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

Chapter 8. Testing for Differences in Mean and Proportion: 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.