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2-SAMPLE DIFFERENCE OF PROPORTIONS TEST

Geographers often work with proportions as well as means.

For example, a political geographer might want to determine if differences exist in the president’s approval rating among the voting population in two neighboring states.

Random samples of voters in each state could be polled and the proportion of potential voters giving the president a favorable rating tabulated.

A two-sample difference of proportions test would then assess the likelihood that a statistically significant difference in approval ratings exists.

In problems testing for significant differences with a dichotomous (binary) variable, one of the two variable categories is selected as the focus for the analysis.

The proportion of the sample in the focus category is termed p. while the other proportion is termed q, where q = 1- p.

objective of the difference test is to determine whether the proportion of population 1 (D1) having the focus attribute differs significantly from the corresponding proportion of population 2 (D2).

H0: P1 = P2

the alternate hypothesis (HA) can take two forms.

If no direction is hypothesized for difference in proportion between the two samples, a two-tailed procedure is applied

 

 

if one of the sample proportions is expected to exceed the other, a one-tailed test is appropriate.

The test statistic for the difference of proportion procedure (ZD) has a form similar to other two sample difference tests:

where D1 = proportion of sample 1 in the category of focus

D2 = proportion of sample 2 in the category of focus

FD1-D2 = standard error of the difference of proportions

where = pooled estimate of the focus category for the population

The pooled estimate, is the proportion in the focus category if the two samples were combined into one sample.

Operationally, the pooled estimate is the weighted proportion from the two samples:

The two-sample difference of proportions test can be illustrated using information collected from the 1995 General Social Survey (GSS).

A series of questions from the GSS investigates attitudes toward a variety of controversial issues, including whether those sampled are in favor of or oppose gun permits. The question can be asked: Does the proportion of those in favor of gun permits differ from one type of place to another? For example, do attitudes differ between residents of large cities, suburbs of large cities, unincorporated areas/smaller cities, and small towns/rural areas? To answer these questions and have sufficient sample sizes to conduct two-sample difference of proportions tests, the original "Expanded NORC Size Code" classification (with ten different categories) is collapsed to four combined categories. Table 9.4 lists the actual GSS data from each of the ten original "Expanded NORC Size Code" categories and combines these data into four "Summary Categories."