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Sampling:Design and Procedures
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SamplingThe researcher generally seeks to draw conclusions about largenumber of individuals- i.e. population or universe.
Since researchers operate with limited time, energy, and economicresources, they rarely study each and every member of a givenpopulation.
Instead, researchers study only a sample that is a small number ofobservations from the population.
Through the sampling process, the researchers seek to generalize froma sample (a small group) to the entire population from which it was
taken (a large group).
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11-3
Basic Concepts in Samples and Sampling
Population:the entire group under study asdefined by research objectives. Sometimescalled the universe.
Researchers define populations in specific termssuch as heads of households, individual persontypes, families, types of retail outlets, etc.Population geographic location and time of studyare also considered.
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11-4
Basic Concepts in Samples and Sampling
Sample:a subset of the population that shouldrepresent the entire group
Sample unit:the basic level ofinvestigationconsumers, store managers, shelf-facings, teens, etc. The research objectiveshould define the sample unit
Census:an accounting of the completepopulation
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11-5
Reasons for Taking a Sample
Practical considerations such as cost andpopulation size
Inability of researcher to analyze large quantitiesof data potentially generated by a census
Samples can produce sound results if properrules are followed for the draw
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11-6
Basic Sampling Classifications
Probability samples:ones in which members ofthe population have a known chance (probability)of being selected
Non-probability samples:instances in which thechances (probability) of selecting members fromthe population are unknown
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11-7
Classification of Sampling Techniques
Fig. 11.2
Sampling Techniques
NonprobabilitySampling Techniques
ProbabilitySampling Techniques
Convenience
Sampling
Judgmental
Sampling
Quota
Sampling
Snowball
Sampling
Systematic
Sampling
Stratified
Sampling
Cluster
Sampling
Other Sampling
Techniques
Simple Random
Sampling
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11-8
Nonprobability Sampling
Techniques
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11-9
Convenience Sampling
Convenience sampling attempts to obtaina sample of convenient elements. Often,respondents are selected because theyhappen to be in the right place at the right
time.
use of students, and members of socialorganizations
people on the street interviews
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11-10
Judgmental Sampling
Judgmental sampling is a form of conveniencesampling in which the population elements areselected based on the judgment of the researcher.
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11-11
Quota Sampling
Quota sampling may be viewed as two-stage restricted judgmentalsampling.
The first stage consists of developing control categories, or quotas,of population elements.
In the second stage, sample elements are selected based onconvenience or judgment.
Population Samplecomposition composition
ControlCharacteristic Percentage Percentage Number
SexMale 48 48 480Female 52 52 520
____ ____ ____100 100 1000
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11-12
Snowball Sampling
In snowball sampling, an initial group ofrespondents is selected, usually at random.
After being interviewed, these respondents areasked to identify others who belong to the targetpopulation of interest.
Subsequent respondents are selected based onthe referrals.
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11-13
Probability
Sampling Techniques
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Random sampling gives each and every member ofpopulation an equal chance of being selected for the
sample.
One way to conduct a simple random sample is to assign anumber to each element in the population, write thesenumber on individual slips of paper, toss them into a hat,
and draw the required member of slip (the sample size, n)from the hat.
Sometimes the elements of the population are already numbered. Forexample population of drivers have driving license number, allemployees of a firm have employee number and all university Student
have student card number.
Simple Random Sampling
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11-15
Simple Random Sampling
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11-16
Systematic Sampling
The sample is chosen by selecting a random starting
point and then picking every ith element insuccession from the sampling frame.
The sampling interval, i, is determined by dividing the
population size N by the sample size n and roundingto the nearest integer.
For example, there are 100,000 elements in thepopulation and a sample of 1,000 is desired. In thiscase the sampling interval, i, is 100. A randomnumber between 1 and 100 is selected. If, forexample, this number is 23, the sample consists ofelements 23, 123, 223, 323, 423, 523, and so on.
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11-17
Systematic Sampling
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11-18
This method is used when the populationdistribution of items is skewed. It allows usto draw a more representative sample.Hence if there are more of certain type of
item in the population the sample has moreof this type and if there are fewer of anothertype, there are fewer in the sample.
Stratified Sampling
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A stratified random sample is obtained by dividing thepopulation into more homogeneous groups or strata from
which simple random samples are then taken. Examples ofcriteria for separating a population into strata are:
1. Sex : male, female
2. Age: under 20, 21-30, 31-40, 41-50, 51-60, over 603. Household income: under Rs. 8,000, Rs. 8,000-
19,999, Rs.20,000-50,000, over Rs. 50,000
Stratified Sampling
11-20
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11-20
Stratified Sampling
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A cluster sample is a simple random sample of group or
clusters of elements.
Suppose we want to estimate the average annualhousehold income in a large city.
A less expensive alternative would be to let each blockwithin the city represent a cluster. By reducing the distance,cluster sampling reduces the cost.
Cluster Sampling
11-22
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Cluster Sampling
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Sampling ErrorSampling error is an error that we expect to occur when wehave a statement about a population that is based only onthe observation contained in a sample taken from thepopulation.
We can use statistical inference to estimate the mean ofthe population, if we are willing to accepts less than 100%accuracy. The difference between the true (unknown)value of the population mean and its sample estimate isthe sampling error.
Then only way we can reduce the expected size of thiserror is to take a larger sample.
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Sampling Error
Population
70 80 93
86 85 90
56 52 67
40 78 57
89 49 48
99 72 3096 94
=
96 40 72
99 86 96
56 56 49
52 67 56
Sample A Sample B Sample C
Final Examination Grade
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Sampling Error
75.75 25.62 25.68
Final Examination Grade
96 40 72
99 86 96
56 56 49
52 67 56
303 249 273
Population
Sample A Sample B Sample C
70 80 93
86 85 90
56 52 67
40 78 57
89 49 48
99 72 30
96 94
= 71.55
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Non Sampling Error
Non sampling error is more serious, because
taking a larger sample wont diminish the size, orthe possibility of occurrence, of this error. Even
census can contain non sampling error.
Non sampling errors are reporting error, nonresponse error, data entry error.
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Sampling Distribution of Mean
Consider the population created by throwing
a fair die many times, with the randomvariable x indicating the number of spotsshowing on any one throw.
Find the probability distribution of therandom variable x. Its mean andVariance.
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Probability distribution of X
x 1 2 3 4 5 6P(x) 1/6 1/6 1/6 1/6 1/6 1/6
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Mean and Variance of X
)(xE )(xPx
6
16
6
15
6
14
6
13
6
12
6
11
5.3
)(2
xV )(.)( 2 xPx
6
15.36
6
15.35
6
15.34
6
15.33
6
15.32
6
15.31
222222
92.2
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Sampling Distribution of Mean
XX
All Samples of Size Two and Their Means
Sample Sample X Sample
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Sampling Distribution of Mean
XX
All Samples of Size Two and Their Means
Sample
1, 1 1.0 3, 1 2.0 5, 1 3.0
1, 2 1.5 3, 2 2.5 5, 2 3.5
1, 3 2.0 3, 3 3.0 5, 3 4.0
1, 4 2.5 3, 4 3.5 5, 4 4.5
1, 5 3.0 3, 5 4.0 5, 5 5.01, 6 3.5 3, 6 4.5 5, 6 5.5
2, 1 1.5 4, 1 2.5 6, 1 3.5
2, 2 2.0 4, 2 3.0 6, 2 4.0
2, 3 2.5 4, 3 3.5 6, 3 4.5
2, 4 3.0 4, 4 4.0 6, 4 5.0
2, 5 3.5 4, 5 4.5 6, 5 5.5
2, 6 4.0 4, 6 5.0 6, 6 6.0
Sample X Sample
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Sampling Distribution of
P( )
1.0 1/36
1.5 2/362.0 3/36
2.5 4/36
3.0 5/36
3.5 6/36
4.0 5/36
4.5 4/36
5.0 3/36
5.5 2/36
6.0 1/36
X X
X
Calculate the mean and
variance of this sampling Distribution.
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Mean and Variance of the sampling Distribution)(xE )(xPx
36
10.6.....
36
25.1
36
10.1
5.3
)(2
xV )(.)( 2 xPx X
36
15.36...........
36
15.35.1
36
15.30.1
222
46.1
X
92.22 46.1
2x
nx
22
nx
xn22
.
X
Z
X
XZ
n
XZ
11-34
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Central Limit Theorem
1. The random variablexhas a distribution
(which may or may not be normal) with mean and standard deviation .
2. Samples all of the same size nare randomly
selected from the population ofxvalues.
Given:
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Central Limit Theorem
Conclusions:
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Central Limit Theorem
1. The distribution of samplex will, as the
sample size increases, approach a normal
distribution.
Conclusions:
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Central Limit Theorem
1. The distribution of samplex will, as the
sample size increases, approach a normal
distribution.2. The mean of the sample means will be the
population mean .
Conclusions:
11-38
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Central Limit Theorem
1. The distribution of samplex will, as the
sample size increases, approach a normal
distribution.2. The mean of the sample means will be the
population mean .
3. The standard deviation of the sample means
will approach n
Conclusions:
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Practical Rules Commonly Used:
1. For samples of sizen
larger than 30, thedistribution of the sample means can be
approximated reasonably well by a normal
distribution. The approximation gets better as
the sample size nbecomes larger.
2. If the original population is itself normally
distributed, then the sample means will be
normally distributed for any sample size n(not just the values of nlarger than 30).
11-40
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Notation
the mean of the sample means
x=
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Notation
the mean of the sample means
the standard deviation of sample mean
x=
x= n
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Notation
the mean of the sample means
the standard deviation of sample mean
(often called standard error of the mean)
x=
x= n
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Example
A population consists of the five numbers
2,3,6,8, and 11. Consider all possible samples ofsize 2 that can be drawn with replacement fromthis population.
Find (a) the mean of the population,(b) the standard deviation of the population,(c) the mean of the sampling distribution of
means, and
(d) the standard deviation of the samplingdistribution of means (i.e., the standarderror of means).