Download The Central Limit Theorem

The Central Limit Theorem
The Statement that follows is a special case
of a more general theorem.
If X is any random variable then X , the
distribution of all possible sample means of
size n, will have an approximate normal
distribution so long as n is large enough.
 2 
That is X ~ N   ,

n 

• The approximation will be good for n > 30
• The approximation will be good for n < 30
for distributions which are not far from
Normal
 2 
• It could be stated that X ~ N   ,  as

n 
n →  whatever the distribution of X.
Example
The scores in an aptitude test taken by a large
number of students are normally distributed with
a mean of 100 and variance 120.
If random samples of 40 students were picked what
would the mean, the variance and standard
deviation (standard error) of X , the mean scores
of the students samples?
X = score on the test
X ~ N(100, 120)
Let X be the distribution of possiblesample means with size n  40
 2 

Using the CentralLimit Theorem: X  N  ,
n 

Mean  μ  100
σ 2 120
Variance 

3
n
40
σ2
StandardError 
 3
n
A random variable X has f x   0.5
for
0
0 x2
elsewhere
(i) Obtain the mean and variance of X
A random sample of 48 observations is obtained
from this distribution and the mean is computed.
 
 
(ii) Obtain E X , var X and the standard error of X
(iii) Describe the approximate distribution of X
f x   0.5 for
(i)
ab 20


1
2
2
0 x2
1
1
1
2
2
  b  a   2  0 
12
12
3
2
(ii)
Let X be the distribution of possiblesample means with size n  48
 2 

Using the CentralLimit Theorem: X  N  ,
n 


E x   1
1
1
VAR x   2  3 
48 144


STANDARD ERR0R x 
2
N

1
1

144 12
1 

Using the CentralLimit Theorem: X  N 1,

 144 
Example
The continuous random variable X is distributed
with mean 25.8 and standard deviation 2.4. Use a
distributional approximation to evaluate the
probability that the mean of a random sample of
50 observations of X will be less than 26.
X ~ N(25.8, 2.42)
Let X be the distribution of possiblesample means with size n  50
 2 

Using the CentralLimit Theorem: X  N  ,
n 

Mean  μ  25.8
σ 2 2.42
Variance 

 0.1152
n
50
Using the CLT: X  N 25.8, 0.1152 approx
Test Statistic: x  26


P x  26 approx = P(z < 0.59)
= 0.7724
X  μ 26  25.8 = 0.59
Z

σ
0.1152
Example
The discrete random variable X has the following
distribution.
X
1
2
3
P(x)
0.6
0.3
0.1
Find an approximate value for the probability that the
mean of a random sample of 80 observations of X will be
greater than 1.65.
Example
Find an approximate value for the probability
that the mean score obtained in 30 throws of a
fair cubical die will be 4 or more.