Basic Concepts Of Sample Surveys (Essay Sample)

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STA308/SMA344/TMA344 SAMPLE SURVEYS
LECTURE 1: BASIC CONCEPTS OF SAMPLE SURVEYS
Introduction: The enumeration of population by sampling method, proposed by Laplace around 1783, came into wide spread use only by mid nineteen thirties. From the onset, the following basic questions arose;
* How should the observations be made?
* How many observations should be made?
* How should the total sample be made?
* How should the data so obtained be analyzed?
The answers to these questions were sought and in the process, a number of different techniques and methods were developed. These methods were tested to determine whether the above questions were adequately answered or not. In course of time, the concept of generalization through the introduction of inductive logic caught the attention of statisticians engaged in developing suitable techniques of sampling. In this context, the question “how should this generalization be made?” cropped up. The answer to this question was sought with the help of the technique of probability. Thus the concept of probability sampling originated. The use of probability in sampling theory came to be recognized as a reliable tool in drawing inferences about the populations, whether finite or not.
In sample surveys, the experimenter introduces the probability element by adopting the technique of randomization. Fisher (1935) gave the idea of probability structure in planned experiments, and showed that deliberately introduced randomization in the selection of a part from the whole provides a valid method of obtaining an estimate of the amount of error committed. He demonstrated practically that the randomization not only gives a procedure for valid selection of the part from the whole, but also gives an expression to the amount of risk committed in doing so. Thus the problem was reduced to determining the methods for selection and estimation, which would minimize the risk involved.
Mahalanobis (1944) introduced “the concept of cost function”. The problem now became that of finding the combinations of selection and estimation which would minimize the cost function.
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Fields of application of sampling techniques and limitations
Sample surveys are widely used as a means of collecting information to meet a definite need in government, industry and trade, physical and life sciences and technology, social, educational and economic problems among other areas.
Sampling techniques can successively be employed;
* When results with maximum accuracy or reliability with a fixed budget, or with the minimum number of units with a specified degree of reliability are required.
* When the units under investigation show considerable variation for the characteristic under study.
* When a total count of the population is not possible or is very costly or destructive.
* When the scope of the investigation is very wide and the population is not completely known.
* When time money and other resources are limited.
Sampling theory has the following limitations:
* In spite of the fact that a proper choice of design is employed, a sample does not fully cover the parent population and consequently results are not exact.
* Sampling theory and its application in the field need the services of trained and qualified personnel without whom results of sample surveys are not dependable.
* The planning and execution of sample surveys should be done very carefully, or the data may provide misleading results.
Definitions and Preliminaries
We are usually faced with a collection (called population):
U=(u1, u2, …, uN) with u1, u2, …, uN being the elements or units of which some property (called characteristic) yi is defined for every unit ui.
Sampling theory is mainly concerned with ways of obtaining samples, that is sequences or sets of units taken from U in order to estimate population parameters such as; population total, mean, variance, ratio of two population characteristics and so on.
We shall consider a random experiment, the outcome of which depends on chance. The results of a random experiment will be called sample points and the totality of all sample points consistent with the method of sampling adopted will be called sample space.
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Every outcome of the experiment is described by one, and only one, sample point. The method of sampling must also define the probability that a particular sample is drawn such that and.
A sampling unit may be taken as a well-defined and identifiable element or group of elements on which observations can be made. A collection of such units is usually called population. A population is said to be finite if the number of units contained in it is finite otherwise it is infinite. For practical purpose, we shall mainly deal with finite populations. Usually a list has to be prepared or a serial order has to be given to all units in the population. This creates the sampling frame.
A sample is a part or fraction of the population. The number of units, not necessarily distinct, included in the sample is known as the sample size. The number of distinct units in the sample is termed as the effective sample size. Any function of sample values is called a statistic. If it is used to estimate any parameter, it is called an estimator. An estimator is a random variate and may take different values from sample to sample. The value the estimator takes on in any particular sample is then its estimate. The difference between the estimator and the parameter is called error. An estimator is said to unbiased estimator for the parameter if Bias is given by A relative measure of bias is
The mean of squares of error taken from is called mean square error (MSE). Symbolically, MSE () =
The sampling variance of is defined by;
. Thus,
MSE () = =
.
The value is called the standard error of the estimator and the value
is called the relative standard error of the estimator.
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Given two estimators and of a parameter, the estimator is said to be more efficient than if the mean square error of is less than the mean square error of. The relative efficiency of as compared to , which differs in respect of sample size or sampling method or both, may be defined as the reciprocal of the ratio of the sampling variances of the estimators given by both techniques when same type of sampling units are taken.
Census and Sample Surveys; Advantages and Disadvantages
The total count of all units of the population for a certain characteristic is known as complete enumeration, also termed census survey. When only part, called a sample is selected from the population and examined , it is called sample survey. The advantages of sample surveys over census surveys are:
* Reduced cost of the survey.
* Greater speed of getting results.
* Greater accuracy of results.
* Greater scope and adaptability.
Despite these advantages, sample surveys are not always preferred to census surveys. Sampling theory has its own limitation and the advantages of sampling over complete enumeration can be derived only if:
* The units are drawn in a scientific manner.
* An appropriate sampling technique is used.
* The size of units selected in the sample is adequate.
If information is required for each unit , census is the only answer.
Principles of sampling theory
The theory of sampling is based on three important basic principles namely:
i) Principle of Validity.
ii) Principle of Statistical regularity, and
iii) Principle of Optimization.
Principle of Validity – The sampling design should provide valid estimates of population parameters. It ensures that there is some definite and pre-assigned probability for each individual in the sampling design.
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Principle of Statistical regularity- This principle stress upon the desirability and importance of selecting sample designs where inclusion of sampling units in the sample is based on a probability theory.
Principle of Optimization- This Principle takes into account the desirability of obtaining a sampling design which gives optimum results. Optimization is meant to develop methods of sample selection and estimation that provide;
i) a given level of efficiency with the minimum possible resources or
ii) a given value of cost with maximum possible efficiency.
The principle stresses upon obtaining optimum results with minimization of total loss in terms of cost and mean square error.
Principle Steps in a Sample Survey.
The main steps that are involved in a sample survey are;
i) Statement of objectives.
ii) Definition of the population to be studied.
iii) Determination of sampling frame and sampling units.
iv) Selection of proper sampling design.
v) Organization of field work.
vi) Summary and analysis of data.
The analysis of data collected in a survey may be broadly classified as follows:-
* Scrutiny and editing of data.
* Tabulation of data.
* Statistical analysis.
* Reporting and conclusions.
Finally, a repor