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Sampling
1. Unit 2: Sampling theory, Sampling
distribution and Estimation 12 hours
2.1 Define terms used in sampling: population, study population, reference population,
sample, sampling unit, sampling frame, parameter and statistic
2.2 Census and sample survey; merits and demerits
2.3 Criteria for selection of appropriate sampling technique in survey
2.4 Differentiation between probability and non-probability sampling
2.5 Describing probability sampling technique: simple random, stratified, systematic,
cluster, multistage and probability proportionate to size sampling (PPS sampling)
2.6 Describing non-probability sampling technique: convenience, purpose, judgmental,
quota sampling, and snowball
2.7 Lot quality assurance sampling
2.8 Sampling errors and non-sampling errors
2.9 Sampling distributions
a. Central limit theorem
b. Estimation: point and interval estimation of the mean, proportion of distribution and confidence
interval
c. Standard error of mean and proportion for finite and infinite case
d. Distribution of sample mean, sample proportion and difference between two sample means and
two sample proportions
e. Determination of sample size by appropriate using formulas
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2. 2.1 Define terms used in sampling:
population, study population, reference
population, sample, sampling unit,
sampling frame, parameter and statistic
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3. Why sample?
Save time and money
More effort to ensure high-quality measurement
if smaller sample
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4. Criteria for a Good Sample
Samples can be selected in two ways
• Purposive Sample
• Random sample
Purposive sample or Non-probability Sample
Sample units are selected from the population to
suit a specific purpose as per the desire of the
investigator
These samples serves very limited purpose
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5. Sample Size
• The question remains as to what constitutes an
adequate sample size.
• Samples should be as large as a researcher can obtain
with a reasonable expenditure of time and energy.
• The recommended minimum number of subjects are
as follows for the following types of studies:
– 100 for a Descriptive Study
– 50 for a Correlational Study
– 30 in each group for Experimental and Causal-Comparative
Study
The use of 15 subjects per group should probably be replicated
6. Examples of population and samples
Situation Population Sample
Sex ratio of births the world’s birth some hospital records
Is my well water safe? Water in well Vial in lab
Medical study people in Nepal some subjects
6
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7. Technical Terms
• A sampling frame is a list of sampling units.
• A sample is a collection of sampling units drawn from a
sampling frame.
• Parameter: numerical characteristic of a population
• Statistic: numerical characteristic of a sample
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8. Sampling Terms
• Target Population:
– Specific pool of cases or sample that researchers
wants to study.
• Sampling Ratio:
– Size of the sample / size of the target population.
– For example
– The population has 50,000 people, and a researcher
draws a sample of 150 from it. Researchers
sampling ratio is 150/50,000 = 0.03 or 0.3 percent.
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9. Sampling
• Sampling is a process of systematically
selecting cases for inclusion in a research
project.
• Sampling involves the selection of a number of
study units from a defined study population.
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10. What is Sampling?
• Sampling is the process of selecting observations (a sample) to
provide an adequate description and robust inferences of the
population
• It is the process of selecting a sufficient number of elements
from the population so that by studying the sample, and
understanding the properties or characteristics of the sample
subjects, it would be possible to generalise the properties or
characteristics to the population elements.
• The more representative the sample is of the population, the
more generalizable are the findings of the research
10
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11. What is Sampling….
Population
Sample
Using data to say something (make an inference) with confidence, about
a whole (population) based on the study of a only a few (sample).
Sampling
Frame
Sampling Process
What you
want to talk
about
What you
actually
observe in
the data
Inference
11
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13. Sampling Frame
• List of population units from which the sample units
are to be selected.
• If Sampling frame - not available, Prepare it –
• From - Telephone Directories, Tax Records, Driver’s
License Records.
• A good sampling frame is crucial to good sampling.
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17. Steps in Sampling
• Defining the population to be covered
• Defining sampling units
• Acquiring frame / list of the population
elements
• Deciding about the size of the sample
• Deciding about the type of the sample to
be used and
• Testing the reliability of the sample
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18. Levels of sample selection
Target population(s) - population(s)
to which the results can be applied
Source population - population(s)
from which eligible subjects are drawn
Eligible population - population(s) of
subjects eligible for inclusion in study
Study participants - individuals who
contribute data to the study: results apply
directly only to these subjects
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20. Factors to be considered in sampling process
* Study Objectives
- Descriptive vs. Analytic
* Selection Criteria
- Inclusion & Exclusion
- Probability vs. Non-probability
* Sampling Frame & Sampling Units
- Unit
- Time & Place
* Strategies in approaching sampling units
- Identification & Classification
- Willing/Consent to Participate
O P D
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21. 1 ) Is the population from which the sample is
drawn consistent with the population of interest
for the study? (generalizability/external validity)
2) Have the methods for selecting subjects or
units
biased the sample? (bias/internal validity)
3) Are the estimates or sample statistics
sufficiently precise for the study purpose?
(power/sample size/precision)
Sampling Issues
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22. Sample Size Determination
The researchers should consider various factors as follows;
1) Expense, time, labor and data collection tools.
2) Population size.
3) Similarity; the more of similar population, the usage of
samples size will be small. If the population is very
different, there will be plenty of variances, therefore, the
usage of sample size will be large.
4) Accuracy
5) Sampling error
6) Reliability
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23. Requirements for estimation of Size of the sample
An approximate idea of the estimate of the characteristic
under observation
Variability of this characteristic from unit to unit in the
population
Initial knowledge of the desired accuracy of the estimate
of the characteristic under study
Probability level within which the desired precision of
estimates is to be maintained
Availability of the experimental material, resources and
other practical considerations
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24. Factors to determine sample size
• Size of population
• Resources – subjects, financial, manpower
• Method of Sampling- random, stratified
• Degree of difference to be detected
• Variability (S.D.) – pilot study, historical
• Degree of Accuracy (or errors)
- Type I error (alpha) p<0.05
- Type II error (beta) less than 0.2 (20%)
- Power of the test : more than 0.8 (80%)
• Statistical Formulae
• Dropout rate, non-compliance to Rx
24
25. Sample size determination:
1. Sample size determination by using criteria
- Hundreds of populations; use 15-30% of sample size
- Thousands of populations; use 10-15% of sample size
- Tens Thousands of populations; use 5-10% of sample
size
- Hundreds Thousands of population; use 1-5% of sample
size
2. Sample size determination by calculation formulas.
3. Sample size determination by using tables.
4. Sample size determination by calculating computer
programs.
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26.
27. Calculation of sample size for field surveys
For Field surveys to estimate the prevalence rates
n = ( 4pq / L2
)
where n is the required sample size
p is the approximate prevalence rate
q= (1-p)
L is the permissible error in the estimate of p
stimates calculated with this sample size would be
true in 95 out of 100 samples 100 samples
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Cochran Formula (Cochran, 1977)
28. p = 40%
q = (100-40 ) = 60%
L = 5% of 40% = 2%
n = ( 4 x 40 x 60) / ( 22
)
= (4 x 40 x 60 ) / 4
= 2400
2,400 persons are to be examined to estimate the
prevalence rate with 5% error.
If we increase the error percentage to 10%
L=10% of 40% = 4
n = ( 4 x 40 x 60 ) / 16
= 600
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31. For Quantitative data, the sample size is
calculated from the formula.
n = (tα
2
x s2
) / e2
n is the desired sample size
s is the standard deviation of observations
e is the permissible error in the estimation
of mean difference
tα is the value of t at 5% level from t tables
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32. In a survey to estimate the haemoglobin level
If mean Hb% level is approximately 12gm%
Standard deviation 1.5gm%
Permissible error 0.5gm%
s=1.5gms
e = 0.5gms
t0.05 can be taken as 2, as it is conventional to use 5% level
of significance
n = { 22
x (1.5) 2
} / (0.52
)
= (4 x 2.25 ) / (2.25 )
= 36 persons
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33. 9/13/2016 Ashok Pandey 33
Formulas for Sample Size (SS)
For Infinite Sample Size SS = [Z2p (1 − p)]/ C2
For Finite Sample Size SS/ [1 + {(SS − 1)/Pop}]
Where,
•SS = Sample size
•Z = Given Z value
•p = Percentage of population
•C = Confidence level
•Pop = Population
Check: Z Score Table
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Question: Find the sample size for a finite and infinite
population when the percentage of 4300 population is 5,
confidence level 99 and confidence interval is 0.01?
Solution:
Z = From the z-table, we have the value of confidence
level, that is 2.58 by applying given data in the formula:
SS=(2.58)2×0.05×(1−0.05)0.012=316
Sample size for finite population
=3161+316−14300=294
New SS = 294
35. Probability proportionate to size
sampling (PPS sampling)
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Probability proportional sampling, usually
known as probability proportional to size (PPS)
sampling, is an unequal probability sampling
technique, in which the probability of selection
for each sampling unit in the population is
proportional to an auxiliary variable.
36. 36
Sampling with Probability Proportional
to Size (PPS)
• Probability of selection is related to an auxiliary variable, Z,
that is a measure of “size”
Example
Number of households
Area of farms
• “Larger” units are given higher chance of selection than
“smaller” units
• Selection probability of ith unit is
i = 1,2, … , N
PPS Sampling
∑
=
= N
i
i
i
i
Z
Z
p
1
37. 37
PPS Selection Procedures
• Cumulative total method: with replacement
• Cumulative total method: without replacement
• PPS systematic sampling
• Lahiri’s method
PPS Sampling
38. 38
Cumulative Total Method
• Sampling unit: village
• Measure of size: number of
households in village
• Selection probability:
Select a sample of 5 villages using
varying probability WR sampling,
the size being the number of
households
Solution
PPS Selection
pi
39. • Write down cumulative total
for the sizes Zi, i=1,2..N
• Choose a random number r
such that 1 ≤ r ≤ Z
• Select ith population unit if
• Ti-1 ≤ r ≤ Ti where
Ti-1 = Z1 + Z2 + .. + Zi-1
and
Ti = Z1 + Z2 + .. + Zi
Cumulative Total Method (Contd.)
PPS Selection
40. 40
• To select a village, a random
number r, 1 ≤ r ≤ 700, is selected.
• Suppose r = 259,
Since 231 ≤ 259 ≤ 288, the 7th
village is therefore selected. The
next 4 random numbers to be
considered are 548, 170, 231, 505.
Hence the required sample selected
using PPS with replacement are
16th, 5th, 7th, 15th .
Note: The 7th village is selected twice.
Cumulative Total Method (Contd.)
PPS Selection
41. 41
• For a PPSWR selection therefore
the sample would be: 16th, 5th, 7th,
15th , with 7th village repeated.
• For a PPSWOR selection, we have
to continue further to get 5 distinct
units in the sample.
• Suppose the next random selected is
r = 375,
The required PPSWOR sample would
be 16th, 5th, 7th, 15th & 11th .
Cumulative Total Method (Contd.)
PPS Selection
42. 42
• Derive cumulative totals for the sizes
Zi, i=1,2..N, and allot random
numbers to different units.
• Calculate interval k = ZN /n (in this
case 700/5 = 140)
• Select a random number r (say 101)
from 1 to k; and obtain r+k, r+2k,
r+3k, …, r+(n-1)k
• In this case, the selected cumulative
sizes are 101, 241, 382, 523 & 664.
PPS Systematic
PPS Selection
43. 43
• Thus the selected units are:
3rd (for 101),
7th (for 241),
11th (for 382),
15th (for 523) &
20th (for 664)
• Note: If any unit has size greater
than k, it may be selected more than
once.
PPS Systematic (Contd.)
PPS Selection
44. 44
Lahiri’s Method
• A procedure which avoids the need of calculating cumulative
totals for each unit has been given by Lahiri (1951)
Steps involved;
1. Select a random number i from 1 to N
2. Select another random number j, such that 1 ≤ j ≤ M, where
M is either equal to the maximum of sizes Zi, i =1,2,.. N, or is
more than the maximum size in the population.
3. If j ≤ Zi , the ith unit is selected, otherwise, the pair (i, j) of
random numbers is rejected and another pair is chosen by
repeating the steps (1) and (2)
PPS Selection
45. 45
Select a sample of 2 villages using
varying probability WR sampling, the
size being the number of households
Solution
• N =20, n=2 , M =58
• Select a random number i, 1 ≤ i ≤ 14,
• Then a second random number j,
1 ≤ j ≤ 58,
• Suppose the 1st pair of random
number is (2, 30). Since 30 ≤ 45 thus
2nd village is selected .
PPS Sampling
Lahiri’s Method
46. 46
Solution (continued)
• Similarly we find the next pair
of random number (12, 47)
since 47 >30, the 12th village is
not selected The 3rd pair ot
random numbers (7, 40) results
in the selection of 7th village
since 40 ≤ 58
• Hence, the selected sample are
2nd and 7th villages.
PPS Sampling
Lahiri’s Method
49. Probability sampling
• Is the random selection of elements from the population.
• Probability sampling is a technique in which the researcher chooses
samples from a larger population using a method based on probability
theory. For a participant to be considered as a probability sample, he/she
must be selected using a random selection.
• This statistical method used to select a sample from a population in such a
way that each member of the population has a known, non-zero chance of
being selected. The most critical requirement of probability sampling is that
everyone in your population has a known and equal chance of getting
selected.
• Probability sampling uses statistical theory to randomly select a small
group of people (sample) from an existing large population and then predict
that all their responses will match the overall population.
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51. Advantages of probability sampling
1. It’s Cost-effective: This process is cost and time
effective. A larger sample can also be chosen based on
numbers assigned to the samples. Then you can choose
random numbers from the more significant sample.
2. It’s simple and straightforward: Probability
sampling is an easy way as it does not involve a
complicated process. It’s quick and saves time. The time
saved can thus be used to analyze the data and draw
conclusions.
3. It is non-technical: This sampling method doesn’t
require any technical knowledge because of its
simplicity. It doesn’t require intricate expertise and is
not at all lengthy. You can also avoid sampling errors.
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52. Methods of Sampling - Probability Sampling
·
·
1. Simple Random Sampling:
Objective: To select n units out of N such that each NCn has
an equal chance of being selected.
Procedure: Use a table of random numbers, a computer
random number generator, or a mechanical device to select
the sample.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
N = 50
n = 10 Sampling Frame
1 2
3 4
5 ….. 49 50
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53. Simple Random Sampling
• Lottery Method
– With replacement (Unrestricted random sampling )
– Without replacement (Restricted random sampling)
• Random number table method
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55. Mixed Sampling
• Notice that from the first interval the choice of an
element is on a random basis but the choice of the
elements from subsequent intervals is dependent upon
the choice from the first, and hence cannot be classified
as a random sample. For this reason it has been
classified here as mixed sampling.
• Although the general procedure for selecting a sample
by the systematic sampling technique is described
above, one can deviate from it by selecting a different
element from each interval with the Simple Random
Sampling technique. By adopting, systematic sampling
can be classified under probability sampling design.
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57. Systematic random sampling
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Procedure: Use a table of random numbers,
a computer random number generator, or a
mechanical device to select the sample.
Example
k = N/n
=1920/210=9
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
19 20 21 22 23 24 25 26 27
28
….. 190
58. Systematic Sampling
Advantages:
1) Do not need to know the entire sampling frame in
advance, just the total number of sampling units;
can be constructed as the study progresses, so ordering
is by time of accrual
2) Often simpler to implement under field conditions than
other sampling methods (e.g. easier to have interviewer
to visit every 5th house on the block rather than to
determine which houses are to be visited by means of a
table of random numbers)
3) If a trend is present in the sampling frame, units will
small values to units with large values, than a systematic
sample will ensure coverage of the spectrum of units
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59. Systematic Sampling
Disadvantage:
1) If cyclical trends exist in the data, a poor estimate of
the mean will be obtained
(e.g., the prevalence of bronchitis would be considerable
higher if one sampled every 12th month starting in January
than every 12th month starting in July)
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60. Methods of Sampling - Probability Sampling
·
·
2. Stratified Random Sampling,
(quota random sampling)
Procedure:
1. Divide the population into non-overlapping homogeneous
subgroups (i.e., strata) N1, N2, N3, ... Ni, such that N1 + N2 + N3
+ ... + Ni = N.
2. Do a simple random sample of f = n/N in each strata.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
Sampling Frame
.. 12 ….. 20 ..
1
30
31 … 50
N = 50 (N1 30; N2 =20)
n = 10
f = 10/50 = 0.2
thus
n1 = 0.2 x 30 = 6
n2 = 0.2 x 20 = 4
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61. Stratified Sampling
Advantages:
1) Investigator can make certain that each subgroup in the
population is represented; guarantees mean heights of each
subgroup can be estimated separately in addition to the
overall mean
2) When population divided into subgroups that are more
homogeneous than population as a whole, a more
precise
estimate of population parameters are obtained than
when
a simple random sample is taken, because variance
computed from the entire sample is based on each
within-stratum variance
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62. Stratified Sampling
Advantages:
3) Strata can be constructed so that those that are least
expensive to study or have the largest variances or largest
number of individuals can be sampled most heavily
4) Administratively it may be easier to deal with strata
Disadvantage:
Loss of precision can occur if very small numbers of units a
sampled within individual strata; although under most
circumstances, greater precision is attained by stratum-spec
estimates of a homogeneous group
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64. Cluster Sampling
• A researcher first samples clusters, each of which contains
elements, then draws a second sample from within the clusters
selected in the first stage of sampling. Clusters are often
geographic units (e.g. districts, villages) or organizational units
(e.g. clinics, training groups).
• Cluster sampling is usually less expensive than simple random
sampling, but it is less accurate. A researcher who uses cluster
sampling must decide the numbers of clusters and the number of
elements within clusters.
• For example
– In a study of the KAP related to family planning in rural communities of a
region, a list is made of all the villages. Using this list, a random sample
of villages is chosen and all the adults in the selected villages are
interviewed.
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65. Cluster Sampling
Advantages:
1) One need not enumerate the entire population in
advance, just the total number of clusters; then just
the units of selected clusters
2) More economical than simple random sampling
Disadvantage:
Factor representing cluster effect must be accounted for
in analysis, complex (violates assumption of independence)
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66. Multistage (Multi-cluster) Sampling
Primary (larger) sampling units are first selected from a population
Secondary (smaller) sampling units (e.g. city blocks) are sampled
from within each chosen primary unit
Can be extended so that tertiary units (e.g. households) or further
(e.g. individuals) are selected within these secondary units
Differs from clustering in that secondary units are sampled,
whereas in cluster sampling all secondary units are included.
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67. Non-probability sampling
Non-probability sampling is a method of
selecting units from a population using a
subjective (i.e. non-random) method. Since
non-probability sampling does not require a
complete survey frame, it is a fast, easy and
inexpensive way of obtaining data. However, in
order to draw conclusions about the
population from the sample, it must assume
that the sample is representative of the
population.
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68. Non-probability sampling
There are five key reasons behind this trend:
• the decline in response rates in probability
surveys;
• the high cost of data collection;
• the increased burden on respondents;
• the desire for access to real-time statistics,
and
• the surge of non-probability data sources
such as web surveys and social media.
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69. Example
1. An example of convenience sampling would be using
student volunteers known to the researcher. Researchers
can send the survey to students belonging to a particular
school, college, or university, and act as a sample.
2. In an organization, for studying the career goals of 500
employees, technically, the sample selected should have
proportionate numbers of males and females. Which
means there should be 250 males and 250 females. Since
this is unlikely, the researcher selects the groups or strata
using quota sampling.
3. Researchers also use this type of sampling to conduct
research involving a particular illness in patients or a rare
disease. Researchers can seek help from subjects to refer
to other subjects suffering from the same ailment to
form a subjective sample to carry out the study.
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70. Convenience or Haphazard Sampling
• Cheap and quick
• Study units that happen to be available at the time of data
collection are selected in the sample
• Choose when population is not clearly defined, sampling
units are not clear.
• When a researcher haphazardly selects cases that are
convenient researcher can easily get a sample that seriously
misrepresents the population.
– Some units over selected, other under selected or missed altogether
– Causes ineffective, unrepresentative samples.
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71. Convenience or Haphazard Sampling
• Examples:
• The person-on-the-street interview conducted by television
programs is an example of a haphazard sample. Television
interviewers go out on the street with camera and microphone
to talk to a few people who are convenient to interview.
• A researcher wants to study the attitudes of villagers toward
family planning services provided by MCH clinic. He decides
to interview all adult patients who visit the out-patient clinic
during one particular day. This is more convenient than taking
a random sample of people in the village, and it gives a useful
first impression.
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72. Purposive or Judgmental Sampling
• Researcher deliberately selects certain units for study
from the population
• Choice of the selection is supreme and nothing is left to
chance
Appropriate in 3 situations:
• For unique cases selection that are especially informative
• For selection of members of difficult to reach, specialized
population
• Another situation for purposive sampling occurs when a
researcher wants to identify particular types of cases for
in-depth investigation.
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73. Types of purposive sampling
• Maximum variation sampling.
• Homogeneous sampling.
• Typical case sampling.
• Extreme (or deviant) case sampling.
• Critical case sampling.
• Total population sampling.
• Expert sampling.
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74. Quota Sampling
• Pre plan number of subjects in specified
categories (e.g. 100 men and 100 women)
• Once the quota sample fixes the categories and
number of cases in each category, researcher uses
convenient sampling
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75. Snowball Sampling
• Selecting the cases in a network
• Begins with one or a few people or cases and
spread out on the basis of links to the initial cases
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76. Snowball
• Recruiting people based
on recommendation of
people you have just
interviewed
• Useful for studying
invisible/illegal
populations, such as drug
addicts
Main
person
Friend
Friend
Friend
Friend
Friend
Friend
Friend
Friend
Friend Friend Friend Friend
Friend Friend Friend Friend
Friend
Friend
Friend
77. Respondent-driven sampling (RDS)
RDS is a type of snowball sampling used for
analyzing characteristics of hidden or hard-
to-reach populations.
It was developed in 1997 by Dr. Douglas
Heckathorn, a professor of Sociology at
Cornell and has been applied to groups
ranging from men who have sex with men,
injection drug users, and children living on
the street
79. Others
• Crowdsourcing: Crowdsourcing has
been defined slightly differently by
researchers from various areas.
• Web panels: A web panel (or online or
internet panel) could be defined as an
access panel of people willing to respond
to web questionnaires.
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80. 80
What is LQAS
(Lot Quality Assurance Sampling)?
• LQAS is a sampling method developed to control the
quality of manufactured goods produced in ‘lots.’
• LQAS takes a small random sample and tests the sample
for quality.
• The sample will tell if program activities (agriculture,
health, etc.) are meeting/not a performance benchmark
• The sample size is chosen so that there is a high
probability of determining what indicators in a given
activity are meeting or not meeting the performance
benchmark.
83. ERRORS IN RESEARCH
Non-sampling errors
• Coverage errors
• Response errors
• Non-response errors
• Processing errors
• Measurement errors
• Estimation errors
• Analysis errors
Sampling errors
• Sample size
• Population size
• Variability of the characteristic of interest
• Sample plan
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Ashok Pandey
84. Types of Survey Errors
• Coverage error
• Non response error
• Sampling error
• Measurement error
Excluded from
frame.
Follow up on
non responses.
Chance
differences from
sample to sample.
Bad Question!
85. 1. Sampling error
– random error- the sample selected is not
representative of the population due to chance
– The uncertainty associated with an estimate that is
based on data gathered from a sample of the
population rather than the full population is known as
sampling error.
– Sampling errors are the random variations in the
sample estimates around the true population
parameters.
85
86. Sampling error cont’d…
the level of it is controlled by sample size
a larger sample size leads to a smaller sampling
error. it decreases with the increase in the size of the
sample, and it happens to be of a smaller magnitude in
case of homogeneous population.
When n = N ⇒ sampling error = 0
Can not be avoided or totally eliminated
86
87. Sampling error cont’d…
why do sample estimates have uncertainty associated
with them? There are two reasons.
Estimates of characteristics from the sample data can
differ from those that would be obtained if the entire
population were surveyed.
Estimates from one subset or sample of the
population can differ from those based on a different
sample from the same population (sample to sample
variations). 87
88. The cause of sampling error
Chance: main cause of sampling error and is the error that
occurs just because of bad luck.
Sampling bias: Sampling bias is a tendency to favor the
selection of participants that have particular characteristics.
The chance component (sometimes called random error)
exists no matter how carefully the selection procedures are
implemented, and the only way to minimize chance-
sampling errors is to select a sufficiently large
sample.
88
89. 2. Non Sampling Error
It is a type of systematic error in the design or conduct of a
sampling procedure which results in distortion of the sample, so
that it is no longer representative of the reference population.
We can eliminate or reduce the non-sampling error (bias) by
careful design of the sampling procedure and not by increasing
the sample size.
It can occur whether the total study population or a sample is
being used.
89
90. Non-sampling Error……
o The basic types of non-sampling error
– Non-response error
– Response or data error
o A non-response error occurs when units selected as part of the
sampling procedure do not respond in whole or in part
– If non-respondents are not different from those that did
respond, there is no non-response error
– When non-respondents constitute a significant proportion of
the sample (about 15% or more
90
91. Non-sampling Error…….
o A response or data error is any systematic bias
that occurs during data collection, analysis or
interpretation
– Respondent error (e.g., lying, forgetting, etc.)
– Interviewer bias
– Recording errors
– Poorly designed questionnaires
91
92. Non-Sampling Error cont’d …
Systematic error makes survey results unrepresentative of the
target population by distorting the survey estimates in one
direction.
Random error can distort the results in any given direction but
tend to balance out on average
Thus, the total survey error
92
sampling error + non-sampling error
94. 2.9 Sampling distributions
a. Central limit theorem
b. Estimation: point and interval estimation of
the mean, proportion of distribution and
confidence interval
c. Standard error of mean and proportion for
finite and infinite case
d. Distribution of sample mean, sample
proportion and difference between two sample
means and two sample proportions
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95. Central limit theorem
For simple random samples from any population
with finite mean and variance, as n becomes
increasingly large, the sampling distribution of
the sample means is approximately normally
distributed.
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96. n↑
Central Limit Theorem
As the
sample
size gets
large
enough…
the sampling
distribution of
the sample
mean becomes
almost normal
regardless of
shape of
population
x
97. Normal Uniform Skewed
Population
n = 2
n = 30
X
X
X
X
General
The Central Limit Theorem Applies to Sampling
Distributions from Any Population
100. Central Limit Theorem: Proportions AND Means
RULE: If many samples or repetitions of the SAME SIZE are taken, the
frequency curve made from STATISTICS from the SAMPLES will be
approximately normally distributed
Categorical (2 outcomes)
PROPORTIONS (𝒑’s):
• Assumptions:
1. Population w/fixed proportion
2. Random sample from population
3. np5 and n(1-p)5 (“large” samples)
• MEAN of samples 𝒑’s will be
population proportion (p)
• STANDARD DEVIATION of the
sample proportions (𝒑′s) will be:
Quantitative (Measurement)
MEANS (𝑿’s ):
• Conditions/Assumptions
1. If population bell-shaped (normal),
random sample of any size
2. If population not bell-shaped, a large
random sample ( 30)
– MEAN of sample means (𝑿’s) will be
population mean (𝝁)
– STANDARD DEVIATION of the sample
means (𝑿’s) will be:
𝒑
𝜇𝒑
= 𝒑 𝜇𝒙
= 𝝁
101. Estimation: point and interval
estimation of the mean, proportion of
distribution and confidence interval
• An estimator of a population parameter is
– a random variable that depends on sample
information . . .
– whose value provides an approximation to this
unknown parameter
• A specific value of that random variable is
called an estimate
9/13/2016 Ashok Pandey 101
102. We can estimate a
Population Parameter …
Point Estimates
with a Sample
Statistic
(a Point Estimate)
Mean
Proportion
p
x
μ
p̂
Variance
Variance σ2 s2
103. Point and Interval Estimates
• A point estimate is a single number,
• a confidence interval provides additional
information about variability
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Width of
confidence interval
104. Confidence Intervals
• How much uncertainty is associated with a
point estimate of a population parameter?
• An interval estimate provides more
information about a population
characteristic than does a point estimate
• Such interval estimates are called
confidence intervals
105. Confidence Interval Estimate
• An interval gives a range of values:
– Takes into consideration variation in sample
statistics from sample to sample
– Based on observation from 1 sample
– Gives information about closeness to unknown
population parameters
– Stated in terms of level of confidence
• Can never be 100% confident
106. Confidence Interval and Confidence
Level
• If P(a < < b) = 1 - then the interval from a
to b is called a 100(1 - )% confidence
interval of .
• The quantity (1 - ) is called the confidence
level of the interval ( between 0 and 1)
– In repeated samples of the population, the true value
of the parameter would be contained in 100(1 - )%
of intervals calculated this way.
– The confidence interval calculated in this manner is
written as a < < b with 100(1 - )% confidence
107. Estimation Process
(mean, μ, is
unknown)
Population
Random Sample
Mean
X = 50
Sample
I am 95%
confident that
μ is between 40
& 60.
108. Confidence Level, (1-)
• Suppose confidence level = 95%
• Also written (1 - ) = 0.95
• A relative frequency interpretation:
– From repeated samples, 95% of all the confidence
intervals that can be constructed will contain the
unknown true parameter
• A specific interval either will contain or will
not contain the true parameter
– The procedure used leads to a correct interval in
95% of the time...
– ...but this does not guarantee anything about a
particular sample.
(continued)
109. General Formula
• The general formula for all confidence
intervals is:
• The value of the reliability factor depends
on the desired level of confidence
Point Estimate (Reliability Factor)(Standard deviation)
111. Confidence Interval for μ
(σ2 Known)
• Assumptions
– Population variance σ2 is known
– Population is normally distributed...
– ....or large sample so that CLT can be used.
• Confidence interval estimate:
(where z/2 is the normal distribution value for a probability of /2 in each
tail)
n
σ
z
x
μ
n
σ
z
x α/2
α/2
112. Margin of Error
• The confidence interval,
• Can also be written as
where ME is called the margin of error
• The interval width, w, is equal to twice the margin of error
n
σ
z
x
μ
n
σ
z
x α/2
α/2
ME
x
n
σ
z
ME α/2
113. Finding the Reliability Factor, z/2
• Consider a 95% confidence interval:
z = -1.96 z = 1.96
.95
1
.025
2
α
.025
2
α
Point Estimate
Lower
Confidence
Limit
Upper
Confidence
Limit
Z units:
X units:
0
Find z.025 = 1.96 from the standard normal distribution table
114. Common Levels of Confidence
• Commonly used confidence levels are 90%,
95%, and 99%
Confidence
Level
Confidence
Coefficient, Z/2 value
1.28
1.645
1.96
2.33
2.58
3.08
3.27
.80
.90
.95
.98
.99
.998
.999
80%
90%
95%
98%
99%
99.8%
99.9%
1
116. • If the population standard deviation σ is
unknown, we can substitute the sample
standard deviation, s
• This introduces extra uncertainty, since s is
variable from sample to sample
• Therefore we use the t distribution instead
of the normal distribution
Confidence Interval for μ
(σ2 Unknown)
117. Student’s t Distribution
• Consider a random sample of n observations
– with mean x and standard deviation s
– from a normally distributed population with mean μ
• Then the variable
follows the Student’s t distribution with (n - 1) degrees of
freedom
n
s/
μ
x
t
118. Student’s t Distribution
• The t is a family of distributions
• The t-value depends on degrees of freedom
(d.f.)
– Number of observations that are free to vary after sample
mean has been calculated
d.f. = n - 1
119. Student’s t Distribution
t
0
t (df = 5)
t (df = 13)
t-distributions are bell-
shaped and symmetric, but
have ‘fatter’ tails than the
normal
Standard
Normal
(t with df = ∞)
Note: t Z as n increases
120. • Assumptions
– Population standard deviation is unknown
– Population is normally distributed
• Use Student’s t Distribution
• Confidence Interval Estimate:
where tn-1,α/2 is the critical value of the t distribution with n-1 d.f. and an
area of α/2 in each tail:
Confidence Interval for μ
(σ Unknown)
n
s
t
x
μ
n
s
t
x α/2
1,
-
n
α/2
1,
-
n
(continued)
α/2
)
t
P(t α/2
1,
n
121. Student’s t Table
Upper Tail Area
df .10 .025
.05
1 12.706
2
3 3.182
t
0 2.920
The body of the table
contains t values, not
probabilities
Let: n = 3
df = n - 1 = 2
= .10
/2 =.05
/2 = .05
3.078
1.886
1.638
6.314
2.920
2.353
4.303
122. Distribution of sample mean, sample
proportion and difference between two
sample means and two sample proportions
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124. The sample proportion is the percentage of
successes in n binomial trials. It is the
number of successes, X, divided by the
number of trials, n.
p
X
n
As the sample size, n, increases, the sampling
distribution of approaches a normal
distribution with mean p and standard
deviation
p
p p
n
( )
1
Sample proportion:
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0.2
0.1
0.0
P(X)
n=15, p = 0.3
X
14
15
13
15
12
15
11
15
10
15
9
15
8
15
7
15
6
15
5
15
4
15
3
15
2
15
1
15
0
15
15
15 ^
p
2
1
0
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
X
P(X)
n=2, p = 0 .3
10
9
8
7
6
5
4
3
2
1
0
0.3
0.2
0.1
0.0
P(X)
n=10,p=0.3
X
The Sampling Distribution of the Sample
Proportion,
p
