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Chi-Square andF Distributions Slide 1 of 54
Chi-Square and F Distributions:Tests for Variances
Edpsy 580
Carolyn J. AndersonDepartment of Educational Psychology
University of Illinois at Urbana-Champaign
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions
Chi-Square andF Distributions Slide 2 of 54
Outline
■ Introduction, motivation and overview
■ Chi-square distribution
◆ Definition & properties
◆ Inference for one variance
■ F distribution
◆ Definition & properties
◆ Inference for two variances
■ Relationships between distributions: “The BIG Five”.
Introduction
● Introduction● Chi-Square &F
Distributions
● Motivation
● Uses for Chi-Square &F
● Overview
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions
Chi-Square andF Distributions Slide 3 of 54
IntroductionChi-Square & F Distribution and Inferences about Variances
■ The Chi-square Distribution
◆ Definition, properties, tables of, density calculator
◆ Testing hypotheses about the variance of a singlepopulation(i.e., Ho : σ2 = K).
◆ Example.
■ The F Distribution
◆ Definition, important properties, tables of
◆ Testing the equality of variances of two independentpopulations(i.e., Ho : σ2
1 = σ22).
◆ Example.
Introduction
● Introduction● Chi-Square &F
Distributions
● Motivation
● Uses for Chi-Square &F
● Overview
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions
Chi-Square andF Distributions Slide 4 of 54
Chi-Square & F Distributions
. . . and Inferences about Variances
■ Comments regarding testing the homogeneity of varianceassumption of the two independent groups t–test (andANOVA).
■ Relationship among the Normal, t, χ2, and F distributions.
Introduction
● Introduction● Chi-Square &F
Distributions
● Motivation
● Uses for Chi-Square &F
● Overview
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions
Chi-Square andF Distributions Slide 5 of 54
Motivation
■ The normal and t distributions are useful for tests ofpopulation means, but often we may want to makeinferences about population variances.
■ Examples:
◆ Does the variance equal a particular value?
◆ Does the variance in one population equal the variance inanother population?
◆ Are individual differences greater in one population thananother population?
◆ Are the variances in J populations all the same?
◆ Is the assumption of homogeneous variances reasonablewhen doing a t–test (or ANOVA) of two (or more) means?
Introduction
● Introduction● Chi-Square &F
Distributions
● Motivation
● Uses for Chi-Square &F
● Overview
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions
Chi-Square andF Distributions Slide 6 of 54
Uses for Chi-Square & F
■ To make statistical inferences about populations variance(s),we need
◆ χ2 −→ The Chi-square distribution (Greek “chi”).
◆ F−→ Named after Sir Ronald Fisher who developed themain applications of F .
■ The χ2 and F–distributions are used for many problems inaddition to the ones listed above.
■ They provide good approximations to a large class ofsampling distributions that are not easily determined.
Introduction
● Introduction● Chi-Square &F
Distributions
● Motivation
● Uses for Chi-Square &F
● Overview
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions
Chi-Square andF Distributions Slide 7 of 54
Overview
■ The Big Five Theoretical Distributions are the Normal,Student’s t, χ2, F , and the Binomial (π, n).
■ Plan:
◆ Introduce χ2 and then the F distributions.
◆ Illustrate their uses for testing variances.
◆ Summarize and describe the relationship among theNormal, Student’s t, χ2 and F .
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 8 of 54
Chi-Square Distributions
■ Suppose we have a population with scores Y that arenormally distributed with mean E(Y ) = µ and variance= var(Y ) = σ2 (i.e., Y ∼ N (µ, σ2)).
■ If we repeatedly take samples of size n = 1 and for each“sample” compute
z2 =(Y − µ)2
σ2= squared standard score
■ Define χ21 = z2
■ What would the sampling distribution of χ21 look like?
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 9 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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 10 of 54
The Chi-Square Distribution, ν = 1
■ χ21 are non-negative Real numbers
■ Since 68% of values from N (0, 1) fall between −1 to 1, 68%of values from χ2
1 distribution must be between 0 and 1.
■ The chi-square distribution with ν = 1 is very skewed.
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 11 of 54
The Chi-Square Distribution, ν = 2
■ Repeatedly draw independent (random) samples of n = 2from N (µ, σ2).
■ Compute Z21 = (Y1 − µ)2/σ2 and Z2
2 = (Y2 − µ)2/σ2.
■ Compute the sum: χ22 = Z2
1 + 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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 12 of 54
The Chi-Square Distribution, ν = 2
■ 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. . .
P (χ21 ≤ 1) = .68
P (χ22 ≤ 1) = .39
■ Note that mean ≈ ν = 2. . . .
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 13 of 54
Chi-Square Distributions
■ Generalize: For n independent observations from aN (µ, σ2), the sum of squared values has a Chi-squaredistribution with n degrees of freedom.
■ Chi–squared distribution only depends on degrees offreedom, which in turn depends on sample size n.
■ The standard scores are computed using population µ andσ2; however, we usually don’t know what µ and σ2 equal.When µ and σ2 are estimated from the sampled data, thedegrees 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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 14 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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 15 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.
Why?
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 16 of 54
Properties of Family of χ2 Distributions
■ E(χ2ν) = mean = ν = degrees of freedom.
■ E[(χ2ν − E(χ2
ν))2] = var(χ2ν) = 2ν.
■ Mode of χ2ν is at value ν − 2 (for ν ≥ 2).
■ Median is approximately = (3ν − 2)/3 (for ν ≥ 2).
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 17 of 54
Properties of Family of χ2 Distributions
IF
■ A random variable χ2ν1
has a chi-squared distribution with ν1
degrees of freedom, and
■ A second independent random variable χ2ν2
has achi-squared distribution with ν2 degrees of freedom,
THEN
χ2(ν1+ν2)
= χ2ν1
+ χ2ν2
their sum has a chi-squared distribution with (ν1 + ν2) degreesof freedom.
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 18 of 54
Percentiles of χ2 Distributions
Note: .95χ21 = 3.84 = 1.962 = z2
.95
■ Tables
■ http://calculator.stat.ucla.edu/cdf/
■ pvalue.f program or the executable version, pvalue.exe, onthe course web-site.
■ SAS: PROBCHI(x,df<,nc>)
where
◆ x = number◆ df = degrees of freedom◆ If p=PROBCHI(x,df), then p = Prob(χ2
df ≤ x)
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 19 of 54
SAS Examples & Computations
Input to program editor to get p-values:
DATA probval;pz=PROBNORM(1.96);pzsq=PROBCHI(3.84,1);output;
RUN;
PROC PRINT data=probval;RUN;
Output:pz pzsq
0.97500 0.95000
What are these 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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 20 of 54
SAS Examples & Computations
. . . To get density values. . .
Probability Density;data chisq3;do x=0 to 10 by .005;
pdfxsq=pdf(’CHISQUARE’,x,3);output;
end;run;
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 21 of 54
Inferences about a Population Variance
or the sampling distribution of the sample variance from anormal population.
■ Statistical Hypotheses:
Ho : σ2 = σ2o versus Ha : σ2 6= σ2
o
■ Assumptions: Observations are independently drawn(random) from a normal population; i.e.,
Yi ∼ N (µ, σ2) i.i.d
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 22 of 54
Inferences about σ2
Test Statistic:
■ We know
n∑
i=1
(Yi − µ)2
σ2=
n∑
i=1
z2i ∼ χ2
n
if z ∼ N (0, 1).
■ We don’t know µ, so we use Y as an estimate of µ
n∑
i=1
(Yi − Y )2
σ2∼ χ2
n−1
or∑n
i=1(Yi − Y )2
σ2=
(n − 1)s2
σ2∼ χ2
n−1
■ So
s2 ∼ σ2
(n − 1)χ2
n−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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 23 of 54
Test Statistic for Ho : σ2 = σ2o
■ Putting this all together, this gives us our test statistic:
X 2ν =
∑ni=1(Yi − Y )2
σ2o
where Ho : σ2 = σ2o .
■ Sampling distribution of Test Statistic: If Ho is true, whichmeans that σ2 = σ2
o , then
X 2ν =
(n − 1)s2
σ2o
=
∑ni=1(Yi − Y )2
σ2o
∼ χ2n−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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 24 of 54
Decision and Conclusion, Ho : σ2 = σ2o
■ Decision: Compare the obtained test statistic to thechi-squared distribution with ν = n − 1 degrees of freedom.
or find the p-value of the test statistic and compare to α.
■ Interpretation/Conclusion: What does the decision mean interms of what you’re investigating?
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 25 of 54
Example of Ho : σ2 = σ2o
■ High School and Beyond: Is the variance of math scores ofstudents from private schools equal to 100?
■ Statistical Hypotheses:
Ho : σ2 = 100 versus Ha : σ2 6= 100
■ Assumptions: Math scores are independent and normallydistributed in the population of high school seniors whoattend private schools and the observations areindependent.
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 26 of 54
Example of Ho : σ2 = σ2o
■ Test Statistic: n = 94, s2 = 67.16, and set α = .10.
X 2 =(n − 1)s2
σ2=
(94 − 1)(67.16)
100= 62.46
with ν = (94 − 1) = 93.
■ Sampling Distribution of the Test Statistic:Chi-square with ν = 93.
Critical values: .05χ293 = 71.76 & .95χ
293 = 116.51.
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 27 of 54
Example of Ho : σ2 = σ2o
■ Critical values: .05χ293 = 71.76 & .95χ
293 = 116.51.
■ Decision: Since the obtained test statistic X 2 = 62.46 is lessthan .05χ
293 = 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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 28 of 54
Confidence Interval Estimate of σ2
■ Start with
Prob(
(α/2)χ2ν ≤ (n − 1)s2
σ2≤ (1−α/2)χ
2ν
)
= 1 − α
■ After a little algebra. . .
Prob[(
1
(1−α/2)χ2ν
)
≤ σ2
(n − 1)s2≤
(
1
(α/2)χ2ν
)]
= 1 − α
■ and a little more
Prob[(
(n − 1)s2
(1−α/2)χ2ν
)
≤ σ2 ≤(
(n − 1)s2
(α/2)χ2ν
)]
= 1 − α
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
● Chi-Square Dist: Varying ν
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Properties of Family of χ2
Distributions
● Percentiles of χ2
Distributions● SAS Examples &
Computations
● SAS Examples &
Computations
● Inferences about a Population
Variance
● Inferences about σ2
● Test Statistic for
Ho : σ2= σ2
o● Decision and Conclusion,
Ho : σ2 = σ2o
● Example of
Ho : σ2= σ2
o
Chi-Square andF Distributions Slide 29 of 54
90% Confidence Interval Estimate of σ2
■ (1 − α)% Confidence interval,
(n − 1)s2
(1−α/2)χ2ν
≤ σ2 ≤ (n − 1)s2
α/2χ293
■ So,
(94 − 1)(67.16)
116.51,
(94 − 1)(67.16)
71.76−→ (53.61, 87.04),
which does not include 100 (the null hypothesized value).
■ s2 = 67.16 isn’t in the center of the interval.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 30 of 54
The F Distribution
■ Comparing two variances: Are they equal?
■ Start with two independent populations, each normal andequal variances.. . .
Y1 ∼ N (µ1, σ2) i.i.d.
Y2 ∼ N (µ2, σ2) i.i.d.
■ Draw two independent random samples from eachpopulation,
n1 from population 1
n2 from population 2
■ Using data from each of the two samples, estimate σ2.
s21 and s2
2
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 31 of 54
The F Distribution
■ Both S21 and S2
2 are random variables, and their ratio is arandom variable,
F =estimate of σ2
estimate of σ2=
s21
s22
■ Random variable F has an F distribution.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 32 of 54
Testing for Equal Variances
■ F gives us a way to test Ho : σ21 = σ2
2(= σ2).■ Test statistic:
F =
(
s21
s22
)
=1
n1−1
∑n1
i=1(Yi1 − Y1)2(
1σ2
)
1n2−1
∑n2
i=1(Yi2 − Y2)2(
1σ2
)
=χ2
ν1/ν1
χ2ν2
/ν2
■ A random variable formed from the ratio of two independentchi-squared variables, each divided by it’s degrees offreedom, is an “F–ratio” and has an F distribution.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 33 of 54
Conditions for an F Distribution
■ IF
◆ Both parent populations are normal.
◆ Both parent populations have the same variance.
◆ The samples (and populations) are independent.
■ THEN the theoretical distribution of F is Fν1,ν2where
◆ ν1 = n1 − 1 = numerator degrees of freedom
◆ ν2 = n2 − 1 = denominator degrees of freedom
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 34 of 54
Eg of F Distributions: F2,ν2
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 35 of 54
Eg of F Distributions: F5,ν2
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 36 of 54
Eg of F Distributions: F50,nu2. . .
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 37 of 54
Important Properties of F Distributions■ The range of F–values is non-negative real numbers (i.e., 0
to +∞).
■ They depend on 2 parameters: numerator degrees offreedom (ν1) and denominator degrees of freedom (ν2).
■ The expected value (i.e, the mean) of a random variable withan F distribution with ν2 > 2 is
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 considerablywith changes in ν1 and ν2.
■ In most applications of the F distribution (at least in thisclass), ν1 < ν2, which means that F is positively skewed.
■ When ν2 > 2, theF distribution is uni-modal.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 38 of 54
Percentiles of the F Dist.
■ http://calculators.stat.ucla.edu/cdf■ p-value program
■ SAS probf
■ 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 takingreciprocals.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 39 of 54
Selected F values from Table V
Note: all values are for upper α = .05
ν1 ν2 Fν1,ν2which is also . . .
1 1 161.00 t211 20 4.35 t2201 1000 3.85 t210001 ∞ 3.84 t2
∞= z2 = χ2
1
ν1 ν2 Fν1,ν2
1 20 4.354 20 2.87
10 20 2.3520 20 2.12
1000 20 1.57
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 40 of 54
Test Equality of Two VariancesAre students from private high schools more homogeneouswith respect to their math test scores than students from publichigh schools?
■ Statistical Hypotheses:
Ho : σ2private = σ2
public or σ2public/σ
2private = 1
versus Ha : σ2private < σ2
public ,(1-tailed test).
■ Assumptions: Math scores of students from private schoolsand public schools are normally distributed and areindependent both between and within in school type.
■ Test Statistic:
F =s21
s22
=91.74
67.16= 1.366
with ν1 = (n1 − 1) = (506 − 1) = 505 andν2 = (n2 − 1) = (94 − 1) = 93.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 41 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 s2
1 in the numerator.
■ Sampling Distribution of Test Statistic isF distribution with ν1 = 505 and ν2 = 93.
■ Decision: Our observed test statistic, F505,93 = 1.366 has ap–value= .032. Since p–value < α = .05, reject Ho.
Or, we could compare the observed test statistic,F505,93 = 1.366, with the critical value ofF505,93(α = .05) = 1.320. Since the observed value of thetest statistic is larger than the critical value, reject Ho.
■ Conclusion: The data support the conclusion that studentsfrom private schools are more homogeneous with respect tomath test scores than students from public schools.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 42 of 54
Example Continued
■ Alternative question: “Are the individual differences ofstudents in public high schools and private high schools thesame with respect to their math test scores?”
■ Statistical Hypotheses: The null is the same, but thealternative hypothesis would be
Ha : σ2public 6= σ2
private (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 largerthan what we got) is larger than α/2 = .025.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 43 of 54
Example Continued
■ Given α = .05, Retain the Ho, because our obtained p–value(the probability of getting a test statistic as large or largerthan what we got) is larger than α/2 = .025.
■ Or the rejection region (critical value) would be anyF–statistic greater than F505,93(α = .025) = 1.393.
■ Point: This is a case where the choice between a 1 and 2tailed test leads to different decisions regarding the nullhypothesis.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 44 of 54
Test for Homogeneity of Variances
Ho : σ21 = σ2
2 = . . . = σ2J
■ These include
◆ Hartley’s Fmax test
◆ Bartlett’s test
◆ One regarding variances of paired comparisons.
■ You should know that they exist; we won’t go over them inthis class. Such tests are not as important as they once(thought) they were.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 45 of 54
Test for Homogeneity of Variances
■ Old View: Testing the equality of variances should be apreliminary 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 notwork well. With large samples, we don’t have to assumeσ2
1 = σ22 .
◆ Test critically depends on population normality.
◆ If n1 = n2, t-tests are robust.
Introduction
Chi-Square Distributions
TheF Distribution
● TheF Distribution
● TheF Distribution
● Testing for Equal Variances
● Conditions for anF
Distribution● Eg ofF Distributions:
F2,ν2
● Eg ofF Distributions:
F5,ν2
● Eg ofF Distributions:
F50,nu2. . .
● Important Properties ofF
Distributions
● Percentiles of theF Dist.● SelectedF 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
Relationship Between
Distributions
Chi-Square andF Distributions Slide 46 of 54
Test for Homogeneity of Variances
■ For small or moderate samples and there’s concern withpossible heterogeneity −→ perform a Quasi-t test.
■ In an experimental settings where you have control over thenumber of subjects and their assignment togroups/conditions/etc. −→ equal sample sizes.
■ In non-experimental settings where you have similarnumbers of participants per group, t test is pretty robust.
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 47 of 54
Relationship Between Distributions
Relationship between z, tν , χ2ν , and Fν1,ν2
. . . and the centralimportance of the normal distribution.
■ Normal, Student’s tν , χ2ν , and Fν1,ν2
are all theoreticaldistributions.
■ We don’t ever actually take vast (infinite) numbers ofsamples from populations.
■ The distributions are derived based on mathematical logicstatements of the form
IF . . . . . . . . . Then . . . . . . . . .
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 48 of 54
Derivation of Distributions◆ THEN Y is approximately normal.
■ Assumptions are part of the “if” part, the conditions used todeduce sampling distribution of statistics.
■ The t, χ2 and F distributions all depend on normal “parent”population.
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 49 of 54
Chi-Square Distribution
■ χ2ν = sum of n(= ν) independent squared normal random
variables with mean µ = 0 and variance σ2 = 1 (i.e.,“standard normal” random variables).
χ2ν =
n∑
i=1
z2i 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.
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 50 of 54
The F Distribution
■ Fν1,ν2= ratio of two independent chi-squared random
variables each divided by their respective degrees offreedom.
Fν1,ν2=
χ2ν1
/ν1
χ2ν2
/ν2
■ Since χ2ν ’s depend on the normal distribution, the F
distribution also depends on the normal distribution.
■ The “limiting” distribution of Fν1,ν2as ν2 → ∞ is
χ2ν1
/ν1.. . . . . . because as ν2 → ∞, χ2ν2
/ν2 → 1.
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 51 of 54
Students t Distribution
Let ν = n − 1, and note that
t2ν =
(
Y − µ
s/√
n
)2
=(Y − µ)2n
∑ni=1(Yi − Y )2/(n − 1)
=(Y − µ)2n
∑ni=1(Yi − Y )2/(n − 1)
( 1σ2
1σ2
)
=
(Y −µ)2
σ2/nni=1
(Yi−Y )2
σ2(n−1)
=z2
χ2/ν
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 52 of 54
Students t Distribution (continued)
■ Student’s t based on normal,
t2ν =z2
χ2ν/ν
or t =z
√
χ2ν/ν
■ A squared t random variable equals the ratio of squaredstandard normal divided by chi-squared divided by itsdegrees of freedom. So. . .
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 53 of 54
Students t Distribution (continued)
Since
t2ν =z2
χ2ν/ν
or t =z
√
χ2ν/ν
■ As ν → ∞, tν → N (0, 1) because χ2ν/ν → 1.
■ Since z2 = χ21,
t2 =z2/1
χ2n/ν
=χ2
1/1
χ2n/ν
= F1,ν
■ Why are the assumptions of normality, homogeneity ofvariance, and independence required for
◆ t test for mean(s)
◆ Testing homogeneity of variance(s).
Introduction
Chi-Square Distributions
TheF Distribution
Relationship Between
Distributions● Relationship Between
Distributions
● Derivation of Distributions
● Chi-Square Distribution
● TheF Distribution
● Students t Distribution● Students t Distribution
(continued)● Students t Distribution
(continued)● Summary of Relationships
Chi-Square andF Distributions Slide 54 of 54
Summary of Relationships
Let z ∼ N (0, 1)
Distribution Definition Parent Limiting
χ2ν
∑νi=1 z2
i normal As ν → ∞,independent z’s χ2
ν → normal
Fν1,ν2(χ2
ν1/ν1)/(χ
2ν2
/ν2) chi-squared As ν2 → ∞,independent χ2’s Fν1,ν2
→ χ2ν1
/ν1
t z/√
χ2/ν normal As ν → ∞,t → normal