Two sample z test  overview
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Two sample $z$ test  Binomial test for a single proportion 


Independent variable  Independent variable  
One categorical with 2 independent groups  None  
Dependent variable  Dependent variable  
One quantitative of interval or ratio level  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
$\mu_1 = \mu_2$
$\mu_1$ is the unknown mean in population 1, $\mu_2$ is the unknown mean in population 2  $\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to the null hypothesis  
Alternative hypothesis  Alternative hypothesis  
Two sided: $\mu_1 \neq \mu_2$ Right sided: $\mu_1 > \mu_2$ Left sided: $\mu_1 < \mu_2$  Two sided: $\pi \neq \pi_0$ Right sided: $\pi > \pi_0$ Left sided: $\pi < \pi_0$  
Assumptions  Assumptions  
 Sample is a simple random sample from the population. That is, observations are independent of one another  
Test statistic  Test statistic  
$z = \dfrac{(\bar{y}_1  \bar{y}_2)  0}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}} = \dfrac{\bar{y}_1  \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}$
$\bar{y}_1$ is the sample mean in group 1, $\bar{y}_2$ is the sample mean in group 2, $\sigma^2_1$ is the population variance in population 1, $\sigma^2_2$ is the population variance in population 2, $n_1$ is the sample size of group 1, $n_2$ is the sample size of group 2. The 0 represents the difference in population means according to H0. The denominator $\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}$ is the standard deviation of the sampling distribution of $\bar{y}_1  \bar{y}_2$. The $z$ value indicates how many of these standard deviations $\bar{y}_1  \bar{y}_2$ 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 $\mu_2 > \mu_1$  $X$ = number of successes in the sample  
Sampling distribution of $z$ if H0 were true  Sampling distribution of $X$ if H0 were true  
Standard normal  Binomial($n$, $p$) distribution
Here $n = N$ (total sample size), and $p = \pi_0$ (population proportion according to the null hypothesis)  
Significant?  Significant?  
Two sided:
 Two sided:
 
$C\%$ confidence interval for $\mu_1  \mu_2$  n.a.  
$(\bar{y}_1  \bar{y}_2) \pm z^* \times \sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}$
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 $\mu_1  \mu_2$ can also be used as significance test.    
Visual representation  n.a.  
  
Example context  Example context  
Is the average mental health score different between men and women? Assume that in the population, the standard devation of the mental health scores is $\sigma_1$ = 2 amongst men and $\sigma_2$ = 2.5 amongst women.  Is the proportion smokers amongst office workers different from $\pi_0 = .2$?  
n.a.  SPSS  
  Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 
n.a.  Jamovi  
  Frequencies > 2 Outcomes  Binomial test
 
Practice questions  Practice questions  