SELECTION OF MIXED SAMPLING PLAN WITH CONDITIONAL REPETITIVE GROUP SAMPLING PLAN AS ATTRIBUTE PLAN INDEXED THROUGH MAPD AND IQL USING IRPD

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

212

SELECTION OF MIXED SAMPLING PLAN WITH

CONDITIONAL REPETITIVE GROUP SAMPLING PLAN

AS ATTRIBUTE PLAN INDEXED THROUGH MAPD

AND IQL USING IRPD

R. Sampath Kumar

1

, R. Vijaya Kumar

2

, R. Radhakrishnan

3

1

_{Assistant Professor, Department of Statistics, Government Arts College, Coimbatore – 641 018, Tamilnadu, India.}

2

_{Assistant Professor, Department of Statistics, SSM College of Arts and Science, Komarapalayam, Namakkal – 638 143,}

Tamil Nadu, India.

3

_{Associate Professor, Department of Statistics, PSG College of Arts and Science, Coimbatore – 641 014, Tamilnadu, India.}

Abstract - This paper presents the procedure for the construction and selection of mixed sampling plan (MSP) using Intervened Random effect Poisson Distribution (IRPD) as a baseline distribution. Having the Conditional Repetitive Group Sampling plan as attribute plan, the plans are constructed through indifference quality level (IQL) and maximum allowable percent defective (MAPD). Tables are constructed for easy selection of the plan

Keywords and phrases

-

indifference quality level, intervention, mixed sampling plan, maximum allowable percent defective, operating characteristic, poisson, intervened random effect poisson distribution.

AMS (2000) Subject Classification Number: Primary: 62P30 Secondary: 62D05

I.

I

NTRODUCTION

Mixed sampling plan is a two stage sampling

procedure involving variables inspection in the first stage

and attributes inspection in the second stage. Use of

variables on the first sample with attributes on the second

sample combines the economy of variables for quick

acceptance on the first sample with broad nonparametric

protection of attributes sampling when a questionable lot

requires second sample. Mixed sampling plans are of

two types, which are independent and dependent plans.

Independent mixed plans do not incorporate first sample

results in the assessment of the second sample.

Dependent mixed plans combine the results of the first

and second samples in making a decision if a second

sample is necessary.

Schilling (1967) proposed a method for determining

the operating characteristics of mixed variables –

attributes sampling plans, single sided specification and

standard

deviation

known

using

the

normal

approximation. Sherman (1965) has introduced a new

acceptance sampling called Repetitive Group Sampling

(RGS).

Kuralmani (1992) made some contribution on

Repetitive Group and certain related conditional

sampling plans. Radhakrishnan (2002) constructed tables

for conditional RGS using MAPD and MAAOQ.

Devaarul (2003) has studied the mixed sampling plans

and reliability based sampling plans. Sampath Kumar

(2007) has constructed mixed variables – attributes

sampling plans indexed through various parameters.

Sampath Kumar et.al (2012)

have made contributions to

mixed sampling plans for independent case

In the product control, the defective units are either

rebuilt or replaced by new units during the sampling

period. Quality engineers are always interested in

improving the quality level of product to enhance the

satisfaction of the customers and hence, they keep

making changes in the production process. These actions

trigger a change in the expected incidence of defective

items in the remaining observational period. Any action

for reducing the number of defectives during the

sampling period is called an intervention and such

intervention parameter ranges from 0 to 1.

In Intervened Random effect Poisson Distribution

(IRPD), Poisson parameter is modified in two ways:

one method is multiplying an intervention parameter ρ (a

constant) and secondly, multiplying an unobserved

random effect which follows Gamma probability

distribution. The IRPD can be very useful to the quality

and reliability engineers, who always make changes in

the production system in the observational period of

quality checking to ensure reliability of the system,

because, the failure rate of the components may vary in

different time intervals. The other areas of application of

IRPD are queuing, demographic studies and process

control and so on.

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Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

213

Radhakrishnan and Sekkizhar (2007a, b, and c)

introduced

Intervened

Random

effect

Poisson

Distribution in the place of Poisson distribution for the

construction of attribute sampling plans.

In this paper, using the operating procedure of mixed

sampling plan (independent case) with conditional

repetitive group sampling plan as attribute plan, tables

are constructed using IRPD as a baseline distribution.

The tables are constructed for mixed sampling plan

(MSP) indexed through i) IQL ii) MAPD. The plan

indexed through MAPD is compared with the plan

indexed through IQL.

II.

C

ONDITIONS

F

OR

A

PPLICATIONS

O

F

IRPD

-

M

IXED

S

AMPLING

P

LAN



Production process is modified during the sampling

inspection by an intervention.



Lots are submitted substantially in the order of their

production.



Inspection is by variable in the first stage and

attribute in the second stage with quality defined as

the fraction defective.



Lot quality variation exists.

III.

G

LOSSARY

O

F

S

YMBOLS

The symbols used in this paper are as follows:

p

: submitted quality of lot or process

P p

a

( )

: probability of acceptance for given quality

„

p

‟

p

0

: submitted quality such that P

a

(

p

0

) = 0.50

(also called IQL)

p

*

: maximum allowable percent defective

(MAPD)

n : sample size for each lot

n

1

: sample size for variable sampling plan

n

2

: sample size for attribute sampling plan

c

1

: first attributes acceptance number

c

2

: second attributes acceptance number

d

: number of defectives in the sample



j

: probability of acceptance for the lot quality

„

p

_j

‟

j





: probability of acceptance under variables plan

for percent defective „

p

_j

‟(with sample size

n

1

)

j





: probability of acceptance under attributes plan

for percent defective „

p

_j

‟(with sample size

n

2

)

z (j) : „z‟ value for the j

th

_{ordered observation}

k : variable factor such that a lot is accepted if

X



  

L k



IV.

O

PERATING

P

ROCEDURE

O

F

M

IXED

S

AMPLING

P

LAN

H

AVING

C

ONDITIONAL

R

EPETITIVE

G

ROUP

S

AMPLING

P

LAN

A

S

A

TTRIBUTE

P

LAN

Schilling (1967) has given the following procedure for

the independent mixed sampling plan with lower

specification limit (L) and standard deviation (



).



Determine the parameters of the mixed sampling

plan n

1

, n

2

, k, c

1

and c

2



Take a random sample of size n

1

from the lot



If a sample average

X



  

L k



, accept the

lot



If a sample average

X

<

  

L k



, go to step

(i)



Take another sample of size n

2



Count the number of defectives „d‟ in the sample



If d ≤ c

1

, accept the lot



If d > c

2

, reject the lot



If c

1

< d ≤ c

2

, utilize the information of the next

preceding „i‟ successive lots (i.e.,) the current lot

is accepted if the preceding „i‟ successive lots

result shows d ≤ c

1

in the sample, otherwise reject

the lot.

V.

C

ONSTRUCTION

O

F

M

IXED

S

AMPLING

P

LAN

H

AVING

C

ONDITIONAL

R

EPETITIVE

G

ROUP

S

AMPLING

P

LAN

A

S

A

TTRIBUTE

P

LAN

U

SING

IRPD.

Schilling (1967) has given the OC function of mixed

sampling plan as

( )

L p

= Pn

1

(

X



A) + Pn

1

(

X

>A)



₂



0

;

c

j

p j n





(1)

The above expression is given as

j



=

 

_j

+ (1-

 

_j

)



_j



(2)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

214

By symmetry, a parallel discussion can be made for

lower specification limits. The procedure for the

construction of mixed variables – attributes sampling

plans is provided by Schilling (1967) for a given „n

1

‟, „k‟

and a point „

p

_j

‟ on the OC curve is given below.



Assume that the mixed sampling plans are

independent



Split

the

probability

of

acceptance

(

j



)

determining the probability of acceptance that will

be assigned to the first stage. Let it be



_j





Decide the sample size n

1

(for variable sampling

plan) to be used



Calculate the acceptance limit for the variable

sampling plan as

1

[ (

_j

) { (

_j

) /

}]

L k



L

z p

z





n



  

 



,

where L is the lower specification limit and z (t) is

the standard normal variate corresponding to „t‟ such that

t =

( )

1

2

z t





_

_











_u2_{/ 2}

e



du



Determine the sample average

X

. If a sample

average

X

<

  

L k



, take a second stage

sample size „n

2

‟ using attribute sampling plan.



Split the probability of acceptance

j



as

j





and



_j



,

such

that

j



=



_j



+

(1-

j





)



_j



and fix the value of



_j



.



Now determine



_j



, the probability of

acceptance assigned to the attributes plan

associated with the second stage sample as

j





=(

j



-



_j



)/(1-



_j



)



Determine the appropriate second stage sample

size „n

2

‟ from

P p

a

( )

=



j



for

p

=

p

j

Using the above procedure, tables can be constructed

to facilitate easy selection of mixed sampling plan with

conditional repetitive group sampling plan as attribute

plan using IRPD as a baseline distribution indexed

through MAPD and AQL.

Radhakrishnan and Sekkizhar (2007a, b, and c)

suggested the probability mass function of the IRPD as

P p

a

( )

=

1 1 3

1

P

P P

-

₍₃₎

where

1

P

=











 



1 0 0

1 !

1

!

!

1 !

1

l

c x x

x l

l

e

l x

l

 













  



_

_

_{ }





_

_



























,



=

, α=1,

2

P

=











 



2 0 0

1 !

1

1

!

!

1 !

1

l

c x x

x l

l

e

l x

l

 













  



_

_

_{ }







_

_



























3

P

=

1

-

P

1

-

P

2

Using the above procedure, tables can be constructed

to facilitate easy selection of MSP using IRPD as a

baseline distribution. The tables furnished in this paper

are for the case when α=1.

VI.

C

ONSTRUCTION

O

F

M

IXED

S

AMPLING

P

LANS

I

NDEXED

T

HROUGH

MAPD

A

ND

MAAOQ

MAPD, introduced by Mayer (1967) and studied by

Soundararajan (1975) is the quality level corresponding

to the inflection point of the OC curve. The degree of

sharpness of inspection about this quality level „

p

_*

‟ is

measured by „

p

_t

‟, the point at which the tangent to the

OC curve at the inflection point cuts the proportion

defective axis.

For designing, Soundararajan (1975) proposed a

selection procedure for single sampling plan indexed

with MAPD and K=

p

t

p

*

.

Using the probability mass function of the IRPD,

given in expression (3), the inflection point (

p

_*

) is

obtained by using

2 2

( )

0

a

d P p

dp

=

and

3 3

( )

a

d P p

dp

≠0. The

n

2

MAPD values are calculated for different values of c

1

,

c

2

and ρ=0.5 for



*

 

0.24 using c++ program and

presented in Table 1.

The

MAAOQ

(Maximum

Allowable

Average

Outgoing Quality) of a sampling plan is defined as the

Average Outgoing Quality (AOQ) at the MAPD.

By definition AOQ =

p

P p

_a

( )

and MAAOQ =

p

_*

*

(

)

a

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ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

215

The values of MAPD and MAAOQ are calculated for

different values of c

1

, c

2

and ρ for



*

 

0.24 and the

ratio

R

MAAOQ

MAPD



is presented in Table 1.

Selection of the plan

For the given values of ρ,



_*



, MAPD and MAAOQ,

the ratio

R

MAAOQ

MAPD



is found and the nearest value

of R is located in Table 1. The corresponding c

1

, c

2

and

n

2

MAPD values are noted and the value of n

2

is obtained

using

2 2

n MAPD

n

MAPD



.

Example 1:

Given ρ=0.5,



_*

 

0.24

,

MAPD=0.0470

and

MAAOQ=0.0102,

the

ratio

0.0102

0.2170

0.0470

MAAOQ

R

MAPD







is computed. In

Table 1 the nearest R value is 0.2173 which is

corresponding to c

1

=5, c

2

=8. The value of

n

2

MAPD=8.8761 is found and hence the value of n

2

is

determined as

n

₂

n MAPD

2

MAPD



8.8761

189

0.0470





.

Thus n

2

=189, c

1

=5 and c

2

=8 are the parameters of

mixed sampling plan having conditional RGS as attribute

plan using IRPD as a baseline distribution for the given

values of ρ=0.5, MAPD=0.0470 and MAAOQ =0.0102.

VII.

C

ONSTRUCTION

O

F

M

IXED

S

AMPLING

P

LANS

I

NDEXED

T

HROUGH

IQL

The procedure given in section 5 is used for

constructing the mixed sampling plan indexed through

IQL (

p

₀

). By assuming the probability of acceptance of

the lot be



₀

=0.50 and



₀

 

0.24

, the

n p

_{2 0}

values

are calculated for different values of c

1

, c

2

and „ρ‟ using

c++ program and is presented in Table 2.

Selection of the plan

Table 2 is used to construct the plans when

p

₀

, ρ, c

1

,

and c

2

are given. For any given values of

p

0

, ρ, c

1

, and

c

2

one can determine n

2

value using

2 0 2

0

n p

n

p



.

Example 2:

Given ρ=0.5,

p

₀

=0.0346, c

1

=5

,

c

2

=8,



0

 

0.24

.

Using

Table

2,

find

₂ 2 0

0

7.6545

221

0.0346

n p

n

p







.

For

a

fixed



₀

 

0.24

, the mixed sampling plan with

conditional RGS as attribute plan is n

2

=221, ρ=0.5, c

1

=5

and c

2

=8.

VIII.

C

OMPARISON

O

F

M

IXED

S

AMPLING

P

LAN

I

NDEXED

T

HROUGH

MAPD

A

ND

IQL

In this section MSP indexed through MAPD is

compared with MSP indexed through IQL by fixing the

parameters c

1

, c

2

and



j



.

For the specified values of ρ, MAPD and MAAOQ

with the assumption for



_*

 

0.24

one can find the

values of c

1

and c

2

indexed through MAPD. By fixing

the values of c

1

and c

2

find the value of

p

0

by

equating

P p

_a

( )

=



₀

=0.50. For



₀

 

0.24

,

c

1

and c

2

one can find the values of n

2

using

2 2 0 0

n p

n

p



from

Table 2. For different combinations of ρ, MAPD and

MAAOQ the values of c

1

, c

2

, and n

2

(indexed through

MAPD) and c

1

, c

2

, and n

2

(indexed through IQL) are

calculated and presented in Table 3.

Construction of OC curve

The OC curves for the plan ρ=0.5, n

2

=189, c

1

=5, c

2

=8

(indexed through MAPD) and n

2

=221, c

1

=5, c

2

=8

(indexed through IQL) based on the different values of

„n

2

p

0

‟ and

P p

a

( )

are presented in Figure 1.

Table III Comparison of the Plans

Given values

Indexed Through MAPD

Indexed Through IQL

MAPD MAAOQ ρ c1 c2 n2 c1 c2 n2

0.0240 0.0037 0.8 7 11 996 7 11 1183 0.0210 0.0057 0.7 2 6 219 2 6 268 0.0140 0.0033 0.6 3 7 443 3 7 532

[image:4.595.312.554.612.704.2]

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

216

*

OC curves are drawn

Fig1: OC curves for the plans (ρ=0.5, c1=5, c2=8, n2=189) and

(ρ=0.5, c1=5, c2=8, n2=221)

IX.

C

ONCLUSION

In this paper the construction of mixed sampling plan

with conditional repetitive group sampling plan as

attribute plan indexed through the parameters MAPD and

IQL are presented by taking IRPD as a baseline

distribution. Further the plan indexed through MAPD is

compared with the plan indexed through IQL. It is

concluded from the study that the second stage sample

size required for conditional repetitive group sampling

plan indexed through MAPD is less than that of second

stage sample size of the conditional repetitive group

sampling plan indexed through IQL. If the floor

engineers know the levels of MAPD or IQL, they can

have their sampling plans on the floor itself by referring

to the tables. This provides the flexibility to the floor

engineers in deciding their sampling plans.

Various plans can also be constructed to make the

system user friendly by changing the first stage

probabilities (



_*



,



₀



) and can also be compared for

their efficiency.

REFERENCES

[1 ] Devaarul, S. Certain Studies Relating to Mixed Sampling Plans and Reliability Based Sampling Plans, Ph.D., Dissertation, Bharathiar University, Coimbatore, Tamil Nadu, India, 2003. [2 ] Kuralmani, V. Studies on Designing Minimum Inspection

Attribute Acceptance Sampling Plans, Ph.D., Thesis, Department of Statistics, Bharathiar University, Coimbatore, Tamilnadu, India, 1992.

[3 ] Mayer, P.L. “A note on sum of Poisson probabilities and an application,” Annals of Institute of Statistical Mathematics, Vol.19, pp.537-542, 1967.

[4 ] Radhakrishnan, R. Contribution to the Study on Selection of Certain Acceptance Sampling Plans, Ph.D., Dissertation, Bharathiar University, Tamilnadu, India, 2002.

[5 ] Radhakrishnan, R. and Sekkizhar, J. “Construction of sampling plans using intervened random effect Poisson distribution,” The International Journal of Statistics and Management Systems, Vol.2, 1-2, pp 88-97, 2007a.

[6 ] Radhakrishnan, R. and Sekkizhar, J. Construction of conditional double sampling plans using intervened random effect Poisson distribution, Proceedings volume of SJYSDNS-2005, Acharya Nagarjuna University, Guntur. pp.57-61. 2007b.

[7 ] Radhakrishnan, R. and Sekkizhar, J. “Application of intervened random effect Poisson distribution in process control plans,” International Journal of Statistics and Systems. Vol. 2. No.1. pp.29-39, 2007c.

[8 ] Sampath Kumar, R. Construction and Selection of Mixed Variables – Attributes Sampling Plans, Ph.D., Dissertation, Bharathiar University, Coimbatore, Tamil Nadu, India, 2007. [9 ] Sampath Kumar, R., Indra, M., and Radhakrishnan, R. “Selection

of mixed sampling plans with QSS-1(n;cN,cT) plan as attribute

plan indexed through MAPD and AQL,” Indian Journal of Science and Technology, Vol.5, No.2, Feb 2012, pp. 2096- 2099, 2012.

[10 ]Sampath Kumar, R., Kiruthika, R., and Radhakrishnan, R. “Construction of mixed sampling plans indexed through MAPD and LQL with Double sampling plan as attribute plan using Weighted Poisson distribution,” International Journal of Electronics and Communication Engineering, Vol.5, No.3, pp. 39-47, 2012.

[11 ]Sampath Kumar, R., Sumithra, S., and Radhakrishnan, R. “Selection of mixed sampling plan with chsp-1 plan as attribute plan indexed through MAPD and MAAOQ,” International Journal of Applied Computational Science and Mathematics, Vol.1, No.1, pp. 37- 44, 2012.Sampath Kumar, R., Vijayakumar, R. and Radhakrishnan, R. “Selection of Mixed Sampling plan with double Sampling Plan as Attribute Plan Indexed through MAPD and IQL Using intervened random effect Poisson distribution,” International Journal of Industries Engineering and Technology. Vol.4, No.1, Jan- June 2012,pp.29-36, 2012. [12 ]Sampath Kumar, R., Vijayakumar, R. and Radhakrishnan, R.

“Selection of Mixed Sampling plan with Conditional Double Sampling Plan as Attribute Plan Indexed through MAPD and LQL using IRPD,” International Journal of Computational Engineering Research. Vol.2, No.2, Mar-Apr 2012, pp.306-313, 2012.

[13 ]Schilling, E.G. A General Method for Determining the Operating Charateristics of Mixed Variables –Attribute Sampling Plans Single Side Specifications, S.D. known, Ph.D., Dissertation – Rutgers – The State University, New Brunswick, New Jersy, 1967.

[14 ]Shanmugam, R. “An intervened Poisson distribution and its medical applications,” Biometrics, 41, 1025-1029, 1985. [15 ]Sherman, R.E. “Design and evaluation of a repetitive group

sampling plan,” Technometrics, Vol.7, No.1, pp.11-21, 1965. [16 ]Soundararajan, V. “Maximum allowable percent defective

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Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

[image:6.595.150.448.160.750.2]

217

Table I

n2MAPD and n2MAAOQ values for a specified values of c1, c2 and different values of ρ for mixed sampling plan when 0.24

ρ c1 c2



*



*



_n

2MAPD n2MAAOQ

MAAOQ

R

MAPD



International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

218

ρ c1 c2



*



*



n2MAPD n2MAAOQ

MAAOQ

R

MAPD



International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

219

ρ c1 c2



*



*



_n

2MAPD n2MAAOQ

MAAOQ

R

MAPD



International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

220

ρ c1 c2



*

*





_n

2MAPD n2MAAOQ

MAAOQ

R

MAPD



0.5 1 3 0.4188 0.2352 3.1599 0.7432 0.2352 4 0.3973 0.2069 3.3911 0.7016 0.2069 5 0.3879 0.1946 3.5085 0.6827 0.1946 2 2 0.5370 0.3907 3.3485 1.3082 0.3907 3 0.4600 0.2894 4.0902 1.1837 0.2894 4 0.4213 0.2385 4.5308 1.0805 0.2385 5 0.3984 0.2084 4.8273 1.0060 0.2084 6 0.3849 0.1906 5.0225 0.9572 0.1906 3 3 0.5169 0.3643 4.6995 1.7120 0.3643 4 0.4574 0.2860 5.4063 1.5462 0.2860 5 0.4234 0.2413 5.8646 1.4151 0.2413 6 0.4012 0.2121 6.1967 1.3143 0.2121 7 0.3867 0.1930 6.4348 1.2419 0.1930 4 4 0.5035 0.3467 6.0397 2.0939 0.3467 5 0.4546 0.2823 6.7222 1.8976 0.2823 6 0.4244 0.2426 7.1891 1.7440 0.2426 7 0.4035 0.2151 7.5420 1.6222 0.2151 8 0.3889 0.1959 7.8086 1.5297 0.1959 5 5 0.4932 0.3331 7.3802 2.4583 0.3331 6 0.4517 0.2785 8.0408 2.2393 0.2785 7 0.4246 0.2428 8.5124 2.0668 0.2428 8 0.4052 0.2173 8.8761 1.9287 0.2173*

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (

ISSN 2250-2459

, Volume 2, Issue 10, October 2012)

[image:10.595.67.537.156.617.2]

221

Table II

n2 IQL values for a specified values of c1, c2 and ρ of mixed sampling plan when =0.50 and =0.24

ρ Values

c1 c2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 1 1.1944 1.2058 1.2203 1.2363 1.2531 1.2700 1.2868 1.3034 1.3196

2 1.2597 1.2720 1.2876 1.3049 1.3228 1.3410 1.3590 1.3767 1.3941

3 1.2880 1.3013 1.3180 1.3364 1.3555 1.3748 1.3940 1.4128 1.4312

4 1.2976 1.3116 1.3291 1.3485 1.3686 1.3889 1.4090 1.4287 1.4481

5 1.3002 1.3145 1.3326 1.3526 1.3733 1.3942 1.4149 1.4354 1.4553

1 1 2.2632 2.2890 2.3221 2.3587 2.3969 2.4356 2.4741 2.5119 2.5490

2 2.3869 2.4144 2.4494 2.4883 2.5288 2.5698 2.6106 2.6508 2.6900

3 2.4770 2.5057 2.5424 2.5829 2.6251 2.6678 2.7103 2.7520 2.7929

4 2.5324 2.5626 2.6009 2.6432 2.6872 2.7315 2.7756 2.8190 2.8614

5 2.5613 2.5932 2.6334 2.6776 2.7234 2.7696 2.8155 2.8606 2.9046

2 2 2.4069 3.4510 3.5073 3.5696 3.6345 3.7000 3.7650 3.8290 3.8915

3 3.5320 3.5779 3.6365 3.7014 3.7689 3.8370 3.9047 3.9713 4.0363

4 3.6328 3.6802 3.7406 3.8074 3.8768 3.9469 4.0166 4.0850 4.1519

5 3.7041 3.7532 3.8156 3.8843 3.9557 4.0277 4.0992 4.1694 4.2381

6 3.7488 3.8001 3.8647 3.9357 4.0093 4.0835 4.1570 4.2292 4.2997

3 3 4.5290 4.5933 4.6753 4.7657 4.8595 4.9541 5.0478 5.1399 5.2296

4 4.6544 4.7207 4.8053 4.8984 4.9951 5.0926 5.1892 5.2840 5.3766

5 4.7611 4.8290 4.9155 5.0107 5.1096 5.2092 5.3079 5.4048 5.4994

6 4.8426 4.9124 5.0009 5.0982 5.1991 5.3007 5.4015 5.5003 5.5968

7 4.8992 4.9713 5.0623 5.1620 5.2653 5.3691 5.4720 5.5730 5.6715

4 4 5.6377 5.7239 5.8335 5.9538 6.0784 6.2037 6.3277 6.4493 6.5677

5 5.7631 5.8515 5.9638 6.0871 6.2148 6.3431 6.4702 6.5947 6.7160

6 5.8734 5.9635 6.0779 6.2034 6.3333 6.4640 6.5933 6.7201 6.8436

7 5.9621 6.0541 6.1705 6.2981 6.4302 6.5630 6.6944 6.8232 6.9487

8 6.0277 6.1221 6.2411 6.3712 6.5056 6.6407 6.7744 6.9054 7.0995

5 5 6.7371 6.8467 6.9854 7.1273 7.2940 7.4514 7.6069 7.7590 7.9071

6 6.8626 6.9743 7.1160 7.2709 7.4309 7.5915 7.7502 7.9054 8.0564

7 6.9752 7.0889 7.2827 7.3900 7.5524 7.7154 7.8764 8.0341 8.1874

8 7.0690 7.1846 7.3305 7.4899 7.6545* 7.8197 7.9830 8.1427 8.2982

9 7.1417 7.2598 7.4081 7.5700 7.7370 7.9046 8.0701 8.2321 8.3898

6 6 7.8297 7.9639 8.1332 8.3178 8.5080 8.6985 8.8863 9.0700 9.2484

7 7.9549 8.0916 8.2639 8.4517 8.6453 8.8391 9.0303 9.2171 9.3986

8 8.0694 8.2079 8.3825 8.5728 8.7688 8.9652 9.1589 9.3482 9.5322

9 8.1671 8.3076 8.4842 8.6767 8.8750 9.0736 9.2696 9.4611 9.6473

10 8.2454 8.3883 8.5674 8.7624 8.9631 9.16400 9.3623 9.5562 9.7447

7 7 8.9170 9.0771 9.2780 9.4965 9.7211 9.9456 10.1666 10.3824 10.5918

8 9.0420 9.2047 9.4088 9.6307 9.8587 10.0866 10.3111 10.5302 10.7428

9 9.1578 9.3224 9.5288 9.7532 9.9838 10.2144 10.4415 10.6632 10.8783

10 9.2586 9.4250 9.6336 9.8603 10.0932 10.3261 10.5555 10.7795 10.9970

11 9.3415 9.5103 9.7213 9.9504 10.1857 10.4210 10.6528 10.8792 11.0990

8 8 10.0001 10.181 10.4207 10.6741 10.9338 11.1931 11.448 11.6965 11.9375

9 10.1250 10.3146 10.5515 10.8084 11.0717 11.3345 11.5929 11.8448 12.0890

10 10.2417 10.4333 10.6727 10.9321 11.1982 11.4637 11.7248 11.9794 12.2262

11 10.3449 10.5384 10.7799 11.0416 11.3100 11.5779 11.8415 12.0984 12.3476