z test for a single proportion  overview
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$z$ test for a single proportion 


Independent variable  
None  
Dependent variable  
One categorical with 2 independent groups  
Null hypothesis  
H_{0}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.  
Alternative hypothesis  
H_{1} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  
Assumptions  
 
Test statistic  
$z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis.  
Sampling distribution of $z$ if H_{0} were true  
Approximately the standard normal distribution  
Significant?  
Two sided:
 
Approximate $C\%$ confidence interval for $\pi$  
Regular (large sample):
 
Equivalent to  
 
Example context  
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? 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  