z test for a single proportion - overview
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$z$ test for a single proportion | Repeated measures ANCOVA |
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Independent variable | Independent variable | |
None | One or more within subjects factors (related groups), one or more quantitative control variables of interval or ratio level (covariates), and possibly one or more between subjects factors (independent groups) | |
Dependent variable | Dependent variables | |
One categorical with 2 independent groups | One quantitative of interval or ratio level | |
Null hypothesis | THIS TABLE IS YET TO BE COMPLETED | |
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 | n.a. | |
H1 two sided: $\pi \neq \pi_0$ H1 right sided: $\pi > \pi_0$ H1 left sided: $\pi < \pi_0$ | - | |
Assumptions | n.a. | |
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Test statistic | n.a. | |
$z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
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 $z$ if H0 were true | n.a. | |
Approximately the standard normal distribution | - | |
Significant? | n.a. | |
Two sided:
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Approximate $C\%$ confidence interval for $\pi$ | n.a. | |
Regular (large sample):
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Equivalent to | n.a. | |
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Example context | n.a. | |
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 | n.a. | |
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
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Jamovi | n.a. | |
Frequencies > 2 Outcomes - Binomial test
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Practice questions | Practice questions | |