chi-square and uji f

8/13/2019 Chi-Square and uji F

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Chi-square and F Distributions

Children of the Normal


2/16

Distributions

There are many theoreticaldistributions, both continuous anddiscrete.

We use 4 of these 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.


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Chi-square Distribution (1)

)(;)(;)(

yzXzSD

XXz

2

22 )(

y

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.


4/16

Chi-square (2)

What if we took 2 values of z2at random and added them?

2

2

22

22

2

12

1

)(;

)(

yz

yz 2

2

2

12

2

2

2

2

12

)2(

)()(zz

yy

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.


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Chi-square 3

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

v). As dfgets large, curve is less skewed,

more normal.


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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


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Distribution of Sample

Variance

1

)( 22

N

yys

Sample estimate of population variance

(unbiased).

2

2

2 )1(

)1(

sNN

Multiply variance estimate by N-1 to

get sum of squares. Divide bypopulation variance to normalize.

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.


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Testing Exact Hypotheses

about a Variance2

0

2

0: H Test the null that the populationvariance has some specific value. Pick

alpha and rejection region. Then:

2

0

2

2)1( )1(

sNN

Plug hypothesized populationvariance and sample variance into

equation along with sample size we

used to estimate variance. Compare

to chi-square distribution.


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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

sN

HH Note: 1 tailed test on

small side. Set alpha=.01.

11.2125.6

)55.4)(29(229

Mean is 29, so its 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

sN

HH

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.


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Confidence Intervals for the

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

95.)1()1(

2)975;.1(

22

2)025;.1(

2

NN

sNsNp

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

2s


11/16

Normality Assumption

We assume normal distributions to figuresampling distributions and thus plevels.

Violations of normality have minor

implications for testing means, especially asN gets large.

Violations of normality are more serious for

testing variances. Look at your data before

conducting this test. Can test for normality.


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The FDistribution (1)

The F distribution is the ratio of two

variance 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

sF

2

2

(

1

2

)(

/)

/

2

1

v

v

Fv

v

In our applications, v2will be larger

than v1and v2will be larger than 2.In such a case, the mean of the F

distribution (expected value) is

v2/(v2-2).


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FDistribution (2)

Fdepends on two parameters: v1andv2(df1and df2). The shape of Fchanges with these. Range is 0 to

infinity. Shaped a bit like chi-square. Ftables show critical values for dfinthe numerator and dfin thedenominator.

Ftables are 1-tailed; can figure 2-tailedif you need to (but you usually dont).


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Testing Hypotheses about 2

Variances Suppose

Note 1-tailed.

We find

Then df1=df2= 15, and

22

211

22

210 :;: HH

7.1;16;8.5;16 2222

11 sNsN

41.37.1

8.5

22

2

1 s

s

F

Going to the Ftable with 15

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

value is 2.40. Therefore the

result is significant.


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A Look Ahead

The Fdistribution 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).


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Relations among Distributions

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

N(0,1) deviates squared and summed.

Fis the ratio of two chi-squares, each

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

F= t2. If you square t, you get an Fwith 1 df in the numerator.

),1(

2

)( vv Ft

chi-square and uji f

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