CHAPTER 9: ONEWAY ANALYSIS OF VARIANCE
9.1 THE
F
DISTRIBUTION
Analysisofvariance procedures rely on a distribution called the
Fdistribution
, named in
honor of Sir Ronald Fisher, who developed the initial techniques of the analysis of
variance in the 1920s and 1930s. A variable is said to have an
F
distribution
if its
distribution has the shape of a special type of rightskewed curve, called an
F
curve
.
There are infinitely many
F
distributions, and we identify an
F
distribution (and
F
curve)
by stating its number of degrees of freedom, just as we did for tdistributions and chi
square distributions. But, as shown below, an
F
distribution has two numbers of degrees
of freedom instead of one.
The first number of degrees of freedom (
df
1
) for an
F
curve
is called the
degrees of
freedom for the numerator
, and the second (
df
2
) is called the
degrees of freedom for
the denominator
.
9.1.1 BASIC PROPERTIES OF
F
CURVES
There are five major properties.
Property 1
: The
F
distribution is a
continuous probability distribution
. The
total area
under an
F
curve equals
1
.
Property 2
: The
F
distribution is
asymptotic
. An
F
curve starts at 0 on the horizontal
axis and extends indefinitely to the right, approaching, but never touching, the horizontal
axis as it does so.
Property 3
: The
F
curve is
positively skewed
. The long tail of the distribution is to the
righthand side. As the number of degrees of freedom increases in both the numerator and
the denominator the
F
distribution approaches a normal distribution.
Property 4
: The
F
distribution
cannot be negative
. The smallest value F can assume is 0.
Property 5
: The mean of the
F
distribution is
2
2
2
df
df
μ
=

for df
2
2, where df
2
is the
degrees of freedom for the denominator.
9.1.2 FINDING THE
F
VALUE HAVING A SPECIFIED AREA TO ITS RIGHT
Dr. LOHAKA: QBA 2305  CHAPTER 9: ONEWAY ANALYSIS OF VARIANCE
Page
24
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Percentages (and probabilities), for a variable having an
F
distribution, are equal to areas
under its associated
F
curve. To perform an analysis of variance test, we need to know
how to find the
F
value having a specified area to its right. The symbol
F
α
is used to
denote the
F
value
having area
α
to its right.
EXAMPLE 9.1
Problem
:
a)
For an
F
curve with df = (4, 12), find
F
0.05
; that is, find the
F
value having area
0.05 to its right.
b)
For an
F
curve with df = (12, 4), find
F
0.95
; that is, find the
F
value having area
0.95 to its right.
Solution
:
a)
To obtain the
F
0.05
value, we use the
Fisher Table
. In this case,
α = 0.05
, the
degrees of freedom for the numerator is
4
, and the degrees of freedom for the
denominator is
12
. We first go down the df column to “
12
”. Next, we go across
the row for
α
labeled
0.05
to the column headed “
4
”. The number in the body of
the table there, 3.26, is the required Fvalue; that is, for an Fcurve with df = (4,
12), the Fcurve having area 0.05 to its right is 3.26:
F
(0.05, 4, 12)
= 3.26
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Hulme
 Normal Distribution, Statistical hypothesis testing, oneway analysis, Dr. LOHAKA

Click to edit the document details