Summary In 6 pages Introduction The objective of a sample survey is to make an i

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Summary In 6 pages Introduction The objective of a sample survey is to make an inference about population parame ters from information contained in a sample. Two factors affect the quantity of infor mation contained in the sample and hence the precision of our inference-making pro- cedure. The first is the size of the sample selected from the population. The second is the amount of variation in the data; variation can frequently be controlled by the method of selecting the sample. The procedure for selecting the sample is called the sample survey design. For a fixed sample size n, we will consider various designs, or sampling procedures, for obtaining the n observations in the sample. Because obser vations cost money, a design that provides a precise estimator of the parameter for a fixed sample size yields a savings in cost to the experimenter. The basic design, or sampling technique, called simple random sampling is discussed in this chapter. DEFINITION 4.1 If a sample of size n is drawn from a population of size N such that every possible sample of size n has the same chance of being selected, the sampling procedure is called simple random sampling. The sample thus obtained is called a simple random sample It is a consequence of this definition that all individual elements in a ulation have the same chance of being selected and that the selection of individual elements is mutually independent: the presence or absence of a given element from the sample

Explanation / Answer

Sample survey is used to make an inference about the population. There are two things on which we need to focus on while studying sample survey:

The procedure of selecting sample is called the sample survey design.

Simple Random Sampling: if a sample of size n is drawn from a population of size N such that every possible sample of size n has the same chance of being selected, the sampling procedure is called Simple Random Sampling.

The sample thus obtained is called a simple random sample.

In this case all individual elements in a population have the same chance of being selected and that the selection of individual elements is mutually independent, the presence or absence of a given element from the sample does not affect the selection probability of any other element. Simple random sampling is used to obtain population estimates like population total, mean as well as proportion.

For example: A federal auditor wants to study open accounts status in a hospital. There are total 28000 open accounts in that hospital. But it is not possible for him to study each and every account under his observation. He will some part of it any will make inference about the whole population. That using sampling techniques.

Out of these 28000 patient records a sample n = 100 is to be drawn. The sample is called simple random sample if every possible sample of n = 100 records has the same chance of being selected. Simple random sampling forms the bases of most sampling designs.

Auditors study simple random samples of accounts in order to check for compliance with audit controls set up by the firm or to verify the actual dollar value of the accounts. Thus they may wish to estimate the proportion of accounts not in compliance with controls or the total value of, say accounts receivable.

While selecting sample two kinds of problems can arise:

There are two factors which affect the quantity of information given in the investigation.

Simple random sampling design does not attempt to reduce the effect of data variation on the error of estimation in which every sample has equal chance of being selected.

Here N= population size, n = sample size

In estimating a population mean and total T, we use sample mean and sample total N respectively. Both estimates are unbiased, that is and . The estimated variance and the bound on the error of estimation are given for both estimates.

Something during the design of an actual survey, the experimenter must decide how much information is desired, that is how large a bound on the error of estimation can be tolerated. Sample size requirements have been presented for estimating and T with a specified bound on the error of estimation.

The third parameter estimated was the population proportion P. the properties of have been presented and related to the properties of the estimate of population parameter .

Selecting the sample size to estimate P with a specified bound on the error of estimation was based on the same principle employed in selecting a sample size for estimating and T.

Stratified Random Sampling

Purpose of this sampling is to maximize the amount of information for a given cost.

Definition: A stratified random sample is one obtained by separating the population elements into nonoverlapping groups, called strata, and then selecting a simple random sample from each strata.

Suppose a public opinion poll designed to estimate the proportion of voters who favour spending more tax revenue on an improved ambulance service is to be conducted in a certain country. The country contains two cities and a rural area. The population elements of interest for the poll are all men and women of voting age who reside in the country. A stratified random sample of adults residing in the country can be obtained by selecting a simple random sample of adults from each city and another simple random sample of adults from the rural area. That is, two cities and the rural area represent three strata from which we obtain simple random samples.

The results of the stratified random sample are combined, the final estimate of the proportion of the voters favouring more expenditure for an ambulance service may have a much smaller bound on the error of estimation than would an estimate from a simple random sample of comparable size.

The cost of obtaining observations varies with the design of the survey.

The estimates of a population parameters may be desired for certain subsets of the population. Stratified random sampling allows for separate estimates of population parameters within each stratum.

The principle reason for using stratified random sampling rather than simple random sampling are as follows:

A stratified random sampling is obtained by separating the population elements into groups, or strata, such that each element belongs to one and only one stratum, and then independently selecting a simple random sample from each stratum.

Three advantages of stratified random sampling over simple random sampling:

An unbiased estimator, of the population mean is a weighted average of the sample means for the strata. An unbiased estimator of the variance of is given in equation (5.2) this estimator is used in placing bounds on the error of estimation. An unbiased estimator of the population total is also given, along with its estimated variance.

Regression:

Regression estimation is another technique for incorporating information on a sub ordinary variable. This method is usually more precise than the ratio estimation if the relationship between the y and x values is a straight line., not necessarily through the origin.

The method of difference estimation is similar in principle to regression estimation. It works well when the plot of y versus x reveals points lying close to a straight line with unit slope.

Systematic Random sampling

The basic idea of systematic sampling is as follows. Suppose a sample of n names is to be selected from a long list. A simple way to make this selection is to choose an appropriate interval.

Definition: A sample obtained by randomly selecting one element from the first k elements in the frame and every kth element thereafter is called a l-in-k systematic sample with random start.

Systematic sampling is easier to perform in the field and hence is less subject selection errors by field works than are either simple random samples or stratified samples.

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