Chapter 11
Goodness-of-Fit and Contingency Tables
11-2
Goodness-of-Fit
1. When digits are randomly generated, they should form a uniform distribution – i.e., a
distribution in which each of the digits is equally likely.
The test for goodness-of-fit is a test of
the hypothesis that the sample data fit the uniform distribution.
2. The calculated
χ
2
is a measure of the discrepancy between the hypothesized distribution and
the sample data.
An exceptionally large value of the
χ
2
test statistic suggests a large
discrepancy between the hypothesized distribution and the sample data – i.e., that there is not
goodness-of–fit, and that the observed and expected frequencies are quite different.
An
exceptionally small value of the
χ
2
test statistic suggests an extremely good fit – i.e., that the
observed and expected values are almost identical.
3. O represents the observed frequencies.
The twelve values for O are 5, 8, 7, 9, 13, 17, 11, 10,
10, 12, 8, 10.
E represents the expected frequencies.
The twelve values for E dependent on
the expected (i.e., the hypothesized) distribution.
If the hypothesized distribution is that
weddings occur in different months with the same frequency, we expect the
Σ
O = 120
observations to be equally distributed among the 12 months so that each of the twelve values
for E is 120/12 = 12.
4. The P-value indicates that the given observed frequencies have a 0.477 probability of
occurring by chance if the population distribution is as hypothesized (i.e., if weddings do occur
in the 12 months with equal frequency).
Because this probability is high, the hypothesis
cannot be rejected.
There is not sufficient evidence to reject the claim that weddings occur in
the 12 months with equal frequency.
NOTE FOR EXERCISES 5 AND 6: The principle that the null hypothesis must contain the equality
still applies.
When the original claim is given in words, it will reduce either to a statement that the
observed results fit or match (i.e., are equal to) some specified distribution or to a statement that
the observed results are different from some specified distribution. In the first case the original
claim is the null hypothesis, in the second case the original claim is the alternative hypothesis.
5. original claim: the actual outcomes agree with the expected frequencies
H
o
: the actual outcomes agree with the expected frequencies
H
1
: at least one outcome is not as expected
α
= 0.05 and df = 9
C.V.
χ
2
=
2
χ
α
=
2
0.05
χ
= 16.919
calculations:
χ
2
=
Σ
[(O – E)
2
/E] = 8.185
P-value =
χ
2
cdf (8.815,99,5) = 0.5156
conclusion:
Do not reject Ho; there is not sufficient evidence to reject the claim that the actual
outcomes agree with the expected frequencies. There is no reason to say the slot machine
is not functioning as expected.