Binomial test for a single proportion - overview

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Binomial test for a single proportion
Independent variable
None
Dependent variable
One categorical with 2 independent groups
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
Alternative hypothesis
Two sided: $\pi \neq \pi_0$
Right sided: $\pi > \pi_0$
Left sided: $\pi < \pi_0$
Assumptions
Sample is a simple random sample from the population. That is, observations are independent of one another
Test statistic
$X$ = number of successes in the sample
Sampling distribution of $X$ if H0 were true
Binomial($n$, $p$) distribution

Here $n = N$ (total sample size), and $p = \pi_0$ (population proportion according to the null hypothesis)
Significant?
Two sided:
  • Check if $X$ observed in sample is in the rejection region or
  • Find two sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Right sided:
  • Check if $X$ observed in sample is in the rejection region or
  • Find right sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Left sided:
  • Check if $X$ observed in sample is in the rejection region or
  • Find left sided $p$ value corresponding to observed $X$ and check if it is equal to or smaller than $\alpha$
Example context
Is the proportion smokers amongst office workers different from $\pi_0 = .2$?
SPSS
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
  • Put your dichotomous variable in the box below Test Variable List
  • Fill in the value for $\pi_0$ in the box next to Test Proportion
Jamovi
Frequencies > 2 Outcomes - Binomial test
  • Put your dichotomous variable in the white box at the right
  • Fill in the value for $\pi_0$ in the box next to Test value
  • Under Hypothesis, select your alternative hypothesis
Practice questions