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Instructor: Dr. Ozan ERUYGUR
e-mail:
[email protected]
Lecture Notes
14
2
( )
Var Y
.
Then, when
t
is large, the sampling distribution of
Y
converges
to
2
(
,
)
Y
Y
N
where
Y
and
2
2
Y
T
.
As a result, the sampling distribution of
2
Y
Y
Y
which is
equal to
/
Y
T
converges to the
standard normal
N(0,1)
as
T
converges to infinity.
The central limit theorem is very important because it holds even
when the distribution from which the observations are drawn is not
normal.
This means that if we make sure that the sample size is large,
then we can use the random variable
/
Y
T
defined above
to answer questions about the population from which the
observations are drawn, and we need not know the precise
distribution
from
which
the
observations
are
drawn
(Ramanathan, 1998, p.37).
APPENDIX
A. Sampling Distribution of Mean
Population:
A
B
C
D
E
3
1
5
6
2
Population mean,
: (3+1+5+6+2)/5=17/5=3.4
Population Size,
N
: 5
From this population of five, list every possible sample of size 2 (n=2) and
calculate the mean for each sample.
Repeat this for samples of size 3 (n=3).

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ECON 301 (01) - Introduction to Econometrics I
March, 2012
METU - Department of Economics
Instructor: Dr. Ozan ERUYGUR
e-mail:
[email protected]
Lecture Notes
15
How many samples of size 2?
5
5
4
10
2
1
2
How many samples of size 3?
5
5
4
3
10
3
1
2
3
Samples
Sample of
Size 2 (n=2)
Sample
Mean
Samples
Sample of
Size 3 (n=3)
Sample
Mean
1
AB=3,1
(y
1
=3, y
2
=1)
for Y
1
and Y
2
Y
=
2.0
Y
1
ABC=3,1,5
(y
1
=3, y
2
=1, y
3
=5)
for Y
1
, Y
2
and Y
3
3.00
2
AC=3,5
4.0
Y
2
ABD=3,5,6
4.67
3
AD=3,6
4.5
Y
3
ABE=3,1,2
2.00
4
AE=3,2
2.5
Y
4
ACD=3,5,6
3.67
5
BC=1,5
3.0
Y
5
ACE=3,5,2
3.33
6
BD=1,6
3.5
Y
6
ADE=3,6,2
3.67
7
BE=1,2
1.5
Y
7
BCD=1,5,6
4.00
8
CD=5,6
5.5
Y
8
BCE=1,5,2
2.67
9
CE=5,2
3.5
Y
9
BDE=1,6,2
3.00
10
DE=6,2
4.0
Y
10
CDE=5,6,2
4.33
Average
3.4
3.4
Means of all sample
means
Y
Y
=3.4
For n=2
Y
=3.4
For n=3
Conclusion
The mean of all the sample means is the same as the population
mean:
Y
=
B. Process Going Into the Sampling Distribution Model
o
We start with a population model, which can have any shape. It
can even be bimodal or skewed (as this one is). We label the mean
of this model
and its standard deviation,
.
o
We draw one real sample (solid line) of size n and show its
histogram and summary statistics. We imagine (or simulate)
drawing many other samples (dotted lines), which have their own
histograms and summary statistics.
o
We (imagine) gathering all the means into a histogram