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Yr
Yes
Fr
15 (15%) 85
100
E: 30
E:70
Chi: 7.5
Chi: 3.2
So
25 (25%) 75
E:
E:
Chi:
Chi:
100
Jr
30 (30%) 70
E:
E:
Chi:
Chi:
100
Se
50 (50%) 50
E:
E:
Chi:
Chi:
100
120
400
Total
No
280
Total
Expected count =
Chi-square Contribution =
Get Chi-square statistic by
summing contributions
from all the cells
How many df for the chiSquare statistic?
The chi-square statistic follows a chisquare distribution with a specific
number of degrees of freedom (df ).
Yr
Yes
Fr
15
85
(15%)
100
So
25
75
(25%)
100
Jr
30
70
(30%)
100
Se
50
50
(50%)
100
Total 120
No
280
Total
400
df = (r – 1)(c – 1)
r = # row variable categories
d = # column variable categories
For our example, df =
A. 1 B. 2
C. 3 D. 4
Example 2: Two Different Categorical
Variables
PSU First
Choice
(1 – 2) (3 – 5) (6 – 8) (≥ 9)
Total
No
12
60
29
19
120
Yes
100
128
41
11
280
Total
112
188
70
30
401
Double all counts to get:
PSU First
Choice
(1 – 2) (3 – 5) (6 – 8) (≥ 9)
Total
No
24
120
58
38
240
Yes
200
256
82
22
560
Total
224
376
140
60
802
Pearson chisquare =
40.392
For fixed proportions,
increasing the sample
size strengthens the
evidence against H0.
Pearson chisquare =
80.784,
which is
exactly
doubled!
Hypotheses for ANOVA:
Remember: There are groups within the population, as
defined by their values of the categorical variable.
H0: Population means are the same for each group.
Ha: Not all population group means are the same.
H0: µ1 = µ2 = … = µk
Ha: Not all µ’s are the same.
In our particular situation…
H0: Each class (F, So, J, Se) has the same mean study time.
Ha: The mean study times are not the same for each class.
ANOVA output is summarized in a single table:
Analysis of Variance
Source
CollYear
Error
Total
DF
3
283
286
Adj SS
186.7
25087.6
25274.2
The second row gives the
“within” variation.
Adj MS
62.23
88.65
F-Value
0.70
P-Value
0.552
The F statistic is merely the
ratio of the MS between to the
MS within. It is the test
statistic we use for ANOVA!
One more ANOVA table fact: The MS error, or MS
within, is also called the pooled sample variance. You
can take its square root to get the pooled standard
deviation.
The p-value is based on
the F statistic and the two
DF values for between
and within.