Two sample t test  equal variances assumed  overview
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Two sample $t$ test  equal variances assumed  Pearson correlation 


Independent/grouping variable  Variable 1  
One categorical with 2 independent groups  One quantitative of interval or ratio level  
Dependent variable  Variable 2  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  
H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  H_{0}: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  H_{1} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  
Assumptions  Assumptions of test for correlation  

 
Test statistic  Test statistic  
$t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ 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, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. 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 $\mu_2 > \mu_1$.  Test statistic for testing H0: $\rho = 0$:
 
Pooled standard deviation  n.a.  
$s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$    
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $t$ and of $z$ if H_{0} were true  
$t$ distribution with $n_1 + n_2  2$ degrees of freedom  Sampling distribution of $t$:
 
Significant?  Significant?  
Two sided:
 $t$ Test two sided:
 
$C\%$ confidence interval for $\mu_1  \mu_2$  Approximate $C$% confidence interval for $\rho$  
$(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
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.  First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 
Effect size  Properties of the Pearson correlation coefficient  
Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1  \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other. 
 
Visual representation  n.a.  
  
Equivalent to  Equivalent to  
One way ANOVA with an independent variable with 2 levels ($I$ = 2):
 OLS regression with one independent variable:
 
Example context  Example context  
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.  Is there a linear relationship between physical health and mental health?  
SPSS  SPSS  
Analyze > Compare Means > IndependentSamples T Test...
 Analyze > Correlate > Bivariate...
 
Jamovi  Jamovi  
TTests > Independent Samples TTest
 Regression > Correlation Matrix
 
Practice questions  Practice questions  