Warm-up


Researchers want to cross two yellowgreen tobacco plants with genetic
makeup (Gg). See the Punnett square
below.
When the researchers perform the
experiment, the resulting offspring are
22 green (GG), 50 yellow-green (Gg),
and 12 albino (gg) seedlings. Use a
chi-square goodness of fit test to
assess the validity of the researchers’
genetic model.
G
g
G
GG
Gg
g
Gg
gg
Section 11.2 Second Day
The Chi-Squared Test
for Homogeneity of
Proportions
Review…

Recall that there are three types of χ2 tests:
–
Goodness of fit test

–
Homogeneity of proportions

–
Tests whether a sample distribution matches a
hypothesized distribution
Tests whether p1 = p2 = p3 = …
Today’s task
Association/Independence

Tests whether two categorical variables are related
Tomorrow’s task
Homogeneity of Proportions



We’ve already performed hypothesis tests to
determine if p1 = p2.
Two-proportion z-test
Which type of test did we use?
Now, we’ll look to compare more than two
proportions using a χ2 test statistic.
Example

Chronic users of cocaine need the drug to feel
pleasure. Perhaps giving cocaine addicts antidepressants will help them stay off cocaine. A threeyear study compared the anti-depressant
desipramine with lithium (which was already being
used to treat cocaine addiction). A placebo was also
used in the experiment. The subjects were 72
chronic users of cocaine. Twenty-four of the
subjects were randomly assigned to each of the
three treatment groups. The variable of interest is
the proportion of users who did not experience a
relapse.
The data
Treatment
Subjects
No relapse
Desipramine
24
14
Lithium
24
6
Placebo
24
4
First Step



The first step is to arrange the data in a twoway table. This table must account for
everyone (so our example would list success
and failure).
This is called a 3 x 2 table because it has 3
rows and 2 columns. You’ll need to know
this to be able to use your calculator for a χ2
test.
What does this remind you of?
The hypotheses…


H0: p1 = p2 = p3 (Be sure you have defined 1,
2, and 3)
Ha: Not all of the proportions are equal.
Using your calculator

Once you have the two-way table created,
you will need to enter the data in Matrix A.
–
–
–
–
–
Press 2nd MATRX.
Choose EDIT.
Hit ENTER.
Type the dimensions of the matrix (3 x 2 in this
case).
Enter the data.
Performing the test

Once you have the data entered, perform the χ2 test
on your calculator.
–
–
–
–
–
Press STAT.
Choose TESTS.
Choose C: χ2 test
The calculator asks you where you have the observed data.
You should always have it in Matrix A so that you don’t have
to change this screen.
The calculator calculates the expected counts (yay!) and
puts them in Matrix B.

IN OTHER WORDS, DON’T CHANGE ANYTHING ON THIS
SCREEN!
Information Received




You can look at your graph or you can just
“calculate”.
If you conclude that there ARE differences
(reject null hypothesis), you need to look at
where the major differences are.
View Matrix B to find expected values.
Compare these to your observed values.
You MUST copy down your expected values!
The assumptions

Recall the assumptions for a χ2 test.
–
–
–
Independent SRSs
All of the expected counts are at least 1.
No more than 20% of the expected counts are
less than 5.
 Since this is step 2, leave space to check your
conditions to fill in after you have completed
your test in your calculator and note your
expected values.
Finishing up the test

Recall the 4 steps
–
–
–
–
State your hypotheses. Define any subscripts!
State your assumptions/conditions.
Name the test statistic and calculate its value.
Graph the test statistic. State the p-value in
symbols and calculate its value.
State the conclusion in the context of the problem.
Final notes…


Degrees of freedom for the homogeneity of
proportions: (r – 1)(c – 1)
Reading generic computer output is
essential! Look at p. 753 for an example.
Example

The nonprofit group Public Agenda conducted
telephone interviews with 3 randomly selected
groups of parents of high school children. There
were 202 black parents, 202 Hispanic parents, and
201 white parents. One question asked was, “Are
the high schools in your state doing an excellent,
good, fair, or poor job, or don’t you know enough to
say.” You want to know if there is evidence that
there is a difference in response based on race.
Results
Excellent
Black
Parents
12
Hispanic
Parents
34
White
Parents
22
Good
69
55
81
Fair
75
61
60
Poor
24
24
24
Don’t Know
22
28
14
Homework
Chapter 11
#27,29, 31, 38, 40, 42