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Determination of the Sampling Technique to Estimate the Performance at Grade Five-Scholarship Examination

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Determination of the Sampling Technique to Estimate the Performance at Grade Five-Scholarship Examination Tropical Agricultural Research Vol. 18 Determination of the Sampling Technique to Estimate the Performance at Grade Five Scholarship Examination W.M.R. Kumari and S. Samita1 Postgraduate Institute of Agriculture University of Peradeniya Peradeniya, Sri Lanka ABSTRACT. Sampling techniques have been given much emphasis since the precision of estimates greatly depends on it. Different methods of sampling have to be investigated to find out the best sampling technique for a given situation. This study was conducted to determine the best sampling technique to estimate performance at the grade five-scholarship examination. The study was done using results of the grade five-scholarship examination held in years 2001, 2002, 2003 and 2004. Simple random sampling and stratified random sampling with different allocations were investigated in the study. Stratification of the population was based on district, gender, medium and two-way stratification by gender and district. The random samples were drawn from each stratum by three types of allocations; proportional allocation, equal allocation and Neyman allocation. The variance of sample mean was calculated for each sampling procedure for the sample sizes 1, 2, 3, 4, 5 and 10% of the population. There was 2-10% gain in precision from simple random sampling to stratified random sampling except for equal allocation. Two-way stratification (by gender and district) was more effective (on average 3% increase in precision) than one-way stratification. Overall, the proportional allocation with two-way stratification was the most appropriate sampling technique to estimate the performance. INTRODUCTION The precision of an estimate made from a sample depends both on the procedure by which the estimate is calculated and on the plan of sampling (Cochran, 1977). Since, there are various sampling procedures, investigation of these techniques is important to find out the best sampling technique for a given situation. Large number of sample surveys have been conducted in the area of education in Sri Lanka. Some of these surveys were to study students’ performance at national examinations. Estimates from those sample surveys had been useful to implement various new educational policies (www.moe.lk). In Sri Lanka, about 4 million children attend government schools. Sri Lanka has near universal gender equity in access to primary and secondary education. The primary education cycle takes student through grades 1-5 while the junior secondary and senior secondary cycles involve grades 6-8 and 9-13, respectively. The grade five-scholarship examination is one of the national examinations conducted by Department of Examinations. 1 Department of Crop Science, Faculty of Agriculture, University of Peradeniya, Peradeniya, Sri Lanka. Kumari & Samita The purpose of this examination is to offer scholarships to the best-performed students to study in popular schools for their secondary education. Students of government and semi government schools are eligible to sit for this examination. In each year there are about 300,000 students from all 25 districts who sit for this examination. It consists of two papers to evaluate students' knowledge on Mathematics and General Intelligence. At present it has become a highly competitive examination in the country. The role of sampling theory is that it can be used to develop a method of sample selection, which is economical and easy to operate. Thereby it makes sampling more efficient and provides estimates that are precise enough for our purposes (Barnett, 1974). Basically there are two approaches to sampling i.e. probability sampling and non-probability sampling. Probability sampling makes conclusions that apply to the population as a whole. Simple random sampling and stratified random sampling are some common probability sampling procedures. Therefore, these sampling procedures were given much emphasis in the present study. The objective of the study was to determine the most appropriate sampling procedure to study the performance at the grade-five scholarship examination. MATERIALS AND METHODS Secondary data The present study was carried out using grade five scholarship examination results (complete population data) of years 2001, 2002, 2003 and 2004. The data set consisted of the variables; school number, serial number, district number, marks of subject-1, marks of subject-2, gender and medium. Table 1 shows the basic information. There is virtually an equal male to female ratio while the Sinhala to Tamil students’ ratio is about 4:1. The dataset was summarized and analyzed using MINITAB statistical software (Minitab, 1991). Table 1. Basic information about the data. Total No. of No. of No. of No. of No. of Year No. males females Sinhala Tamil absentees 2001 292579 144613 147966 230875 61704 17957 2002 306252 153042 153239 240764 65517 22839 2003 311044 156045 154999 242570 68474 18599 2004 306234 154166 152068 237375 68859 16459 Sampling techniques investigated In the present study, sampling methods were investigated under two main techniques. They were simple random sampling and stratified random sampling. Both techniques were investigated for the sample sizes 1, 2, 3, 4, 5, and 10% of the population. Since there was a significant positive correlation (r=0.8; p<0.05) between subject 1 and subject 2, the total of the both subjects was used in the analysis. Sampling Technique to Estimate the Examination Performance Under simple random sampling, samples were drawn from each of the year 2001 to 2004 using MINITAB statistical software. Sampling units were drawn from population without replacement. The precision of estimates under simple random sampling was investigated using the measure, variance of sample mean, V() y , where; ( 1− f ) V() y = S 2 (1) n and N 2 ∑ ( yi − Y ) S 2 = i = 1 (2) N − 1 Y is population mean, n is the sample size, N is the size of the population and f is sampling fraction (n/N). Under stratified random sampling, the population was first divided into strata. Strata in this study were based on the criteria, district (25 strata), gender (2 strata), medium (2 strata) and two-way stratification by gender and district (50 strata). In order to choose stratum sample size, three types of allocations were considered namely, proportional allocation, equal allocation and Neyman allocation. The notations used in the computation th under stratified random sampling were Nh-Total number of units in h stratum, N- th Population size, nh-Number of units in sample in the h stratum, n-Sample size, yhi-Value obtained from the ith sampling unit in the hth stratum, th n h - sampling fraction for the h stratum, (3) f h = N h N W = h - stratum weight, (4) h N n h th y hi - sample mean in the h stratum, (5) y h = ∑ h = 1 n h N h th y hi - true mean of the h stratum, (6) Y h = ∑ h = 1 N h N h ( y− Y ) 2 th ∑ hi - True variance of the h stratum. (7) 2 i = 1 S h = N h − 1 The precision of estimates under stratified random sampling were investigated using the measureV() yst , where; L 2 (8) V() y st = ∑ Wh V() y h h =1 Kumari & Samita L N y ∑ h h L h = 1 ( ) 9 y st = = ∑ Wh y h N h = 1 L N y ∑ h h L h = 1 (10) y st = = ∑ Wh y h N h = 1 th L is the number of strata, V() yh is the variance of sample mean from h stratum given by; S 2 ⎛ N− n ⎞ (11) h ⎜ h h ⎟ V() y h = ⎜ ⎟ n h ⎝ N h ⎠ In the proportional allocation, the specific number of sampling units was drawn from each stratum proportional to the stratum size i.e.; nN n = h . h N Variance of sample mean under proportional allocation is given by; ⎛ 1 − f ⎞ 2 (12) V() y st = ⎜ ⎟∑ WSh h ⎝ n ⎠ In the case of equal allocation, the same number of sample units were drawn from each stratum, i.e. n1= n2 = ………= nL Variance of the sample mean for equal allocation is given by; L 2 2 S h (13) V() y st = ∑ W h (1− f h ) h = 1 n h The Neyman allocation is a special case of optimum allocation. If cost per sample unit is the same for all strata, then optimum allocation for fixed cost reduces to fixed sample size. Accordingly, NSh h (14) nh = n ∑ NSh h and the variance of sample mean under Neyman allocation is given by, 2 2 ()WSh h WSh h (15) V() y = ∑ − ∑ st n N Sampling Technique to Estimate the Examination Performance RESULTS AND DISCUSSION Population statistics The basic statistics of marks for subject-1, subject-2 and subject-total are shown in Table 2. The population mean of the subject-1 during 2001-2004 varied from 41 to 48. For subject 2 mean varied from 26 to 31. It is clear that marks for subject-1 was greater than that of subject-2. The maximum marks for subject-1 was 100 in all four years and for subject-2 the maximum was around 97. The population mean of subject-total was around 75. The maximum marks for subject-total varied 191 to 193. Overall population statistics were consistent in all four years. Table 2. Population statistics of grade 5-scholarship examination results. Year Variable Mean Median CV Minimum Maximum 2001 Subject 1 45.10 45 37.01 0 100 Subject 2 30.99 26 66.53 0 98 Subject Total* 76.10 73 46.58 0 191 2002 Subject 1 48.63 53 45.44 0 100 Subject 2 26.02 21 69.55 0 96 Subject Total* 74.66 73 51.39 3 192 2003 Subject 1 41.38 43 45.53 0 100 Subject 2 31.71 28 68.34 0 98 Subject Total* 73.1 70 53.43 0 193 2004 Subject 1 45.45 48 38.78 0 100 Subject 2 30.48 27 65.57 0 97 Subject Total* 75.94 74 47.31 0 192 Note: CV-Coefficient of variation, *out of 200 Basic statistics of marks by gender The basic statistics of marks (subject total) according to gender is given in Table 3.
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