Goodness of fit test  overview
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Goodness of fit test 


Independent variable  
None  
Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  
 
Alternative hypothesis  
 
Assumptions  
 
Test statistic  
$X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here the expected cell count for one cell = $N \times \pi_j$, the observed cell count is the observed sample count in that same cell, and the sum is over all $J$ cells.  
Sampling distribution of $X^2$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  
Significant?  
 
Example context  
Is the proportion of people with a low, moderate, and high social economic status in the population different from $\pi_{low} = 0.2,$ $\pi_{moderate} = 0.6,$ and $\pi_{high} = 0.2$?  
SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 
Jamovi  
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 
Practice questions  