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 proportionsGoodness of fit testChi-squared test for the relationship between two categorical variablesOne sample $z$ test for the meanOne sample $t$ test for the meanPaired sample $t$ testTwo sample $z$ testTwo sample $t$ test - equal variances not assumedTwo sample $t$ test - equal variances assumedOne way ANOVATwo way ANOVAPearson correlationRegression (OLS)Logistic regressionMann-Whitney-Wilcoxon testKruskal-Wallis testSign testMcNemar's testCochran's Q testMarginal Homogeneity test / Stuart-Maxwell testFriedman testWilcoxon signed-rank testOne sample Wilcoxon signed-rank testSpearman's rho
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