z test for the difference between two proportions  overview
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$z$ test for the difference between two proportions  McNemar's test  MannWhitneyWilcoxon test 
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Independent/grouping variable  Independent variable  Independent/grouping variable  
One categorical with 2 independent groups  2 paired groups  One categorical with 2 independent groups  
Dependent variable  Dependent variable  Dependent variable  
One categorical with 2 independent groups  One categorical with 2 independent groups  One of ordinal 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.  Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
 If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
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: $\pi_1 < \pi_2$  The alternative hypothesis H_{1} is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
 If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
 
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 $\pi_2 > \pi_1.$  $X^2 = \dfrac{(b  c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0.  Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2.  
Sampling distribution of $z$ if H_{0} were true  Sampling distribution of $X^2$ if H_{0} were true  Sampling distribution of $W$ and of $U$ if H_{0} were true  
Approximately the standard normal distribution  If $b + c$ is large enough (say, > 20), approximately the chisquared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$.  Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated.  
Significant?  Significant?  Significant?  
Two sided:
 For test statistic $X^2$:
 For large samples, the table for standard normal probabilities can be used: Two sided:
 
Approximate $C\%$ confidence interval for $\pi_1  \pi_2$  n.a.  n.a.  
Regular (large sample):
     
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. 
 If there are no ties in the data, the two sided MannWhitneyWilcoxon test is equivalent to the KruskalWallis test with an independent variable with 2 levels ($I$ = 2).  
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.  Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders?  Do men tend to score higher on social economic status than women?  
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 > 2 Related Samples...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
 
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 > Paired Samples  McNemar test
 TTests > Independent Samples TTest
 
Practice questions  Practice questions  Practice questions  