AMA 4432:Design and Analysis of Sample Surveys
Course Outline
✓ Basic Concepts
✓ Simple Random Sampling
✓ Sampling for Proportions and Percentages
✓ Stratified Sampling
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The Basic Concepts
1.1 Introduction
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Statistics is the science of data. Data are the numerical values containing some
information. Statistical tools can be used on a data set to draw statistical
inferences.
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These statistical inferences are in turn used for various purposes. For example,
government uses such data for policy formulation for the welfare of the people,
marketing companies use the data from consumer surveys to improve the
company and to provide better services to the customer, etc.
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There are two ways of obtaining the information:
1. Sample surveys
2. Complete enumeration or census
1.2 Definitions
Definition 1.1 Sample survey: is a method of drawing an inference about the
characteristics of a population or universe by observing only a part of the
population.
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Ideally, total information about the population is obtained through census,
where every individual in the population is involved in giving out information.
However, most of the times due to certain constraints, it is not always
possible to carry out a census.
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In a sample survey the purpose of the survey statistician is to estimate some
functions of the population parameter, θ (y), say, by choosing a sample(part of
the population) and by observing the values of y only on units selected in the
sample.
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The statistician therefore wants to make an inference about the population by
observing only a part of it. This is essential and perhaps the only practical
method of inference about the characteristics of the population since in many
socio-economic investigations the survey population may be very large,
containing say hundreds or thousands of units.
Definition 1.2. Survey population: A finite(survey) population is a collection of
known number N of identifiable units labelled 1,2,3,...,i,...,N where i stands for
the label as well as the physical unit labelled i. The number N is the size of the
population. A population can be finite or infinite.
Definition 1.3. Sample: A sample is a part of the population/subset of the
population selected for study. A sample may be drawn from a population either
with replacement(wr) or without replacement(wor). After a sample is selected ,
data are collected from the sampled units.
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Definition 1.4. Sampling units: This refers to the individual items whose characteristics
are to be measured in the sample survey.
Definition 1.5. Sampling frame: This is the list of all sampling units. It may be a list of
units with identification and particulars or a map showing the boundaries of sampling
unis e.g. a manufacturing firm may want to determine how popular a newly
manufactured product is within the community suggests a possible frame for the
survey. The firm may decide to concentrate its surveys in urban residential areas only.
In this case, you have a complete list of estates in urban areas. The residents in those
chosen estates will be interviewed and inferences are made.
Definition 1.6. Sampled population : It is the set of individuals in the sampling frame.
Its actually the subset of the target population. Target population: The entire group
about which we want to draw conclusions.
Definition 1.7 Sampling scheme: Its the technique by which the elements which
constitute the sampled are obtained from the population
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1.3 Merits of Sample Survey
Sample surveys have potential advantages over complete enumeration(census).
They include;
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Reduced cost. If data are secured from only a small fraction of the aggregate,
expenditures may be expected to be smaller than if a complete census is
attempted
Greater speed. For the same reason, the data can be collected and
summarized more quickly with a sample than with a complete count. This may
be a vital consideration when the information is urgently needed.
Greater scope. In certain types of inquiry, highly trained personnel or
specialized equipment, limited in availability, must be used to obtain the
data. A complete census may then be impracticable: the choice lies between
obtaining the information by sampling or not at all. Thus, surveys which rely
on sampling have more scope and flexibility as to the types of information
that can be obtained.
Greater accuracy. Because personnel of higher quality can he employed and
can be given intensive training, a sample may actually produce more accurate
results than
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1.4 Conditions for Sample Survey
The investigations through sample survey are appropriate in the following
conditions:
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Broad Area: When the investigation area is broad, for example, testing of the
effect of a drug for disease by a drug company, then they have to adopt the
sample survey method.
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When the Population is Infinite: When the number of elements in the
population is infinite then this method is suitable.
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Insufficient Resources: Where the money, time and employee/workers are in
insufficient numbers, then this method can be adopted.
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When Units are of Destructive Nature: In some situations, if the units of the
population are of destructive nature and if the census method is used then all
the population would be destroyed. In this situation, the sample survey is
advisable, for example, testing of sound of crackers.
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Homogeneity: If the elements of a population are homogeneous then the
sample units would be of same characteristics as of the population. In short a
sample should be a true representative of population.
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1.5 Principles Of Sample Survey
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Three basic principles for the design of a sample survey are:
1.
Principle of Optimization The principle of optimization takes into account the
factors of (a) Efficiency and (b) cost.
Efficiency: Efficiency is measured by the inverse of sampling variance of the
estimator. The principle of optimization ensures that a given level of efficiency
will be reached with the minimum possible resources and minimum cost.
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Cost: Cost is measured by expenditure incurred in terms of money or man
powers. So, the term optimization means that, it is based on developing methods
of sample selection and of estimation; that provide a given value of cost with the
maximum possible efficiency.
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2. Principle of Validity By validity of a sample design, we mean that the sample
should be so selected that the results could be interpreted objectively in terms of
probability. According to this, sampling provides valid estimates about population
parameters. This principle ensures that there is some definite and preassigned
probability for each individual of the aggregate (population) to be included in the
sample.
3. Principle of Statistical Regularity According to the principle of statistical regularity
we mean that a moderately large number of items chosen at random from a large
group are almost sure on the average to possess the characteristics of the large
group. This principle has also its origin in the law of large numbers of the theory of
probability.
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1.6 Essentials of Sampling
For obtaining the unbiased and real result by a sampling method, a sample should
have the following factors (characteristics):
1.
Homogeneity: The nature of each and every unit of the population should not
contain much difference. If two or more samples are selected then they should
be similar in nature not in their response/output.
2.
Representativeness: The sample should represent all the characteristics of the
population that can be possible only when the selection of items or units has
been done unbiased and each and every unit have an equal probability of
chance to be selected in the sample.
3.
Independency: Each and every unit of the population should be independent.
In other words, the selection of a unit in the sample should not be dependent
on the selection of other units.
4.
Adequacy: The number of units or elements which are to be selected in the
sample should be sufficient. If the sample size is not sufficient then results
cannot be reliable. The more the sample units in the sample, more reliable
results would occur.
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1.7 Principal steps involved in planning and execution of a sample
survey.
The broad steps to conduct any sample surveys are as follows:
1.
Objective of the survey: The objective of the survey must be defined in clear
and concrete terms. The investigation team should be quite clear in what they
want and how they are going to use the results. Some of the objectives may be
immediate and some far-reaching. The investigator should take care of these
objectives with the available resources in terms of money, manpower and the
time limit required for the availability of the survey.
2.
Population to be sampled: Based on the objectives of the survey, decide the
population from which the information can be obtained. The geographical,
demographic and other boundaries of the population must be specified so that
no ambiguity arises regarding the coverage of the survey.
3.
Data to be collected: Collection of data must be done in conformity with the
objectives of the survey and the nature of the data. After it is decided upon,
one must prepare a questionnaire or a schedule of enquiry. A schedule or a
questionnaire contains a list of items of which information is sought, but the
exact form of the questions to be asked is not standardized but left to the
judgment of the investigators. A questionnaire should be in a specified order.
The questions should be clear, brief, collaborative, non offending and
unambiguous and to the point so that not much scope of speculation is left on
the part of the respondent or interviewer.
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4. Selection of Proper Sampling Design This is the most important step in planning a
sample survey. There is a group of sampling designs (to be discussed later) and selection
of the proper one is an important task. The design should take into account the
available resources and the time-limit, if any, besides the degree of accuracy desired.
The cost and precision should also be considered before the final selection of sampling
design.
5.Method of data collection. The choice of measuring instrument and the method to
measure the data from the population needs to be specified clearly. For example, the
data has to be collected through interview, questionnaire, personal visit, combination
of any of these approaches, etc. The forms in which the data is to be recorded so that
the data can be transferred to mechanical equipment for easily creating the data
summary etc. is also needed to be prepared accordingly In the cases, where the
information is to be collected by observation they must decide upon the method of
measurement.
6. The sampling frame and sampling units: The sampling frame has to be clearly
specified. The population is divided into sampling units such that the units cover the
whole population and every sampling unit is tagged with identification. The frame must
cover the whole population and the units must not overlap each other in the sense that
every element in the population must belong to one and only one unit.
7. Selection of sample: The size of the sample needs to be specified for the given
sampling plan. This helps in determining and comparing the relative cost and time of
different sampling plans. The method and plan adopted for drawing a representative
sample should also be detailed.
8. The Pre-test: It is advised to try the questionnaire and field methods on a small
scale. This may reveal some troubles and problems beforehand which the surveyor may
face in the field in large scale surveys.
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9. Organization of the field work: How to conduct the survey, how to handle
business administrative issues, providing proper training to surveyors, procedures,
plans for handling the non-response and missing observations etc. are some of the
issues which need to be addressed for organizing the survey work in the fields. The
procedure for early checking of the quality of return should be prescribed. It
should be clarified how to handle the situation when the respondent is not
available.
10. Summary and analysis of data. This is the last step wherein inference is to be
made on the basis of collected data. This step again consists of the following
steps:
a)
The filled in questionnaires should be carefully scrutinized to find out whether
the data furnished are plausible and consistent;
b)
Depending upon the quantity of data, a hand-tabulation or machine tabulation
is to be drawn;
c)
After the data has been properly scrutinized, edited and tabulated, a very
careful statistical analysis is to be made; and
d)
Finally, a report incorporating detailed statement of the different stages of the
survey should be prepared. In the presentation of the result, it is advisable to
report technical aspects of the design. .
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1.8 Scope of sample surveys
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Descriptive surveys: Aim to describe the characteristics of a population at a
given point in time.
Example: A survey to determine the average income of households in a city.
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Analytical surveys: Aim to explore relationships between variables and test
hypotheses.
Example: A survey to investigate the relationship between exercise habits and
health outcomes.
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Exploratory surveys: Used for exploratory research to generate hypotheses.
Example: A survey to identify potential factors affecting employee satisfaction.
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Causal surveys: Aim to identify cause-and-effect relationships.
Example: A survey to assess the impact of a training program on employee
performance.
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1.9 Types Of Sampling
According to the method of selection of sample, the sampling schemes can be
categorized as :
1.
Non-probability sampling
2.
Probability sampling
Probability sampling :Definition Probability sampling is a sampling technique where
each member of the population has a known, non-zero chance of being selected in
the sample.
Types of probability sampling
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Simple random sampling (SRS) Every member of the population has an equal
chance of being selected. • Example: Selecting 10 students from a class of 50 by
drawing names from a hat.
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Systematic sampling Every k-th member of the population is selected after a
random start. • Example: Selecting every 5th person on an alphabetical list of
employees after starting at a random point.
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Stratified sampling: The population is divided into homogeneous subgroups
(strata) and random samples are taken from each stratum. • Example:
Dividing a population into male and female groups and then randomly
selecting an equal number of participants from each group.
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Cluster sampling: The population is divided into clusters, some clusters are
randomly selected, and all members of selected clusters are included in the
sample. • Example: Dividing a city into blocks and then randomly selecting
certain blocks, including all households within those blocks.
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Multistage sampling: A combination of different sampling methods used in
various stages. • Example: Using cluster sampling to select schools and then
using simple random sampling to select students within those schools.
Non-Probability sampling: Non-probability sampling is a sampling technique
where some elements of the population have no chance of being selected, or
the probability of selection is unknown.
Types of non probability sampling
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Convenience sampling Samples are chosen based on their ease of access. •
Example: Surveying people at a shopping mall because it is easy to find
participants.
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Judgmental/Purposive sampling Samples are selected based on the researcher’s
knowledge and judgment. • Example: Choosing experts in a field to participate
in a study.
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Quota sampling The population is segmented into mutually exclusive subgroups, and then a non-random set of observations is chosen from each
subgroup. • Example: Ensuring that a survey includes a certain number of men
and women based on their proportion in the population.
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Snowball sampling. Existing study subjects recruit future subjects from among
their acquaintances. • Example: A researcher studying drug abuse starts with
known users who then refer other users.
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Simple Random Sampling
2.1 Introduction
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The samples can be drawn in two possible ways.
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Simple random sampling with replacement (SRSWR)
The probability of drawing a sample in SRSWR is given by:
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Simple random sampling without replacement (SRSWOR)
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Remarks
1)
2)
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3)
4)
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2.3. Procedure of Selecting a random Sample
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2.4.Notations and Terminology
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2.5. Estimation of the population mean and Variance
2.5 .1 Simple Random Sampling Without Replacement
Theorem 1
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Proof
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Therefore
Theorem 2
Proof
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Consider
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Consider
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Therefore
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Therefore
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Therefore
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2.5.2.Simple Random Sampling With Replacement
Theorem 3
Proof
We define random variable 𝑡𝑖 as number of times 𝑖𝑡ℎ unit is included
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Alternatively
Theorem 4
Proof
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But we know that
Remarks: When N is infinite or large enough
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Theorem 5
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Thus
2.5.3. Estimation of variance from a sample
Theorem 6 (SRSWOR)
Exercise: Prove the above theorem
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Proof ( Answer to the Exercise)
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In the case of SRSWOR
In the case of SRSWR
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An unbiased estimate of 𝒗𝒂𝒓(ഥ
𝒚)
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2.5.4. Sampling Errors
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Theorem 7
Variance of sample total is given by
Proof
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2.5. 5. Estimator of population total
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ഥ
2.5.6. Confidence Intervals for Population Mean 𝒀
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When 𝝈𝟐 is unknown
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2.5.7. Sample size determination
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The size of the sample is needed before the survey starts and goes into operation. One
point to be kept is mind is that when the sample size increases, the variance of estimators
decreases but the cost of survey increases and vice versa.
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So there has to be a balance between the two aspects. The sample size can be determined
on the basis of prescribed values of standard error of sample mean, error of estimation,
width of the confidence interval, coefficient of variation of sample mean, relative error
of sample mean or total cost among several others.
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An important constraint or need to determine the sample size is that the information
regarding the population standard derivation S should be known for these criterion. The
reason and need for this will be clear when we derive the sample size in the next section.
A question arises about how to have information about S before hand?
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The possible solutions to this issue are to conduct a pilot survey and collect a
preliminary sample of small size, estimate S and use it as known value of S.
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Alternatively, such information can also be collected from past data, past experience,
long association of experimenter with the experiment, prior information etc.
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Now we find the sample size under different criteria assuming that the samples have
been drawn using SRSWOR. The case for SRSWR can be derived similarly.
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1.
Prespecified variance
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N/B
2. Pre-specified estimation error
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If N is large then
3. Prespecified width of confidence interval
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4. Prespecified coefficient of variation
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5. Prespecified relative error
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6. Prespecified cost
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Example 1
Example 2
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Solution
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Sampling for Proportions and Percentages
Introduction
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In many situations, the characteristic under study on which the observations are collected
is qualitative in nature. For example, customers' responses in many marketing surveys are
based on replies like ‘yes’ or ‘no’, ‘agree’ or ‘disagree’ etc.
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Sometimes, the respondents are asked to arrange several options in the order like the first
choice, second choice, etc. Sometimes, the objective of the survey is to estimate the
population proportion or the percentage of brown-eyed persons, unemployed persons,
graduate persons, or persons favoring a proposal, etc.
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In such situations, the first question arises: how to do the sampling and secondly, how to
estimate the population parameters like population mean, population variance, etc
Sampling Procedure
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The same sampling procedures that are used for drawing a sample in case of quantitative
characteristics can also be used for drawing a sample for qualitative characteristics.
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So, the sampling procedures remain the same irrespective of the nature of the
characteristic under study - either qualitative or quantitative. For example, the SRSWOR
and SRSWR procedures for drawing the samples remain the same for qualitative and
quantitative characteristics.
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Similarly, other sampling schemes like stratified sampling, two-stage sampling, etc. also
remain the same.
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Estimation of population proportion
The population proportion in the case of qualitative characteristics can be estimated
similarly as the estimation of the population mean in the case of quantitative characteristics.
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Now the population total is:
The population mean is
Suppose a sample of size n is drawn from a population of size N by simple random
sampling
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Since
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SRSWOR
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SRSWR
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Estimation of population total or total number of count
Confidence interval estimation of P
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Stratified Sampling
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The objective of any sampling method is usually to estimate the unknown population
parameters with the highest precision i.e. the variance of the estimators should be
minimized.
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If the population is heterogeneous as will be in most situations, then a sample taken via
SRS might yield high levels of variability . As a result, in a survey where precision is a
main factor to be considered, then a strategy that addresses heterogeneity must be found.
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One way of achieving higher precision is to divide the population which is originally
heterogeneous into sub population which are to a big extent homogeneous with respect to
survey characteristics.
The basic idea behind stratified sampling is to:
❖ Divide the whole heterogeneous population into smaller groups or subpopulations such
that the sampling units are homogeneous with respect to the characteristic under study
within the subpopulation
❖ Heterogeneous with respect to the characteristic under study between/among the
subpopulations. Such subpopulations are termed as strata.
❖Treat each subpopulation as a separate population and draw a sample by SRS from each
stratum.
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Example
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In order to find the average height of the students in a school of class 1 to class 12, the
height varies a lot as the students in class 1 are of age around 6 years, and students in
class 10 are of age around 16 years.
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So, one can divide all the students into different subpopulations or strata, such as
Students of classes 1, 2, and 3: Stratum 1
Students of classes 4, 5, and 6: Stratum 2
Students of classes 7, 8, and 9: Stratum 3
Students of classes 10, 11, and 12: Stratum 4
Now draw the samples by SRS from each of the strata 1, 2, 3 and 4. All the drawn samples
combined together will constitute the final stratified sample for further analysis.
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Notations
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Procedure of stratified sampling
Difference between stratified and cluster sampling schemes
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In stratified sampling, the strata are constructed such that they are within
homogeneous and among heterogeneous.
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In cluster sampling, the clusters are constructed such that they are within
heterogeneous and among homogeneous.
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Issues in the estimation of parameters in stratified sampling
We now consider the estimation of population mean and population variance from a stratified
sample
Estimation of population mean and its variance
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Estimation of population mean
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Based on this, one can modify 𝑦ത so as to obtain an unbiased estimator of 𝑌ത
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Consider the stratum mean, which is defined as the weighted arithmetic mean of
strata sample means with strata sizes as weights given by:
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ഥ𝒔𝒕
Variance of 𝒚
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The total variation in the population is fixed and can be orthogonally partitioned into
between and within strata variations, i.e
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For example, the units in geographical proximity will tend to be more closer. The
consumption patterns in the households will be similar within a lower-income group housing
society and within a higher-income group housing society, whereas they will differ a lot
between the two housing societies based on income.
Estimate of Variance
SRSWOR
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SRSWR
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Allocation problem and choice of sample sizes in different strata
Question: How to choose the sample sizes 𝑛1 , 𝑛2 , … , 𝑛𝑘 so that the available resources are
used in an effective way?
There are two aspects of choosing the sample sizes:
i) Minimize the cost of survey for a specified precision.
(ii) Maximize the precision for a given cost.
Note: The sample size can not be determined by minimizing both the cost and variability
simultaneously. The cost function is directly proportional to the sample size whereas variability
is inversely proportional to the sample size.
1.
Equal allocation
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2. Proportional allocation
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3. Neyman or optimum allocation
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Choice of sample size based on cost of survey and variability
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How to determine  ?
(i) Minimize variability for fixed cost
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(ii) Minimize cost for given variability
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Sample size under proportional allocation
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Variances under different allocation
(i)
Proportional allocation
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(ii) Optimum allocation
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Comparison of variance of sample mean under SRS
(a.) Proportional allocation
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(b.) Optimum allocation
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Estimate of variance and confidence intervals
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Example
A population of size 800 is divided into three strata. Their sizes and standard deviations
are as given below.
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Solution
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