Paired sample t test  overview
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Paired sample $t$ test  Chisquared test for the relationship between two categorical variables 


Independent variable  Independent /column variable  
2 paired groups  One categorical with $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent /row variable  
One quantitative of interval or ratio level  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  
H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.  H_{0}: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  H_{1}: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). 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 $\mu_0$.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells.  
Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  
$t$ distribution with $N  1$ degrees of freedom  Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  
Significant?  Significant?  
Two sided:

 
$C\%$ confidence interval for $\mu$  n.a.  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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.    
Effect size  n.a.  
Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$    
Visual representation  n.a.  
  
Equivalent to  n.a.  
   
Example context  Example context  
Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?  Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  
SPSS  SPSS  
Analyze > Compare Means > PairedSamples T Test...
 Analyze > Descriptive Statistics > Crosstabs...
 
Jamovi  Jamovi  
TTests > Paired Samples TTest
 Frequencies > Independent Samples  $\chi^2$ test of association
 
Practice questions  Practice questions  