Chi-Square and F Distributions:
Tests for Variances
Edpsy 580
Carolyn J. Anderson
Department of Educational Psychology
University of Illinois at Urbana-Champaign
Chi-Square and F Distributions
Slide 1 of 54
Outline
■
Introduction, motivation and overview
■
Chi-square distribution
Introduction
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
■
■
Chi-Square and F Distributions
◆
Definition & properties
◆
Inference for one variance
F distribution
◆
Definition & properties
◆
Inference for two variances
Relationships between distributions: “The BIG Five”.
Slide 2 of 54
Introduction
Chi-Square & F Distribution and Inferences about Variances
■
Introduction
● Introduction
● Chi-Square & F
Distributions
The Chi-square Distribution
◆
Definition, properties, tables of, density calculator
◆
Testing hypotheses about the variance of a single
population
(i.e., Ho : σ 2 = K).
◆
Example.
● Motivation
● Uses for Chi-Square & F
● Overview
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
■
Chi-Square and F Distributions
The F Distribution
◆
Definition, important properties, tables of
◆
Testing the equality of variances of two independent
populations
(i.e., Ho : σ12 = σ22 ).
◆
Example.
Slide 3 of 54
Chi-Square & F Distributions
Introduction
● Introduction
● Chi-Square & F
. . . and Inferences about Variances
Distributions
● Motivation
● Uses for Chi-Square & F
■
Comments regarding testing the homogeneity of variance
assumption of the two independent groups t–test (and
ANOVA).
■
Relationship among the Normal, t, χ2 , and F distributions.
● Overview
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
Chi-Square and F Distributions
Slide 4 of 54
Motivation
■
The normal and t distributions are useful for tests of
population means, but often we may want to make
inferences about population variances.
■
Examples:
Introduction
● Introduction
● Chi-Square & F
Distributions
● Motivation
● Uses for Chi-Square & F
● Overview
Chi-Square Distributions
◆
Does the variance equal a particular value?
◆
Does the variance in one population equal the variance in
another population?
◆
Are individual differences greater in one population than
another population?
◆
Are the variances in J populations all the same?
◆
Is the assumption of homogeneous variances reasonable
when doing a t–test (or ANOVA) of two (or more) means?
The F Distribution
Relationship Between
Distributions
Chi-Square and F Distributions
Slide 5 of 54
Uses for Chi-Square & F
■
Introduction
To make statistical inferences about populations variance(s),
we need
● Introduction
● Chi-Square & F
Distributions
● Motivation
◆
χ2 −→ The Chi-square distribution (Greek “chi”).
◆
F −→ Named after Sir Ronald Fisher who developed the
main applications of F.
● Uses for Chi-Square & F
● Overview
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
Chi-Square and F Distributions
■
The χ2 and F–distributions are used for many problems in
addition to the ones listed above.
■
They provide good approximations to a large class of
sampling distributions that are not easily determined.
Slide 6 of 54
Overview
Introduction
■
The Big Five Theoretical Distributions are the Normal,
Student’s t, χ2 , F, and the Binomial (π, n).
■
Plan:
● Introduction
● Chi-Square & F
Distributions
● Motivation
● Uses for Chi-Square & F
● Overview
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
Chi-Square and F Distributions
◆
Introduce χ2 and then the F distributions.
◆
Illustrate their uses for testing variances.
◆
Summarize and describe the relationship among the
Normal, Student’s t, χ2 and F.
Slide 7 of 54
Chi-Square Distributions
■
Suppose we have a population with scores Y that are
normally distributed with mean E(Y ) = µ and variance
= var(Y ) = σ 2 (i.e., Y ∼ N (µ, σ 2 )).
■
If we repeatedly take samples of size n = 1 and for each
“sample” compute
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
(Y − µ)2
= squared standard score
z =
2
σ
2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
■
Define χ21 = z 2
■
What would the sampling distribution of χ21 look like?
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 8 of 54
The Chi-Square Distribution, ν = 1
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 9 of 54
The Chi-Square Distribution, ν = 1
Introduction
■
χ21 are non-negative Real numbers
■
Since 68% of values from N (0, 1) fall between −1 to 1, 68%
of values from χ21 distribution must be between 0 and 1.
■
The chi-square distribution with ν = 1 is very skewed.
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 10 of 54
The Chi-Square Distribution, ν = 2
■
Repeatedly draw independent (random) samples of n = 2
from N (µ, σ 2 ).
■
Compute Z12 = (Y1 − µ)2 /σ 2 and Z22 = (Y2 − µ)2 /σ 2 .
■
Compute the sum: χ22 = Z12 + Z22 .
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
■
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 11 of 54
The Chi-Square Distribution, ν = 2
Introduction
■
All value non-negative
■
A little less skewed than χ21 .
■
The probability that χ22 falls in the range of 0 to 1 is smaller
relative to that for χ21 . . .
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
P (χ21 ≤ 1) =
2
● Properties of Family of χ
Distributions
P (χ22 ≤ 1) =
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
2
● Inferences about σ
■
.68
.39
Note that mean ≈ ν = 2. . . .
● Test Statistic for
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 12 of 54
Chi-Square Distributions
■
Generalize: For n independent observations from a
N (µ, σ 2 ), the sum of squared values has a Chi-square
distribution with n degrees of freedom.
■
Chi–squared distribution only depends on degrees of
freedom, which in turn depends on sample size n.
■
The standard scores are computed using population µ and
σ 2 ; however, we usually don’t know what µ and σ 2 equal.
When µ and σ 2 are estimated from the sampled data, the
degrees of freedom are less than n.
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 13 of 54
Chi-Square Dist: Varying ν
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 14 of 54
Properties of Family of χ2 Distributions
■
They are all positively skewed.
■
As ν gets larger, the degree of skew decreases.
■
As ν gets very large, χ2ν approaches the normal distribution.
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Why?
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 15 of 54
Properties of Family of χ2 Distributions
Introduction
■
E(χ2ν ) = mean = ν = degrees of freedom.
■
E[(χ2ν − E(χ2ν ))2 ] = var(χ2ν ) = 2ν.
■
Mode of χ2ν is at value ν − 2 (for ν ≥ 2).
■
Median is approximately = (3ν − 2)/3 (for ν ≥ 2).
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 16 of 54
Properties of Family of χ2 Distributions
IF
Introduction
■
A random variable χ2ν1 has a chi-squared distribution with ν1
degrees of freedom, and
■
A second independent random variable χ2ν2 has a
chi-squared distribution with ν2 degrees of freedom,
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
THEN
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
χ2(ν1 +ν2 ) = χ2ν1 + χ2ν2
their sum has a chi-squared distribution with (ν1 + ν2 ) degrees
of freedom.
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 17 of 54
Percentiles of χ2 Distributions
Note:
2
.95 χ1
2
= 3.84 = 1.962 = z.95
Introduction
Chi-Square Distributions
■
Tables
■
http://calculator.stat.ucla.edu/cdf/
■
pvalue.f program or the executable version, pvalue.exe, on
the course web-site.
■
SAS: PROBCHI(x,df<,nc>)
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
where
◆
◆
◆
x = number
df = degrees of freedom
If p=PROBCHI(x,df), then p = P rob(χ2df ≤ x)
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 18 of 54
SAS Examples & Computations
Input to program editor to get p-values:
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
DATA probval;
pz=PROBNORM(1.96);
pzsq=PROBCHI(3.84,1);
output;
RUN;
PROC PRINT data=probval;
RUN;
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
2
● Inferences about σ
Output:
pz
0.97500
pzsq
0.95000
What are these values?
● Test Statistic for
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 19 of 54
SAS Examples & Computations
. . . To get density values. . .
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
Probability Density;
data chisq3;
do x=0 to 10 by .005;
pdfxsq=pdf(’CHISQUARE’,x,3);
output;
end;
run;
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 20 of 54
Inferences about a Population Variance
or the sampling distribution of the sample variance from a
normal population.
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
■
Statistical Hypotheses:
ν = 1
● The Chi-Square Distribution,
Ho : σ 2 = σo2
ν = 2
● The Chi-Square Distribution,
ν = 2
versus
Ha : σ 2 6= σo2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
■
Assumptions: Observations are independently drawn
(random) from a normal population; i.e.,
Yi ∼ N (µ, σ 2 )
i.i.d
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 21 of 54
Inferences about σ 2
Test Statistic:
■
Introduction
n
X
(Yi − µ)2
We know
σ2
i=1
=
n
X
i=1
zi2 ∼ χ2n
Chi-Square Distributions
if z ∼ N (0, 1).
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
■
We don’t know µ, so we use Ȳ as an estimate of µ
● The Chi-Square Distribution,
ν = 2
n
X
(Yi − Ȳ )2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
i=1
Distributions
or
2
● Properties of Family of χ
Distributions
2
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
Pn
2
(Y
−
Ȳ
)
(n − 1)s2
i
2
i=1
=
∼
χ
n−1
σ2
σ2
2
● Properties of Family of χ
Distributions
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
σ2
∼ χ2n−1
■
So
σ2
χ2n−1
s ∼
(n − 1)
2
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 22 of 54
Test Statistic for Ho : σ 2 = σo2
■
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
Putting this all together, this gives us our test statistic:
Pn
2
(Y
−
Ȳ
)
i
Xν2 = i=1 2
σo
where Ho : σ 2 = σo2 .
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
■
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
Sampling distribution of Test Statistic: If Ho is true, which
means that σ 2 = σo2 , then
Pn
2
2
(Y
−
Ȳ
)
(n − 1)s
i
2
2
i=1
Xν =
=
∼
χ
n−1
σo2
σo2
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 23 of 54
Decision and Conclusion, Ho : σ 2 = σo2
■
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
Decision: Compare the obtained test statistic to the
chi-squared distribution with ν = n − 1 degrees of freedom.
ν = 1
● The Chi-Square Distribution,
or find the p-value of the test statistic and compare to α.
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
■
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Interpretation/Conclusion: What does the decision mean in
terms of what you’re investigating?
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 24 of 54
Example of Ho : σ 2 = σo2
■
High School and Beyond: Is the variance of math scores of
students from private schools equal to 100?
■
Statistical Hypotheses:
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
Ho : σ 2 = 100
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
versus
Ha : σ 2 6= 100
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
■
Assumptions: Math scores are independent and normally
distributed in the population of high school seniors who
attend private schools and the observations are
independent.
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 25 of 54
Example of Ho : σ 2 = σo2
■
Introduction
Test Statistic: n = 94, s2 = 67.16, and set α = .10.
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
(n − 1)s2
(94 − 1)(67.16)
X =
= 62.46
=
2
σ
100
ν = 1
2
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
with ν = (94 − 1) = 93.
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
■
Sampling Distribution of the Test Statistic:
Chi-square with ν = 93.
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
Critical values:
2
.05 χ93
= 71.76 & .95 χ293 = 116.51.
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 26 of 54
Example of Ho : σ 2 = σo2
2
.05 χ93
= 71.76 & .95 χ293 = 116.51.
■
Critical values:
■
Decision: Since the obtained test statistic X 2 = 62.46 is less
than .05 χ293 = 71.76, reject Ho at α = .10.
Introduction
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 27 of 54
Confidence Interval Estimate of σ 2
■
Start with
Introduction
Prob
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
2
(α/2) χν
2
(n − 1)s
≤
≤
σ2
2
(1−α/2) χν
=1−α
ν = 1
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
■
After a little algebra. . .
2
1
σ
1
Prob
≤
=1−α
≤
2
2
2
χ
χ
(n
−
1)s
(1−α/2) ν
(α/2) ν
■
and a little more
2
2
(n − 1)s
(n − 1)s
2
≤
σ
≤
=1−α
Prob
2
2
(1−α/2) χν
(α/2) χν
ν = 2
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 28 of 54
90% Confidence Interval Estimate of σ 2
■
Introduction
(1 − α)% Confidence interval,
(n − 1)s2
(n − 1)s2
2
≤σ ≤
2
2
χ
(1−α/2) ν
α/2 χ93
Chi-Square Distributions
● Chi-Square Distributions
● The Chi-Square Distribution,
ν = 1
● The Chi-Square Distribution,
ν = 1
■
● The Chi-Square Distribution,
ν = 2
So,
(94 − 1)(67.16)
,
116.51
● The Chi-Square Distribution,
ν = 2
● Chi-Square Distributions
ν
2
● Properties of Family of χ
● Chi-Square Dist: Varying
Distributions
(94 − 1)(67.16)
−→ (53.61, 87.04),
71.76
which does not include 100 (the null hypothesized value).
2
● Properties of Family of χ
Distributions
2
● Properties of Family of χ
Distributions
2
● Percentiles of χ
Distributions
● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about
● Test Statistic for
■
s2 = 67.16 isn’t in the center of the interval.
σ2
2
Ho : σ 2 = σo
● Decision and Conclusion,
2
Ho : σ 2 = σo
●
Example of
Chi-Square
and F Distributions
Ho : σ 2 = σ 2
Slide 29 of 54
The F Distribution
■
Comparing two variances: Are they equal?
■
Start with two independent populations, each normal and
equal variances.. . .
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
Y1
● Testing for Equal Variances
● Conditions for an F
Y2
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
■
● Important Properties of F
Distributions
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
■
i.i.d.
∼ N (µ2 , σ 2 )
i.i.d.
Draw two independent random samples from each
population,
n1
n2
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
∼ N (µ1 , σ 2 )
from population
from population
1
2
Using data from each of the two samples, estimate σ 2 .
s21
and
s22
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 30 of 54
The F Distribution
■
Introduction
Both S12 and S22 are random variables, and their ratio is a
random variable,
Chi-Square Distributions
The F Distribution
● The F Distribution
F
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
=
estimate of σ 2
s21
= 2
2
estimate of σ
s2
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
■
Random variable F has an F distribution.
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 31 of 54
Testing for Equal Variances
■
Introduction
■
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
F gives us a way to test Ho : σ12 = σ22 (= σ 2 ).
Test statistic:
Pn1
2
1
2
(Y
−
Ȳ
)
i1
1
s1
i=1
n1 −1
P
F =
=
n2
1
2
s22
i=1 (Yi2 − Ȳ2 )
n2 −1
=
■
χ2ν1 /ν1
χ2ν2 /ν2
1
σ2
1
σ2
A random variable formed from the ratio of two independent
chi-squared variables, each divided by it’s degrees of
freedom, is an “F –ratio” and has an F distribution.
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 32 of 54
Conditions for an F Distribution
■
IF
Introduction
Chi-Square Distributions
◆
Both parent populations are normal.
◆
Both parent populations have the same variance.
◆
The samples (and populations) are independent.
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
■
THEN the theoretical distribution of F is Fν1 ,ν2 where
● Important Properties of F
Distributions
◆
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
ν1 = n1 − 1 = numerator degrees of freedom
◆
ν2 = n2 − 1 = denominator degrees of freedom
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 33 of 54
Eg of F Distributions: F2,ν2
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 34 of 54
Eg of F Distributions: F5,ν2
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 35 of 54
Eg of F Distributions: F50,nu2 . . .
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 36 of 54
Important Properties of F Distributions
Introduction
■
The range of F–values is non-negative real numbers (i.e., 0
to +∞).
■
They depend on 2 parameters: numerator degrees of
freedom (ν1 ) and denominator degrees of freedom (ν2 ).
■
The expected value (i.e, the mean) of a random variable with
an F distribution with ν2 > 2 is
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
E(Fν1 ,ν2 ) = µFν1 ,ν2 = ν2 /(ν2 − 2).
■
For any fixed ν1 and ν2 , the F distribution is non-symmetric.
■
The particular shape of the F distribution varies considerably
with changes in ν1 and ν2 .
■
In most applications of the F distribution (at least in this
class), ν1 < ν2 , which means that F is positively skewed.
■
When ν2 > 2, theF distribution is uni-modal.
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 37 of 54
Percentiles of the F Dist.
■
http://calculators.stat.ucla.edu/cdf
p-value program
■
SAS probf
■
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
■
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
■
Tables textbooks given the upper 25th , 10th , 5th , 2.5th , and
1st percentiles. Usually, the
◆
Columns correspond to ν1 , numerator df.
◆
Rows correspond to ν2 , denominator df.
Getting lower percentiles using tables requires taking
reciprocals.
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 38 of 54
Selected F values from Table V
Note: all values are for upper α = .05
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
ν1
1
1
1
1
ν2
1
20
1000
∞
Fν1 ,ν2
161.00
4.35
3.85
3.84
ν1
1
4
10
20
1000
ν2
20
20
20
20
20
Fν1 ,ν2
4.35
2.87
2.35
2.12
1.57
which is also . . .
t21
t220
t21000
t2∞ = z 2 = χ21
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 39 of 54
Test Equality of Two Variances
Are students from private high schools more homogeneous
with respect to their math test scores than students from public
high schools?
Introduction
Chi-Square Distributions
■
Statistical Hypotheses:
The F Distribution
2
2
2
2
Ho : σprivate
= σpublic
or σpublic
/σprivate
=1
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
2
2
,(1-tailed test).
< σpublic
versus Ha : σprivate
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
■
Assumptions: Math scores of students from private schools
and public schools are normally distributed and are
independent both between and within in school type.
■
Test Statistic:
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
s21
91.74
= 1.366
F = 2 =
s2
67.16
with ν1 = (n1 − 1) = (506 − 1) = 505 and
ν2 = (n2 − 1) = (94 − 1) = 93.
Slide 40 of 54
Test Equality of Two Variances
■
Since the sample variance for public schools, s21 = 91.74, is
larger than the sample variance for private schools,
s22 = 67.16, put s21 in the numerator.
■
Sampling Distribution of Test Statistic is
F distribution with ν1 = 505 and ν2 = 93.
■
Decision: Our observed test statistic, F505,93 = 1.366 has a
p–value= .032. Since p–value < α = .05, reject Ho .
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
Or, we could compare the observed test statistic,
F505,93 = 1.366, with the critical value of
F505,93 (α = .05) = 1.320. Since the observed value of the
test statistic is larger than the critical value, reject Ho .
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
■
Conclusion: The data support the conclusion that students
from private schools are more homogeneous with respect to
math test scores than students from public schools.
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 41 of 54
Example Continued
■
Alternative question: “Are the individual differences of
students in public high schools and private high schools the
same with respect to their math test scores?”
■
Statistical Hypotheses: The null is the same, but the
alternative hypothesis would be
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
2
2
Ha : σpublic
6= σprivate
■
(a 2–tailed alternative)
Given α = .05, Retain the Ho , because our obtained p–value
(the probability of getting a test statistic as large or larger
than what we got) is larger than α/2 = .025.
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 42 of 54
Example Continued
Introduction
■
Given α = .05, Retain the Ho , because our obtained p–value
(the probability of getting a test statistic as large or larger
than what we got) is larger than α/2 = .025.
■
Or the rejection region (critical value) would be any
F –statistic greater than F505,93 (α = .025) = 1.393.
■
Point: This is a case where the choice between a 1 and 2
tailed test leads to different decisions regarding the null
hypothesis.
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 43 of 54
Test for Homogeneity of Variances
Introduction
Ho : σ12 = σ22 = . . . = σJ2
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
■
These include
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
◆
Hartley’s Fmax test
◆
Bartlett’s test
◆
One regarding variances of paired comparisons.
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
■
You should know that they exist; we won’t go over them in
this class. Such tests are not as important as they once
(thought) they were.
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 44 of 54
Test for Homogeneity of Variances
■
Old View: Testing the equality of variances should be a
preliminary to doing independent t-tests (or ANOVA).
■
Newer View:
◆ Homogeneity of variance is required for small samples,
which is when tests of homogeneous variances do not
work well. With large samples, we don’t have to assume
σ12 = σ22 .
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
◆
Test critically depends on population normality.
◆
If n1 = n2 , t-tests are robust.
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 45 of 54
Test for Homogeneity of Variances
■
For small or moderate samples and there’s concern with
possible heterogeneity −→ perform a Quasi-t test.
■
In an experimental settings where you have control over the
number of subjects and their assignment to
groups/conditions/etc. −→ equal sample sizes.
■
In non-experimental settings where you have similar
numbers of participants per group, t test is pretty robust.
Introduction
Chi-Square Distributions
The F Distribution
● The F Distribution
● The F Distribution
● Testing for Equal Variances
● Conditions for an F
Distribution
● Eg of F Distributions:
F2,ν
2
● Eg of F Distributions:
F5,ν
2
● Eg of F Distributions:
F50,nu . . .
2
● Important Properties of F
Distributions
● Percentiles of the F Dist.
● Selected F values from
Table V
● Test Equality of Two
Variances
● Test Equality of Two
Variances
● Example Continued
● Example Continued
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
● Test for Homogeneity of
Variances
Chi-Square and
F Distributions
Relationship
Between
Distributions
Slide 46 of 54
Relationship Between Distributions
Introduction
Chi-Square Distributions
Relationship between z, tν , χ2ν , and Fν1 ,ν2 . . . and the central
importance of the normal distribution.
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
■
Normal, Student’s tν , χ2ν , and Fν1 ,ν2 are all theoretical
distributions.
■
We don’t ever actually take vast (infinite) numbers of
samples from populations.
■
The distributions are derived based on mathematical logic
statements of the form
● Derivation of Distributions
● Chi-Square Distribution
● The F Distribution
● Students t Distribution
● Students t Distribution
(continued)
● Students t Distribution
(continued)
● Summary of Relationships
IF . . . . . . . . . Then . . . . . . . . .
Chi-Square and F Distributions
Slide 47 of 54
◆
THEN Ȳ is approximately normal.
Derivation of Distributions
Introduction
Chi-Square Distributions
■
Assumptions are part of the “if” part, the conditions used to
deduce sampling distribution of statistics.
■
The t, χ2 and F distributions all depend on normal “parent”
population.
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● The F Distribution
● Students t Distribution
● Students t Distribution
(continued)
● Students t Distribution
(continued)
● Summary of Relationships
Chi-Square and F Distributions
Slide 48 of 54
Chi-Square Distribution
Introduction
■
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
χ2ν =
● Derivation of Distributions
● Chi-Square Distribution
(continued)
● Students t Distribution
(continued)
n
X
i=1
● The F Distribution
● Students t Distribution
● Students t Distribution
χ2ν = sum of n(= ν) independent squared normal random
variables with mean µ = 0 and variance σ 2 = 1 (i.e.,
“standard normal” random variables).
■
zi2
where
zi ∼ N (0, 1)
i.i.d.
Based on the Central Limit Theorem, the “limit” of the χ2ν
distribution (i.e., ν = n → ∞) is normal.
● Summary of Relationships
Chi-Square and F Distributions
Slide 49 of 54
The F Distribution
Introduction
■
Fν1 ,ν2 = ratio of two independent chi-squared random
variables each divided by their respective degrees of
freedom.
χ2ν1 /ν1
Fν1 ,ν2 = 2
χν2 /ν2
■
Since χ2ν ’s depend on the normal distribution, the F
distribution also depends on the normal distribution.
■
The “limiting” distribution of Fν1 ,ν2 as ν2 → ∞ is
χ2ν1 /ν1 .. . . . . . because as ν2 → ∞, χ2ν2 /ν2 → 1.
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● The F Distribution
● Students t Distribution
● Students t Distribution
(continued)
● Students t Distribution
(continued)
● Summary of Relationships
Chi-Square and F Distributions
Slide 50 of 54
Students t Distribution
Introduction
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
Let ν = n − 1, and note that
2
Ȳ − µ
√
t2ν =
s/ n
=
● Derivation of Distributions
● Chi-Square Distribution
● The F Distribution
● Students t Distribution
● Students t Distribution
2
=
(continued)
● Students t Distribution
(continued)
● Summary of Relationships
Chi-Square and F Distributions
(Ȳ − µ)2 n
Pn
2
i=1 (Yi − Ȳ ) /(n − 1)
=
(Ȳ − µ) n
2
i=1 (Yi − Ȳ ) /(n − 1)
Pn
(Ȳ −µ)2
σ 2 /n
n
2
i=1 (Yi −Ȳ )
σ 2 (n−1)
1
σ2
1
σ2
z2
= 2
χ /ν
Slide 51 of 54
Students t Distribution (continued)
Introduction
■
Chi-Square Distributions
Student’s t based on normal,
2
z
t2ν = 2
χν /ν
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
or
z
t= p
χ2ν /ν
● Derivation of Distributions
● Chi-Square Distribution
● The F Distribution
● Students t Distribution
● Students t Distribution
(continued)
● Students t Distribution
(continued)
■
A squared t random variable equals the ratio of squared
standard normal divided by chi-squared divided by its
degrees of freedom. So. . .
● Summary of Relationships
Chi-Square and F Distributions
Slide 52 of 54
Students t Distribution (continued)
Since
2
z
t2ν = 2
χν /ν
Introduction
Chi-Square Distributions
or
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
t= p
χ2ν /ν
■
As ν → ∞, tν → N (0, 1) because χ2ν /ν → 1.
■
Since z 2 = χ21 ,
● Derivation of Distributions
● Chi-Square Distribution
z
● The F Distribution
● Students t Distribution
● Students t Distribution
χ21 /1
z 2 /1
= 2
= F1,ν
t = 2
χn /ν
χn /ν
2
(continued)
● Students t Distribution
(continued)
● Summary of Relationships
■
Chi-Square and F Distributions
Why are the assumptions of normality, homogeneity of
variance, and independence required for
◆
t test for mean(s)
◆
Testing homogeneity of variance(s).
Slide 53 of 54
Summary of Relationships
Introduction
Let z ∼ N (0, 1)
Chi-Square Distributions
The F Distribution
Relationship Between
Distributions
● Relationship Between
Distributions
Distribution
χ2ν
● Derivation of Distributions
● Chi-Square Distribution
● The F Distribution
● Students t Distribution
● Students t Distribution
(continued)
● Students t Distribution
(continued)
● Summary of Relationships
Chi-Square and F Distributions
Fν1 ,ν2
t
Definition
Pν
2
z
i=1 i
independent z’s
(χ2ν1 /ν1 )/(χ2ν2 /ν2 )
independent χ2 ’s
p
z/ χ2 /ν
Parent
normal
chi-squared
normal
Limiting
As ν → ∞,
χ2ν → normal
As ν2 → ∞,
Fν1 ,ν2 → χ2ν1 /ν1
As ν → ∞,
t → normal
Slide 54 of 54