Distribution
SUNY F at Albany - Department of Economics
Eco 320 Economic Statistics
F-Distribution
3.0
2.5
F(1, 1)
2.0
1.5
1.0
F(2, 2)
0.5
0.0
0.0
1.0
SUNY at Albany - Department of Economics
2.0
3.0
4.0
5.0
6.0
7.0
Eco 320 Economic Statistics
F-Distribution
1.2
1.0
F(2, 10)
0.8
0.6
0.4
F(10, 2)
0.2
0.0
0.0
1.0
SUNY at Albany - Department of Economics
2.0
3.0
4.0
5.0
6.0
7.0
Eco 320 Economic Statistics
F-Distribution
1.6
1.4
F(50, 50)
1.2
F(2, 2)
1.0
0.8
0.6
0.4
F(10, 2)
0.2
0.0
0.0
1.0
SUNY at Albany - Department of Economics
2.0
3.0
4.0
5.0
6.0
7.0
Eco 320 Economic Statistics
1. F Distribution
A.
Let 1 , 2 , , be a random sample of size m from a normal
2
distribution X~N( , ), and let 1 , 2 , , be a random sample of
2
size n from a normal distribution Y~N( , ). Assume that these two
2
2
samples are independent. We want to find out whether = .
B.
Since we cannot directly observe the two population variances, we use
2
( )2
2 = ( )
their sample estimator
=
and
1
1
2
2
Statistical theory shows that if = , the following ratio
2
( )2 /( 1)
= 2 =
( )2 /( 1)
2
C.
follows the F distribution with (m-1)(numerator) d.f. and (n-1)
(denominator) d.f., designated as 1 ,2 .
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
2. Properties of F Distribution
A.
Properties of the F Distribution
F distribution takes only positive values and ranges between 0 and
+.
It is skewed to the right and approaches the normal distribution as
k1 and k2 go to .
If W ~ Fm, n, then (1/W) ~ Fn, m. More specifically,
1
F ,m,n
F(1 ),n,m
2
= 1,
(,) =
2
SUNY at Albany - Department of Economics
as n
Eco 320 Economic Statistics
3. F Distribution Table
F(, k1, k2)
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 1
A.
X ~ F(12, 5) with k1 = 12 and k2 = 5.
Find 10% critical value of this F distribution. That is,
P(X > F(0.10, 12, 5)) = 0.10.
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 1
A.
X ~ F(12, 5) with k1 = 12 and k2 = 5.
Find 10% critical of value this F distribution. Since, P(X > 3.27) = 0.10.
F(0.10, 12, 5) = 3.27.
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 1
A.
X ~ F(12, 5) with k1 = 12 and k2 = 5.
Find 10% critical value of this F distribution. Since, P(X > 3.27) = 0.10.
F(0.10, 12, 5) = 3.27.
P(X > F(0.05, 12, 5)) = 0.05.
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 1
A.
X ~ F(12, 5) with k1 = 12 and k2 = 5.
Find 10% critical value of this F distribution. Since, P(X > 3.27) = 0.10.
F(0.10, 12, 5) = 3.27
P(X > 4.68) = 0.05. F(0.05, 12, 5) = 4.68
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 2
A.
Illustrate
1
F( ,m,n )
F((1 ), n,m )
F(0.10, 12, 5) = 3.27.
F(0.90, 5, 12) = 1/3.27 = 0.306.
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 2
A.
Illustrate
1
F( ,m,n )
F((1 ), n,m )
F(0.10, 12, 5) = 3.27.
F(0.90, 5, 12) = 1/3.27 = 0.306.
1/ F(0.10, 12, 5) = 1/3.27 = 0.306 = F(0.90, 5, 12). Or,
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example] 2
A.
Illustrate
1
F( ,m,n )
F((1 ), n,m )
F(0.10, 12, 5) = 3.27.
F(0.90, 5, 12) = 1/3.27 = 0.306.
1/ F(0.10, 12, 5) = 1/3.27 = 0.306 = F(0.90, 5, 12). Or,
1/ F(0.90, 5, 12) = 1/0.306 = 3.27 = F(0.10, 12, 5).
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[Example]
A.
An instructor gives that same econometrics exam to two classes, one
consisting of 100 students and the other consisting of 150 students. She
draws a random sample of 25 students from the first class and a random
sample of 31 students from the other class and observes that the sample
variance of the grade-point average in the two classes are 100 and 132,
respectively. It is assumed that the grade-point average in the two
classes is normally distributed. Can we assume that the variances of GPA
in the two classes are the same?
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics
[In-class Problem]
SUNY at Albany - Department of Economics
Eco 320 Economic Statistics