One sample t test for the mean  overview
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One sample $t$ test for the mean  McNemar's test 


Independent variable  Independent variable  
None  2 paired groups  
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  For each pair of scores, the data allow four options:
Other formulations of the null hypothesis are :
 
Alternative hypothesis  Alternative hypothesis  
Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  Alternative hypothesis is that for each pair of scores:
Other formulations of the alternative hypothesis are that, for each pair of scores:
 
Assumptions  Assumptions  
 Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another  
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 $\mu_0$  $X^2 = \dfrac{(b  c)^2}{b + c}$
$b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0  
Sampling distribution of $t$ if H0 were true  Sampling distribution of $X^2$ if H0 were true  
$t$ distribution with $N  1$ degrees of freedom  If $b + c$ is large enough (say, > 20), approximately a chisquared distribution with 1 degree of freedom. If $b + c$ is small, the binomial($n$, $p$) distribution should be used, with $n = b + c$ and $p = 0.5$. In that case the test statistic becomes equal to $b$.  
Significant?  Significant?  
Two sided:
 For test statistic $X^2$:
 
$C\%$ 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 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?  Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?  
SPSS  SPSS  
Analyze > Compare Means > OneSample T Test...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  Jamovi  
TTests > One Sample TTest
 Frequencies > Paired Samples  McNemar test
 
Practice questions  Practice questions  