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8/13/2019 Chi-Square and uji F
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Chi-square and F Distributions
Children of the Normal
8/13/2019 Chi-Square and uji F
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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.
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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
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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