APPLICATION OF MARKOVIAN MODEL TO . SCHOOL ENROLMENT PROJECTION

This paper deals with the application of Markov chain model to student's enrolment projectioli in a u~iversity in Nigeria. Specifically, the paper contains the estimation of the constant rate of increase in the number of new intake into a programme of study. The estimators proposed by previous authors are used for the estimation and subsequently, the enrolment projection.

Uche (2000) proposed the use of Markov chain model for the.estimation of future enrolment irl schools in developing countries.However, there is no practiesl illustration of the methods of Uche (2000).Adeyemi (1998) considered enrolment structure for primary schools.There is a problem of finding an acceptable method for estimating the number of new intake into the school system in the works of these authors.In line with this problem, Osagiede and Omosigho (2004) proposed methods for estimating new intake into the first grade.The pro,blem of the methods of Osagiede and Omosigho (2004) is the arbitrary choice of a constant rate of increase in the number of new intake into the first grade, which makes prediction to vary from one researcher to another. .New and better methods were proposed by Osagiede and Ekhosuehi (2006).However, there is no practical illustration of the new methodsoin .Osagiede and Ekhosuehi (2006).Based on the suggestion of Osagiede and Ekhosuehi (2006), this study provides a practical illustration to justify the use of the methods proposed in Osagiede and Ekhosuehi (2006).]

The Model
The Markov Chain Model for manpewer planning is given by Osagiede and Omosigho (2004) as Ei = E;:\-+~,E;-' (1 where EL = enrolment in grade g, year t, N i = new entrants to grade g, year t, W , is wastage rate from grade g, year t, P,', r,' are promotion and repetition rates,respectively.Osagiede and,Omosigho (2004) introduced the method for estimating the new intake in grade 1 as where p is the rate of increasefn new entrant into grade 1 and N: is the new entrant figure in the base year.In Osagiede and Omosigho (2004), p value in equation ( 2) is assumed or chosen arbitrarily.Osagiede and Ekhosuehi (2006) derived a method for estimating the rate of increase, P , in new intake 2s n Where N,' is the number of new intake into level i , year t, i = 1, 2, and n is the number of years for which data is available.?o"

V. U. EKHOSUEHI and A. A. OSAGlEDE
The number of new entrants in the base year is given by Osagiede and Ekhosuehi (2006) as We assume that the students' flow over time is 'stable and orderly so that the transition probability matrix (TPM)&stationary.That is P,,(t) = P, for all t, and for given i, and for all i, j = 1, 2. ..., 6.
The maximum likelihood-estimates, P, is given by and the equation of projection is given by where PIZ + (PlO) (PO2) is the probability of promotion from level 1 to 2, and the probabilityJ'that the withdiawals from level 1 would be replaced by candidates admitted through direct entry into level 2; Q is the transpose of the projection matrix, Q ~; .

Po2 Y" (1 -Y ) N,2
where A* is the probabilistic difference arising from fresh students admitted into.levelj, j = 1, 2. (See Osagiede and Ekhosuehi, 2006).Now, Ni dehotih the number of fresh students into level1 in the base year: and

Q =
N: is the number of fresh students into level 2'in the base year.20011L002 session did not exist in the university 2.
In the academic system under consideration, students who were unregistered in year (t -1) have ihe opportunity to register in year t, so, the number of withdrawal cannot be ascertained in year (t -1) for enrolment level i.

3.
Withdrawal (voluntary withdrawal) can be detected when unregistered student(s) in year (t -1) level i, fail to register in year (t + 1) for that level i, i.e. two from the course may be due to illness, expulsion, death, academic deficiency, financial insolvency and so on; , 5 : w6 denotes the number graduates.

Estimation of Parameters
To estimate the parameters of the model, we first state the assumptions of the model as given in Osagiede and Ekhosuehi (2006) as follows;

1.
There are six levels in the course of study 2.
Students enter the school syslem through level one (100 level) or through direct entry into level two (200 level).

3.
The change in the number of new intake is proportional to the previous new intake.4.
Promotion from one level,to the next is based on attaining a minimum of 10 credit course loa& othenrvise, the student is withdrawn from the university.In other words no repetition of classes or levels is allowed, except irr level six.That is , PI, (t)=O for i=1,2 ,..., 5.

5.
No double promotion and no demotion.That is, e,(t)=O f o r a l l j > i + l a n d j s i -1 6.

7.
The probability estimates are stationary;

8.
It is assumed there is no withdrawal in level 6 for whatever reason.
Using equation (5), the pooled probability estimates are given in Table VI (See list of tables).By the assumption PI, = (PI o) (Po,), i = 1, 2, ... , 6, j = 1, 2, which is the probability that students who leave the academic system in level i will be replaced by fresh students who enter the system In level j, we obtain the matrix Q as:  From !be data collected we can estimate P and y as follows using equations ( 7) and ( 9).This is presented in Table VII.(Seelist of tables).
Recall equations ( 7) and ( 8) we have that The absolute change in the number of new entrants is shown in Table VIII, where t = 0 is the base year.(Seelist of Tables).

Table VIII: Difference in New Entrant Projection
Highertransition probabilities, t,!") , n 5 4 of matrix Q are given below.
INTRODUCTION This study focuses on enrolment profile of a new course of study in the Department of Mathematics, University of Benin, Nigeria.It .employsMarkov Chain 'Models used for analysing manpower planning'systems as in Raghavendra (1 951) and McClean (1 991).

Application
To estimate the parameters of this inodel, data for B.Sc Statistics with Computer Science Part -Time programme in the ljniversity of Benin, Nigeria, for the period between 199811999 and 200312004 sessi&ns.$erecoilected,The data are presented in Tables I -V (See list of tables).
Values of N: and N:Assuming that the estimated probabilities in matrix Q are stattrmmy, and taking the session-200312004 as the base year, we obtain the equation of projection as follows:

Table II :
Enrolment Data for 199912000 Sesslon Table I: Enrolment Data for 1998H999 ~e s s i o n Tahlo Ill: Enrolment Data for 200012001 Sesslon

Table VI :
Pooled Probability Estimates