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Rudy Moser’s Husker Probability Page |
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Goodness of
Fit Tests[i]
I figured it would be useful to include some goodness-of-fit
tests in order to validate some of the claims made.
One such goodness of fit test is a chi-squared test.
This gives us a test statistic to see how far off our
observed values are off from the expected values. If the observed frequencies are close to the
corresponding expected frequencies, the X2-value will be small,
indicating a good fit. If the observed
frequencies differ considerably from the expected frequencies, the X2-value
will be large and the fit is poor. The critical value of a Chi-squared
distribution with a .05 level of significance is 21.026. That is, that if the Chi-squared statistic is
greater than 21.026 then we cannot say we have a good fit.
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Spread (Expected) |
Observed |
(oi-ei)^2/ei |
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Western Michigan |
14 |
23 |
5.785714 |
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San Jose St. |
26.5 |
23 |
0.462264 |
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New Mexico St. |
25.5 |
31 |
1.186275 |
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Virginia Tech |
6.5 |
-5 |
20.34615 |
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Missouri |
-10 |
-35 |
-62.5 |
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Texas Tech |
-20.5 |
-6 |
-10.2561 |
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Iowa State |
7 |
28 |
63 |
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Baylor |
11 |
12 |
0.090909 |
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Oklahoma |
-22 |
-34 |
-6.54545 |
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Kansas |
1 |
10 |
81 |
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Kansas State |
6 |
28 |
80.66667 |
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Colorado |
18 |
9 |
4.5 |
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Gator Bowl (Clemson) |
-2.5 |
5 |
-22.5 |
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Chi-Squared Statistic |
155.2364 |
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However it is not recommended to use a Chi-squared test if some of the expected frequencies are less than 5.
There are other tests however. We are making an assumption that our data follows a normal distribution. To test normality we can use Geary’s Test. The test statistic for Geary’s Test can be computed by the following formula.
The denominator gives a reasonable estimate of σ, std. deviation, whether the distribution is normal or otherwise. The numerator is a good estimator of σ if the distribution is normal, but overestimates or underestimates when there are departures from normality. Thus values of u differing considerably from 1.0 represent that the hypothesis of normality should be rejected. A standardization of U is given by
Geary’s Test statistic then follows a normal distribution and we use a two-tailed test for significance. The table for the husker data is as follows.
|
Result vs. Spread |
|Xi - X| |
(Xi - X)^2 |
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Western Michigan |
9 |
6.807692 |
46.34467 |
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San Jose St. |
-3.5 |
5.692308 |
32.40237 |
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New Mexico St. |
5.5 |
3.307692 |
10.94083 |
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Virginia Tech |
-11.5 |
13.69231 |
187.4793 |
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Missouri |
-25 |
27.19231 |
739.4216 |
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Texas Tech |
14.5 |
12.30769 |
151.4793 |
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Iowa State |
21 |
18.80769 |
353.7293 |
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Baylor |
1 |
1.192308 |
1.421598 |
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Oklahoma |
-12 |
14.19231 |
201.4216 |
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Kansas |
9 |
6.807692 |
46.34467 |
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Kansas State |
22 |
19.80769 |
392.3447 |
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Colorado |
-9 |
11.19231 |
125.2678 |
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Gator Bowl (Clemson) |
7.5 |
5.307692 |
28.1716 |
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Average |
2.192308 |
Geary's Test Statistic |
1.056608 |
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Std. Deviation |
13.89475 |
p-value |
0.443069 |
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The test statistic is very close to 1 and the p-value is large, more than .05, which means we fail to reject the hypothesis that the data follows a normal distribution.
Rudy Moser: rmoser@lps.org