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  Spearman's rho 


Independent variable  Variable 1  
None  One of ordinal level  
Dependent variable  Variable 2  
One categorical with 2 independent groups  One of ordinal level  
Null hypothesis  Null hypothesis  
H_{0}: $\pi = \pi_0$
$\pi$ is the population proportion of 'successes'; $\pi_0$ is the population proportion of successes according to the null hypothesis  H_{0}: $\rho_s = 0$
$\rho_s$ is the unknown Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H_{0}: there is no monotonic relationship between the two variables in the population  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  H_{1} two sided: $\rho_s \neq 0$ H_{1} right sided: $\rho_s > 0$ H_{1} left sided: $\rho_s < 0$  
Assumptions  Assumptions  

 
Test statistic  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, and $\pi_0$ is the population proportion of successes according to the null hypothesis.  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ where $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.  
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Approximately the standard normal distribution  Approximately the $t$ distribution with $N  2$ degrees of freedom  
Significant?  Significant?  
Two sided:
 Two sided:
 
Approximate $C\%$ confidence interval for $\pi$  n.a.  
Regular (large sample):
   
Equivalent to  n.a.  
   
Example context  Example context  
Is the proportion of smokers amongst office workers different from $\pi_0 = .2$? Use the normal approximation for the sampling distribution of the test statistic.  Is there a monotonic relationship between physical health and mental health?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 Analyze > Correlate > Bivariate...
 
Jamovi  Jamovi  
Frequencies > 2 Outcomes  Binomial test
 Regression > Correlation Matrix
 
Practice questions  Practice questions  