Download The Central Limit Theorem

• Displaying data & lying with statistics
• Summarizing data
– Measures of location & dispersion
• Probability
– Binomial, Poisson, & Normal distributions
• Functions of Random Variables
– e.g., mean & variance of a portfolio
• Estimation / Statistical inference
– Making educated guesses about population
parameters
– Saying how confident you are in those guesses
Population
Sample
All conceivable units of interest
µ = Mean
s² = Variance
s = Standard deviation
N = Size (Number of Units)
Parameters
A part of the population
X
s²
s
n
= Mean
= Variance
= Standard deviation
= Size (Number of Units)
Statistics
Population
x1, x2, ..., xN
Random sample
of size n
Parameters
m = Population mean
s = standard deviation
Sample
X
X is a random variable. It has mean
m X and standard deviation s X .
The Central Limit Theorem*
For a random sample of size n taken from a
population with mean m and standard deviation s:
• X is a Normal random variable
• mX = m, i.e.
E(X) = E(X)
• sX = s/ n
* Applies if n  30. Holds for any distribution of X.
An Illustration of the
derivation of 95%
confidence intervals.
m
s
m - 1.96
n
-3
95%
-2
0
1.96
m  1.96
2
s
n
3
X
Z
95% Confidence Intervals
From the diagram, we can write:
s
s 

P  m - 1.96
 X  m  1.96
  0.95
n
n

Algebra
s
s 

P  X - 1.96
 m  X  1.96
  0.95
n
n

100(1-a)%
m - Za 2
s
n
m
0
Za/2
m  Za 2
s
n
X
Z
100(1-a)% Confidence Intervals
From the diagram, we can write:
s
s 

P  m - Za 2
 X  m  Za 2
  1- a
n
n

Algebra
s
s 

P  X - Za 2
 m  X  Za 2
  1- a
n
n

95% Confidence Interval Width as a Function of Sample Size
s  15000
Interval Width = 2E
7000
6000
5000
4000
3000
2000
1000
0
0
2000
4000
6000
Sample Size
8000
10000