Binomial test for a single proportion overview

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Binomial test for a single proportion	Binomial test for a single proportion $z$ test for a single proportion $z$ test for the difference between two proportions Goodness of fit test Chi-squared test for the relationship between two categorical variables One sample $z$ test for the mean One sample $t$ test for the mean Paired sample $t$ test Two sample $z$ test Two sample $t$ test - equal variances not assumed Two sample $t$ test - equal variances assumed One way ANOVA Two way ANOVA Pearson correlation Regression (OLS) Logistic regression Mann-Whitney-Wilcoxon test Kruskal-Wallis test Sign test McNemar's test Cochran's Q test Marginal Homogeneity test / Stuart-Maxwell test Friedman test Wilcoxon signed-rank test One sample Wilcoxon signed-rank test Spearman's rho
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

Binomial test for a single proportion - overview