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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Chapter 7
Sampling
Prepared by Group 6
Eureka Enterprises
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Members of EUREKA (Group 6)
AmritaVijaykumar 43 Amit Talreja 40Himanshu Prabhu 32Jharna Serai 36Shweta Patel 29Shreyance Shah 37
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
SAMPLING
Census versus Sample
• Census in simple terms means to measure each element in the group or
population of interest.
• A part of a population, or a subset from a set of units, which is provided by
some process or other, usually by deliberate selection with the object of
investigating the properties of the parent population or set.
• Surveys of industrial consumers or of distributors of consumer products are
frequently in the form of a census.
• However there are certain reasons, which make census impractical or even
impossible. The reasons are as follows:
1. Cost: Cost is an obvious constraint on the determination of whether a census
should be taken. If information is desired on grocery purchase and use
behaviour (frequencies and amounts of purchase of each product category,
average amount kept at home and the like) and the population of interest is all
households in a country, the cost will preclude a census being taken. Thus a
sample is the only logical way of obtaining new data from a population of this
size.
2. Time : The kind of cost we have just considered is an outlay cost. The time
involved in obtaining information from either a census or a sample involves
the possibility of also incurring an opportunity cost. That is, the decision until
information is obtained may result in a smaller gain or a larger loss than would
have been the case from making the same decision earlier. The opportunity to
make more (or save more, as the case may be) is, therefore, foregone.
3. Accuracy : A study using a census, by definition, contains no sampling error. A
study using a sample may involve sampling error in addition to other types of
error. Other things being equal, a census will provide more accurate data than
a sample.
However it has been argued that a more accurate estimate of the population
of a country could be made from a sample than from a census. Taking a
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
census of a population on a “mail out – mail back” basis requires that the
names and addresses of almost all households be obtained, census
questionnaires mailed, and interviews conducted of those not responding.The questionnaires are sent to a population of which only about half have
completed high school. The potential for errors in a returned questionnaire is
therefore high.
4. Destructive nature of the measurement : Measurements are sometimes
destructive in nature. When they are, it is apparent that taking a census would
usually defeat the purpose of a measurement. If one were producing
firecrackers, electrical fuses, or gas seed, performing a functional use test on
all products for quality control purposes would not be considered from an
economic standpoint. A sample is then the only practical choice. On the other
hand, if the light bulbs, bicycles, or electrical appliances are to be tested, a
100% sample (census) may be entirely reasonable.
Advantages of Sampling
1. Sampling is cheaper than a census survey. It is obviously more economical, for
instance, to cover a sample of households than all households in a territory
although the cost per unit of study may be higher in a sample survey than in a
census.
2. Since magnitude of operations involved in a sample survey is small, both the
execution of the fieldwork and the analysis of the results can be carried out
speedily.
3. Sampling results in greater economy of effort as relatively small staffs is
required to carry out the survey and to tabulate and process the survey data.
4. A sample survey enables the researcher to collect more detailed information
than would otherwise be possible in a census survey. Also, information of a
more specialised type can be collected, which would not be possible in a
census survey on account of availability of a small number of specialists.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
5. Since the scale of operations involved in a sample survey is small, the quality of
interviewing, supervision and other related activities can be better than the
quality in a census survey.Limitations of Sampling
1. When the information is needed on every unit in the population such as
individuals, dwelling units or business establishments, a sample survey cannot
be of much help for it fails to provide information on individual count.
2. Sampling gives rise to certain errors. If these errors are too large, the results of
the sample survey will be of extremely limited use.
3. While in a census survey it may be easy to check the omissions of certain unitsin view of complete coverage, this is not so in the case of sample survey.
The Sampling Process
Step Description
1. Define the population The population is defined in terms of a) element, b)
units, c) extent and d) time.
2. Specify sampling frame The means of representing the elements of the
population – for example telephone book, map, or
city directory – are described.
3. Specify sampling unit The unit for sampling – for example, city block,
company, or household – is selected. The sampling
unit may contain one or several population
elements.
4. Specify sampling method The method by which sampling units are to be
selected is described.
5. Determine sample size The number of elements of the population to be
sampled is chosen.
6. Specify sampling plan The operational procedures for selection of the
sampling units are selected.
7. Select the sample The office and fieldwork necessary for the selection
of the sample are carried out.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Step 1: Define the population
It is the aggregate of all elements defined prior to selection of sample. A population
must be defined in terms of • elements,
• sampling units,
• extent and
• time.
Eliminating any one of these specifications leaves an incomplete definition of the
population that is to be sampled.
Step 2: Specify the Sampling frame
If a probability sample is to be taken, a sampling frame is required. A sampling frame
is a means of representing the elements of the population. A sampling frame may be
a telephone book, city directory, an employee roster, a listing of all students
attending a university, or a list of possible phone numbers.
Maps also serve frequently as sampling frames. A sample of areas within a city may
be taken and another sample of household then be taken within each area. City
blocks are sometimes sampled and all households on each sample block are
included. A sampling of street intersections may be taken and interviewers given
instructions as to how to take “Random walks”. From the intersection and select the
households to be interviewed.
A perfect sampling frame is one in which every element of the population is
represented once but only once. One does not need a sampling frame to take a non-
probability sample.
Step 3: Specify the sampling Unit
The sampling unit is the basic unit containing the elements of the population to be
sampled. It may be the element itself or a unit in which the element is contained. For
example, if one wanted a sample of males over 13 years of age, it might be possible
to sample them directly. In this case, the sampling unit would be identical with the
element. However, it might be easier to select households as the sampling unit and
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
interview all males over 13 years of age in each household. Here the sampling unit
and the population element are not the same.
Step 4: Specify the Sampling Methods
It indicates how the sample units are selected. One of the most important decisions
in this regard is to determine which of the two –probability and non-probability
sample –is to be chosen. Probability samples are also known as random samples
and non-probability samples as non-random samples.
There are various types of sample designs, which can be covered under two broad
groups – random or probability samples and non-random, or non-probability
samples.
Step 5: Determination of the Sample size
Traditional sampling theory generally ignores the concept of the cost versus the
value of the information to be provided by various sized samples. The problem of
determination of sample size is dealt later on in depth.
Step 6: Specify the Sampling Plan
The sampling plan involves the specification of how each of the decisions made thus
far is to be implemented. It may have been decided that the household will be the
element and the block the sampling unit. How is a household defined operationally?
How is the interviewer to be instructed to distinguish between families and
households in instances where two families and some distant relatives of one of
them are sharing the same apartment? How is the interviewer to be instructed to
take a systematic sample of households on the block? What should the interviewer
do when a housing unit selected is vacant? What is the callback procedure for
households at which no one is at home? What age respondent speaking for the
household is acceptable?
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Step 7: Select the Sample
The final step in the sampling process is the actual selection of the sample elements.
This requires a substantial amount of office and fieldwork particularly if personalinterview are involved.
Characteristics of a good Sample Design
A good sample design requires the judicious balancing of four broad criteria –goal
orientation, measurability, practicality and economy.
1. Goal orientation: This suggests that a sample design “should be oriented to the
research objectives, tailored to the survey design, and fitted to the surveyconditions”. If this is done, it should influence the choice of the population, the
measurement as also the procedure of choosing a sample.
2. Measurability: A sample design should enable the computation of valid
estimates of its sampling variability. Normally, this variability is expressed in the
form of standard errors in surveys. However, this is possible only in the case of
probability sampling. In non-probability samples, such a quota sample, it is not
possible to know the degree of precision of the survey results.
3. Practicality: This implies that the sample design can be followed properly in the
survey, as envisaged earlier. It is necessary that complete, correct, practical,
and clear instructions should be given to the interviewer so that no mistakes are
made in the selection of sampling units and the final selection in the field is not
different from the original sample design. Practicality also refers to simplicity of
the design, i.e. it should be capable of being understood and followed in actual
operation of the field work.
4. Economy: Finally, economy implies that the objectives of the survey should be
achieved with minimum cost and effort. Survey objectives are generally spelt
out in terms of precision, i.e. the inverse of the variance of survey estimates. For
a given degree of precision, the sample design should give the minimum cost.
Alternatively, for a given per unit cost, the sample design should achieve
maximum precision (minimum variance).
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
It may be pointed out that these four criteria come into conflict with each other in
most of the cases, and the researcher should carefully balance the conflicting
criteria so that he is able to select a really good sample design.
Sampling Techniques
Sampling techniques may be broadly classified as non-probability and probability
sampling techniques.
Non-probability sampling techniques:
1. It relies on the personal judgment of the researcher rather than t he chance toselect sample elements.
2. The researcher can arbitrarily or consciously decide which element to include in
the sample.
3. Non-probability may yield good estimates of the population characteristic.
However they do not allow for objective evaluation of the precision of the
sample results.
4. Since there is no way of determining the probability of selecting any particular
element for inclusion in the sample, the estimates obtained are not statistically
projectable to the population.
Probability sampling techniques:
1. Sampling units are selected by chance.
2. It is possible to pre-specify every potential sample of a given size that could be
drawn from the population, as well as the probability of selecting each sample.
3. Every potential sample need not have the same probability of selection, but it is
possible to specify the probability of selecting any particular sample of a given
size.
4. This requires not only a precise definition of the target population, but also a
general specification of the sampling frame. Because sample elements are
selected by chance.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
5. It is possible to determine the precision of the sample estimated of the
characteristics of interest. Confidence intervals, which contain the true
population value with a given level of certainty, can be calculated. This permitsthe researcher to make inferences of projections about the target population
from which the sample was drawn. Probability sampling techniques are
classified based on :
− Element versus cluster sampling
− Equal unit probability versus unequal probabilities
− Unstratified versus stratified selection
− Random versus systematic selection
− Single-stage versus multistage techniques
Diagrammatic representation of the sampling techniques.
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Sampling techniques
Non probabilitysampling
Probability samplingtechniques
QuotaSampling
ConvenienceSampling
JudgmentalSampling
SimpleRandomSampling
SystematicSampling
StratifiedSampling
Cluster Sampling
MultistageSampling
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Non-probability techniques:
Convenience Sampling
Definition
A non-probability sampling technique that attempts to obtain a sample of convenient
elements. The selection of sampling units is left primarily to the interviewer .
Explanation
1. It is a form of Non-Probability sampling.
2. It is mainly used for Dipstick studies. This type of sampling is normally used to
get basic information to take elementary decisions.
3. Convenience samples are often used in exploratory situations when there is a
need to get only an approximation of the actual value quickly and inexpensively.
4. Commonly used Convenience samples are associates and “the man on the
street”. Such samples are often used in the pre-test phase of the study, such as
pre-testing of a questionnaire.
Examples:
•Use of students, church groups, and members of social organizations,
• Mall-intercept interviews without qualifying the respondents,
• Department stores using charge account lists
• Tear out questionnaire included in a magazines, and
• People on the street interviews
Advantages
• Convenience sampling is the least expensive and least time consuming of all
sampling techniques.
• The sampling units are accessible, easy to measure and co-operative.
• This technique is used in exploratory research for generating ideas, insight or
hypothesis.
Disadvantages
• Convenience samples contain unknown amounts of both variables and
systematic selection errors.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
• These errors can be very large when compared to the variable error in a simple
random sampling of the same size.
Convenience samples are not representatives of any definable population. So theyare not recommended for descriptive or casual research.
Judgmental sampling
Definition
A form of convenience sampling in which the population elements are purposively
selected based on the judgment of the researcher.
Explanation A judgment sample is one in which there is an attempt to draw a representative
sample of the population using judgmental selection procedures. Judgment samples
are common in industrial market research.
Example
A sample of addresses taken by the municipal agency to which questionnaires on
bicycle riding habits were sent. A judgment sample was taken after researchers
looked at traffic maps of the city, considered the tax assessment on houses and
apartment buildings (per unit), and kept location of schools and parks in mind.
Advantages
• Judgmental sampling is low cost, convenient and quick.
• Judgmental sampling is subjective and its value depends entirely on the
researchers judgment, expertise and creativity.
• It is useful if broad population inferences are not required.
Disadvantage• It does not allow direct generalization to a specific population, usually because
the population is not defined explicitly.
Quota Sampling
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Example
If one wants to select a Quota sample of persons for a test of flavored tea and wants
to control (control variables are the parameters based on which he would like toclassify the universe) it by ethnic background, income bracket, age group and
geographical area. Then the sample taken would have the same proportion of
people in each ethnic background, income bracket, age group and geographical area
as the population.
Disadvantages
• Scope for high variances
•
Scope for sizable selection errors.
• Selection errors arise from the way interviewers select the persons/ variables to
fill the quota. Incorrect information of the proportions of the population in each of
the control variables, biases in the relationship of the control variables to the
variables being measured, and from other sources.
Probability Techniques:
Probability sampling techniques vary in terms of sampling efficiency. Sampling
efficiency is a concept that reflects a trade-offs between sampling cost and precision.
Precision refers to the level of uncertainty about the characteristic being measured.
The greater the precision, the greater the cost and most studies require trade-off.
Simple Random Sampling
Definition
A probability sampling technique in which each element in the population has aknown and equal probability of selection is known as simple random sampling
(SRS). Every element is selected independently of every other element and the
sample is drawn by a random procedure from a sampling frame.
Explanation
In random sampling, each element in the population has a known and equal
probability of selection. Furthermore, each possible sample of a given size (n) has a
known and equal probability of being the sample actually selected. This implies that
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
every other element is selected independently of every other element. The sample is
drawn by a random procedure from a sampling frame. This method is equivalent to a
lottery system in which names are placed in a container, the container is shaken,and the names of the winners are then drawn out in an unbiased manner.
To draw a simple random sample, the researcher first compiles a sampling frame in
which each element is assigned a unique identification number. Then random
numbers are generated to determine which element to include in the sample. The
random numbers may be generated with a computer routine or a table.
Advantages
• It is easy to understand
• The sample result may be projected to the target population.
Disadvantages
• It is often difficult to construct a sampling frame that will permit a simple random
sample to be drawn.
• SRS can result in samples that are very large or spread over large geographic
areas, thus increasing the time and cost of data collection.
• SRS often results in lower precision with larger standard errors than other
probability sampling techniques.
• SRS may or may not result in a representative sample. Although samples drawn
will represent the population well on average, a given simple random sample
may grossly misrepresent the target population. This more likely if the size of
the sample is small.
Systematic sampling
Definition
A probability sampling technique in which the sample is chosen by selecting a
random starting point and then picking every ith element in succession from the
sampling frame.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Explanation
In systematic sampling, the sample is chosen by selecting a random starting point
and then picking every i th
element in succession from the sampling frame. Thesampling interval, i , is determined by dividing the population size N by the sample
size n and rounding to the nearest integer.
Example
Suppose there are 100,000 elements in the population and a sample of 1000
desired. In this case the sampling interval, i , is 100. A random number between 1 to
100 is selected. If say number 23 is selected, the sample will then consists of
elements 23, 123, 223, 323, 423, 523, and so on.
Systematic sampling is similar to SRS in that each population element has a known
and equal probability of selection. However, it is different from SRS in that only the
permissible samples of size n that can be drawn have a known and equal probability
of selection. The remaining samples of size n have a zero probability of being
selected.
For systematic sampling, the researcher assumes that the population elements are
ordered in some respect. In some cases the ordering (alphabetic listing in a
telephone book) is unrelated to the characteristic of interest. In other instances, the
ordering is directly related to the characteristic under investigation. (Credit card
customers may be listed in order of outstanding balances. If the population elements
are arranged in a manner unrelated to the characteristic of interest, systematic
sampling will yield result quite similar to SRS.
On the other hand, when the ordering of the element is related to the characteristic
of interest, systematic sampling increases the representatives of the sample.
Advantages
• Systematic sampling is less costly and easier that SRS, because random
selection is done only once.
• The random numbers do not have to be matched with individual element as in
SRS. Since some lists contains millions of elements, considerable time can be
saved. This in turn again reduces the cost.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
• If the information related to the characteristic of interest is available for the
population, systematic sampling can be used to obtain a more representative
and reliable sample than SRS.
• Systematic sampling can even be used without knowledge of the composition
(elements) of the sampling frame.
Stratified Random Sampling
Definition
A probability sampling technique that uses a two-step process to partition the
population into subpopulations, or strata is known as stratified random sampling.Elements are selected from each stratum by a random procedure.
Explanation
Stratified Random Sampling emerges from the word Stratum. A Stratum in a
population is a segment of that population having one or more characteristics. E.g.
people in the age strata of 35-40, people in the income strata to Rs. 20000 p.m. etc
Stratified Sampling involves treating each stratum as a separate subpopulation for
sampling purposes, and from each stratum sampling units would be drawn randomly.
The reasons for conducting Stratified Random Sampling are:
• To reduce sampling error by ensuring representation from the population.
• The required sample size for the same level of sampling error will usually be
smaller.
As compared to other methods of sampling, in Stratified Random Sampling
representativeness to a certain degree is forced.
The greater degree to which there is similarity within stratum, smaller is the sample
size required to provide information about that stratum.
Thus the more homogeneous each stratum is with respect to the variable of interest
the smaller is the sample required.
Example
If the head of the household age strata (18-34, 35-49, 50+) are of interest in a study
on household spending habits on household furnishings, then each of these groups
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
would be taken separately for sampling purposes. That is, the total population could
be divided into age groups and a separate sample is drawn from each group.
Cluster Sampling
Definition
The target population is divided into mutually exclusive and collectively exhaustive
subpopulation called clusters. Then a random sample of clusters is selected based
on probability sampling techniques such as simple random sampling. For each
selected clusters, either all the elements are included in the sample or a sample of
elements is drawn probabilistically.
Explanation
• If all the elements in each selected cluster are included in the sample, the
procedure is called one stage cluster sampling.
• If a sample of elements is drawn probabilistically from each selected cluster, the
procedure is called two-stage cluster sampling.
• The key distinction between cluster sampling and stratified sampling is that in
cluster sampling only a sample of subpopulations (clusters) is chosen, whereasin stratified sampling all the subpopulations are selected.
• The objective of the cluster sampling is to increase the sampling efficiency by
decreasing costs.
Example
If the study requires studying the households in the city then in cluster sampling the
whole city is divided into Blocks and to take each household on each block selected.
Thus to get a representative whole of the universe.
Advantages
• Low population heterogeneity / high population homogeneity
• Low expected cost of errors.
• The main advantage of cluster sampling is the low cost per sampling unit as
compared to other sampling methods.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Disadvantage
• High potential of sampling error as compared to other methods.
• For eg: The lower cost per unit and higher sampling error potential of a cluster
sample is illustrated by considering a sample of 100 households to be selected
for personal interviews from a particular city. In this method the city would be
divided in blocks and 10 households from 10 selected blocks would be selected
and interviewed. Thus the cost of personal interview per unit will be low
because of the close proximity of the units in the cluster. This sample may not
be the exact representation of the entire city. Thus there is a possibility of
sampling error.
Single Stage V/s Multistage Sampling
Explanation
The number of stages involved in the sampling method is partially a function of the
number of sampling frame available. If a perfect frame were always available
complete with all the associated information one might want for purposes of
clustering and / or stratifying, there would be far fewer multiple samples taken than
there are now. In practice, it is not uncommon to have a first stage area sample of,
say, census tracts, followed by a second stage sample of blocks, and completed with
a systematic sample of households within each block. These stages would not be
necessary if a complete listing of households were available.
Example
AC Nielsen’s Multistage Sampling Procedure to select its PeopleMeter Panel
The first stage involves the selection of counties using a stratified random sample
based on population. Next within the selected counties there is a random selection of
blocks or enumeration districts. These blocks then go through a process called
prelisting. A trained field representative visits the selected blocks and creates a list of
all the individual hosing units. This list is then returned to the home office where it is
checked for internal consistency and external agreement with other data. Finally,
individual household units are randomly selected from each block.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
STRENGTHS AND WEAKNESS OF BASIC SAMPLING TECHNIQUES
Techniques Strengths Weaknesses
Non probability sampling
Convenience sampling Least expensive, least
time consuming, most
convenient
Selection bias; sample not
representative; not
recommended for
descriptive or casual
research.
Judgmental sampling Low cost, convenient, not
time consuming
Does not allow
generalization subjective
Quota sampling Sample can be controlled
for certain characteristics
Selection bias, no
assurance of
representativeness.
Snowball sampling Can estimate rare
characteristics
Time consuming
Probability Sampling
Simple Random Sampling Easily understood Difficult to construct
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
(SRS) Result projectable sampling frame;
expensive lower precison;
no assurance of representativeness
Systematic Sampling Can increase representa-
tiveness. Easier to
implement than SRS
sampling frame not
necessary
Can decrease
representativeness
Stratified sampling Includes all important
subpopulations; precision
Difficult to select relevant
stratification variables; not
feasible to stratify on
many variable; expensive
Cluster sampling Easy to implement, cost
effective
Imprecise; difficult to
compute and interpretresults
Choosing Non probability versus Probability Sampling
The choice between non probability and probability samples should be based on
considerations such as the nature of the research, relative magnitude of non
sampling versus sampling errors, variability in the population, as well as statistical
and operational considerations. For example,
Conditions favoring the use of
Non probability Probability
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
Factors Sampling Sampling
Nature of research Exploratory Conclusive
Relative magnitude of
sampling and non-sampling
errors
Non sampling errors
are larger
Sampling errors are
larger.
Variability in the population Homogenous (low) Heterogeneous (high)
Statistical consideration Unfavorable Favorable
Operational consideration Favorable Unfavorable
In exploratory research the findings are treated as preliminary and the use of
probability sampling may not be warranted. On the other hand, in conclusive
research in which the researcher wishes to use the results to estimate overall market
shares or the size of the total market, probability sampling is favored. Probability
samples allow statistical projection of the results to a target population.
For some research problems, highly estimates of population characteristic are
required. In these situations, the elimination of selection bias and the ability to
calculate sampling error make probability sampling desirable. However probability
sampling will not always result in more accurate results. If nonsampling errors are
likely to be an important factor, then non-probability sampling may be preferable, as
the use of judgment may allow greater control over the sampling process.
Another consideration is the homogeneity of the population with respect to the
variables of interest. A more heterogeneous population would favor probability
sampling, because it would be important to secure a representative sample.
Probability sampling is preferable from a statistical viewpoint, as it is the basis of
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
However, probability sampling is sophisticated and requires statistically trained
researcher. It generally costs more and takes longer than does nonprobabilitysampling. In many marketing research projects, it is difficult to justify the additional
time and expense. Therefore, in practice, the objectives of the study dictate which
sampling method will be used.
Methods of determining sample size
There are six methods of determining sample size in market research. They are
1. Unaided Judgement: When no specific method is used to determine sample
size, it is called Unaided Judgement. Such approach when used to arrive at
sample size gives no explicit considerations to either the likely precision of the
sample results or the cost of obtaining them (characteristics in which client
should have interest). It is an approach to be avoided.
2. All –You –Can –Afford : In this method, a budget for the project is set by some
(generally unspecified) process and, after the estimated fixed costs of designing
the project, preparing a questionnaire (if required), analysing the data, andpreparing the report are deducted, the remainder of the budget is allocated to
sampling. Dividing this remaining amount by the estimated cost per sampling
unit gives the sample size.
This method concentrates on the cost of the information and is not concerned
about its value. Although cost always has to be considered in any systematic
approach to sample size determination, one also needs to give consideration to
how much the information to be provided by the sample will be worth. This
approach produces sample sizes that are larger than required as well as sizes
that are smaller than optimal.
3. Required Size Per Cell: This method of determining sample size can be used on
simple random, stratified random, purposive and quota samples. For example,
in a study of attitudes with respect to fast food establishments in a local
marketing area it was decided that information was desired for two occupational
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
groups and for each of the four age groups. This resulted in 2 x 4 = 8 sample
cells. A sample size of 30 was needed per cell for the types of statistical
analyses that were to be conducted. The overall sample size was therefore 8 x30 = 240.
4. Use of Traditional Statistical Model: The formula for traditional statistical model
depends upon the type of sample to be taken and it always incorporates three
common variables
• an estimate of the variance in the population from which the sample is to be
drawn,
•
the error from sampling that the researcher will allow, and• the desired level of confidence that the actual sampling error will be within the
allowable limits.
The statistical models for simple random sampling include estimation of means
and estimation of proportion.
5. Use of Bayesian Statistical Model: The Bayesian model involves finding the
difference between the expected value of the information to be provided by the
sample size. This difference is known as expected net gain from sampling (ENGS). The sample size with the largest positive ENGS is chosen.
The Bayesian model is not as widely used as the traditional statistical models
for determining sample size, even though it incorporates the cost of sampling
and the traditional models do not. The reasons for the relative infrequent use of
Bayesian model are related to greater complexity and perceived difficulty of
making the estimates required for Bayesian model as compared to the
traditional models.
The Sampling Distribution
Sampling theory rests on the concept of a sampling distribution. Sampling
distribution includes
• Sampling distribution of the mean
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
• Sampling distribution of the proportion
Simulated sampling distribution of the mean
A sampling distribution of the mean is the relative frequency distribution of the
means of all possible samples of size n taken from a population of size N. The
definition specifies that all possible samples of size n from population of size N
should be taken, and the mean of each sample should be calculated and plotted in
relative frequency table.
A sampling distribution of the mean for simple random samples that are large(30 or more) has
• a normal distribution
• a mean equal to the population (M)
• a standard deviation, called the standard error of the mean( ), that is equal to
the population standard deviation( ) divided by the square root of the sample
size
FORMULA:
Standard deviation is called standard error of the mean to indicate to indicate that it
applies to a distribution of sample means and not to a single sample or a population.
A basic characteristic of a sampling distribution is that the area under it
(between any two points) can be calculated so long as each point is defined by the
number of standard errors it is away from the mean. The number of standard error, apoint is away from the mean is referred as the Z value for that point.
Sampling Distribution of the Proportion
A sampling distribution of the proportion is the relative frequency distribution of the
proportion (p) of all possible samples of size n taken from population of size N . A
sampling distribution of a proportion for a simple random sample has a
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
• normal distribution
• a mean equal to the population proportion (P)
• a standard error ( ) equal to
FORMULA:
The estimated standard error of the proportion (given a large sample size that is a
small proportion of the population) is
FORMULA:
where p represents the sample population.
Traditional Statistical Methods of Determining Sample
Determination of Sample Size in Problem Involving Means
Three kinds of specifications have to be made before the sample size necessary to
estimate the population mean can be determined. These are
1. Specification of error (e) that can be allowed –how close must the estimate be
(how accurate do we need to be)?
2. Specification of confidence coefficient –what level of confidence is required that
the actual sampling error does not exceed that specified (how sure do we want
to be that we have achieved our desired accuracy)?
3. Estimate of the population standard deviation( ) –what is the standard deviation
of the population (how “spread out” or diverse is the population)?
The three specifications are related in the following way:
Number of standard errors implied by confidence coefficient = allowable error
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
or in symbols,
FORMULA:
The only unknown variable is sample size (n). A simpler formula for the size of
simple random samples can be derived from the above equation.
FORMULA:
Determination of Sample Size in Problem Involving Proportions
The specifications that must be made to determine the sample size for an estimation
problem involving a proportion are very similar to those for a mean. They are
1. Specification of error (e) that can be allowed –how close must the estimate be?
2. Specification of confidence coefficient –what level of confidence is required that
the actual sampling error does not exceed that specified?
3. Estimate of the population proportion (P) using prior information –what is the
approximate or estimated population proportion?
Specifications, along with the sample size, collectively determine the sampling
distribution for the problem. Because sample size is the only remaining unknown, it
can be calculated. The above mentioned three specifications are related as follows:
Number of standard errors implied by confidence coefficient = allowable error
standard error
or in symbols,
FORMULA:
The formula for determining n that is sample size directly is
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
FORMULA:
Determination of Sample Size in Problems Involving Hypothesis Testing
A hypothesis is a proposition, which the researcher wants to verify. It may be
mentioned that while a hypothesis is useful, it is not always necessary. Many a time,
the researcher is interested in collecting and analysing data, indicating the main
characteristics without a hypothesis excepting the one, which he may suggest
incidentally during the course of his study. However, in a problem-oriented research,
it is necessary to formulate a hypothesis. In such research, hypothesis are generally
concerned with the causes of a certain phenomenon or a relationship between two
or more variables under investigation.
In order to determine the sample the size in a hypothesis testing problem involving
proportion, the following specifications must be made:
1. the hypotheses to be tested: A null and an alternate hypothesis are involved in
each hypothesis test. A null hypothesis, designated by Ho, is one that, if
accepted, will result in no option being formed and/or action being taken that is
different from those currently held or being used. The null hypothesis in the
problem just described is
Ho: order rate = 3.5%
The alternate hypothesis, designated by H1, is one that will lead to opinions
being formed and/or actions being taken that are different from those currently
held or being used. The alternate hypothesis here is
H1: order rate = 5.0%
Although null hypothesis is always explicitly stated, this is sometimes not true of
the alternate hypothesis. In those instances when it is not stated it is understood
that it consists of all values of the proportion not reserved by the null hypothesis.
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Chapter 7 –Sampling V.E.S College of Arts, Science & Commerce
In this situation if the alternate hypothesis were not explicitly stated, it would be
understood that it would be
H1: order rate (is not equal to) 3.5%2. the level of sampling error permitted in the test of each hypothesis: Two types of
error can be made in hypothesis testing problems. An error is made when null
hypothesis is true but the conclusion is reached that the alternate hypothesis
should be accepted. This is known as Type I error. The Type II error is made
when the alternate hypothesis is accepted
3. the test statistic to be used.
In order to determine the sample the size in a hypothesis-testing problem involving
means, the following specifications must be made:
1. the hypotheses to be tested,
2. the level of sampling error permitted in the test of each hypothesis,
3. the standard deviation of population, and
4. the test statistic to be used.
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