z test for the difference between two proportions  overview
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$z$ test for the difference between two proportions  $z$ test for a single proportion  Paired sample $t$ test 
You cannot compare more than 3 methods 

Independent/grouping variable  Independent variable  Independent variable  
One categorical with 2 independent groups  None  2 paired groups  
Dependent variable  Dependent variable  Dependent variable  
One categorical with 2 independent groups  One categorical with 2 independent groups  One quantitative of interval or ratio level  
Null hypothesis  Null hypothesis  Null hypothesis  
H_{0}: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2.  H_{0}: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.  H_{0}: $\mu = \mu_0$
Here $\mu$ is the population mean of the difference scores, and $\mu_0$ is the population mean of the difference scores according to the null hypothesis, which is usually 0. A difference score is the difference between the first score of a pair and the second score of a pair.  
Alternative hypothesis  Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\pi_1 \neq \pi_2$ H_{1} right sided: $\pi_1 > \pi_2$ H_{1} left sided: $\pi_1 < \pi_2$  H_{1} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  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  


 
Test statistic  Test statistic  Test statistic  
$z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2  p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$  $z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis.  $t = \dfrac{\bar{y}  \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean of the difference scores, $\mu_0$ is the population mean of the difference scores according to the null hypothesis, $s$ is the sample standard deviation of the difference scores, and $N$ is the sample size (number of difference scores). The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$.  
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $t$ if H_{0} were true  
Approximately the standard normal distribution  Approximately the standard normal distribution  $t$ distribution with $N  1$ degrees of freedom  
Significant?  Significant?  Significant?  
Two sided:
 Two sided:
 Two sided:
 
Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  Approximate $C\%$ confidence interval for $\pi$  $C\%$ confidence interval for $\mu$  
Regular (large sample):
 Regular (large sample):
 $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N1}$ 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 $\mu$ can also be used as significance test.  
n.a.  n.a.  Effect size  
    Cohen's $d$: Standardized difference between the sample mean of the difference scores and $\mu_0$: $$d = \frac{\bar{y}  \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean of the difference scores $\bar{y}$ is removed from $\mu_0.$  
n.a.  n.a.  Visual representation  
    
Equivalent to  Equivalent to  Equivalent to  
When testing two sided: chisquared test for the relationship between two categorical variables, where both categorical variables have 2 levels. 

 
Example context  Example context  Example context  
Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic.  Is the average difference between the mental health scores before and after an intervention different from $\mu_0 = 0$?  
SPSS  SPSS  SPSS  
SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
 Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 Analyze > Compare Means > PairedSamples T Test...
 
Jamovi  Jamovi  Jamovi  
Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chisquared test instead. The $p$ value resulting from this chisquared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples  $\chi^2$ test of association
 Frequencies > 2 Outcomes  Binomial test
 TTests > Paired Samples TTest
 
Practice questions  Practice questions  Practice questions  