z test for a single proportion - overview
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$z$ test for a single proportion | Kruskal-Wallis test | Marginal Homogeneity test / Stuart-Maxwell test |
You cannot compare more than 3 methods |
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Independent variable | Independent/grouping variable | Independent variable | |
None | One categorical with $I$ independent groups ($I \geqslant 2$) | 2 paired groups | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with $J$ independent groups ($J \geqslant 2$) | |
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 all $I$ populations:
Formulation 1:
| H0: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$ | |
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 all $I$ populations:
Formulation 1:
| H1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group. | |
Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | |
$z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}$
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. | $H = \dfrac{12}{N (N + 1)} \sum \dfrac{R^2_i}{n_i} - 3(N + 1)$ | Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand. | |
Sampling distribution of $z$ if H0 were true | Sampling distribution of $H$ if H0 were true | Sampling distribution of the test statistic if H0 were true | |
Approximately the standard normal distribution | For large samples, approximately the chi-squared distribution with $I - 1$ degrees of freedom. For small samples, the exact distribution of $H$ should be used. | Approximately the chi-squared distribution with $J - 1$ degrees of freedom | |
Significant? | Significant? | Significant? | |
Two sided:
| For large samples, the table with critical $X^2$ values can be used. If we denote $X^2 = H$:
| If we denote the test statistic as $X^2$:
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Approximate $C\%$ confidence interval for $\pi$ | n.a. | n.a. | |
Regular (large sample):
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Equivalent to | n.a. | n.a. | |
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Example context | Example context | Example context | |
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. | Do people from different religions tend to score differently on social economic status? | Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best? | |
SPSS | SPSS | SPSS | |
Analyze > Nonparametric Tests > Legacy Dialogs > Binomial...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Independent Samples...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | Jamovi | n.a. | |
Frequencies > 2 Outcomes - Binomial test
| ANOVA > One Way ANOVA - Kruskal-Wallis
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Practice questions | Practice questions | Practice questions | |