z test for the difference between two proportions: overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the righthand column. To practice with a specific method click the button at the bottom row of the table
$z$ test for the difference between two proportions 


Independent variable  
One categorical with 2 independent groups  
Dependent variable  
One categorical with 2 independent groups  
Null hypothesis  
$\pi_1 = \pi_2$
$\pi_1$ is the unknown proportion of "successes" in population 1; $\pi_2$ is the unknown proportion of "successes" in population 2  
Alternative hypothesis  
Two sided: $\pi_1 \neq \pi_2$ Right sided: $\pi_1 > \pi_2$ Left sided: $\pi_1 < \pi_2$  
Assumptions  
 
Test statistic  
$z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
$p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2 Note: we could just as well compute $p_2  p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1$  
Sampling distribution of $z$ if H0 were true  
Approximately standard normal  
Significant?  
Two sided:
 
Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  
Regular (large sample):
 
Equivalent to  
When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels  
Example context  
Is the proportion smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  
Pratice questions  