11-2 Goodness-of-Fit
In this section, we consider sample data consisting of
observed frequency counts arranged in a single row or
column (called a one-way frequency table).
We will use a hypothesis test for the claim that the
observed frequency counts agree with some claimed
distribution, so that there is a good fit of the observed
data with the claimed distribution.
Definition
A goodness-of-fit test is used to test the hypothesis that an
observed frequency distribution fits (or conforms to) some
claimed distribution.
Goodness-of-Fit Test
Notation
O
represents the observed frequency of an outcome, found
from the sample data.
E
represents the expected frequency of an outcome, found
by assuming that the distribution is as claimed.
k
represents the number of different categories or cells.
n
represents the total number of trials.
Goodness-of-Fit
Hypotheses and Test Statistic
H 0 : The frequency counts agree with the claimed distribution.
H A : The frequency counts do not agree with the claimed distribution.
(O  E )
x 
E
Critical Values
2
2
1. Found in Table A-4 using k – 1 degrees of
freedom, where k = number of categories.
2. Goodness-of-fit hypothesis tests are always righttailed.
Finding Expected Frequencies
If all expected frequencies are assumed equal:
n
E
k
If all expected frequencies are assumed not equal:
E  np for each individual category
Goodness-of-Fit Test
A close agreement between observed and expected values
will lead to a small value of χ2 and a large P-value. (Do Not
Reject Ho.)
A large disagreement between observed and expected
values will lead to a large value of χ2 and a small P-value.
A significantly large value of χ2 will cause a rejection of the
null hypothesis of no difference between the observed and
the expected. (Reject Ho)
GoodnessOf-Fit Tests
Example
A random sample of 100 weights of Californians is
obtained, and the last digit of those weights are
summarized on the next slide.
When obtaining weights, it is extremely important to
actually measure the weights rather than ask people to
self-report them.
By analyzing the last digit, we can verify the weights
were actually measured since reported weights tend to
be rounded to something ending with a 0 or a 5.
Test the claim that the sample is from a population of
weights in which the last digits do not occur with the
same frequency.
.
.
Example - Continued
.
.
Example - Continued
The hypotheses can be written as:
H 0 : p0  p1  p2  p3  p4  p5  p6  p7  p8  p9
H1 : At least one of the probabilities is different.
No significance level was specified, so we select α =
0.05.
.
.
Example - Continued
The calculation of the test statistic is given:
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.
Example - Continued
The test statistic is χ2 = 212.800 and the critical value is
χ2 = 16.919 (Table A-4).
.
.
Example - Continued
2
2
Since the TS
= 212.8 > CV
=16.919 we have SE to
reject H O and support H A . We conclude there is
sufficient evidence to support the claim that the last
digits do not occur with the same relative frequency.
In other words, we have evidence that the weights were
self-reported by the subjects, and the subjects were not
actually weighed.
.
.