Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Section 11.2-1
Chapter 11
Goodness-of-Fit and
Contingency Tables
11-1 Review and Preview
11-2 Goodness-of-Fit
11-3 Contingency Tables
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Section 11.2-2
Key Concept
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.
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Section 11.2-3
Definition
A goodness-of-fit test is used to test the hypothesis that an
observed frequency distribution fits (or conforms to) some
claimed distribution.
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Section 11.2-4
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.
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Section 11.2-5
Goodness-of-Fit Test
Requirements
1. The data have been randomly selected.
2. The sample data consist of frequency counts for each
of the different categories.
3. For each category, the expected frequency is at least 5.
(The expected frequency for a category is the
frequency that would occur if the data actually have the
distribution that is being claimed. There is no
requirement that the observed frequency for each
category must be at least 5.)
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Section 11.2-6
Goodness-of-Fit
Hypotheses and Test Statistic
H 0 : The frequency counts agree with the claimed distribution.
H1 : The frequency counts do not agree with the claimed distribution.
(O  E )
x 
E
2
2
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Section 11.2-7
P-Values and Critical Values
P-Values
P-values are typically provided by technology, or a range
of P-values can be found from Table A-4.
Critical Values
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 right-tailed.
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Section 11.2-8
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
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Section 11.2-9
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.
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.
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Section 11.2-10
GoodnessOf-Fit Tests
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Section 11.2-11
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.
.
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.
Section 11.2-12
Example - Continued
.
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Section 11.2-13
Example - Continued
Requirement Check:
1. The data come from randomly selected subjects.
2. The data do consist of counts.
3. With 100 sample values and 10 categories that are
claimed to be equally likely, each expected
frequency is 10, which is greater than 5.
All requirements are met to proceed.
.
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.
Section 11.2-14
Example - Continued
Step 1: The original claim is that the digits do not occur
with the same frequency. That is:
at least one of the probabilities p0 , p1 ,
, p9
is different from the others
Step 2: If the original claim is false, then all the
probabilities are the same:
p0  p1  p2  p3  p4  p5  p6  p7  p8  p9
.
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Section 11.2-15
Example - Continued
Step 3: 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.
Step 4: No significance level was specified, so we
select α = 0.05.
Step 5: We use the goodness-of-fit test with a χ2
distribution.
.
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Section 11.2-16
Example - Continued
Step 6: The calculation of the test statistic is given:
.
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Section 11.2-17
Example - Continued
Step 6: The test statistic is χ2 = 212.800 and the critical
value is χ2 = 16.919 (Table A-4). The P-value was
found to be less than 0.0001 using technology.
.
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Section 11.2-18
Example - Continued
Step 7: Reject the null hypothesis, since the P-value is
small and the test statistic is in the critical region.
Step 8: 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.
.
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Section 11.2-19