markov chain analysis of succession for a university admission test of bangladesh
TRANSCRIPT
This article was downloaded by: [New York University]On: 06 December 2014, At: 20:51Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Journal of Statistics and Management SystemsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tsms20
Markov chain analysis of succession for a universityadmission test of BangladeshM. Aminul Hoque a & M. Sayedur Rahman aa Department of Statistics , Rajshahi University , Rajshahi , 6205 , Bangladesh E-mail:Published online: 14 Jun 2013.
To cite this article: M. Aminul Hoque & M. Sayedur Rahman (2001) Markov chain analysis of succession for auniversity admission test of Bangladesh, Journal of Statistics and Management Systems, 4:3, 313-325, DOI:10.1080/09720510.2001.10701045
To link to this article: http://dx.doi.org/10.1080/09720510.2001.10701045
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Markov chain analysis of succession for a university admission test of Bangladesh
M. Aminul Hoque
M. Sayedur Rahman
Department of Statistics
Rajshahi University
Rajshahi- 6205
Bangladesh
Email: [email protected]
ABSTRACT
One hundred students are selected from a large number of applicants on the basis of an admission test in the Department of Statistics, Rajshahi University, Bangladesh every year. The academic performance of the students vary from one to another. Markov chain analysis that sacrifice all information about the position of observation within the succession. The statistic -2 log I, has an asymptotic X2.distribution with (m - 1)2 d.f. and the hypothesis of independence of successive state is correct. There is a statistically significant tendency for certain states not be preferentially followed by certain other states. This study has therefore, revealed the need for undertaking a variety of further investigations, if we wish to gain a deeper insight into the influencing teachers which cause variation. The results suggest that Markov chain analysis is important for policy making in education system and in designing future monitoring programs.
Keywords: Admission succession, Markov chain, Asymptotic, Statistical Inference, Maximum Likelihood
1. INTRODUCTION
Student assessment is an area which has been investigated by a very large number of researchers. The documented studies in this area run into as many as 127 at the doctoral and post doctoral levels (Passi and Sansanwal, 1979). An equally good number of studies have been conducted at the post graduate level also. Inter-examiner variability in assessing students performance has been one of the major areas of research in examination and evaluation. Harper (1962), Jhaveri
Journal of Statistics & Management Systems Vol. 4 (2001), No.3, pp. 313-325 © Academic Forum
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
314 M. A. HOQUE AND M. S. RAHMAN
and Patel (1968), Mishra (1970), Taylor (1963, 1964a, b), Nath (1972), Deo (1974), conducted studies related to inter-examiner and intra examiner variability.
The University autonomy is desirable for generating free environment. The autonomy gives the university its distinctive character. The desirability of university autonomy is a function of our vision which Hoffman (1970) has so distinctly and beautifully persecuted "The university secreted a kind of ideal or mythology of itself as a temple of learning, for above the market place, distinct from political world, but neutral as a institution and devoted to the normative function of evaluating human achievement". University education system is today faced with grave massive problem of overcrowding causing frustration of unemployment among graduates at all levels and an almost anarchic condition in the university campus.
The demand for education is increasing at an unprecedented rate in all regions of the world. All indications are that this trend will gather momentum. It seems to be irreversible. Education is now drawing closer and closer to sections of populations usually excluded from educational circuits, providing the system with new clients workers, professional men, executives, technicians and adults. Every position is entitled to the right to education. Equality of educational opportunity should aim not at an equal inputs but at equal educational outcomes. It really means making certain that each individual receives a suitable education at a pace and through methods adopted to his particular person. Equal access to education is only a necessary not a sufficient condition for justice. It must comprise equal chance of success (Ganesam, 1990).
Education is an important media of economic growth. Economic growth requires proper planning. Education occupies an important position in planning for development from poverty to plenty education as levels (Venkata, 1980). Level indicate pattern which leads to comprehensive manpower planning. Bangladesh wish to prosper in the knowledge based economy of the future, it must develop its most abundant but neglected resource: its people. As a nation, we must provide our citizens with a dynamic and relevant education system, especially related to information technology. Ahmed et. al. (1985) show insignificant effect of H.S.C. results on the admission test performance. Ahmed (1987) also shows that S.S.C. and H.S.C. results together explain only 10% of the variation in their performance at the Bangladesh Agricultural University admission test.
The quality of science or IT teachers in our school, colleges and other types of formal and non-formal institutes is also poor. This is the
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
315 MARKOV CHAIN ANALYSIS
major barrier for advancement in information technology has a resource development. The high quality of science and math education, beginning from middle school and extending to the H.S.C. level is usually required for eventual advancement in information technology, While these subjects are introduced in our Government educational system, the quality of such education is very poor. The manner in which students receive this education is not sufficient for generating interest and stimulating analytical or problem solving skills (Bose, 1977).
Regarding the role of the evaluator it can be marked that in a formal set up of evaluation where the hypotheses are specified, designed is predetermined and the criteria of judgement are explicitly mentioned the evaluator may proceed with an indifferent attitude. His role may be treated as a, "stimulator of subjects with a view to testing critical performances" (Borg and Gall, 1983).
The country has a need to have a plan which should consist of three parts: (a) Family Planning and Population Control; (b) Economic development ensuring greater use of modern technology and science, an increase in the productivity of labour and (c) Educational reconstruction (Khan, 1990).
If this policy is adopted the environment in higher education will be formed by no means excessive. It is really a dilemma that the country has unemployment of educational persons when she has not yet been able to fight successfully the basic problems of food, clothing and housing. The only thing which is lacking is a meaningful and effective meeting of the human resources on the one hand and natural resources on the other hand (Witmer, 1973), Mollah et. al. (1998) found that better performance of the students in statistics depends on their previous academic results and some socio-economic characteristics such as father's education and profession, mother's education, marital status, status of previous academic institution and the attendance of the students in the class.
The marks of Secondary School Certificate (S.S.C.), Higher Secondary Certificate (H.S.C.) and Admission Test of Department of Statistics, Rajshahi University used in this study pertained to a student with average capabilit:es. It would be worthwhile studying the variability in the case of highly competent students on the one hand and students of very low abilities on the other hand. The findings could then be compared with the result of this study.
This study has therefore, revealed the need for undertaking a variety of further investigations, if we wish to gain a deeper insight into the influencing teachers which cause variation.
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
-----------------------
316 M. A. HOQUE AND M. S. RAHMAN
2. METHODOLOGY
Admission succession coded into twelve mutually exclusive and exhaustive states shown in Tables 1 and 2. The data may be created that consist of sequences of ordered succession of mutually exclusive states A to L. The contents in these sequences are essentially same as those used for autoassociation and corss-association. However, the interest of this study is in the nature of transitions from one state to another, rather than in the relative positions of states in the sequence. Let us consider Markov chain techniques that sacrifice all information about the position of observations within the succession, but that provide in return information on the tendency of one state to follow another. This type of study is not found in Bangladesh as well as in literature.
Table 1 Coded value of the test score of under graduate students
Test scores are coded in the following manner -S-.s-.-C-.-&-H.s'C~d';:: 60% marks = 1
59% - 45% marks = 2
Admission test sco~e .(First Succession = 1 year honours StatistICs) INot Succession but got required marks =2
. Below Pass marks = 3
Table 2 Mutually exclusive and exhaustive state of Test Scores
S.S.C. H.s.C. Admission Test Code --
1 1 1 A 1 1 2 B 1 1 3 C 1 2 1 D 1 2 2 E 1 2 3 F 2 1 1 G 2 1 2 H 2 1 3 I 2 2 1 J 2 2 2 K 2 2 3 L
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
317 MARKOV CHAIN ANALYSIS
2.1. Statistical Inference for Markov Chain Model
Markov chains have been considered by several authors Barlett (1955), Whittle (1955), Anderson and Goodman (1957), Billingsley (1961) not only because of their theoretical interest but also for their applications in diverse areas. Methods put forward for estimation of transition probabilities under different situations include a recent one involving linear and quadratic programming procedures to produce least squares estimates (Lee et. aL, 1970). We shall discuss here the maximum-likelihood method of estimation of transition probabilities from individual or micro-unit data. Some tests based on these estimates will also be discussed.
2.1.1 Maxintum-likelihood estimation for Markov Chain Model
Consider a time-homogenou Markov chain with a finite number, nt, of states (1,2, ... , nt) and having transition probability matrix P =(Pi)' i, j = i, 2, ... , nt. Suppose that the number of observed direct transitions from the state j to the state h is njk, and that the total number of observations is (N + 1).
IlL
Put L njk = nj. and L njh ILk, j, k = 1, 2, ... , 1H.
k=l j=l
That there is a striking similarity between a sample from a Markov chain and one from a set of independent multinomial trials has been observed, among others, by Whittle (1955) who obtained the exact probability of the observed njh in the form:
II (nj)1
j IIII nT(njh) II II p]t ; (1)
(njk)! j k j k
the factor T(nj0 which involves the joint distribution of the n/s is indep~ndent of the Pjk's.
The logarithm of the likelihood function can be put as lit lit
L(Pj0 = C + L L njh logpjh, (2) )=1 k=1
where C contains all terms independent of Pjk'
Since LPjk == 1, equation 2 can be written as k
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
318 M. A. HOQUE AND M. S. RAHMAN
III litm-I [/It-I 1 L(PjW C +E~ njk logpjk +Enjm log 1-~Pjk . (3)
Let r be a special value of j. The maximum-likelihood estimates Prk are given by the solutions of the equations
k = 1, 2, .. _, (m - 1).
These equations give
nnn -,~==O, k = 1, 2, ... ) (Ill 1)_ (4)
1- LPrk 11=1
To fix our ideas, let us take a specified value, s, of k. Then
nrm m-l k = 1, 2, _. _, s, ... , (m - 1).
PI'S Prk 1- LPrk
k=l
m-I nrm
Thus (5)-1' PI'S "1'5
k == 1,2, _.. , s, ... , (m - 1). (6)
Summing (6) OVer all k and adding (5), we get II!
2:nrk 1- k=l --n- Prs
rs
and hence the estimate PI'S is given by
A nrsPrs=-m--- (7)
Lnrk 11=1
Now r, s are two arbitrary values of j, k respectively. Hence, for j, k 1, 2, _.. , (m - 1),
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
319 MARKOV CHAIN ANALYSIS
A
Pjk (8)
are the maximum-likelihood estimates of Pjk'
2.1.2 Hypothesis testing for Markov chain model
Some of the tests developed (Anderson and Goodman, 1957) on the above estimates are given below.
(i) Suppose that one wishes to test the null hypothesis that the observed realisation comes from a Markov chain with a given transition, i.e. matrix pO. Suppose that the null hypothesis is
Ho: P pO.
Then, for large N, and for Pjk given by (8), the statistic
A 0 ? 111 n. (p. p:.\~L J. }k }IIJ, (9)
k=1
is distributed as l with (m 1) d.f. Here pJk'S which are equal to zero are excluded and the d.f. is reduced by the number of pJk'S equal to O.
Alternatively, a test for all Pjk can be obtained by adding over all j, and the statistic
i: i nj. <Pjk pJk) (10)
j=1 k=1
has an asymptotic X2-distribution with m(m - 1) d.f. (the number of d.f. being reduced by the number of pJk'S equal to zero, if any, for j, k = 1, 2, ... , m).
The likelihood ratio criterion for Ho is given by
Thus, under the null hypothesis, the statistic
-2 log A 2 L L njk log 0 ' (ll) (n;)Pjk
has an asymptotic X2-distribution with m(m - 1) d.f.
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
320 M. A. HOQUE AND M. S. RAHMAN
The maximum-likelihood estimates could also be used to test the order of a Markov chain. For testing the null hypothesis that the chain is of order zero, i.e. Ho : Pjk = Pk for allj, against the alternative that the chain is of order 1, the test criterion is
A n Ill' !k Jnjk , j k Pjk
where
( ~ I ILjk I' J k J
Under the null hypothesis, the statistic
has an asymptotic X2-distribution with (m - 1)2 dJ. Similar test can be constructed for testing the null hypothesis that the chain is of order one against the alternative that it is of order two.
3. RESULTS AND DISCUSSION
Admission to the higher education through merit is again the strictly required criteria. A uniform entrance examination throughout the country has been introduced and students are selected and placed in the institutions of higher learning in accordance to their performance in the test. Indeed good students go to schools and then the lowest out of them to the best universities (Tluanga, 1974).
A 12 x 12 matrix constructed and showing the number of times a given test scores are succeeded or overlain by another. A matrix of this type is called a transition frequency matrix (Table 3). The measured admission seeker contains 3714 observations, so there are (n - 1) == 3713 transitions. Since the twelve states are mutually exclusive and so the row total is 100% (Table 4).
If the occurrence of States A and B are independent then we can write,
P(BIA) P(B).
Similarly, if the occurrence of all the states A to L is independent, relationship holds for all possible transitions,
p(BIA) P(BIB) = P(BIC) P(BID) = ... == P(BIL) == P(B).
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
321 MARKOV CHAIN ANALYSIS
Table 3 The transition frequency matrix of tests score of under
graduate students
A B C D E F G H I J K L Total "-""-
A 6 56 31 2 28 20 .0 5 3 .0 0 2 153
B 63 478 229 28 3.0.0 154 0 31 15 .0 12 7 1313
C 29 226 138 8 136 89 .0 12 12 0 8 3 661
D .0 30 9 2 12 7 1 1 1 .0 0 1 64
E 31 299 133 15 218 99 .0 28 15 1 9 7 855
F 16 149 83 12 109 65 1 9 6 1 3 3 457
G 0 1 0 0 1 1 .0 .0 .0 0 .0 .0 3
H 3 34 16 1 22 8 .0 5 1 0 .0 2 92
I 4 14 11 0 12 10 1 1 .0 .0 1 0 54
J 0 1 .0 0 1 .0 0 .0 0 0 .0 .0 2
K 1 14 3 .0 12 3 0 .0 1 0 1 0 35
L .0 11 8 0 4 1 0 .0 .0 0 1 0 25
Total 153 1313 661 64 855 457 3 92 54 2 35 25 3714 -'
Table 4
The transition probability matrix of a test score for under graduate students
.04 .37 .20 .01 .18 .14 0 .03 ..02 0 .0 .01
.05 .36 .17 ..02 .23 .12 0 .02 .01 .0 .01 .01
.04 .34 .21 .01 .21 .13 .0 .02 .02 .0 .01 .01
.0 .46 .14 .03 .18 .11 .02 ..02 .02 0 0 .02
.04 .35 .16 .02 .24 .12 .0 ..03 ..02 0 .01 ..01
..04 .32 .18 .03 .24 .14 0 .02 ..01 0 ..01 .01
0 .33 .0 0 .34 .33 .0 0 0 0 0 0
.03 .36 .17 .01 .24 ..09 0 ..05 ..01 .0 0 .02
.07 .26 .20 .0 .22 .19 .02 .02 0 .0 .02 0
0 .50 .0 .0 .50 .0 0 .0 -0 .0 0 0
.03 .39 .09 .0 .34 ..09 .0 0 .03 0 .03 0
.0 .44 .32 0 .16 ..04 .0 .0 0 0 ..04 0
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
322 M. A. HOQUE AND M. S. RAHMAN
We can compare this expected transition probability matrix (Table 5) to the transition probability matrix (Table 4), we actually observe to test the hypothesis that all states are independent of the im
X2mediately preceding states. This is done by using test first converting the probabilities to expected numbers of occurrences by multiplying each row by the corresponding total numb8r of occurrences (Table 6).
Table 5 The expected transition probability matrix of test score for
under graduate students
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
.04 .35 .18 .02 .23 .12 .0008 .02 .01 .0005 .009 .006
The row totals and the column totals will be the same, provided it begins and ends with the same state otherwise rows and columns will differs by one. Unlike most matrices we have calculated before, the transition frequency matrix is asymmetric and in general aij '" aji if the sequence begins and ends with a different state.
The tendency for one state to succeed another can be emphasized in the matrix by coverting the frequencies do decimal fractions or percentages. If each element in the ith row is divided by the total of the ith row, the resulting fractions express the resulting number of times state i is succeeded by the other states. In a probabilistic sense, these are estimates of the conditional probability P(iIj), the probability that state j will be the next state to occur, given that the present state is i.
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
Tab
le 6
~ i:l:
IT
he e
xp
ecte
d t
ran
siti
on
fre
qu
en
cy
matr
ix o
f a
test
sco
re f
or
un
derg
rad
uate
stu
den
t in
~
Ban
gla
desh
<:
0
Tot
al
0 ~ 6.
3 54
.09
27.2
3 2.
63
35.2
2 18
.82
12
3.79
2.
22
.08
1.43
1.
03
152.
91
z > Z54
.1
464.
45
233.
7 22
.58
302.
25
161.
5 1.
05
32.5
6 19
.04
.66
12.3
4 8.
8 13
13.2
~
27.2
3 23
3.66
11
7.66
11
.4
152.
16
81.3
0 .5
3 16
.39
9.58
.3
3 6.
21
4.43
66
1.08
rJ
l >-
< rJ
l 2.
64
22.6
2 11
.39
1.10
14
.73
7.87
.0
5 1.
59
.93
.03
.60
.43
63.9
9
35.1
3 30
2.24
15
2.19
14
.7
196.
82
105.
17
.68
21.2
0 12
.40
.46
8.04
5.
73
854.
94
18.8
3 16
1.55
81
.35
7.86
10
5.20
56
.21
.37
11.3
3 6.
63
.25
4.30
3.
16
457.
03
.12
1.06
.5
3 .0
5 .6
9 .3
7 .0
02
.07
.04
.002
.0
3 .0
2 2.
964
3.79
32
.52
16.3
8 1.
68
21.1
8 11
.31
.07
2.28
1.
33
.05
.86
.62
92.0
9
2.22
19
.09
9.61
.9
3 12
.43
6.64
.0
4 1.
34
.78
.03
.51
.36
53.8
5
.08
.71
.36
.03
.46
.35
.002
.0
5 .0
3 .0
01
.02
.01
1.92
3
1.44
12
.37
6.23
.6
0 8.
05
4.41
.0
3 .8
7 .5
1 .0
2 .3
3 .2
3 34
.91
1.03
8.
84
4.45
.4
3 5.
75
3.08
.0
2 .6
2 .3
6 .0
1 .2
4 .1
7 24
.99
152.
96
1313
.03
660.
88
63.9
8 85
4.76
45
7.04
2.
984
92.0
7 53
.98
2.10
3 35
.09
25
3713
.877
w
'-'='
w
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
324 M. A. HOQUE AND M. S. RAHMAN
The statistic -2 log 'A has 1m asymptotic l distribution with (m - 1)2 degrees of freedom. The resulting -2 log 'A statistic is 52.35. The critical value of X2 for 121 d.f. and 5% level of significance is not exceeded. So we may conclude that the hypothesis of independence of successive states is correct. There is a statistically significant tendency for certain states not to be preferentially followed by certain other states.
4. CONCLUSIONS
The interest of this study is in the nature of transitions from one state to another, rather than in the relative positions of states in the sequence. Markov chain techniques that sacrifice all information about the position of observations within the succession, but that provide in return information on the tendency of one state to follow another. The statistic -2 log 'A == 52.35 has an asymptotic l- distribution with (m 1)2 degrees of freedom and the hypothesis of independence of successive state is correct. There is a statistically significant tendency for certain states not be preferentially followed by certain other states.
This study has therefore, revealed the need for undertaking a variety of further investigations, if we wish to gain a deeper insight into the influencing teachers which cause variation. The results suggest that MarYov chain analysis is important for p~licy making in education system and in designing future monitoring programs.
Acknowledgements. The author thank Dr. S. K. Bhattacharjee, Professor and Chairman, Department of Statistics, Rajshahi University, Bangladesh for supplying the data base. A great thank s due to computer personnel for providing us with various computational steps. We would like to thank Professor Dr. M. G. Mostafa, professor Dr. M. A. Basher Main and Professor M. A. Razzaque, Dean, Faculty of Science for their valuable suggestions and comments.
REFERENCES
1. T. W. Anderson and L. A. Guodman (1957), Statistical inference about Markov chains, Ann. Math. Stat., Vol. 28, pp. 89·110.
2. A. R. Ahmed, P. C. Modak and M. 1. Hossain (1985), An analysis of Agricultural University admission test data, Graduate training Institute, Bangladesh Agricltitltral University, Mymensingh, GTI Pltblicatiol! No. 54.
3. A. R. Ahmed (1987), Bangladesh Agricultural University admission test and S.S.C. and H.S.C. examinations, Proceedings of the second National Statistical Conference, ISRT, Dhaka University, Bangladesh, pp. 245·248.
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014
325 MARKOV CHAIN ANALYSIS
4. M. S. Bartlett (1955), Introduction to Stochastic Processes, Cambridge University Press, New York.
5. P. Billingsley (1961), Statistical methods of Markov chains, Ann. Math. Stat., VoL 32, pp. 12-40.
6. P. K Bose (1977), Higher education at aoss mads, Worlds Trades Pvt. Ltd. Calcutta. 7. W. R. Borg and M. D. Gall (1983), Educational Research - An introdlLclion, 4th
ed., New York, London.
8. P. Deo (1974), Effects of revaluation on the results of candidates appearing at the University examinations, DeiJa.rtment of EdlLcatiol!, Bombay University.
9. D. R. Ganesam (1990), Admission to Universities - A miscellany.
10. A. E. Harper (1962), Objective and traditional e.wmination: Some I'esearch, Ewing Christian College, Allahabad.
11. S. Hoffman (1970), Participation in the perspective, Dea.dalns, VoL 99(1), pp. 167-221:
12. B. J. Jhavel'i and B. C. Patel (1968), A study of Inter and Intra-examiner reliability in marking essays with and without using the marking scheme, New Arts College, SP.U.
13. Q. U. Khan (1990), Higher Education in India - Some issues, D. Thakur, (ed,), Education and Manpower Planning, Deep & Deep Populations, New Delhi.
14. T. C. Lee, G. G. Judge and A. Zellner (1970), Estimating the parameters of Markov probability model fOI: aggregate time series data, North Holland, Amsterdam.
15. M. N. H. Mollah, M. G. Hossain and M. N. Islam (1998), Factors associated with the performance of the students in statistics: A case study, Rajshahi University stlLdies, Part-B (in press).
16. V. S. Mishra (1970), A aitical study of essay type examination, D, Phil. Edu. Gan. U.
17. B. Nath (1972), Inter-Zonal analysis of P.U. results, 1969 examination research, Unit. Gan. U.
18. B. K. Passi and D. N. Sansanwal (1979), Educational evaluation and examination in Buch. M.B. (Ed.), in Second Survey of Research in Education, Society for educational research and development, Baroda.
19. H. J. Taylor (1963), Operation passmark An account of the method used in the Matriculation examination of 1962, Gan. U.
20. H. J. Taylor (1964a), An examination of examiner, Gan. U.
21. H. J. Taylor (1964b), Supplimentary examinations, Gan. U.
22. L. V. Tluanga (1974), Examination as a mode of measurement, Ph.D., Edu., Ran. U.
23. S. K. Venkata (1980), Education and Economic Development in Ind:(l, Ph.D. Dissertation, Frank Bros. Delhi.
24. P. Whittle (1955), Some distributions and moment formulae for the Markov chains, J.R.S.S., Vol. B 17, pp. 235-242.
25. D. R. Witmer (1973), Cost Studies in Higher Education, Review of Educational Research, pp, 99-127.
Received November, 2000
Dow
nloa
ded
by [
New
Yor
k U
nive
rsity
] at
20:
51 0
6 D
ecem
ber
2014