Chisquared test for the relationship between two categorical variables  overview
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Chisquared test for the relationship between two categorical variables  One sample $z$ test for the mean 


Independent /column variable  Independent variable  
One categorical with $I$ independent groups ($I \geqslant 2$)  None  
Dependent /row variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  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  
Alternative hypothesis  Alternative hypothesis  
 Two sided: $\mu \neq \mu_0$ Right sided: $\mu > \mu_0$ Left sided: $\mu < \mu_0$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
where 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  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to H0, $\sigma$ is the population standard deviation, $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$  
Sampling distribution of $X^2$ if H0 were true  Sampling distribution of $z$ if H0 were true  
Approximately a chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  Standard normal  
Significant?  Significant?  
 Two sided:
 
n.a.  $C\%$ confidence interval for $\mu$  
  $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where $z^*$ is the value under the normal curve with the area $C / 100$ between $z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval) The confidence interval for $\mu$ can also be used as significance test.  
n.a.  Effect size  
  Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{\sigma}$$ Indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0$  
n.a.  Visual representation  
  
Example context  Example context  
Is there an association between economic class and gender? Is the distribution of economic class different between men and women?  Is the average mental health score of office workers different from $\mu_0$ = 50? Assume that the standard deviation of the mental health scores in the population is $\sigma$ = 3.  
SPSS  n.a.  
Analyze > Descriptive Statistics > Crosstabs...
   
Jamovi  n.a.  
Frequencies > Independent Samples  $\chi^2$ test of association
   
Practice questions  Practice questions  