4/27/2011
1
STAT 350
Lecture 27
8.3 Tests Concerning Hypotheses About a
Categorical Population
8.3 Tests Concerning Hypotheses
About a Categorical Population
Examples of Categorical Data
◦
Gender
◦
Color or Type of Transmissions
of
automobiles
◦
Political Party Registration
Chisquare Test
◦
Invented in 1900 by Karl Pearson
Example 1
A gambler is accused of using a loaded
die, but he pleads innocent.
A record has
kept of the last 60 throws. There is a
disagreement about how to interpret the
data and a statistician is called in.
value
1
2
3
4
5
6
sum
Observed
4
6
17
16
8
9
60
expected
Example 1
A gambler is accused of using a loaded
die, but he pleads innocent.
A record has
kept of the last 60 throws. There is a
disagreement about how to interpret the
data and a statistician is called in.
value
1
2
3
4
5
6
sum
Observed
4
6
17
16
8
9
60
expected
10
10
10
10
10
10
60
Chisquare test
idea:
combine all these differences into
one overall measure of the distance
between observed and expected value.
χ
2
= ∑(observed –
expected )
2
/expected
χ
2
has chisquare distribution with degree
of freedom = #categories 1
Example 1
χ
2
= ∑(observed –
expected )
2
/expected
χ
2
= 6
2
/10 + 4
2
/10 + 7
2
/10
+ 6
2
/10 + 2
2
/10+ 1
2
/10
= 14.2
df = 5
value
1
2
3
4
5
6
sum
Observed
4
6
17
16
8
9
60
expected
10
10
10
10
10
10
60
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Example 1
Hypothesis:
◦
H0: π1=π2=π3=π4=π5=π6=1/6
◦
Ha: die is not fair (H0 is incorrect)
Test Statistic:
◦
χ
2
= 14.2, with df=5
Pvalue < 0.015 => reject H0 at
α
=0.05
◦
Table VII (page 571):
◦
Right tail area = 0.015 when
χ
2
= 14.09
ChiSquared Distributions
A family of distributions, each member is
identified by the degree of freedom
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 Fall '08
 Staff
 Statistics, ChiSquare Test, Hebrew numerals, Chisquare distribution, Pearson's chisquare test, ARCO

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