Test of Two Independent Proportions
z-test of Two Independent Proportions
Assume that you have two proportions, p1 and p2. Are they significantly different? There are two hypothesis tests, a z-test and a chi-square, and they are numerically exactly equivalent. The z-test is slightly easier to compute.
Let,
X1 = the number of people selected in group 1
N1 = the total number of people in group 1
X2 = the number of people selected in group 2
N2 = the total number of people in group 2
Then,
[pic]
and
[pic]
and
[pic].
The obtained value of z is compared to a normal distribution. For example, using the common alpha level of 0.05 and a two-tailed test, the critical value of z is about 1.96.
Numerical Example
For example, assume that we select 304 of 500 applicants from the majority group and 50 out of 100 for a minority group. Then,
N1 = 100, X1 = 50
N2 = 500, X2 = 304
therefore p1 = .50, p2 = .61, [pic] = .59 and z = -2.00. This value is more extreme than -1.96 and thus we conclude that these sample proportions were, indeed, drawn from populations with differing proportions selected. If the assumptions of the test are met, then we know that across all identical tests, we have less than 5% chance of being incorrect in this conclusion. Also, as can be seen from the proportions, the group impact favors the majority group (the observed z is negative).
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