Paired sample t test: overview
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Paired sample $t$ test 


Independent variable  
2 paired groups  
Dependent variable  
One quantitative of interval or ratio level  
Null hypothesis  
$\mu = \mu_0$
$\mu$ is the unknown population mean of the difference scores; $\mu_0$ is the population mean of the difference scores according to H0, which is usually 0  
Alternative hypothesis  
Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  
Assumptions  
 
Test statistic  
$t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
$\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to H0, $s$ is the sample standard deviation of the difference scores, $N$ 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 $\mu_0$  
Sampling distribution of $t$ if H0 were true  
$t$ distribution with $N  1$ degrees of freedom  
Significant?  
Two sided:
 
$C\%$ confidence interval for $\mu$  
$\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
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  
Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0$  
Visual representation  
Equivalent to  
One sample $t$ test on the difference scores
Repeated measures ANOVA with one dichotomous within subjects factor  
Example context  
Is the average difference between the mental health scores before and after an intervention different from $\mu_0$ = 0?  
SPSS  
Analyze > Compare Means > PairedSamples T Test...
 
Jamovi  
TTests > Paired Samples TTest
 
Practice questions  