z test for the difference between two proportions - overview
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$z$ test for the difference between two proportions | Multivariate regression |
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Independent/grouping variable | Independent variables | |
One categorical with 2 independent groups | One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | |
Dependent variable | Dependent variables | |
One categorical with 2 independent groups | Two or more quantitative of interval or ratio level | |
Null hypothesis | THIS TABLE IS YET TO BE COMPLETED | |
H0: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | - | |
Alternative hypothesis | n.a. | |
H1 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | - | |
Assumptions | n.a. | |
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Test statistic | n.a. | |
$z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | - | |
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 $\pi_1 - \pi_2$ | n.a. | |
Regular (large sample):
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Equivalent to | n.a. | |
When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. | - | |
Example context | n.a. | |
Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | - | |
SPSS | n.a. | |
SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
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Jamovi | n.a. | |
Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | |