1. 14-04-2012
1
Research Methodology
Dr. NimitChowdhary,Professor
© Dr. Nimit Chowdhary Research Methodology Workshop p. 2 Saturday, April 14, 2012
 There exists a populationwith its parameters
 A number of samples can be drawn from this
population
 Each sample will have its own sample
statistics like sample’s mean standard
deviation,etc.
2. 14-04-2012
2
© Dr. Nimit Chowdhary Research Methodology Workshop p. 3 Saturday, April 14, 2012
 Distributionof a sample statistic is called a
sampling distribution
 Samplingdistribution is different from
sampledistribution- distribution of variables
withinthe sample
© Dr. Nimit ChowdharySaturday, April 14, 2012
A baby sitter has five children under her supervision.
The average age of these five children is 6 years.
However, the age of each child individually is as
follows:
Child (X) Age Child (X) Age
1 2 4 8
2 4 5 10
3 6
3. 14-04-2012
3
© Dr. Nimit Chowdhary Research Methodology Workshop p. 5
N = 5,
5
1
5
2 4 6 8 10
5
30
6
5
i
i
X
years





   

 

© Dr. Nimit ChowdharySaturday, April 14, 2012
2
5
( )
40
8
5
2.83
N
i
i
X
N
years







 

 X
2 6 16
4 6 4
6 6 0
8 6 4
10 6 16
 2
( )X 
2
( ) 40X   
4. 14-04-2012
4
© Dr. Nimit Chowdhary Research Methodology Workshop p. 7
Let us take all the
samples of size 2
from this
population.
There will be 10
samples
1 2 1
1 3 2
1 4 3
1 5 4
2 3 5
2 4 6
1 5 7
3 4 8
3 5 9
4 5 10
, (2, 4) 3
, (2, 6) 4
, (2,8) 5
, (2,10) 6
, (4, 6) 5
, (4,8) 6
, (4,10) 7
, (6,8) 7
, (6,10) 8
, (8,10) 9
X X X
X X X
X X X
X X X
X X X
X X X
X X X
X X X
X X X
X X X
  
  
  
  
  
  
  
  
  
  
© Dr. Nimit Chowdhary Research Methodology Workshop p. 8 Saturday, April 14, 2012
10
1
10
3 4 5 6 5 6 7 7 8 9
10
60
6
10
i
i
X
X
X
X years


        

 

5. 14-04-2012
5
© Dr. Nimit ChowdharySaturday, April 14, 2012
Sample
mean Frequency
Relative
Frequency Probability
3 1 1/10 .1
4 1 1/10 .1
5 2 2/10 .2
6 2 2/10 .2
7 2 2/10 .2
8 1 1/10 .1
9 1 1/10 .1
1.0
© Dr. Nimit ChowdharySaturday, April 14, 2012
Sample mean Probability
3 .1
4 .1
5 .2
6 .2
7 .2
8 .1
9 .1
1.0
X ( )P x
6. 14-04-2012
6
© Dr. Nimit Chowdhary Saturday, April 14, 2012
. ( )
(3 .1) (4 .1) (5 .2) (6 .2) (7 .2) (8 .1) (9 .1)
6
x
x
x
x P x
years



 
             

© Dr. Nimit Chowdhary
Research Methodology Workshop p.
12 Saturday, April 14, 2012
 Samplingdistribution of means
 Samplingdistribution of proportions
7. 14-04-2012
7
© Dr. Nimit Chowdhary Saturday, April 14, 2012
1. Mean of sampling distribution and mean of
populationis the same
2. Spread of the sample means in the
distributionis smaller than the spread in the
samplevalues
3. Samplingdistribution of samplingmeans
tend to be bell shaped
© Dr. Nimit Chowdhary Research Methodology Workshop p. 14
Regardless of the shape of the distribution
of the population, the distribution of the
sample means approaches the normal
probability distribution as the sample size
increases.
Thus, we can use our knowledge of normal
distributions to arrive at conclusions about
distribution of sample means
8. 14-04-2012
8
© Dr. Nimit Chowdhary Research Methodology Workshop p. 15
x

Standard error of mean is the standard
deviation (a measure of dispersion) of the
distribution of sample means ( ) around
mean of sampling distribution.
This mean can be considered as same as
population mean
x
© Dr. Nimit Chowdhary
Research Methodology Workshop p.
16 Saturday, April 14, 2012
2
( )x
x
x
N


 

N= is the number of samples
(not individual sample size)
9. 14-04-2012
9
3 6 9
4 6 4
5 6 1
6 6 0
7 6 1
8 6 4
9 6 9
X x
 2
( )x
X 
2
( )x
x   = 28
2
( )
28
4
7
2
x
x
x
x
x
N




 

 

Note: Standard error (2)
is smaller than
population standard
deviation (2.83)
10. 14-04-2012
10
© Dr. Nimit ChowdharySaturday, April 14, 2012
1x
x
N n
Nn
n








1
N n
N


x

For large samples when N>>n
would approach 1
Then,
N = Population size
N = sample size
 = populations standard
deviation
= standard error of the
mean
Note: decreases as
sample size increases
x

© Dr. Nimit Chowdhary Research Methodology Workshop p. 20
1
N n
N


Is the finite correction factor
• Must be used in case of large
samples
• Used when sample size is more
that 5% of the population size
11. 14-04-2012
11
The IQ scores of college students are normally
distributed with a mean  of 120 and standard
deviation of 10.
 What is the probability that the IQ scores
of any one student chosen at random is
between 120 and 125
120
 = 120
 = 10
125
( ) (125 120)
10
5
0.5
10
x
Z
Z


 
 
 
Calculating for Z, we get Area for Z=0.5, from Z tables
is 0.1915.
Therefore, there are 19.15 %
chance that student picked
up randomly will have an IQ
score between 120 and 125.
12. 14-04-2012
12
 If a random sample of 25 students is
taken, what is the probability that the
mean of this sample will be between 120
and 125.
We know,
1.This is the distribution of sample means. One of
these samples (of size 25) has a mean of 120
2.The mean of the distribution will be same as
population mean 120
3.The standard deviation of means around
population mean (S.E.) will have to be calculated
© Dr. Nimit Chowdhary Research Methodology Workshop p. 24
. . x
S E
n

 
10 10
2
525
x
   
2.5
( ) 125 120
2
5
2.5
2
0.4938
x
x
z
x
Z
Z
Area



 
 
 

2x
  Therefore,
120
125
120x
 
13. 14-04-2012
13
© Dr. Nimit Chowdhary Research Methodology Workshop p. 25 Saturday, April 14, 2012
 Thus there are 49.38% chance that the
samplemean would be between 120 and 125.
 One can see that as sample size increases,
s.d. reduces further and this chance will
further increase.
 The previous case can be considered as a
limiting casewhen sample size = 1 (the
samplesize reduces and so does the chance).
…can be defined as a distribution of
proportions of all possible random samples of
a fixed size n…
p = sample proportion
 = populationproportion
Be the first to comment