Transcript LSSG Black Belt Training

Hypothesis Testing
Goodness-of-fit & Independence
Chi-Squared Tests
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Goodness-of-fit Test
A managed forest has the
following distribution of
trees:
Mannan & Meslow (1984)
made 156 observations of
foraging by red-breasted
nuthatches and found the
following:
Douglas Fir
54%
Ponderosa Pine
40%
Grand Fir
5%
Western Larch
1%
Douglas Fir
70
Ponderosa
Pine
79
Grand Fir
3
Western Larch
4
Mannan, R.W., and E.C. Meslow. 1984. “Bird populations and vegetation characteristics in
managed and old-growth forests, northeastern Oregon.” J. Wildl. Manage. 48: 1219-1238.
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Hypotheses
Do the birds forage randomly, without regard
to what species of tree they are in? To be true,
the observed and expected distributions should
be alike.
Null: The distributions are alike (good fit,
meaning birds forage randomly)
Alternate: The distributions are different (lack
of fit, or birds prefer certain vegetation)
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Expected Values
Based on the percentage distribution
of trees, the expected counts for each
type (out of 156) are:
Douglas Fir
84.24
Ponderosa Pine
62.40
Grand Fir
7.80
Western Larch
1.56
(54% of 156 = 84.24)
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Chi-Square Statistic
Expected
Observed
o-e
(o-e)Sq
(o-e)Sq/e
2.41
Douglas Fir
84.24
70
-14.24
202.78
4.42
Ponderosa Pine
62.40
79
16.60
275.56
2.95
Grand Fir
7.80
3
-4.80
23.04
3.82
Western Larch
1.56
156.00
4
156
2.44
5.95
Chi-square =
p-value =
13.593
0.003514
For p-value in Excel, type =CHIDIST(13.593,3),
for 3 degrees of freedom (n groups -1)
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Conclusion
Given the small p-value, we reject the null.
These birds are not foraging randomly –
they prefer certain types of trees.
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Test of Independence
Demographic data on 111 students
is available. We wish to study
gender differences, in this case
pertaining to dog ownership.
No
Dog
Have
Dog
Female
29
23
Male
35
24
Data Set: Student
Variables: Gender, Dog (Yes/No)
Are Gender and Dog Ownership
independent of each other?
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The Hypotheses
Null: The two variables are independent of each
other (the occurrence of one does not
influence the probability of the occurrence
of the other.)
Alternate: They are not independent (one
influences the other)
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Chi Square Test Results
Tabulated statistics: Gender, Dog
Rows: Gender
Columns: Dog
No
Yes
All
Female
29
29.98
23
22.02
52
52.00
Male
35
34.02
24
24.98
59
59.00
All
64
64.00
47
47.00
111
111.00
Cell Contents: Count
Expected count
Pearson Chi-Square = 0.143, DF = 1, P-Value = 0.705
Likelihood Ratio Chi-Square = 0.143, DF = 1, P-Value = 0.705
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