# 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
H0: $\pi = \pi_0$

Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis.
Alternative hypothesis
H1 two sided: $\pi \neq \pi_0$
H1 right sided: $\pi > \pi_0$
H1 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 of smokers amongst office workers different from $\pi_0 = 0.2$?
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