AP Statistics Section 14.2 A
The two-sample z procedures of chapter 13
allowed us to compare the proportions of
successes in two groups (either two populations
or two treatment groups in an experiment). We
need a new statistical test if we want to
compare more than two groups.
A contingency table (or two-way frequency
table) is a table in which frequencies correspond
to two variables. One variable categorizes rows
and the other columns.
Discussed earlier in section 4.2.
Example 14.1: Market researchers know that
background music can influence the mood and
purchasing behavior of customers. One study in
a supermarket in Northern Ireland compared
three treatments: no music, French accordion
music and Italian string music. Under each
condition, the researchers recorded the
numbers of bottles of French, Italian and other
wine purchased. Here is a table that summarizes
the data:
Music
None
French
Italian
Total
French
30
39
30
99
Wine
Italian
11
1
19
31
Chosen
Other
43
35
35
113
Total
84
75
84
243
Section 14.2 presents two types of hypothesis
testing based on contingency tables.
Tests of homogeneity are used to determine
whether different populations have the same
proportion of some characteristic.
Tests of independence are used to determine
whether a contingency table’s row variable is
independent of its column variable.
Both types of tests use the same basic methods from section
14.1.
Test Statistic:
2


O

E
2  
E
(row total)(column total)
where E =
grand total
We find one primary test statistic by finding the sum of the test
statistics for each cell in the table
(# rows - 1)(# of columns - 1)
The degrees of freedom equal ___________________________
Conditions:
Data must come from independent SRS’s of the populations of
interest.
All expected cell counts are greater than 1 and no more than
20% are less than 5
Use a  test to compare the
distribution of wines selected for
each type of music.
2
Hypothesis: The populations of interest are
French, Italian and other wines and
_______________________
____________________
French, Italian and no music
distributions of wine selected is the same for each music type
H0:__________________________________________
Ha:__________________________________________
distributions of wine selected are not all the same
Conditions:
Not unreasonable to view the data as an SRS.
All expected counts are greater than 5, the smallest
being 9.57.
Seems reasonable to assume sales are independent.
Must also assume N  10n since sampling w/o replacement.
Calculations:
2
(
30

34
.
22
)
2 
     18.28
34.22
D of F  (3 - 1)(3 - 1)  4
P - value  .001
18.28
TI83/84 : Input data in a matrix
STATS TESTS C :   Test
2
Conclusions:
Our p - value of .001 is less than any common significance level,
so we reject the H 0 and conclude that the type of music played
has an effect on wine sales.