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  
$\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to H0  
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  
When testing two sided: goodness of fit test, with categorical variable with 2 levels  
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.  
Pratice questions  