Goodness of fit test  overview
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Goodness of fit test  Two sample $t$ test  equal variances assumed  One sample $z$ test for the mean  Goodness of fit test 


Independent variable  Independent/grouping variable  Independent variable  Independent variable  
None  One categorical with 2 independent groups  None  None  
Dependent variable  Dependent variable  Dependent variable  Dependent variable  
One categorical with $J$ independent groups ($J \geqslant 2$)  One quantitative of interval or ratio level  One quantitative of interval or ratio level  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  Null hypothesis  Null hypothesis  
 H_{0}: $\mu_1 = \mu_2$
Here $\mu_1$ is the population mean for group 1, and $\mu_2$ is the population mean for group 2.  H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. 
 
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
 H_{1} two sided: $\mu_1 \neq \mu_2$ H_{1} right sided: $\mu_1 > \mu_2$ H_{1} left sided: $\mu_1 < \mu_2$  H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$ 
 
Assumptions  Assumptions  Assumptions  Assumptions  



 
Test statistic  Test statistic  Test statistic  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.  $t = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$
Here $\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $s_p$ is the pooled standard deviation, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to the null hypothesis. The denominator $s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the standard error of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $t$ value indicates how many standard errors $\bar{y}_1  \bar{y}_2$ is removed from 0. Note: we could just as well compute $\bar{y}_2  \bar{y}_1$ in the numerator, but then the left sided alternative becomes $\mu_2 < \mu_1$, and the right sided alternative becomes $\mu_2 > \mu_1$.  $z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ 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 $\mu_0$.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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.  
n.a.  Pooled standard deviation  n.a.  n.a.  
  $s_p = \sqrt{\dfrac{(n_1  1) \times s^2_1 + (n_2  1) \times s^2_2}{n_1 + n_2  2}}$      
Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  
Approximately the chisquared distribution with $J  1$ degrees of freedom  $t$ distribution with $n_1 + n_2  2$ degrees of freedom  Standard normal distribution  Approximately the chisquared distribution with $J  1$ degrees of freedom  
Significant?  Significant?  Significant?  Significant?  
 Two sided:
 Two sided:

 
n.a.  $C\%$ confidence interval for $\mu_1  \mu_2$  $C\%$ confidence interval for $\mu$  n.a.  
  $(\bar{y}_1  \bar{y}_2) \pm t^* \times s_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$
where the critical value $t^*$ is the value under the $t_{n_1 + n_2  2}$ 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 $\mu_1  \mu_2$ can also be used as significance test.  $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ 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  Effect size  n.a.  
  Cohen's $d$: Standardized difference between the mean in group $1$ and in group $2$: $$d = \frac{\bar{y}_1  \bar{y}_2}{s_p}$$ Cohen's $d$ indicates how many standard deviations $s_p$ the two sample means are removed from each other.  Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$    
n.a.  Visual representation  Visual representation  n.a.  
    
n.a.  Equivalent to  n.a.  n.a.  
  One way ANOVA with an independent variable with 2 levels ($I$ = 2):
     
Example context  Example context  Example context  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 $\pi_{high} = 0.2$?  Is the average mental health score different between men and women? Assume that in the population, the standard deviation of mental health scores is equal amongst 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 $\sigma = 3.$  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  SPSS  n.a.  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 Analyze > Compare Means > IndependentSamples T Test...
   Analyze > Nonparametric Tests > Legacy Dialogs > Chisquare...
 
Jamovi  Jamovi  n.a.  Jamovi  
Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 TTests > Independent Samples TTest
   Frequencies > N Outcomes  $\chi^2$ Goodness of fit
 
Practice questions  Practice questions  Practice questions  Practice questions  