Chi-Square & F Distributions

8/12/2019 Chi-Square & F Distributions

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Chi-Square and F

Distributions



Questions

• What is the chi-square distribution?

How is it related to the Normal?

• How is the chi-square distribution

related to the sampling distribution ofthe variance?

• Test a population value of the variance;

put confidence intervals around a population value.



Questions

• How is the F distribution related the

Normal? To Chi-square?



Distributions

• There are many theoreticaldistributions, both continuous anddiscrete. Howell calls these test

statistics• We use 4 test statistics a lot: z (unit

normal), t , chi-square ( ), and F .

• Z and t are closely related to thesampling distribution of means; chi-square and F are closely related to thesampling distribution of variances.

2



Chi-square Distribution (1)

)(;)(

X z

SD X X z

2

22 )(

X

z

z score

z score squared

2)1(

2 z Make it Greek

What would its sampling distribution look like?

Minimum value is zero.

Maximum value is infinite.

Most values are between zero and 1;

most around zero.



Chi-square (2)

What if we took 2 values of z2 at random and added them?

2

2

22

22

2

12

1

)(;

)(

X z

X z 2

2

2

12

2

2

2

2

12

)2(

)()( z z

X X

Chi-square is the distribution of a sum of squares.

Each squared deviation is taken from the unit normal: N(0,1). The shape of the chi-square distribution

depends on the number of squared deviates that are

added together.

Same minimum and maximum as before, but now averageshould be a bit bigger.



Chi-square 3

The distribution of chi-square depends on1 parameter, its degrees of freedom (df or

v). As df gets large, curve is less skewed,

more normal.



Chi-square (4)

• The expected value of chi-square is df .

– The mean of the chi-square distribution is its

degrees of freedom.

• The expected variance of the distribution is2df .

– If the variance is 2df , the standard deviation must

be sqrt(2df ).

• There are tables of chi-square so you can find5 or 1 percent of the distribution.

• Chi-square is additive.2

)(

2

)(

2

)( 2121 vvvv



Distribution of Sample

Variance

1

)( 2

2

N

y y s

Sample estimate of population variance

(unbiased).

2

2

2 )1(

)1(

s N N

Multiply variance estimate by N-1 to

get sum of squares. Divide by population variance to stadnardize.

Result is a random variable distributed

as chi-square with (N-1) df .

We can use info about the sampling distribution of the

variance estimate to find confidence intervals and

conduct statistical tests.



Testing Exact Hypotheses

about a Variance20

20 : H Test the null that the population

variance has some specific value. Pick

alpha and rejection region. Then:

2

0

2

2)1( )1(

s N N

Plug hypothesized population

variance and sample variance into

equation along with sample size we

used to estimate variance. Compare

to chi-square distribution.



Example of Exact Test

Test about variance of height of people in inches. Grab 30

people at random and measure height.

55.4;30

.25.6:;25.6:

2

2

1

2

0

s N

H H Note: 1 tailed test on

small side. Set alpha=.01.

11.2125.6

)55.4)(29(2

29

Mean is 29, so it’s on the small

side. But for Q=.99, the value

of chi-square is 14.257.

Cannot reject null.

55.4;30

.25.6:;25.6:2

2120

s N

H H

Now chi-square with v=29 and Q=.995 is 13.121 and

also with Q=.005 the result is 52.336. N. S. either way.

Note: 2 tailed with alpha=.01.



Confidence Intervals for the

VarianceWe use to estimate . It can be shown that:2 s 2

95.)1()1(

2)975;.1(

22

2)025;.1(

2

N N

s N s N p

Suppose N=15 and is 10. Then df =14 and for Q=.025

the value is 26.12. For Q=.975 the value is 5.63.

95.

63.5

)10)(14(

12.26

)10)(14( 2

p

95.87.2436.5 2 p

2 s



Normality Assumption

• We assume normal distributions to figuresampling distributions and thus p levels.

• Violations of normality have minor

implications for testing means, especially as N gets large.

• Violations of normality are more serious for

testing variances. Look at your data before

conducting this test. Can test for normality.



Review

• You have sample 25 children from anelementary school 5th grade class and

measured the height of each. You

wonder whether these children are morevariable in height than typical children.

Their variance in height is 4. Compute

a confidence interval for this variance.If the variance of height in children in

5th grade nationally is 2, do you

consider this sample ordinary?



The F Distribution (1)

• The F distribution is the ratio of twovariance estimates:

• Also the ratio of two chi-squares, each

divided by its degrees of freedom:

2

2

2

1

2

2

2

1

.

.

est

est

s

s F

2

2

(

1

2

)(

/)

/

2

1

v

v

F v

v

In our applications, v2 will be larger

than v1 and v2 will be larger than 2.In such a case, the mean of the F

distribution (expected value) is

v2 /(v2 -2).



F Distribution (2)

• F depends on two parameters: v1 andv2 (df 1 and df 2). The shape of F changes with these. Range is 0 to

infinity. Shaped a bit like chi-square.• F tables show critical values for df inthe numerator and df in thedenominator.

• F tables are 1-tailed; can figure 2-tailedif you need to (but you usually don’t).



F table – critical values

Numerator df: df B

df W 1 2 3 4 5

5 5%1%

6.6116.3

5.7913.3

5.4112.1

5.1911.4

5.0511.0

10 5%

1%

4.96

10.0

4.10

7.56

3.71

6.55

3.48

5.99

3.33

5.64

12 5%

1%

4.75

9.33

3.89

6.94

3.49

5.95

3.26

5.41

3.11

5.06

14 5%

1%

4.60

8.86

3.74

6.51

3.34

5.56

3.11

5.04

2.96

4.70

e.g. critical value of F at alpha=.05 with 3 & 12 df =3.49



Testing Hypotheses about 2

Variances• Suppose

– Note 1-tailed.

• We find

• Then df 1=df 2 = 15, and

22

211

22

210 :;: H H

7.1;16;8.5;16 2

22

2

11 s N s N

41.37.1

8.5

22

2

1 s

s

F

Going to the F table with 15

and 15 df , we find that for alpha= .05 (1-tailed), the critical

value is 2.40. Therefore the

result is significant.



A Look Ahead

• The F distribution is used in manystatistical tests

– Test for equality of variances.

– Tests for differences in means in ANOVA.

– Tests for regression models (slopes

relating one continuous variable to another

like SAT and GPA).



Relations among Distributions

– the Children of the Normal• Chi-square is drawn from the normal.

N(0,1) deviates squared and summed.

• F is the ratio of two chi-squares, each

divided by its df. A chi-square divided by its df is a variance estimate, that is,a sum of squares divided by degrees offreedom.

• F = t 2. If you square t , you get an F with 1 df in the numerator.

),1(

2

)( vv F t



Review

• How is F related to the Normal? Tochi-square?

• Suppose we have 2 samples and we

want to know whether they were drawnfrom populations where the variances

are equal. Sample1: N =50, s2=25;

Sample 2: N =60, s2

=30. How can wetest? What is the best conclusion for

these data?

Chi-Square & F Distributions

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