STA 6126 Chap 8, Page 1 of 20
Chapter 8 The ChiSquare Tests
Chapter 8 deals with association between two categorical variables. We will cover three tests that
are very similar in nature but differ in the conditions when they can be used. These are
A)
Goodnessoftests
B)
Tests of homogeneity
and
C)
Test of independence.
Let’s start with the easiest one.
A) Goodnessoffit Test
This is an extension of the onepopulation, onesample, oneparameter problem where the
random variable of interest is a categorical variable with 2 categories and the hypotheses were
Ho: π = π
o
versus Ha: π ≠ π
0
.
We now extend the above test to the case of a categorical random variable with k (k ≥ 2)
categories.
Suppose we have a random variable that has k = 3 categories. Then the hypotheses of interest
will be
Ho: π
1
= π
10
, π
2
= π
20
, π
3
= π
30
vs.
Ha: At least one of π
i
≠ π
i0
Where π
i
are the proportion of population units in the i
th
category and π
i0
are the values of π
i
specified by the null hypothesis.
To test these hypotheses we select a random sample of size n and count the number of
sample units
observed
in each category (denoted by O
i
).
Next, we calculate the
expected
number of observations (E
i
) in each category
assuming
Ho to be true,
using E
i
= n×π
i0
.
Finally we compare the observed frequencies with the expected frequencies using the test
statistic
2
22
()
1
~
k
ii
df
i
i
OE
E
, where df = k – 1.
Other steps of hypothesis testing are the same as before:
1)
Assumptions
a)
Simple random samples from the population
b)
Categorical variable with k categories
c)
Large samples (O
i
≥ 5 for all i)
2)
Hypotheses:
Ho: π
1
= π
10
, π
2
= π
20
, π
3
= π
30
vs. Ha: At least one of π
i
≠ π
i0
3)
Test Statistic:
2
1
~
k
df
i
i
E
, with df = (k–1)
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View Full DocumentSTA 6126 Chap 8, Page 2 of 20
4)
The pvalue =
22
()
df
cal
P
5)
Decision
Same rule as ever,
Reject Ho if the pvalue ≤ α.
6)
Conclusion
Same as before, explain the decision in simple English for the layman.
Example:
Suppose we suspect that a die (used in a Las Vegas Casino) is loaded. To see if this
suspicion is warranted we roll the die 600 times and observe the frequencies given in Table 8.1.
The hypotheses of interest are
Ho: π
1
= π
2
= π
3
= π
4
= π
5
= π
6
= 1/6 vs. Ha: At least one of the π
i
≠ 1/6.
Let’s test these hypotheses. But first we need to check if all of the conditions are satisfied:
1)
Assumptions
Satisfied?
a)
Simple random samples from the population
Yes
b)
Categorical variable with k categories
Yes, k = 6
c)
Large samples (O
i
≥ 5 for all i)
Yes, look at Table 8.1
2) Hypotheses: Ho: π
1
= π
2
= π
3
= π
4
= π
5
= π
6
= 1/6 vs. Ha: At least one of the π
i
≠ 1/6.
3. Test Statistic:
2
6
(6 1)
1
~
ii
i
i
OE
E
4. The pvalue:
For this we need to find the calculated value of the test statistic first. This is
done in the following table (worksheet):
Table 8.1 Observed and Expected Values of 600 rolls of a die
Category
Observed
(O
i
)
Expected
(E
i
)
2
2
i
E
1
115
100
15
225
2.25
2
97
100
– 3
9
0.09
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 Summer '09
 Statistics, ChiSquare Test, Inferential Statistics, Probability theory, Statistical hypothesis testing

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