One sample z test for the mean  overview
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One sample $z$ test for the mean  Pearson correlation 


Independent variable  Variable 1  
None  One quantitative of interval or ratio level  
Dependent variable  Variable 2  
One quantitative of interval or ratio level  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  
H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis.  H_{0}: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level.  
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} two sided: $\rho \neq \rho_0$ H_{1} right sided: $\rho > \rho_0$ H_{1} left sided: $\rho < \rho_0$  
Assumptions  Assumptions of test for correlation  

 
Test statistic  Test statistic  
$z = \dfrac{\bar{y}  \mu_0}{\sigma / \sqrt{N}}$
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 $\mu_0$.  Test statistic for testing H0: $\rho = 0$:
 
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $t$ and of $z$ if H_{0} were true  
Standard normal distribution  Sampling distribution of $t$:
 
Significant?  Significant?  
Two sided:
 $t$ Test two sided:
 
$C\%$ confidence interval for $\mu$  Approximate $C$% confidence interval for $\rho$  
$\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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.  First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
 
Effect size  Properties of the Pearson correlation coefficient  
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 $\mu_0.$ 
 
Visual representation  n.a.  
  
n.a.  Equivalent to  
  OLS regression with one independent variable:
 
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 a linear relationship between physical health and mental health?  
n.a.  SPSS  
  Analyze > Correlate > Bivariate...
 
n.a.  Jamovi  
  Regression > Correlation Matrix
 
Practice questions  Practice questions  