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BUS12009
1
Part 9
Chisquare Tests
1.
Goodnessoffit Tests
2.
Tests of Independence: Contingency Tables
Section 9.1
Goodnessoffit Tests
Example 1
A journal reported that, in a bag
of M&M chocolate peanut candies, there are
30% brown, 30% yellow, 10% blue, 10% red,
10% green and 10% orange candies. Suppose
you purchase a bag of M&M chocolate
peanut candies at a nearby store and find 17
brown, 20 yellow, 13 blue, 7 red, 6 green and
9 orange candies, for a total of 72 candies.
Does the bag purchased agree with the
distribution suggested by the journal?
Objective of a
goodnessoffit test
: to find
out how well a set of observed frequencies
fits a set of expected frequencies calculated
according to a specified distribution. The
hypotheses considered are
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H
0
: the population follows a specified
distribution,
H
1
: the population does not follow the
specified distribution stated in
H
0
.
Suppose the whole population can be divided
into
k
categories. We reject
H
0
if and only if
∑
=
−
k
i
i
i
i
e
e
o
1
2
)
(
≥
χ
2
α
,
k
−
1
where the lefthand side is the value of the
test statistic for the test, the righthand side is
the critical value,
is the significance level,
o
i
is the observed frequency for the
i
th
category of the population and
e
i
is the
corresponding expected frequency calculated
by
e
i
=
Sample size
×
Corresponding probability in the
distribution specified by
H
0
.
The value of
2
,
n
can be obtained from the
chisquare distribution table
.
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Graphs of Density Functions of the
2
Distribution (“df” for degrees of freedom)
The rule introduced here is only an
0
0.2
0.4
0.6
0.8
246
8
df
=
1
df
=
2
df
=
3
df
=
4
df
=
5
df
=
10
df
=
15
df
=
20
df
=
25
df
=
30
0.02
0.04
0.06
0.08
0
10
20
30
40
50
Density function of the
2
distribution with
n
degrees of freedom
Area
=
2
,n
0
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approximate result. It is suggested that all
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 Spring '07
 Wood

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