Chi-Square andF Distributions: Tests for Variances …courses.education.illinois.edu/EdPsy580/lectures/6ChiSq_Fdist_ha.pdf · Chi-Square andFDistributions Slide 1 of 54 Chi-Square

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


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


TheF Distribution


Distributions


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


Distributions

● Motivation


● Overview


TheF Distribution


Distributions


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


Distributions

● Motivation


● Overview


TheF Distribution


Distributions


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


Distributions

● Motivation


● Overview


TheF Distribution


Distributions


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


Distributions

● Motivation


● Overview


TheF Distribution


Distributions


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

● The Chi-Square Distribution,

ν = 1● The Chi-Square Distribution,



ν = 2


● Chi-Square Dist: Varying ν

● Properties of Family of χ2

Distributions


Distributions


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



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


The Chi-Square Distribution, ν = 1

Introduction







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


Chi-Square Dist: Varying ν

Introduction







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


Example of Ho : σ2 = σ2o

■ High School and Beyond: Is the variance of math scores ofstudents from private schools equal to 100?


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o



■ 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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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







ν = 2




Distributions


Distributions


Distributions



Computations

● SAS Examples &

Computations


Variance



Ho : σ2= σ2


Ho : σ2 = σ2o

● Example of

Ho : σ2= σ2

o


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


TheF Distribution

● TheF Distribution


● Testing for Equal Variances

● Conditions for anF

Distribution● Eg ofF Distributions:

F2,ν2

● Eg ofF Distributions:

F5,ν2


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


● Test for Homogeneity of

Variances● Test for Homogeneity of


Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


Eg of F Distributions: F2,ν2

Introduction


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


Eg of F Distributions: F5,ν2

Introduction


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


Eg of F Distributions: F50,nu2. . .

Introduction


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


Test Equality of Two VariancesAre students from private high schools more homogeneouswith respect to their math test scores than students from publichigh schools?


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions


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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions



■ 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


TheF Distribution






F2,ν2


F5,ν2


F50,nu2. . .


Distributions




Variances






Variances


Distributions



■ 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


TheF Distribution


Distributions● Relationship Between

Distributions

● Derivation of Distributions

● Chi-Square Distribution


● Students t Distribution● Students t Distribution

(continued)● Students t Distribution

(continued)● Summary of Relationships


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


TheF Distribution



Distributions








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


TheF Distribution



Distributions








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


TheF Distribution



Distributions








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


TheF Distribution



Distributions








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


TheF Distribution



Distributions








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


TheF Distribution



Distributions








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


TheF Distribution



Distributions








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

Chi-Square andF Distributions: Tests for Variances …courses.education.illinois.edu/EdPsy580/lectures/6ChiSq_Fdist_ha.pdf · Chi-Square andFDistributions Slide 1 of 54 Chi-Square

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