Page 273 - Statistics II for Dummies

Chapter 14: Being Independent Enough for the Chi-Square Test 257

Comparing Two Tests for Comparing

Two Proportions

You can use the Chi-square test to check whether two population propor-
tions are equal. For example, is the proportion of female cellphone users the
same as the proportion of male cellphone users?

You may be thinking, “But wait a minute, don’t statisticians already have a
test for two proportions? I seem to remember it from my Stats I course . . .
I’m thinking . . . yeah, it’s the Z-test for two proportions. What’s that test got
to do with a Chi-square test?” In this section, you get an answer to that ques-
tion and practice using both methods to investigate a possible gender gap in
cellphone use.

Getting reacquainted with the Z-test
for two population proportions

The way that most people figure out how to test the equality of two popula-
tion proportions is to use a Z-test for two population proportions. With this
test, you collect a random sample from each of the two populations, find and
subtract their two sample proportions, and divide by their pooled standard
error (see your Stats I textbook for details on this particular test).
This test is possible to do as long as the sample sizes from the two popula-
tions are large — at least five successes and five failures in each sample.

The null hypothesis for the Z-test for two population proportions is
Ho: p = p , where p is the proportion of the first population that falls into the
1 2 1
category of interest, and p is the proportion of the second population that
2
falls into the category of interest. And as always, the alternative hypothesis is
one of the following choices, Ha: Not equal to, greater than, or less than.
Suppose you want to compare the proportion of male versus female cell-
phone users, where p is the proportion of males who own a cellphone, and
1
p is the proportion of all females who own a cellphone. You collect data, find
2
the sample proportions from each group, take their difference and make a

Z-statistic out of it using the formula , where .

Here, x and x are the number of individuals from samples one and two,
1 2
respectively, with the desired characteristic; n and n are the two sample
1 2
sizes.

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