620
Chapter 12
Inference on Categorical Data
12.1 Goodness of Fit Test
1.
These procedures are for testing whether sample data are a good fit with a hypothesized
distribution.
2.
The
2
χ
goodness of fit tests are always right tailed because the numerator in the test
statistic is squared, making every test statistic other than a perfect fit positive. So, we are
measuring if
2
2
0
α
χ
χ
>
. Large values of
2
0
χ
indicate that the sample data are far from their
expected values and so lead one to reject the null hypothesis that the data fit a specified
distribution.
3.
The sample data must be obtained one by random sampling, all expected frequencies must
be greater than or equal to 1, and no more than 20% of the expected frequencies should be
less than 5.
4.
If the expected count of a category is less than one, two or more categories can be
combined together so that all expected values are at least one. Alternatively, the sample
size can be increased to achieve the desired expected counts.
5.
Each expected count is
i
n p
⋅
where
500
n
=
.
This gives expected counts of 100, 50, 225
and 125 respectively.
6.
Each expected count is
i
n p
⋅
where
700
n
=
.
This gives expected counts of 105, 210, 245
and 140 respectively.
7. (a)
(
)
(
)
(
)
2
2
2
0
Observed (
)
Expected
/
30
25
25
1
20
25
25
1
28
25
9
0.36
22
25
9
0.36
2.72
i
i
i
i
i
i
i
O
E
O
E
O
E
E
χ
−
−
=
(b)
df = 4 – 1 = 3
(c)
2
0.05
7.815
χ
=
(d)
The test statistic is not in the (right-tailed) critical region so we do not reject
0
H
.

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
Section 12.1
Goodness of Fit Test
621
8. (a)
(
)
(
)
(
)
2
2
2
0
Observed (
)
Expected
/
38
40
4
0.1
45
40
25
0.625
41
40
1
0.025
33
40
49
1.225
43
40
9
0.225
2.200
i
i
i
i
i
i
i
O
E
O
E
O
E
E
χ
−
−
=
(b)
df = 5 – 1 = 4
(c)
2
0.05
9.488
χ
=
(d)
The test statistic is not in the (right-tailed) critical region so we do not reject
0
H
.
9. (a)
(
)
(
)
(
)
2
2
2
0
Observed (
)
Expected
/
1
1.6
0.36
0.225
38
25.6
153.76
6.006
132
153.6
466.56
3.038
440
409.6
924.16
2.256
389
409.6
424.36
1.0363
12.561
i
i
i
i
i
i
i
O
E
O
E
O
E
E
χ
−
−
=
(b)
df = 5 – 1 = 4
(c)
2
0.05
9.488
χ
=
(d)
The test statistic is in the (right-tailed) critical region so we reject
0
H
. There is
sufficient evidence to reject the claim that the random variable
X
is binomial with
4
n
=
and
0.8
p
=
.
10. (a)
(
)
(
)
(
)
2
2
2
0
Observed (
)
Expected
/
260
240.1
396.01
1.649
400
411.6
134.56
0.327
280
264.6
237.16
0.896
50
75.6
655.36
8.669
10
8.1
3.61
0.446
11.987
i
i
i
i
i
i
i
O
E
O
E
O
E
E
χ
−
−
=