Download Chapter 11

Chapter 11: Analyzing the
Association Between
Categorical Variables
Section 11.1: What is Independence and What is
Association?
1
Learning Objectives
1. Comparing Percentages
2. Independence vs. Dependence
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Learning Objective 1:
Example: Is There an Association Between
Happiness and Family Income?
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Learning Objective 1:
Example: Is There an Association Between
Happiness and Family Income?
 The percentages in a particular row of a
table are called conditional percentages
 They form the conditional distribution for
happiness, given a particular income level
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Learning Objective 1:
Example: Is There an Association Between
Happiness and Family Income?
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Learning Objective 1:
Example: Is There an Association Between
Happiness and Family Income?
 Guidelines when constructing tables with
conditional distributions:

Make the response variable the column
variable

Compute conditional proportions for the
response variable within each row

Include the total sample sizes
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Learning Objective 2:
Independence vs. Dependence
 For two variables to be independent, the
population percentage in any category of
one variable is the same for all categories
of the other variable
 For two variables to be dependent (or
associated), the population percentages in
the categories are not all the same
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Learning Objective 2:
Independence vs. Dependence
 Are race and belief in life after death independent
or dependent?

The conditional distributions in the table are
similar but not exactly identical

It is tempting to conclude that the variables are
dependent
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Learning Objective 2:
Independence vs. Dependence
 Are race and belief in life after death
independent or dependent?

The definition of independence between
variables refers to a population

The table is a sample, not a population
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Learning Objective 2:
Independence vs. Dependence
 Even if variables are independent, we
would not expect the sample conditional
distributions to be identical
 Because of sampling variability, each
sample percentage typically differs
somewhat from the true population
percentage
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Chapter 11: Analyzing the
Association Between
Categorical Variables
Section 11.2: How Can We Test Whether
Categorical Variables Are Independent?
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Learning Objectives
1. A Significance Test for Categorical Variables
2. What Do We Expect for Cell Counts if the
3.
4.
5.
6.
Variables Are Independent?
How Do We Find the Expected Cell Counts?
The Chi-Squared Test Statistic
The Chi-Squared Distribution
The Five Steps of the Chi-Squared Test of
Independence
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Learning Objectives
7. Chi-Squared is Also Used as a “Test of
Homogeneity”
8. Chi-Squared and the Test Comparing
Proportions in 2x2 Tables
9. Limitations of the Chi-Squared Test
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Learning Objective 1:
A Significance Test for Categorical Variables
 Create a table of frequencies divided into the
categories of the two variables

The hypotheses for the test are:
H0: The two variables are independent
Ha: The two variables are dependent
(associated)
 The test assumes random sampling and a large
sample size (cell counts in the frequency table of
at least 5)
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Learning Objective 2:
What Do We Expect for Cell Counts if the
Variables Are Independent?
 The count in any particular cell is a
random variable

Different samples have different count
values
 The mean of its distribution is called an
expected cell count

This is found under the presumption that
H0 is true
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Learning Objective 3:
How Do We Find the Expected Cell Counts?
 Expected Cell Count:

For a particular cell,
(Row total) (Column total)
Expected cell count 
Total sample size
 The expected frequencies are values that have
the same row and column totals as the observed
counts, but for which the conditional
distributions are identical (this is the assumption
of the null hypothesis).
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Learning Objective 3:
How Do We Find the Expected Cell Counts?
Example
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Learning Objective 4:
The Chi-Squared Test Statistic
 The chi-squared statistic summarizes how far the
observed cell counts in a contingency table fall
from the expected cell counts for a null
hypothesis
2
(observed count - expected count)
 
expected count
2
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Learning Objective 4:
Example: Happiness and Family Income
 State the null and alternative hypotheses
for this test
 H0: Happiness and family income are
independent
 Ha: Happiness and family income are
dependent (associated)
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Learning Objective 4:
Example: Happiness and Family Income
 Report the  statistic and explain how it was
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calculated:
2

 To calculate the
statistic, for each cell,
calculate:
2
(observed count - expected count)
expected count
 Sum the values for all the cells
 The  value is 73.4
2
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Learning Objective 4:
Example: Happiness and Family Income
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Learning Objective 4:
The Chi-Squared Test Statistic
 The larger the

2
value, the greater the
evidence against the null hypothesis of
independence and in support of the alternative
hypothesis that happiness and income are
associated
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Learning Objective 5:
The Chi-Squared Distribution
 To convert the
test statistic to a P-value,
we use the sampling distribution of the
statistic  2
 For large sample sizes, this sampling
distribution is well approximated by the chisquared probability distribution
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Learning Objective 5:
The Chi-Squared Distribution
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Learning Objective 5:
The Chi-Squared Distribution
 Main properties of the chi-squared
distribution:

It falls on the positive part of the real
number line

The precise shape of the distribution
depends on the degrees of freedom:
df = (r-1)(c-1)
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Learning Objective 5:
The Chi-Squared Distribution
 Main properties of the chi-squared
distribution:
 The mean of the distribution equals the
df value
 It is skewed to the right
 The larger the
value, the greater the
evidence against H0: independence
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Learning Objective 5:
The Chi-Squared Distribution
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Learning Objective 6:
The Five Steps of the Chi-Squared Test of
Independence
1. Assumptions:

Two categorical variables

Randomization

Expected counts ≥ 5 in all cells
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Learning Objective 6:
The Five Steps of the Chi-Squared Test of
Independence
2. Hypotheses:
 H0: The two variables are independent
 Ha: The two variables are dependent
(associated)
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Learning Objective 6:
The Five Steps of the Chi-Squared Test of
Independence
3. Test Statistic:
2
(observed count - expected count)
 
expected count
2
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Learning Objective 6:
The Five Steps of the Chi-Squared Test of
Independence
4. P-value: Right-tail probability above the
observed
value, for the chi-squared
distribution with df = (r-1)(c-1)
5. Conclusion: Report P-value and interpret in
context

If a decision is needed, reject H0 when P-value ≤
significance level
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Learning Objective 7:
Chi-Squared is Also Used as a “Test of
Homogeneity”
 The chi-squared test does not depend on which
is the response variable and which is the
explanatory variable
 When a response variable is identified and the
population conditional distributions are
identical, they are said to be homogeneous
 The test is then referred to as a test of
homogeneity
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Learning Objective 8:
Chi-Squared and the Test Comparing Proportions
in 2x2 Tables
 In practice, contingency tables of size 2x2 are very
common. They often occur in summarizing the
responses of two groups on a binary response
variable.


Denote the population proportion of success by p1 in
group 1 and p2 in group 2
If the response variable is independent of the group,
p1=p2, so the conditional distributions are equal
 H0: p1=p2 is equivalent to H0: independence

z 2   2 where
z  pˆ1  pˆ 2  se0
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Learning Objective 8:
Example: Aspirin and Heart Attacks Revisited
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Learning Objective 8:
Example: Aspirin and Heart Attacks Revisited
 What are the hypotheses for the chi-
squared test for these data?
 The null hypothesis is that whether a doctor
has a heart attack is independent of whether he
takes placebo or aspirin
 The alternative hypothesis is that there’s an
association
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Learning Objective 8:
Example: Aspirin and Heart Attacks Revisited
 Report the test statistic and P-value for the chi-
squared test:

The test statistic is 25.01 with a P-value of 0.000
 This is very strong evidence that the population
proportion of heart attacks differed for those
taking aspirin and for those taking placebo
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Learning Objective 8:
Example: Aspirin and Heart Attacks Revisited
 The sample proportions indicate that the
aspirin group had a lower rate of heart
attacks than the placebo group
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Learning Objective 9:
Limitations of the Chi-Squared Test
 If the P-value is very small, strong
evidence exists against the null
hypothesis of independence
But…
 The chi-squared statistic and the P-value
tell us nothing about the nature of the
strength of the association
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Learning Objective 9:
Limitations of the Chi-Squared Test
 We know that there is statistical
significance, but the test alone does not
indicate whether there is practical
significance as well
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Learning Objective 9:
Limitations of the Chi-Squared Test
 The chi-squared test is often misused. Some
examples are:




when some of the expected frequencies are
too small
when separate rows or columns are
dependent samples
data are not random
quantitative data are classified into categories
- results in loss of information
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Learning Objective 10:
“Goodness of Fit” Chi-Squared Tests
 The Chi-Squared test can also be used for testing
particular proportion values for a categorical variable.



The null hypothesis is that the distribution of the
variable follows a given probability distribution; the
alternative is that it does not
The test statistic is calculated in the same manner
where the expected counts are what would be
expected in a random sample from the hypothesized
probability distribution
For this particular case, the test statistic is referred to
as a goodness-of-fit statistic.
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Chapter 11: Analyzing the
Association Between
Categorical Variables
Section 11.3: How Strong is the Association?
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Learning Objectives
1. Analyzing Contingency Tables
2. Measures of Association
3. Difference of Proportions
4. The Ratio of Proportions: Relative Risk
5. Properties of the Relative Risk
6. Large Chi-square Does Not Mean There’s a
Strong Association
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Learning Objective 1:
Analyzing Contingency Tables
 Is there an association?
 The
chi-squared test of
independence addresses this
 When
the P-value is small, we infer
that the variables are associated
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Learning Objective 1:
Analyzing Contingency Tables
 How do the cell counts differ from what
independence predicts?
 To answer this question, we compare
each observed cell count to the
corresponding expected cell count
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Learning Objective 1:
Analyzing Contingency Tables
 How strong is the association?
 Analyzing the strength of the association
reveals whether the association is an
important one, or if it is statistically
significant but weak and unimportant in
practical terms
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Learning Objective 2:
Measures of Association
 A measure of association is a statistic or a
parameter that summarizes the strength of
the dependence between two variables

a measure of association is useful for
comparing associations
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Learning Objective 3:
Difference of Proportions
 An easily interpretable measure of association is
the difference between the proportions making a
particular response
Case (a) exhibits the weakest possible association – no
association. The difference of proportions is 0
Case (b) exhibits the strongest possible association: The
difference of proportions is 1
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Learning Objective 3:
Difference of Proportions
 In practice, we don’t expect data to follow
either extreme (0% difference or 100%
difference), but the stronger the
association, the larger the absolute value
of the difference of proportions
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Learning Objective 3:
Difference of Proportions Example: Do Student
Stress and Depression Depend on Gender?
 Which response variable, stress or
depression, has the stronger sample
association with gender?
 The difference of proportions between females
and males was 0.35 – 0.16 = 0.19 for feeling
stressed
 The difference of proportions between females
and males was 0.08 – 0.06 = 0.02 for feeling
depressed
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Learning Objective 3:
Difference of Proportions Example: Do Student Stress
and Depression Depend on Gender?
 In the sample, stress (with a difference of
proportions = 0.19) has a stronger
association with gender than depression
has (with a difference of proportions =
0.02)
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Learning Objective 4:
The Ratio of Proportions: Relative Risk
 Another measure of association, is the
ratio of two proportions: p1/p2
 In medical applications in which the
proportion refers to an adverse outcome,
it is called the relative risk
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Learning Objective 4:
Example: Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
 Treating the auto accident outcome as the
response variable, find and interpret the
relative risk
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Learning Objective 4:
Example: Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
 The adverse outcome is death
 The relative risk is formed for that outcome
 For those who wore a seat belt, the proportion
who died equaled 510/412,878 = 0.00124
 For those who did not wear a seat belt, the
proportion who died equaled 1601/164,128 =
0.00975
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Learning Objective 4:
Example: Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
 The relative risk is the ratio:

0.00124/0.00975 = 0.127

The proportion of subjects wearing a
seat belt who died was 0.127 times the
proportion of subjects not wearing a
seat belt who died
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Learning Objective 4:
Example: Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
 Many find it easier to interpret the relative
risk but reordering the rows of data so
that the relative risk has value above 1.0
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Learning Objective 4:
Example: Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
 Reversing the order of the rows, we
calculate the ratio:
 0.00975/0.00124 = 7.9

The proportion of subjects not wearing
a seat belt who died was 7.9 times the
proportion of subjects wearing a seat
belt who died
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Learning Objective 4:
Example: Relative Risk for Seat Belt Use and
Outcome of Auto Accidents
 A relative risk of 7.9 represents a strong
association

This is far from the value of 1.0 that would
occur if the proportion of deaths were the
same for each group

Wearing a set belt has a practically
significant effect in enhancing the chance
of surviving an auto accident
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Learning Objective 5:
Properties of the Relative Risk
 The relative risk can equal any
nonnegative number
 When p1= p2, the variables are
independent and relative risk = 1.0
 Values farther from 1.0 (in either direction)
represent stronger associations
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Learning Objective 6:
Large Does Not Mean There’s a Strong
Association
 A large chi-squared value provides strong
evidence that the variables are associated
 It does not imply that the variables have a
strong association
 This statistic merely indicates (through its Pvalue) how certain we can be that the variables
are associated, not how strong that
association is
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Chapter 11: Analyzing the
Association Between
Categorical Variables
Section 11.4: How Can Residuals Reveal The
Pattern of Association?
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Learning Objectives
1. Association Between Categorical Variables
2. Residual Analysis
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Learning Objective 1:
Association Between Categorical Variables
 The chi-squared test and measures of
association such as (p1 – p2) and p1/p2 are
fundamental methods for analyzing contingency
tables
 The P-value for
summarized the strength of
evidence against H0: independence
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Learning Objective 1:
Association Between Categorical Variables
 If the P-value is small, then we conclude that
somewhere in the contingency table the
population cell proportions differ from
independence
 The chi-squared test does not indicate whether
all cells deviate greatly from independence or
perhaps only some of them do so
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Learning Objective 2:
Residual Analysis
 A cell-by-cell comparison of the observed
counts with the counts that are expected
when H0 is true reveals the nature of the
evidence against H0
 The difference between an observed and
expected count in a particular cell is called
a residual
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Learning Objective 2:
Residual Analysis
 The residual is negative when fewer
subjects are in the cell than expected
under H0
 The residual is positive when more
subjects are in the cell than expected
under H0
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Learning Objective 2:
Residual Analysis
 To determine whether a residual is large
enough to indicate strong evidence of a
deviation from independence in that cell
we use a adjusted form of the residual:
the standardized residual
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Learning Objective 2:
Residual Analysis
 The standardized residual for a cell=
(observed count – expected count)/se
 A standardized residual reports the number of


standard errors that an observed count falls from its
expected count
The se describes how much the difference would tend
to vary in repeated sampling if the variables were
independent
Its formula is complex
Software can be used to find its value
A large standardized residual value provides
evidence against independence in that cell


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Learning Objective 2:
Example: Standardized Residuals for Religiosity
and Gender
 “To what extent do you consider yourself
a religious person?”
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Learning Objective 2:
Example: Standardized Residuals for Religiosity
and Gender
 Interpret the standardized residuals in the table
 The table exhibits large positive residuals for the
cells for females who are very religious and for
males who are not at all religious.
 In these cells, the observed count is much larger
than the expected count
 There is strong evidence that the population has
more subjects in these cells than if the variables
were independent
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Learning Objective 2:
Example: Standardized Residuals for Religiosity
and Gender
 The table exhibits large negative residuals for the
cells for females who are not at all religious and
for males who are very religious
 In these cells, the observed count is much
smaller than the expected count
 There is strong evidence that the population has
fewer subjects in these cells than if the variables
were independent
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Chapter 11: Analyzing the
Association Between
Categorical Variables
Section 11.5: What if the Sample Size is
Small?
Fisher’s Exact Test
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Learning Objectives
1. Fisher’s Exact Test
2. Example using Fisher’s Exact Test
3. Summary of Fisher’s Exact Test of
Independence for 2x2 Tables
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Learning Objective 1:
Fisher’s Exact Test
 The chi-squared test of independence is a
large-sample 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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Learning Objective 1:
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
75
Learning Objective 2:
Fisher’s Exact Test Example: Tea Tastes Better
with Milk Poured First?
 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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Learning Objective 2:
Fisher’s Exact Test Example: Tea Tastes Better
with Milk Poured First?
 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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Learning Objective 2:
Fisher’s Exact Test Example: Tea Tastes Better
with Milk Poured First?
Results:
78
Learning Objective 2:
Fisher’s Exact Test Example: Tea Tastes Better
with Milk Poured First?
Analysis:
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Learning Objective 2:
Fisher’s Exact Test Example: Tea Tastes Better
with Milk Poured First?
 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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Learning Objective 2:
Fisher’s Exact Test Example: Tea Tastes Better
with Milk Poured First?
 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
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Learning Objective 3:
Summary of Fisher’s Exact Test of Independence
for 2x2 Tables
 Assumptions:


Two binary categorical variables
Data are random
 Hypotheses:


H0: the two variables are independent (p1=p2)
Ha: the two variables are associated
(p1≠p2 or p1>p2 or p1<p2)
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Learning Objective 3:
Summary of Fisher’s Exact Test of Independence
for 2x2 Tables
 Test Statistic:
 First cell count (this determines the others
given the margin totals)
 P-value:
 Probability that the first cell count equals the
observed value or a value even more extreme
as predicted by Ha
 Conclusion:
 Report the P-value and interpret in context. If
a decision is required, reject H0 when P-value
≤ significance level
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