Binomial test for a single proportion - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Binomial test for a single proportion | One sample $t$ test for the mean | Pearson correlation |
You cannot compare more than 3 methods |
---|---|---|---|
Independent variable | Independent variable | Variable 1 | |
None | None | One quantitative of interval or ratio level | |
Dependent variable | Dependent variable | Variable 2 | |
One categorical with 2 independent groups | One quantitative of interval or ratio level | One quantitative of interval or ratio level | |
Null hypothesis | Null hypothesis | 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. | H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | H0: $\rho = \rho_0$
Here $\rho$ is the Pearson correlation in the population, and $\rho_0$ is the Pearson correlation in the population according to the null hypothesis (usually 0). The Pearson correlation is a measure for the strength and direction of the linear relationship between two variables of at least interval measurement level. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | H1 two sided: $\rho \neq \rho_0$ H1 right sided: $\rho > \rho_0$ H1 left sided: $\rho < \rho_0$ | |
Assumptions | Assumptions | Assumptions of test for correlation | |
|
|
| |
Test statistic | Test statistic | Test statistic | |
$X$ = number of successes in the sample | $t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $s$ is the sample standard deviation, and $N$ is the sample size. The denominator $s / \sqrt{N}$ is the standard error of the sampling distribution of $\bar{y}$. The $t$ value indicates how many standard errors $\bar{y}$ is removed from $\mu_0$. | Test statistic for testing H0: $\rho = 0$:
| |
Sampling distribution of $X$ if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $t$ and of $z$ 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). | $t$ distribution with $N - 1$ degrees of freedom | Sampling distribution of $t$:
| |
Significant? | Significant? | Significant? | |
Two sided:
| Two sided:
| $t$ Test two sided:
| |
n.a. | $C\%$ confidence interval for $\mu$ | Approximate $C$% confidence interval for $\rho$ | |
- | $\bar{y} \pm t^* \times \dfrac{s}{\sqrt{N}}$
where the critical value $t^*$ is the value under the $t_{N-1}$ distribution with the area $C / 100$ between $-t^*$ and $t^*$ (e.g. $t^*$ = 2.086 for a 95% confidence interval when df = 20). The confidence interval for $\mu$ can also be used as significance test. | First compute the approximate $C$% confidence interval for $\rho_{Fisher}$:
Then transform back to get the approximate $C$% confidence interval for $\rho$:
| |
n.a. | Effect size | Properties of the Pearson correlation coefficient | |
- | Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{s}$$ Cohen's $d$ indicates how many standard deviations $s$ the sample mean $\bar{y}$ is removed from $\mu_0.$ |
| |
n.a. | Visual representation | n.a. | |
- | - | ||
n.a. | n.a. | Equivalent to | |
- | - | OLS regression with one independent variable:
| |
Example context | Example context | Example context | |
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Is the average mental health score of office workers different from $\mu_0 = 50$? | Is there a linear relationship between physical health and mental health? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Compare Means > One-Sample T Test...
| Analyze > Correlate > Bivariate...
| |
Jamovi | Jamovi | Jamovi | |
Frequencies > 2 Outcomes - Binomial test
| T-Tests > One Sample T-Test
| Regression > Correlation Matrix
| |
Practice questions | Practice questions | Practice questions | |