z test for a single proportion  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 a single proportion 


Independent variable  
None  
Dependent variable  
One categorical with 2 independent groups  
Null hypothesis  
$\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to the null hypothesis  
Alternative hypothesis  
Two sided: $\pi \neq \pi_0$ Right sided: $\pi > \pi_0$ Left sided: $\pi < \pi_0$  
Assumptions  
 
Test statistic  
$z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
$p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size  
Sampling distribution of $z$ if H0 were true  
Approximately standard normal  
Significant?  
Two sided:
 
Approximate $C\%$ confidence interval for $\pi$  
Regular (large sample):
 
Equivalent to  
 
Example context  
Is the proportion smokers amongst office workers different from $\pi_0 = .2$? Use the normal approximation for the sampling distribution of the test statistic.  
SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 
Jamovi  
Frequencies > 2 Outcomes  Binomial test
 
Practice questions  