Binomial test for a single proportion - overview
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Binomial test for a single proportion | Mann-Whitney-Wilcoxon test | One sample $z$ test for the mean |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent variable | |
None | One categorical with 2 independent groups | None | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with 2 independent groups | One of ordinal 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. | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | If the dependent variable is measured on a continuous scale and the shape of the distribution of the dependent variable is the same in both populations:
Formulation 1:
| H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$X$ = number of successes in the sample | Two different types of test statistics can be used; both will result in the same test outcome. The first is the Wilcoxon rank sum statistic $W$:
Note: we could just as well base W and U on group 2. This would only 'flip' the right and left sided alternative hypotheses. Also, tables with critical values for $U$ are often based on the smaller of $U$ for group 1 and for group 2. | $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$. | |
Sampling distribution of $X$ if H0 were true | Sampling distribution of $W$ and of $U$ if H0 were true | Sampling distribution 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). | Sampling distribution of $W$:
Sampling distribution of $U$: For small samples, the exact distribution of $W$ or $U$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_W$ and $\sigma_U$ is more complicated. | Standard normal distribution | |
Significant? | Significant? | Significant? | |
Two sided:
| For large samples, the table for standard normal probabilities can be used: Two sided:
| Two sided:
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n.a. | n.a. | $C\%$ confidence interval for $\mu$ | |
- | - | $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). The confidence interval for $\mu$ can also be used as significance test. | |
n.a. | n.a. | Effect size | |
- | - | Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | |
n.a. | n.a. | Visual representation | |
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n.a. | Equivalent to | n.a. | |
- | If there are no ties in the data, the two sided Mann-Whitney-Wilcoxon test is equivalent to the Kruskal-Wallis test with an independent variable with 2 levels ($I$ = 2). | - | |
Example context | Example context | Example context | |
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Do men tend to score higher on social economic status than women? | Is the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$ | |
SPSS | SPSS | n.a. | |
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Independent Samples...
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Jamovi | Jamovi | n.a. | |
Frequencies > 2 Outcomes - Binomial test
| T-Tests > Independent Samples T-Test
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Practice questions | Practice questions | Practice questions | |