2937 10.8460 12.3153

3 3.1696 3.5261 4.7893 6.4103 7.9872 8.0580 9.3897 10.7181

4 2.8785 3.2031 4.3328 5.7604 7.2254 8.6730 8.3542 9.5877

5 2.8694 2.9533 3.9730 5.2910 6.6445 7.9872 9.2937 10.5932

6 2.6831 2.9824 3.6982 4.9188 6.1501 7.4019 8.6730 9.9010

7 2.5336 2.8161 3.8880 4.6189 5.7971 6.9493 8.1301 9.2937

8 2.4062 2.6674 3.6832 4.3745 5.4825 6.5488 7.6511 8.7566

9 2.4450 2.5478 3.5125 4.1545 5.2002 6.2344 7.2833 8.3542

10 2.3502 2.4450 3.3693 3.9730 4.9727 5.9488 6.9493 7.9872

0.99 0 5.6019 5.9077 8.8621 11.8245 14.7863 17.7274 18.6741 21.3493

1 4.2319 4.7148 6.5488 8.7566 10.9769 13.2100 15.4560 17.5439

2 3.8066 3.9904 5.9880 7.2833 9.1075 10.9769 12.8370 14.7059

3 3.3944 3.7750 5.2910 6.4103 7.9872 9.5877 11.2486 12.8370

4 3.2605 3.4329 4.8146 5.7604 7.2254 8.6730 10.1215 11.5340

5 3.0321 3.1807 4.4385 5.9102 6.6445 7.9872 9.2937 10.5932

6 2.8425 3.1586 4.1356 5.5157 6.1501 7.4019 8.6730 9.9010

7 2.8249 2.9922 3.8880 5.2002 5.7971 6.9493 8.1301 9.2937

8 2.6911 2.8425 4.0080 4.9188 6.1501 6.5488 7.6511 8.7566

9 2.5767 2.7152 3.8226 4.6904 5.8720 6.2344 7.2833 8.3542

10 2.4783 2.7480 3.6684 4.4823 5.6180 5.9488 6.9493 7.9872

CHAPTER 4 ACCEPTANCE SAMPLING PLANS - …shodhganga.inflibnet.ac.in/bitstream/10603/8131/10/10_chapter 4.pdf · 103 CHAPTER – 4 ACCEPTANCE SAMPLING PLANS 4.1 Introduction Acceptance

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