z test for a single proportion  overview
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$z$ test for a single proportion  $z$ test for the difference between two proportions 


Independent variable  Independent variable  
None  One categorical with 2 independent groups  
Dependent variable  Dependent variable  
One categorical with 2 independent groups  One categorical with 2 independent groups  
Null hypothesis  Null hypothesis  
$\pi = \pi_0$
$\pi$ is the population proportion of "successes"; $\pi_0$ is the population proportion of successes according to the null hypothesis  $\pi_1 = \pi_2$
$\pi_1$ is the unknown proportion of "successes" in population 1; $\pi_2$ is the unknown proportion of "successes" in population 2  
Alternative hypothesis  Alternative hypothesis  
Two sided: $\pi \neq \pi_0$ Right sided: $\pi > \pi_0$ Left sided: $\pi < \pi_0$  Two sided: $\pi_1 \neq \pi_2$ Right sided: $\pi_1 > \pi_2$ Left sided: $\pi_1 < \pi_2$  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$z = \dfrac{p  \pi_0}{\sqrt{\dfrac{\pi_0(1  \pi_0)}{N}}}$
$p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size  $z = \dfrac{p_1  p_2}{\sqrt{p(1  p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
$p_1$ 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, $n_2$ 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 $\pi_2 > \pi_1$  
Sampling distribution of $z$ if H0 were true  Sampling distribution of $z$ if H0 were true  
Approximately standard normal  Approximately standard normal  
Significant?  Significant?  
Two sided:
 Two sided:
 
Approximate $C\%$ confidence interval for $\pi$  Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  
Regular (large sample):
 Regular (large sample):
 
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  
Is the proportion smokers amongst office workers different from $\pi_0 = .2$? Use the normal approximation for the sampling distribution of the test statistic.  Is the proportion smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic.  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 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...
 
Jamovi  Jamovi  
Frequencies > 2 Outcomes  Binomial test
 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
 
Practice questions  Practice questions  