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


Independent variable  Independent /column variable  
None  One categorical with $I$ independent groups ($I \geqslant 2$)  
Dependent variable  Dependent /row variable  
One quantitative of interval or ratio level  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  
H_{0}: $\mu = \mu_0$
$\mu$ is the population mean; $\mu_0$ is the population mean according to the null hypothesis  H_{0}: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  H_{1}: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
 
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
$\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ 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 $\mu_0$.  $X^2 = \sum{\frac{(\mbox{observed cell count}  \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
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  
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  
Standard normal distribution  Approximately the chisquared distribution with $(I  1) \times (J  1)$ degrees of freedom  
Significant?  Significant?  
Two sided:

 
$C\%$ confidence interval for $\mu$  n.a.  
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.    
Effect size  n.a.  
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 $\mu_0$    
Visual representation  n.a.  
  
Example context  Example context  
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 there an association between economic class and gender? Is the distribution of economic class different between men and women?  
n.a.  SPSS  
  Analyze > Descriptive Statistics > Crosstabs...
 
n.a.  Jamovi  
  Frequencies > Independent Samples  $\chi^2$ test of association
 
Practice questions  Practice questions  