Binomial test for a single proportion  overview
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Binomial test for a single proportion  Marginal Homogeneity test / StuartMaxwell test 


Independent variable  Independent variable  
None  2 paired groups  
Dependent variable  Dependent variable  
One categorical with 2 independent groups  One categorical with $J$ independent groups ($J \geqslant 2$)  
Null hypothesis  Null hypothesis  
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}: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$  
Alternative hypothesis  Alternative hypothesis  
H_{1} two sided: $\pi \neq \pi_0$ H_{1} right sided: $\pi > \pi_0$ H_{1} left sided: $\pi < \pi_0$  H_{1}: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.  
Assumptions  Assumptions  

 
Test statistic  Test statistic  
$X$ = number of successes in the sample  Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.  
Sampling distribution of $X$ if H0 were true  Sampling distribution of the test statistic if H_{0} were true  
Binomial($n$, $P$) distribution.
Here $n = N$ (total sample size), and $P = \pi_0$ (population proportion according to the null hypothesis).  Approximately the chisquared distribution with $J  1$ degrees of freedom  
Significant?  Significant?  
Two sided:
 If we denote the test statistic as $X^2$:
 
Example context  Example context  
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$?  Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?  
SPSS  SPSS  
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
 Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
 
Jamovi  n.a.  
Frequencies > 2 Outcomes  Binomial test
   
Practice questions  Practice questions  