ON THE COMPARISON OF SOME METHODS OF ALLOCATION IN ... · ON THE COMPARISON OF SOME METHODS OF...

International Journal of Mathematics and Statistics Studies

Vol.6, No.3, pp.29-42, September 2018

___Published by European Centre for Research Training and Development UK (www.eajournals.org)

29 Print ISSN: 2053-2229, Online ISSN: ISSN 2053-2210

ON THE COMPARISON OF SOME METHODS OF ALLOCATION IN

STRATIFIED RANDOM SAMPLING FOR SKEWED POPULATION

Lawal M1., Salami A2., Obisesan K.O2., Yusuff K.M3. and Owolabi A.A1

1Mathematical & Computer Science Department, Fountain University Osogbo Nigeria 2Department of Statistics University of Ibadan, Nigeria

3Department of Statistics Federal University of Agriculture Abeokuta, Nigeria

ABSTRACT: A study to evaluate and compare some methods of allocation in stratified

random sampling suitable for the estimation of population total of a skewed population was

carried out in this paper. We looked at three methods of allocation in the above scheme

namely; Optimum allocation, N-proportional allocation, and variable (X) proportional

allocation methods. We investigate the condition under which one method of allocation is

better than the other using three sets of real life data on staff and student enrolment,

collected from the record of the Teaching Service Commission (TESCOM) Oyo State Nigeria.

The third set of data is on Income and expenditure of Industrial and General Insurance (IGI)

Plc. We found out that optimum allocation is the least and the best despite variation

observed in the sizes of nh within the strata.

KEYWORDS: Stratification, Allocation, Skewness, Estimation, Sampling

INTRODUCTION

Sampling is the process for selecting a part or a fraction of a population and observing the

selected part with respect to some properties of interest and then drawing some conclusions

about the population. Sampling is carried out every day by making decision concerning the

characteristics of a large number of items based on analysis of a small number selected from

them. Sampling plays a vital role in research design involving human population and

commands increasing attention from social scientists, chemist, engineers, accountants,

biologist and medical practitioners (Kish 1965) and (Hunt & Tyrell 2004).

Stratified random sampling is a technique which attempts to restrict the possible samples to

those which are ``less extreme'' by ensuring that all parts of the population are represented in

the sample in order to increase the efficiency.

Rarely, is a survey carried out without stratification, where a population is divided into a set

nearly homogenous subset called strata and independent samples are drawn from each

stratum. In stratified sampling, the values of sample size nh in the respective strata are chosen

by the sampler, which may be carried out with either the aim of minimizing the variance for a

specified cost or to minimize the cost for a specified value of variance. Adebola et al (2014).

A different approach of minimization of general variance and maximization of sample sizes

using stratification of skewed population has been suggested by different authors among

whom are Dalenius and Hodges (1959), Glasser (1962), Hidiroglou (1986) and Morgan et al

(2006).

Morgan et al (2006) said if stratification boundaries are taken in geometric progression, the

coefficient of variation of strata using skewed distribution are equal however, this method

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does not satisfy Dalenius and Hodges (1959) conditions of variance minimization of the

sample mean but compares favourably with the commonly used cumulative root frequency

approximation of Dalenius and Hodges in terms of precision.

In the practical applications of Neyman’s method of allocation, one is faced with certain

limitations especially when it is concerned with problem of estimation of several population

characteristics. It is often found that these characteristics make conflicting demands on the

design and therefore, allocation of sample to the different strata based on one character using

Neyman’s allocation may vary, and as well lead to loss in precision of estimates of other

characters as compared to proportional allocation, Sukhatme (1966).

It was discovered, by Cochran (1977) that the precision of the estimate is increased if

Neyman’s approach is adopted in preference to methods of proportional sampling, be it N-

proportional or variable (X) proportional sampling.

If the allocation of the sample among the strata is far from optimum, proportional methods

and stratified sampling may have a higher variance, Sukhatme (1935). However,

proportional allocation will be very useful if the strata averages differ from one another, Raj

(1968).

When skewed populations are used in the estimation of N-proportional allocation, a

considerable variance results due to the fact that the stratum containing the very large units

will be found to be much greater than other strata. It is believed that a more considerable

estimate of population totals will be yielded if X-proportional allocation is used in the

estimation of skewed population as its estimate makes use of stratum averages which is

expected to be many times greater than the general average. In estimating population totals

and percentages, Hansen, Hurwitz and Madow (1953) found out that there are cases where a

kind of combined procedure might be useful if optimum is used in estimating population

totals.

It is our aim in this paper to examine and compare some of these methods of allocation in

stratified random sampling to know the one suitable for estimating population totals of a

skewed population

METHODOLOGY

The analysis is based on stratification of skewed population where three sets of real life data

were used. The first set of data was sourced from the record of Oyo State Teaching Service

Commission on number of Staff in each school of the thirty three (33) local governments in

Oyo State for 2008/2009 session, which represent our auxiliary variable (X), as we allow the

variable of interest (Y), to be staff enrolment for 2009/2010 session.

Also from the record of Oyo State Teaching Service Commission is the second set of data on

students’ enrolment for the year 2008/2009 session (X), and 2009/2010 session (Y). The last

data set is on Income (X) and Expenditure (Y) of Industrial and General Insurance Plc.

Definition Of Population Total

The estimate of population in stratified random sampling is;






stY^

= N

sty

=

L

h 1

hyNh

while the variance is

^

YV

=

L

h 1

22

yhhN

hh Nn

11 (Cochran 1977)

Allocation Of Sample To Strata

Stratified random sampling does not specify a particular size of sample to be attached to a

given stratum. The sample can be selected in order to have same size in each stratum or it can

be allocated in some other ways. As long as we select at least one element in a stratum the

specification of a stratified random sampling is satisfied and with two units in a stratum, we

can estimate both its mean and standard error.

N-Proportional Allocation

In this allocation method, sample units are selected from within strata in proportion to their

strata sizes. Its number of sample in hth stratum can be estimated by

nh = N

nNh

Its corresponding variance of total estimate without replacement is

)

^

propNYV = n

nN

L

h 1

2

yhNh (Raj 1968)

Optimum Allocation

For optimum allocation, the number of samples to be selected in the hth stratum (Neyman

1934) is given by

nh =

L

h

yh

yh

Nh

nNh

1

Its corresponding variance of total estimate for without replacement therefore,

optYV

^

= n

12

1

L

h

ynNh -

L

h 1

Nh2

yh (Raj 1968)






X-Proportional Allocation

The sample size per hth stratum, nh can be found as a proportion of Xh rather than to Nh i.e.

nh = X

nX h such that;

X =

L

h

hX1

The variance of population total for without replacement is

propxYV

^

= n

nN

L

h 1

XX

Nh

h

yh

/

2 Raj (1968)

COMPUTATIONAL FRAMEWORK

Statistics On Teachers Population Data

__

X =

6

1h N

X h =

6

1h

N

XN hh

__

= 14.01447777

__

Y =

6

1h

N

Yh =

6

1h

N

YN hh

__

= 967

13751 = 14.22026887

x =

967

11

1

__

N

YYi

= 7.432

Skewness (X) = 3(Mean – Median) = 1.682

x

Y =

967

11

1

__

N

YYi

= 7.379

Skewness (Y) = 3(Mean – Median) = 1.6263

Y

i=1

i=1






Sample Size Allocation to Strata for N-Proportional Allocation

nh h = 1, 2, 3, 4, 5, 6

h 1 2 3 4 5 6

nh 90.0692 127.9658 23.3609 7.0082 1.8169 0.7786

90 + 128 + 23 + 7 + 2 + 1 = 251

i.e.

6

1h

nh = n

^

stY =

6

1h

Nh hy__

= 13587.99138

)

^

propNYV = n

nN

6

1h

2

yhhN

= 967 – 251 X 6534.513685

251

= 18640. 28605

S.E

)

^

propNY = 28605.18640

= 136.5294329

C.V = ^

^

2

st

propN

Y

Y

x 100

= 1.004780097% 1%

Sample Size Allocation to Strata for X-Proportional Allocation

N = 967, and n = 251

nh = X

nX h , X = h

h

h XN__6

1

h 1 2 3 4 5 6

nh 48.0441 136.8164 40.0059 17.2062 5.8341 3.0930

S. E






Thus,

6

1h

hn = n = 251

stY^

=

6

1h

Nh hy__

= 13734.27693

)

^

propxYV = n

nN

6

1h____

2

/ XX

N

h

yhh

= 251

251967 x 7647.005514

= 21813.76872

S.E

)

^

propxY = 76872.21813

= 147.69485

C.V =

st

propX

Y

Y

^

^

.

X 100

= 27693.13734

69485.147

x 100

= 1.0757405% 1%

Sample Size Allocation to Strata for Optimum Allocation

N = 967, n = 251

nh =

6

1h

yhh

yhh

N

nN

, and yh

6

1h

1

__

h

hhi

n

yy

H 1 2 3 4 5 6

nh 70.3420 142.8698 25.7414 8.2094 2.4732 1.5638

Hence,

6

1h

hn = n

^

stY =

6

1h

Nh hy__

= 13657.97692

S.E

^






optYV

^

= n

1

26

1

h

yhhN −

6

1

2

h

yhhN

= 17494.23428

optY

^

2. = 23428.17494

= 132.2657714

C.V = st

opt

Y

Y

^

^

2.

x 100

= 0.968414078% 1%

Statistics On Student Population Data

__

X =

4

1h

h

N

X =

4

1

__

h

hh

N

XN

= 423.9721

__

Y =

4

1h

h

N

Y =

4

1

__

h

hh

N

YN

= 435.7394

X =

967

1i1

__

N

XX i

= 163.7611

Skewness (X) = 3 (Mean – Median) = 1.4892

Y =

967

1i1

__

N

YYi

= 168.3974

Skewness (Y) = 3(Mean – Median) = 1.3626

Y


nh = N

nNh , h = 1, 2, 3, 4

S.E

S.E






h 1 2 3 4

nh 44.6452 179.8790 19.9866 6.4891

Thus,

4

1h

hn = n

^ stY =

4

1h

Nh hy__

= 415665.5944

prop)N

^YV =

n

nN

4

1h

2

yhhN

= 967 – 251 X 5107633.143

251

= 14569981.39

S.E

)

^

propNY = 914569981.3

= 3817.0645

C.V = ^

^

2

st

propN

Y

Y

x 100

= 0.918301767% 1%


N = 967, and n = 251

nh = X

nX h , X = h

h

h XN__4

1

h 1 2 3 4

nh 23.9416 178.0768 33.3858 15.5978

Thus,

4

1h

hn = n = 251

stY^

=

4

1h

Nh hy__

= 418751.3805

S. E






)

^

propxYV = n

nN

4

1h

____

2

/ XX

N

h

yhh

= 251

251967 x 5155503.555

= 14706536.04

S.E

)

^

propxY = 414706536.0

= 3834.910174

C.V =

st

propX

Y

Y

^

^

.

X 100

= 3834.910174 x 100

418751.3805

= 0.915796425% 1%


N = 967, n = 251

nh =

4

1h

yhh

yhh

N

nN

, and yh

4

1h

1

__

h

hhi

n

yy

h 1 2 3 4

nh 35.2752 184.2733 22.6087 9.4426

Hence,

6

1h

hn = n

Yst =

4

1h

Nh hy__

= 416318. 6349

optYV

^ =

n

1

24

1

h

yhhN −

6

1

2

h

yhhN

= 14079204.21

optY

^

2. = 114079204.2

S.E

S.E

^






= 3752.226567

C.V = st

opt

Y

Y

^

^

2.

x 100

= 0.901287199% 1%

Statistics on Income And Expenditure Data

__

X =

5

1h N

X h =

5

1h

N

XN hh

__

= 410439089.99

267 = 1503517.191

__

Y =

5

1h

N

Yh =

5

1h

N

YN hh

__

= 350506667.00

267

= 1312759.052

x =

267

11

1

__

N

XX i

= 1084714.27

Skewness (X) = 3(Mean – Median) = 1.24

x

Y =

267

11

1

__

N

YYi

= 1046044.56

Skewness (Y) = 3(Mea n – Median) = 1.40

Y


nh = N

nNh , h = 1, 2, 3, 4,5

S.E

i= 1

i=1






h 1 2 3 4 5

nh 32.1573 29.1235 9.7078 6.0674 3.9438

Thus,

5

1h

hn = n

^

stY =

5

1h

Nh hy__

= 348102711.8

V

)

^

propNY = n

nN

5

1h

2

yhhN

= 267 – 81 X 10574224790000

81

= 2.428155322 x 1013

S.E

)

^

propNY = 00002428155322

= 4927631.604

C.V = ^

^

2

st

propN

Y

Y

x 100

= 1.41556829% 1%


N = 267, and n = 81

nh = X

nX h , X = h

h

h XN__5

1

h 1 2 3 4 5

nh 12.6897 27.9280 15.3062 13.0797 11.9961

Thus,

5

1h

hn = n = 81

stY^

=

5

1h

Nh hy__

= 342196894.9

S. E






)

^

propxYV = n

nN

5

1h

____

2

/ XX

N

h

yhh

= 81

81267 X 15493007393710

= 3.557653548 x 1013

S.E

)

^

propxY = 0003557653548

= 5964606.901

C.V =

st

propX

Y

Y

^

^

.

X 100

= 1.743033613% 2%


N = 267, n = 81

nh =

5

1h

yhh

yhh

N

nN

, and yh

5

1h

1

__

h

hhi

n

yy

h 1 2 3 4 5

nh 31.2685 32.4703 8.9514 2.5341 5.8755

Hence,

5

1h

hn = n

Y =

5

1h

Nh hy__

= 343846076.6

optYV

^

= n

1

25

1

h

yhhN −

5

1

2

h

yhhN

= 1.428373095 x 1013

optY

^

2. = 00001428373095

= 3779382.35

S.E

S.E

^






C.V = st

opt

Y

Y

^

^

2.

x 100

= 1.099149476% 1%

RESULTS AND DISCUSSION

In the analysis, N-Proportional, X-proportional and optimum method of allocation in

stratified random sampling of population totals of a skewed population were demonstrated on

three sets of real life data. These demonstrations are shown using the three sets of data with

four, five and six strata, and two varieties drawn from different secondary sources. The first

and second sets of data are on teachers’ population and students’ enrolment respectively in

each considered school from the thirty three Local Government Area of Oyo State Nigeria.

While the third set of data is on income and expenditure of IGI.

Summary of the results is as shown in the following table;

TABLE 1: SUMMARY OF THE RESULTS OBTAINED

S/

N

DATA POPULATIO

N SIZE (N)

SAMPLE

SIZE (n)

V (Yopt) V (YX-prop) V (YN-prop) SKEWNE

SS (X)

SKEWNE

SS (Y)

1. Staff Enrolment

(2008/ 2009(X),

2009/ 20010(Y)

867 251 17,

494.23428

21,813.76872 18,640.286

05

1. 6820 1.6263

2. Student

Enrolment

2008/2009(X),

2009/ 2010(Y)

967 251 14,079,20

4.21

14,706536.04 14,569,981.

39

1. 489 2 1. 3626

3. Income (X) and

Expenditure (Y)

267 81 1.4283730

95x1013

3.557653548x

1013

2.42815532

2x1013

1. 2400 1.4000

DISCUSSION

The table above shows the summary of the results obtained from the analysis using

stratification of skewed populations for three methods of allocation in stratified random

sampling. From the table, it was observed that;

optYV

^

< V (YN) < V(Yx) at skewness rate of X > 1 and Y > 1

It then follows that;

1.

optYV

^

dominates the other two methods

2. V (YN) dominates V (Yx) in all the cases when skewness rate of auxiliary variable is

greater than one.

Based on the results obtained and our observations in table 1, we could see that, an optimum

allocation method is the best for estimating population total of a skewed population. Also, as

^ ^

S.E

^ ^

^ ^






it can be observed in table 4.14, that < V(YN) < V(Yx), inferring that N- proportional

allocation provided that the skewness rates of auxiliary variable and variable of interest are

both greater than one.

CONCLUSION

Results from table 1 show that optimum allocation procedure attracted the least variance as

compared to other procedures, despite variation observe in the sizes of nh within the strata.

It is therefore evident in this research work, that for estimating the average and variances of

parameters under stratified random sampling of skewed population, the performance of

Optimum Allocation technique is the best when compared with the two other considered

allocation procedures.

REFERENCES

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Adebola, F. B. & Ajayi O. S. (2014) Comparison of allocation procedures in stratified

random sampling of skewed populations under different distributions and sample

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Adegboye, O. S. & Ipinyomi R.A. (1995), “Statistical Tables for class work &

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Amahia G. N. (2009) “STA 764 note on Design and Analysis of Sample Survey”

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Cochran, W. G. (1977) “Sampling Technique (2nd Edition)” Wiley Publication Ltd.

Dalenius T. (1950) “Problem of optimum stratification” skandinavsk Aktuarietidskrift Vol

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Des Raj (1968), “Sampling Theory (3rd Edition)” MC Graw Hall Book Company.

Hidiroglou, M.A. & Lavellee (1988) “On the stratification of skewed populations” A journal

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Vol. 74.

Hunt, N. & Tyrell, S. (2004) Stratified Sampling, Coventry University Press

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Sethi, V.K. (1963), “A note on optimum stratification of populations for estimating the

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