13.2 Inference for Two Way Tables

Analyze Two Way Tables Using Chi-Squared
Test for Homogeneity and Independence
Goodness of
Fit
1 variable
Homogeneity
Independence
2 variables (2
way table)
2 variables (2
way table)
-distribution
-distribution
-proportions
-association
-dependent
upon
-relationship

Expected Counts=
row  total  column  total
table  total

Degrees of freedom
(r-1)(c-1)
Chi-Squared Test Statistic


H₀:the proportion of ________ is the SAME as
__________
Ha: the proportion of ________ is the
DIFFERENT than __________

Example 1: Do the boys’ preferences for the
following TV programs differ significantly
from the girls’ preferences? Use a 5%
significance level.
House
Grey’s
Anatomy
American
Idol
CSI
Boys
66
78
67
105
Girls
48
130
123
61


H₀:the boys preference for TV programs is the
SAME as the girls
Ha: the boys preference for TV programs is
DIFFERENT than the girls
Assumptions:
-random sample
-all expected counts are ≥ 1
-no more than 20% of the expected counts <5

House
Grey’s
Anatomy
American
Idol
CSI
Boys
53.1
96.9
88.6
77.4
Girls
60.9
111.1
101.4
88.6




Chi-Squared Test (Homogeneity) w/ α=0.05
P(x²>41.08)=0.000000006
df=3
Since p< α, it is statistically significant.
Therefore we reject H₀. There is enough
evidence to say the preference of TV
programs for boys is different than girls.

Example 2: The following data is an SRS of
650 patients at a local hospital. Does the
effect of aspirin significantly differ from a
placebo for these medical conditions?
Aspirin
Placebo
Fatal Heart
Attacks
20
60
Non-Fatal Heart
Attacks
125
220
Strokes
75
150


H₀:the effects of aspirin is the same as the
placebo
Ha: the effects of aspirin is different than the
placebo
Assumptions:
-random sample
-all expected counts are ≥ 1
-no more than 20% of the expected counts <5

Fatal Heart
Attacks
Aspirin
Placebo
27.1
52.9
Non-Fatal Heart 116.8
Attacks
228.2
Strokes
148.8
76.2




Chi-Squared Test (Homogeneity) w/ α=0.05
P(x²>3.70)=0.1573
df=2
Since p∡ α, it is not statistically significant.
Therefore we do not reject H₀. There is not
enough evidence to say the effect of aspirin
differs from the placebo.


H₀: There is no relationship (association)
between ________ and ________.
Ha: There is a relationship (association)
between ________ and ________.


Example 3: An SRS of 1000 was taken
Republican
Democrat
Independent
Male
200
150
50
Female
250
300
50
Is there a relationship between gender and
political parties?


H₀: There is no relationship between gender
and political party
Ha: There is a relationship between gender
and political party
Assumptions:
-random sample
-all expected counts are ≥ 1
-no more than 20% of the expected counts <5

Republican
Democrat
Independent
Male
180
180
40
Female
270
270
60




Chi-Squared Test (Independence) w/ α=0.05
P(x²>16.2)=0.0003
df=2
Since p< α, it is statistically significant.
Therefore we reject H₀. There is enough
evidence to say there is a relationship
between gender and political party
Example 4: An SRS of 592 people
were taken comparing their hair and
eye color.

Black
Brown
Red
Blonde
Brown
68
119
26
7
Green
20
84
17
94
Blue
15
54
14
10
Hazel
8
29
14
16
Is there an association between
hair color and eye color?


H₀: There is no association between hair color
and eye color
Ha: There is an association between hair color
and eye color
Assumptions:
-random sample
-all expected counts are ≥ 1
-no more than 20% of the expected counts <5

Black
Brown
Red
Blonde
Brown
41.0
105.7
26.3
47.0
Green
40.1
103.3
25.7
45.9
Blue
17.3
44.7
11.1
19.9
Hazel
12.5
32.2
8.0
14.3




Chi-Squared Test (Independence) w/ α=0.05
P(x²>134.98)≈0
df=9
Since p< α, it is statistically significant.
Therefore we reject H₀. There is enough
evidence to say there is an association
between hair color and eye color