One sample z test for the mean  overview
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One sample $z$ test for the mean  Logistic regression 


Independent variable  Independent variables  
None  One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
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  Model chisquared test for the complete regression model:
 
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\mu \neq \mu_0$ H_{1} right sided: $\mu > \mu_0$ H_{1} left sided: $\mu < \mu_0$  Model chisquared test for the complete regression model:
 
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 $\mu_0$.  Model chisquared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chisquared test for individual $\beta_k$:
 
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $X^2$ and of the Wald statistic if H_{0} were true  
Standard normal distribution  Sampling distribution of $X^2$, as computed in the model chisquared test for the complete model:
 
Significant?  Significant?  
Two sided:
 For the model chisquared test for the complete regression model and likelihood ratio chisquared test for individual $\beta_k$:
 
$C\%$ confidence interval for $\mu$  Waldtype approximate $C\%$ confidence interval for $\beta_k$  
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.  $b_k \pm z^* \times SE_{b_k}$ 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)  
Effect size  Goodness of fit measure $R^2_L$  
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 $\mu_0$  $R^2_L = \dfrac{D_{null}  D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit.  
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.  Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes?  
n.a.  SPSS  
  Analyze > Regression > Binary Logistic...
 
n.a.  Jamovi  
  Regression > 2 Outcomes  Binomial
 
Practice questions  Practice questions  