Download 7.0 Sampling and Sampling Distribution

7.0 Sampling and Sampling
Distribution
7.2
7.1
Sampling Methods
Introduction to Sampling Distribution
Why?
Types
Sampling
Frame
Plan
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less timeconsuming
more
practical
WHY??
less costly
less
cumbersome
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Sampling Frame
• listing of items that make up the population
• data sources such as population lists, directories, or
maps
Sampling Plan
• The way a sample is selected
• determines the quantity of information in the sample
• allow to measure the reliability or goodness of your
inference
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Types of Samples
Samples
Nonprobability
Samples
Convenient
Probability
Samples
Judgement
Simple
Random
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Systematic
Stratified
Cluster
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Selection of elements is left primarily to the interviewer.
Easy, inexpensive, or convenient to the sample
limitations- not representative of the population.
Recommended for pre testing Q, generating ideas, insight @ hypotheses.
Eg: a survey was conducted by one local TV stations involving a small number
of housewives, white collar workers & blue collar workers. The survey
attempts to elicit the respondents response towards a particular drama series
aired over the channel.
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The population elements are selected based on the judgment of
the researcher.
From the judgment, the elements are representative of the
population of interest.
Eg: testing the consumers’ response towards a brand of instant
coffee, Indocafe at a wholesale market.
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Definition:
If a sample of n is drawn from a population of N in such a way that every
possible sample of size n has the same chance of being selected,
the sample obtained is called a simple random sampling.
N – number of units in the population
n – number of units in sample
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Do not have any bias element (every element treated equally).
Target population is homogenous in nature (the units have
similar characteristics)
Eg: canteen operators in primary school, operators in cyber
cafes, etc..
Disadvantages:
Sampling frame are not updated. Sampling frame are costly to
produce.
Impractical for large study area.
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• Definition:
a sample obtained by randomly selecting one element
from the 1st k elements in the frame & every kth
element there is called a 1-in-k systematic sample,
with a random start.
• k – interval size
• k = population size
sample size
= N/n
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Systematic Sampling
 eg:
Let say, there are a total of N=500 primary school canteen operators in the KlangValley
in 1997 who are registered with the Ministry of Education. We required a sample of
n=25 operators for a particular study.
Step 1: make sure that the list is random(the name sorted
alphabetically).
Step 2: divide the operators into interval contain k operators.
k = population size = 500/25 = 20 for every 20 operators
sample size
selected only one to
represent that interval
Step 3: 1st interval only, select r at random. Let say 7. operators with id
no.7 will be 1st sample. The rest of the operators selected in
remaining intervals will depend on this number.
Step 4: after 7 has been selected, the remaining selection will be
operators with the following id no.
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Stratified Sampling
 Definition:
obtained by separating the population elements into non
overlapping groups, called strata, & then selecting a random
sample from each stratum.
 Large variation within the population.
 Eg: lecturers that can be categorized as lecturers, senior lecturers,
associate prof & prof.
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Stratified Sampling
 Definition:
obtained by separating the population elements into non
overlapping groups, called strata, & then selecting a random
sample from each stratum.
 Large variation within the population.
 Eg: lecturers that can be categorized as lecturers, senior lecturers,
associate prof & prof.
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Cluster Sampling
 Definition:
probability sample in which each sampling unit is a
collection, @ cluster of elements.
 Advantages- can be applied to a large study areas
- practical & economical.
- cost can be reduced-interviewer only need
to stay within the specific area instead
travelling across of the study area.
 Disadvantages – higher sampling error.
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Cluster Sampling
 Definition:
probability sample in which each sampling unit is a
collection, @ cluster of elements.
 Advantages- can be applied to a large study areas
- practical & economical.
- cost can be reduced-interviewer only need
to stay within the specific area instead
travelling across of the study area.
 Disadvantages – higher sampling error.
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7.0 Sampling and Sampling
Distribution
7.2 Introduction to Sampling Distribution
• Sampling distribution is a probability
distribution of a sample statistic based on all
possible simple random sample of the same
size from the same population.
7.2.1 Sampling Distribution of Mean ( )
X1  X2  X3  ...  X n
x
n
Sample mean
will have a theoretical sampling distribution with
 
mean, x  E X  x
variance,  x  Var  X  
2
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 x N  n
2
n
 N  1
and standard errors
 x N  n
of the sample mean X is n  N  1
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 The spread of the sampling distribution of
is smaller than
the spread of the corresponding population distribution.
 The standard deviation of the sampling distribution of
decreases as the sample size increase.
 Consistent estimator ;
 the standard deviation of a sample statistics decrease as the
sample size increased
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Example :
 The mean wage per hour for all 5000 employees who work
at a large company is RM17.50 and the standard deviation is
RM29.90. Let be the mean wage per hour for a random
sample of certain employee selected from this company. Find
the mean and standard deviation of for a sample size of
 30
 75
 200
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 Solution
Population mean, x  17.50
Population standard deviation,  x  29.90
(a)
Mean, x  x  17.50
Standard Deviation  29.90 5000  30
5000  1
30
 5.4590  0.9971
 5.4431
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Central Limit Theorem
 If we are sampling from a population that has an unknown
probability distribution, the sampling distribution of the
sample mean will still be approximately normal with mean
and standard deviation , if the sample size is large
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Example
 An electronic firm has a total of 350 workers with a mean
age of 37.6 years and a standard deviation of 8.3 years. If a
sample of 45 workers is chosen at random from these
workers, what is probability that this sample will yield an
average age less than 40 years?
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Solution
Population mean, x  37.6 years
Population SD,  x  8.3 years
Population size N  350
Random sample size ,n=45
So sample mean,  x  37.6 years
Sample SE, x 
x
n
 N  n
 N  1
8.3 350  45
45 350  1
 1.157

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 Now as the sample size n  45  30 , then by Central Limit
Theorem, we have
N  37.6,1.157 2 
X
X  37.6
1.157
40  37.6 

P X  40  P  Z 

1.157 

Z


 P  Z  2.07 
 0.9808
 The probability that the sample will yield an average age less
than 40 years is 0.9808
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7.2.2Sampling Distribution of Propotion ( )
 The population and sample proportion are denoted by p and p
, respectively, are calculated as,
X and
x
p
p
N
n
where
 N = total number of elements in the population;
 X = number of elements in the population that possess a
specific characteristic;
 n = total number of elements in the sample; and
 x = number of elements in the sample that possess a specific
characteristic.
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Example
In a recent survey of 150 household, 54 had central air
conditioning. Find p and q where p is the proportion of
household that have central air conditioning.
Solution
Since X  54 and n  150
X 54

 0.36  36%
n 150
n  X 150  54 96
q


 0.64  64%
n
150
150
p
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