§16.1–The F -distribution
Tom Lewis
Fall Term 2009
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
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Outline
1
What is ANOVA?
2
The F -distribution
3
Basic properties of F -distributions
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
2/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Suppose that there are three competing methods for teaching reading.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Suppose that there are three competing methods for teaching reading.
Randomly assign students to each of three groups.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Suppose that there are three competing methods for teaching reading.
Randomly assign students to each of three groups.
Teach each group accordingly.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Suppose that there are three competing methods for teaching reading.
Randomly assign students to each of three groups.
Teach each group accordingly.
Test each group at the end with a common exam.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Suppose that there are three competing methods for teaching reading.
Randomly assign students to each of three groups.
Teach each group accordingly.
Test each group at the end with a common exam.
Collect sample means and sample standard deviations for each group?
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
What is ANOVA?
ANOVA
Here is a typical question that the ANOVA (ANalysis Of VAriance)
method is suited to answer: Does the method of teaching reading affect
test scores in reading?
Suppose that there are three competing methods for teaching reading.
Randomly assign students to each of three groups.
Teach each group accordingly.
Test each group at the end with a common exam.
Collect sample means and sample standard deviations for each group?
Use the ANOVA method to test the hypothesis that there is no
difference among the groups, that is, the mean score for each
population is the same.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
3/9
The F -distribution
The Chi-Square Distribution
The sum of k independent, standard normal random variables is said to
have a chi-squared distribution on k degrees of freedom.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
4/9
The F -distribution
The Chi-Square Distribution
The sum of k independent, standard normal random variables is said to
have a chi-squared distribution on k degrees of freedom.
The F -distribution
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
4/9
The F -distribution
The Chi-Square Distribution
The sum of k independent, standard normal random variables is said to
have a chi-squared distribution on k degrees of freedom.
The F -distribution
An F -distribution is the distribution ratio of two independent
chi-square random variables, U1 /U2 .
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
4/9
The F -distribution
The Chi-Square Distribution
The sum of k independent, standard normal random variables is said to
have a chi-squared distribution on k degrees of freedom.
The F -distribution
An F -distribution is the distribution ratio of two independent
chi-square random variables, U1 /U2 .
If U1 has d1 degrees of freedom and U2 has d2 degrees of freedom,
then the resulting F -distribution has (d1 , d2 ) degrees of freedom.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
4/9
The F -distribution
The density of an F with df = (5, 15)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
Tom Lewis ()
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3
§16.1–The F -distribution
4
5
Fall Term 2009
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The F -distribution
The density of an F with df = (15, 5)
0.6
0.5
0.4
0.3
0.2
0.1
1
Tom Lewis ()
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§16.1–The F -distribution
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5
Fall Term 2009
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Basic properties of F -distributions
Basic properties of F -distributions
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
7/9
Basic properties of F -distributions
Basic properties of F -distributions
The total area under an F -curve equals 1.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
7/9
Basic properties of F -distributions
Basic properties of F -distributions
The total area under an F -curve equals 1.
An F -curve is only defined for x ≥ 0.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
7/9
Basic properties of F -distributions
Basic properties of F -distributions
The total area under an F -curve equals 1.
An F -curve is only defined for x ≥ 0.
An F -curve has value 0 at x = 0, is positive for x > 0, extends
indefinitely to the right, and approaches 0 as x → +∞.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
7/9
Basic properties of F -distributions
Basic properties of F -distributions
The total area under an F -curve equals 1.
An F -curve is only defined for x ≥ 0.
An F -curve has value 0 at x = 0, is positive for x > 0, extends
indefinitely to the right, and approaches 0 as x → +∞.
An F -curve is right-skewed.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
7/9
Basic properties of F -distributions
The meaning of Fα
As is customary, the point on the x-axis such that there are α units of area
under the F -curve to the right is denoted by Fα .
0.8
0.6
0.4
0.2
1
Tom Lewis ()
2
FΑ
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§16.1–The F -distribution
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Fall Term 2009
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Basic properties of F -distributions
Problem
An F -curve has df = (22, 30). In each case, find the F -value having the
specified area to its right.
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
9/9
Basic properties of F -distributions
Problem
An F -curve has df = (22, 30). In each case, find the F -value having the
specified area to its right.
.05
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
9/9
Basic properties of F -distributions
Problem
An F -curve has df = (22, 30). In each case, find the F -value having the
specified area to its right.
.05
.01
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
9/9
Basic properties of F -distributions
Problem
An F -curve has df = (22, 30). In each case, find the F -value having the
specified area to its right.
.05
.01
.025
Tom Lewis ()
§16.1–The F -distribution
Fall Term 2009
9/9