One sample t test for the mean  overview
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One sample $t$ test for the mean  Two sample $t$ test  equal variances assumed 


Independent variable  Independent variable  
None  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  
$\mu = \mu_0$
$\mu$ is the unknown population mean; $\mu_0$ is the population mean according to the null hypothesis  $\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  
Alternative hypothesis  Alternative hypothesis  
Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  Two sided: $\mu_1 \neq \mu_2$ Right sided: $\mu_1 > \mu_2$ Left sided: $\mu_1 < \mu_2$  
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 $\mu_0$  $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to H0. The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$  
n.a.  Pooled standard deviation  
  $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$  
Sampling distribution of $t$ if H0 were true  Sampling distribution of $t$ if H0 were true  
$t$ distribution with $N  1$ degrees of freedom  $t$ distribution with $n_1 + n_2  2$ degrees of freedom  
Significant?  Significant?  
Two sided:
 Two sided:
 
$C\%$ confidence interval for $\mu$  $C\%$ confidence interval for $\mu_1  \mu_2$  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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.  $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2  2}$ 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 $\mu_1  \mu_2$ can also be used as significance test.  
Effect size  Effect size  
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 $\mu_0$  Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1  \bar{y}_2}{s_p}$$ Indicates how many standard deviations $s_p$ the two sample means are removed from each other  
Visual representation  Visual representation  
n.a.  Equivalent to  
  One way ANOVA with an independent variable with 2 levels ($I$ = 2):
OLS regression with one categorical independent variable with 2 levels:
 
Example context  Example context  
Is the average mental health score of office workers different from $\mu_0$ = 50?  Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst men and women.  
SPSS  SPSS  
Analyze > Compare Means > OneSample T Test...
 Analyze > Compare Means > IndependentSamples T Test...
 
Jamovi  Jamovi  
TTests > One Sample TTest
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  