z test for a single proportion  overview
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$z$ 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}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.  H_{0}: $\rho_s = 0$
Here $\rho_s$ is the 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  
$z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
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.  $t = \dfrac{r_s \times \sqrt{N  2}}{\sqrt{1  r_s^2}} $ Here $r_s$ 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 $\pi_0 = 0.2$? 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  