Binomial test for a single proportion - overview
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Binomial test for a single proportion | $z$ test for a single proportion |
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Independent variable | Independent variable | |
None | None | |
Dependent variable | Dependent variable | |
One categorical with 2 independent groups | One categorical with 2 independent groups | |
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: $\pi = \pi_0$
Here $\pi$ is the population proportion of 'successes', and $\pi_0$ is the population proportion of successes according to the null hypothesis. | |
Alternative hypothesis | Alternative hypothesis | |
H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | |
Assumptions | Assumptions | |
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Test statistic | Test statistic | |
$X$ = number of successes in the sample | $z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
Here $p$ is the sample proportion of successes: $\dfrac{X}{N}$, $N$ is the sample size, and $\pi_0$ is the population proportion of successes according to the null hypothesis. | |
Sampling distribution of $X$ 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). | Approximately the standard normal distribution | |
Significant? | Significant? | |
Two sided:
| Two sided:
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n.a. | Approximate $C\%$ confidence interval for $\pi$ | |
- | Regular (large sample):
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n.a. | Equivalent to | |
- |
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Example context | Example context | |
Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? | Is the proportion of smokers amongst office workers different from $\pi_0 = 0.2$? Use the normal approximation for the sampling distribution of the test statistic. | |
SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| |
Jamovi | Jamovi | |
Frequencies > 2 Outcomes - Binomial test
| Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | |