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


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