Download Symmetric Population Distribution

Asian School of Business
PG Programme in Management (2005-06)
Course: Quantitative Methods in Management I
Instructor: Chandan Mukherjee
Session 18: Sampling Distribution (a demo)
Definitions
Statistic = A function of the sample values
Example: Sample Mean, Sample Variance are Statistics
Estimator: When a Statistics is used to estimate a parameter of the distribution
from which the sample has been drawn, it is called an Estimator of the
parameter.
Value of a Statistic varies from sample to sample.
Distribution of the all possible values of an Estimator is called the Sampling
Distribution of the Estimator.
An Estimator is called Unbiased if the mean of its Sampling Distribution is
equal to the parameter.
Standard Deviation of the Sampling Distribution of an Estimator is called its
Standard Error.
Symmetric Population Distribution
Population: 1 2 2 3 3 3 3 4 5
X
1
2
3
4
5
All
Mean
F
1
2
4
2
1
10
= 3.0
Variance = 1.2
80%
Symmetric Population Distribution:
Random Sample Without Replacement, Size 2
List of All Possible Sample Means
Sample 1
Sample 2
1
2
2
3
3
3
3
4
4
5
1
1.5
1.5
2.0
2.0
2.0
2.0
2.5
2.5
3.0
X2
2
2
3
3
3
3
4
4
5
1.5
1.5
2.0
2.0
2.5
2.5
2.0
2.5
2.5
3.0
2.0
2.5
2.5
3.0
3.0
2.0
2.5
2.5
3.0
3.0
3.0
2.5
3.0
3.0
3.5
3.5
3.5
3.5
2.5
3.0
3.0
3.5
3.5
3.5
3.5
4.0
3.0
3.5
3.5
4.0
4.0
4.0
4.0
4.5
4.5
2.0
2.5
2.5
2.5
2.5
3.0
3.0
3.5
2.5
2.5
2.5
2.5
3.0
3.0
3.5
3.0
3.0
3.0
3.5
3.5
4.0
3.0
3.0
3.5
3.5
4.0
3.0
3.5
3.5
4.0
3.5
3.5
4.0
4.0
4.5
4.5
Distribution of Sample Mean: The Sampling Distribution
X2
F
RF
1.5
2.0
2.5
3.0
3.5
4.0
4.5
All
4
4.4
10 11.1
20 22.2
22 24.4
20 22.2
10 11.1
4
4.4
90 100.0
Mean
= 3.0
Variance = 0.53
91%
Asymmetric Population Distribution
Population: 1 1 1 1 2 2 2 3 3 4
X
1
2
3
4
All
Mean
F
4
3
2
1
10
= 2.0
Variance = 1.0
90%
Asymmetric Population Distribution:
Random Sample Without Replacement, Size 2
List of All Possible Sample Means
Sample 1
Sample 2
1
1
1
1
2
2
2
3
3
4
1
1.0
1.0
1.0
1.5
1.5
1.5
2.0
2.0
2.5
X2
1
1
1
2
2
2
3
3
4
1.0
1.0
1.0
1.0
1.0
1.0
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
2.0
1.5
1.5
1.5
1.5
2.0
2.0
2.0
2.0
2.0
2.0
2.5
2.5
2.5
2.0
2.0
2.0
2.0
2.5
2.5
2.5
3.0
2.5
2.5
2.5
2.5
3.0
3.0
3.0
3.5
3.5
1.0
1.0
1.5
1.5
1.5
2.0
2.0
2.5
1.0
1.5
1.5
1.5
2.0
2.0
2.5
1.5
1.5
1.5
2.0
2.0
2.5
2.0
2.0
2.5
2.5
3.0
2.0
2.5
2.5
3.0
2.5
2.5
3.0
3.0
3.5
3.5
Distribution of Sample Mean: The Sampling Distribution
X2
1.0
1.5
2.0
2.5
3.0
3.5
ALL
Mean
F
RF
12 13.3
24 26.7
22 24.4
20 22.2
8
8.9
4
4.4
90 100.0
= 2.0
Variance = 0.44
96%
Lessons from the Demo
•Consider a random sample and the sample mean.
•If the distribution of the universe (population) is symmetric then the
sampling distribution of the sample mean is also symmetric around the
population mean.
•If the distribution of the universe (population) is asymmetric then the
sampling distribution of the sample mean is asymmetric around the
population mean.
•In both cases the sample mean is an unbiased estimator of the population
mean.
Theoretical Results
X ~ N(μ, σ2)
Random sample of size n.
Xn
= Sample Mean
Then,
Xn
~ N(μ, σ2/n)
•If X has an arbitrary distribution, then also the above result about the
sample mean holds if n is sufficiently large.
•How large is sufficiently large for the result to be valid?
•Well, that depends on the distribution of X.
Theoretical Results
X ~ N(μ, σ2)
Random sample of size n.
xn  
~ N (0,1)
/ n
xn  
~ t( n 1)
s/ n
Where
n
1
2
s2 
(
x

x
)
 i n
n 1 1
Theoretical Results
In Binomial Distribution in n.p ≥ 5 and n.(1-p) ≥ 5 then
x  n. p
~ N (0,1)
n. p.(1  p)
Which implies
pˆ  p
~ N (0,1)
p.(1  p) / n
Where x is the number of successes in n trials