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Chapter 11
Analyzing the Association
Between Categorical
Variables
Section 11.5
Small Sample Sizes: Fisher’s Exact Test
Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Fisher’s Exact Test
 The chi-squared test of independence is a largesample test.
 When the expected frequencies are small, any of them
being less than about 5, small-sample tests are more
appropriate.
 Fisher’s exact test is a small-sample test of
independence.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Fisher’s Exact Test
 The calculations for Fisher’s exact test are complex.
 Statistical software can be used to obtain the P-value
for the test that the two variables are independent.
 The smaller the P-value, the stronger the evidence
that the variables are associated.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: A Tea-Tasting Experiment
This is an experiment conducted by Sir Ronald Fisher.
 His colleague, Dr. Muriel Bristol, claimed that when
drinking tea she could tell whether the milk or the tea had
been added to the cup first.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: A Tea-Tasting Experiment
Experiment:
 Fisher asked her to taste eight cups of tea:
 Four had the milk added first.
 Four had the tea added first.
 She was asked to indicate which four had the
milk added first.
 The order of presenting the cups was
randomized.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: A Tea-Tasting Experiment
Results:
Table 11.18 Result of Tea-Tasting Experiment. The table cross-tabulates what was actually
poured first (milk or tea) by what Dr. Bristol predicted was poured first. She had to indicate
which four of the eight cups had the milk poured first.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: A Tea-Tasting Experiment
Analysis:
Table 11.19 Result of Fisher’s Exact Test for Tea-Tasting Experiment. The chi-squared Pvalue is listed under Asymp. Sig. and the Fisher’s exact test P-values are listed under Exact
Sig. “Sig” is short for significance and “asymp.” is short for asymptotic.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: A Tea-Tasting Experiment
The one-sided version of the test pertains to the
alternative that her predictions are better than random
guessing.
Does the P-value suggest that she had the ability to
predict better than random guessing?
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Example: A Tea-Tasting Experiment
The P-value of 0.243 does not give much evidence
against the null hypothesis.
The data did not support Dr. Bristol’s claim that she
could tell whether the milk or the tea had been added to
the cup first.
If she had predicted all four cups correctly, the one-sided
P-value would have been 0.014. We might then believe
her claim.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Summary of Fisher’s Exact Test of
Independence for 2x2 Tables
1. Assumptions:
 Two binary categorical variables
 Data are random
2. Hypotheses:
 H 0 : the two variables are independent ( p1  p2 )
 H a : the two variables are associated
( p1  p2 or p1  p2 or p1  p2 )
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.
Summary of Fisher’s Exact Test of
Independence for 2x2 Tables
3. Test Statistic:
 First cell count (this determines the others given
the margin totals).
4. P-value:
 Probability that the first cell count equals the
observed value or a value even more extreme as
predicted by H a .
5. Conclusion:
 Report the P-value and interpret in context. If a
decision is required, reject H 0 when P-value 
significance level.
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Copyright © 2013, 2009, and 2007, Pearson Education, Inc.