One sample t test for the mean  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
One sample $t$ test for the mean  $z$ test for a single proportion 


Independent variable  Independent variable  
None  None  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean 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: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  Two sided: $\pi \neq \pi_0$ Right sided: $\pi > \pi_0$ Left sided: $\pi < \pi_0$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $s$ is the sample standard deviation, $N$ is the sample size. The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$  $z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
$p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size  
Sampling distribution of $t$ if H0 were true  Sampling distribution of $z$ if H0 were true  
$t$ distribution with $N  1$ degrees of freedom  Approximately standard normal  
Significant?  Significant?  
Two sided:
 Two sided:
 
$C\%$ confidence interval for $\mu$  Approximate $C\%$ confidence interval for $\pi$  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ distribution with the area $C / 100$ between $t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20) The confidence interval for $\mu$ can also be used as significance test.  Regular (large sample):
 
Effect size  n.a.  
Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0$    
Visual representation  n.a.  
  
n.a.  Equivalent to  
 
 
Example context  Example context  
Is the average mental health score of office workers different from $\mu_0$ = 50?  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 > Compare Means > OneSample T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 
Jamovi  Jamovi  
TTests > One Sample TTest
 Frequencies > 2 Outcomes  Binomial test
 
Practice questions  Practice questions  