Independent Proportions Z-test: Purpose, Hypotheses, and Assumptions

8. Testing for Differences in Means and Proportions: Independent Proportions Z-test

Independent Proportions Z-test: Purpose, Hypotheses, and Assumptions

Independent proportions Z-test: Purpose and Hypotheses

The independent proportions #\boldsymbol{Z}#-test is used to test hypotheses about the difference between two population proportions #\pi_1 - \pi_2#.

Specifically, the test is used to determine whether or not it is plausible that #\pi_1-\pi_2# differs from some value #\Delta#. In most situations #\Delta=0#, so we will only present this specific setting.

The hypotheses of a independent proportions #Z#-test are:

Two-tailed#^1#	Left-tailed	Right-tailed
#H_0: \pi_1 - \pi_2 = 0# #H_a: \pi_1 - \pi_2 \neq 0#	#H_0:\pi_1 - \pi_2 \geq 0# #H_a: \pi_1 - \pi_2 \lt 0#	#H_0: \pi_1 - \pi_2 \leq 0# #H_a: \pi_1 - \pi_2 \gt 0#

Note that #\pi_1 - \pi_2 = 0# means the same as #\pi_1 = \pi_2#.

Assumptions of the Independent proportions Z-test

The following assumptions are required to hold in order for an Independent proportions #Z#-test to produce valid results:

Random sampling is used to draw the samples.
Independence of observations, meaning:
- No individual can be part of both samples.
- No individual in either sample can influence individuals in the same sample.
- No individual in either sample can influence individuals in the other sample.
The sampling distribution of the difference between the sample proportions is approximately normal. This is the case when the number of failures and the number of successes for both samples is greater than 5 (thus #\hat{p}_1 * n_1 > 5# and #(1 - \hat{p}_1) * n_1 > 5#, this should also apply for sample #n_2#).