CHAPTER THREE Data and Methodology
3.1 Preludes
This research is exploratory and descriptive in nature. Both qualitative and quantitative data from primary and secondary sources are used in this study. The primary data sources include the study area, individuals necessary for this study and information generated through the selected individuals. The secondary data sources are the official statistics, reports, documents, laws, published materials, ordinances, books, articles, periodicals, Annual reports of concerning agencies, different reports of Bangladesh Bureau of Educational Statistics (BANBEIS), Ministry of Education, Ministry of Primary and Mass Education, Directorate of Secondary and Higher Secondary Education and Various NGO‟s working with Bangladesh Govt. for development of education. Data from primary sources both qualitative and quantitative collected through face to face interview using structured questionnaire at the first phase. At the second phase it was collected through written examination among the students of class V of selected schools. Data from secondary sources are collected through content and document analysis. Requesting questionnaire and developed to collect necessary primary data for achieving objectives of the study. Before finalization, questionnaires are pre-tested on a small portion [(5-7) %] of selected respondents in the study area.
A brief discussion of statistical tools that have been used in collecting and measuring data followed by interpretation of this study findings are given in this chapter.
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Simple random sampling is a method of selecting units out of the such that every one of the distinct samples has an equal chance of being drawn. In practice a simple random sample is drawn unit by unit.
The units in the population are numbered from A series of random numbers between is than drawn, either by means of a table of random numbers or by means of a computer program that produces such a table. At any draw the process used must give an equal chance of selection to any number in the population not already drawn.
The units that bear these numbers constitute the sample.
It is easily verified that all distinct samples have an equal chance of being select by this method. Consider one distinct sample, that is, one set of specified units. At the first draw the probability that some one of the specified units is selected is ⁄ . At the second draw the probability that some one of the remaining ( ) specified units is drawn ( ) ( )⁄ , and so on. Hence the probability that all specified units are selected in draws is
( ) ( )
( )
( ) ( ) ( )
………..(1.1)
Since a number that has been drawn is removed from the population for all subsequent draws, this method is also called random sampling without replacement. Random sampling with replacement is entirely feasible: at any draw, all members of the population are given an equal chance of being, no matter how often they have already been drawn. The formulas for the variances and estimated variances of estimates made from the sample are often simpler when sampling is with replacement than when it
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is without replacement. For this reason sampling with replacement is sometimes used in the more complex sampling plans, although at first sight there seems little point in having the same unit two or more times in the sample.
In a sample survey we decide on certain properties that we attempt to measure and record for every unit that comes into the sample. These properties of the units are referred to as characteristics or, more simply, as items.
The values obtained for any specific item in the units that comprise the population are denoted by . The corresponding values for the units in the sample are denoted by
, or if we wish to refer to a typical sample member, by ( ). Note that the sample will not consist of the first units in the population, except in the instance, usually rare, in which these units happen to be drawn. If this point is kept in mind, my experience has been that no confusion needs result.
Capital letters refer to characteristics of the population and lowercase letters to those of the sample. For totals and means we have the following definitions. Population total: ∑ and mean: ̅ ∑ . Sample total: ∑ and mean: ̅
∑
.
Although sampling is undertaken for many purposes, interest centers most frequently on four characteristics of the population.
1. Mean = ̅ (e.g., the average number of children per school).
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2. Total= Y (e.g., the total number of acres of wheat in a region).
3. Ratio of two totals or means ⁄ ̅ ̅⁄ (e.g, ratio of liquid assets to total assets in a group of families).
4. Proportion of units that fall into some defined class (e.g., proportion of people with false teeth).
The symbol ^ denotes an estimate of a population characteristics made from a sample. The simplest estimators are considered population mean ̅ and estimated population mean ̅̂ ̅ = sample mean, population total and estimated population total ̂ ̅ ∑ ⁄ , population ratio and estimated population ratio ̂ ̅ ̅⁄ . In ̂ the factor ⁄ by which the sample total is multiplied is sometimes called the expansion or raising or inflation factor. Its inverse ⁄ , the ratio of the size of the sample to that of the population, is called sample fraction and is denoted by the letter .
3.2.1 Properties of the Estimates
The sample mean ̅ is an unbiased estimate of ̅.
Here ̂ ̅ is an unbiased estimate of the population total.
The variance of the in a finite population is usually defined as
∑ ( ̅)
………...………(1.2) As a matter of notation, results are presented in terms of a slightly different expression, in which the divisor ( ) is used instead of . We take ∑ ( ̅)
……….………...………(1.3)
The variance of the mean ̅ from a simple random sample is
( ̅) ( ̅ ̅) ( ) ( )………..……...(1.4)
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60 Where is the sampling fraction.
The standard error of ̅ is
̅ √ √( ) ⁄
√ √ ………...……..…..(1.5) 3.2.2 Random Sampling with replacement
A similar approach applies when sampling is with replacement. In this event the unit may appear times in the sample.
Let be the number of times that the unit appears in the sample.
Then ̅ ∑ ………...……….…(1.6)
Since the probability that the unit is drawn is ⁄ at each draw, the variate distributed as a binomial number of successes out of trials with ⁄ . Hence
( ) , ( ) ( ) ( ) ……….…….….…..(1.7) Jointly, the variates follow a multinomial distribution. For this,
( ) ……….(1.8)
Using (1.8), (1.9), and (1.10), we have, for sampling with replacement, ( ̅) *∑ ( ) ∑ +……….…….….…(1.9)
∑ ( ̅) ……….…..(1.10) Consequently, ( ̅) in sampling without replacement is only ( ) ( ) times its value in sampling with replacement (Cochran, 1977).