8.1
8.8: F-distribution
The t-distribution will be very useful in Chapters 9 and 10
to make inferences about the actual value of . There
are many times when we would like to make inferences
about 2 as well. The F-distribution described below is
often used for that purpose (among others) in Chapters
9 and 10.
Theorem 8.6: Suppose U and V are two independent
random variables with chi-squared PDFs with 1 and 2
degrees of freedom, respectively. Then
U / 1
F
V / 2
has an F-distribution with 1 degrees of freedom in the
numerator and 2 degrees of freedom in the denominator.
The PDF is
   1  2  / 2  1 / 2 1 / 2
f 1 / 21



0f 
( 1 2 ) / 2
h(f)  
  1 / 2    2 / 2 
1  1f / 2 

0
otherwise

Example: Finding probabilities from a F-distribution
(F_distribution.xls and F_prob.xls)
 2005 Christopher R. Bilder
8.2
The FDIST(f, 1, 2) function in Excel can be used to find
probabilities. To find quantiles, the Excel function is
FINV(area to the right, 1, 2).
Below are part of the results from F_prob.xls.
Below is a diagram of what is being found above.
 2005 Christopher R. Bilder
8.3
Below are part of the results presented in
F_distribution.xls. Note that the book represents
quantiles using f(1, 2). Thus, P[F > f(1, 2) ] = .
Numerator degrees of freedom
Denominator degrees of freedom
F0.1(num df, den df)
F0.05(num df, den df)
F0.01(num df, den df)
F0.001(num df, den df)
5
20
2.1582
2.7109
4.1027
6.4606
F Distribution
0.8
h(f)
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
f
 2005 Christopher R. Bilder
8.4
The plot and f(1, 2) values will update
corresponding to what you enter in as the 1
(numerator degrees of freedom) and 2
(denominator degrees of freedom).
Table A.6 on p. 676-679 can be used to find values
from an F-distribution. Notice there are separate
tables for values of  = 0.05 and 0.01. The columns
in each table denote 1 and the rows denote 2.
Theorem 8.7: Writing f(1,2) for f with 1 and 2 degrees of
1
freedom, we obtain f1 (1, 2 ) 
.
f (2 , 1)
This theorem is useful when using the tables.
Theorem 8.8: If S12 and S22 are the sample variances of
independent random samples of size n1 and n2 taken from
populations with normal PDFs with variances 12 and 22 ,
respectively, then
S12 / 12
F 2 2
S2 / 2
 2005 Christopher R. Bilder
8.5
has an F-distribution with 1 = n1-1 and 2 = n2-1 degrees of
freedom.
The example on p. 226-227 gives a nice illustration about
where the F-distribution is used in a procedure called
analysis of variance (ANOVA). This is the subject of
Chapters 13-15.
Examples of how to use this PDF for hypothesis testing
when examining the equality of two variances will be
discussed in Section 9.12.
 2005 Christopher R. Bilder