Download Theorem 8.2 (Central Limit Theorem) If is the mean of a random

Theorem 8.2 (Central Limit Theorem)
If X is the mean of a random sample of size n
from a population with mean  and finite
variance  , then the distribution of
2
X 
Z
/ n
is approximately the standard normal (i.e.
N(0,1)) if n is “large”.
Notes:
(a) n  30 is usually sufficiently large for
the normal approximation to be good
(b) if the original population is normally
distributed, then Z above is a standard
normal.
Terminology:
The distribution of X is called the “sampling
distribution of X ” .
Facts about “sampling distribution of X ”
from a random sample of size n from a
2
population with mean  and variance  :
(a) E ( X )   (denoted
X )
(b) Var ( X )   / n (denoted  X )
2
2
(c) the distribution of X is approximately
normal if n is sufficiently large
Sampling Distribution of the
Difference Between 2 Means
Setting:
 2 populations
1 and variance  12
2
pop. 2 has mean 2 and variance  2
pop. 1 has mean
 random samples are available from
each population
X 11 , X 12 , , X 1n1 -- from pop. 1
2
sample mean and var. -- X 1, S1
X 21 , X 22 ,, X 2 n2 -- from pop. 2
2
sample mean and var. -- X 2 , S2
 samples are independent
Note:
We will be interested in the distribution of
the random variable X1  X 2
Theorem 8.3:
Facts about “sampling distribution of X1  X 2 ”
(a) E ( X1  X 2 )  1  2 (denoted  X  X )
1
2
(b) Var ( X1  X 2 ) 
12
n1

 22
n2
(denoted  X  X )
1
2
2
(c) the distribution of X1  X 2 is approximately
normal if n1 and n2 are sufficiently large
So,
Z
X 1  X 2  ( 1   2 )
 12  22
n1

n2
is approximately standard normal, i.e.
N(0,1)
Sampling Distribution of S 2
Setting:
A random sample X1, X 2 ,, X n is taken
from a N (  ,  ) distribution
Theorem 8.4:
2
Facts about “sampling distribution of S ”
(or more precisely of
(n  1) S 2

2
)
Given the setting above,
2 
(n  1) S 2

2
n
( X i  X )2
i 1
2

has a chi-squared distribution with n  1
degrees of freedom
Chi-Squared Table (Table A.5)
Tabled value is
2 where P[  2  2 ]  
t - Distribution
The probability density function of a random
variable that has a t -distribution with v
degrees of freedom is given by
[(v  1) / 2]
t 2  (v1) / 2
h(t ) 
(1  )
,
v
(v / 2)  v
 t 
Note: The distribution of this statistic is
symmetric and bell shaped but has “fatter
tails” than a normal distribution.
Theorem 8.5:
Suppose Z has a standard normal
distribution and V has a chi-squared
distribution with v df. If Z and V are
independent rvs, then the rv T given by
Z
T
V /v
has a t -distribution with v df.
Corollary:
Let X1, X 2 ,, X n be a random sample from
a N (  ,  ) distribution. The statistic
X 
T
S/ n
has a t- distribution with n  1 df.
t - Table (Table A.4)
Tabled value is t where P[T  t ]  
Notes:
(a) The t distribution gets closer to the
normal as df increases.
(b) "  " row at bottom of t -table is for
normal distribution
-- i.e. it gives z where P[ Z  z ]  
for a standard normal Z