Question

# Consider a hypothesis test of difference of proportions for two independent populations. Suppose random samples produce $r_{1}$ successes out of $n_{1}$ trials for the first population and $r_{2}$ successes out of $n_{2}$ trials for the second population. What is the formula for the sample test statistic?

Solution

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Given: hypothesis test of difference of proportions for two independent populations $x_1$ and $x_2$.

Formula for the sample test statistic:

$z=\dfrac{\hat{p}_1-\hat{p}_2}{\sqrt{\overline{p}(1-\overline{p})}\sqrt{\dfrac{1}{n_1}+\dfrac{1}{n_2}}}$

with

$\hat{p}_1$ the sample proportion of the first sample $\left( \hat{p}_1=\dfrac{r_1}{n_1} \right)$

$\hat{p}_2$ the sample proportion of the second sample $\left( \hat{p}_2=\dfrac{r_2}{n_2} \right)$

$\overline{p}$ the pooled proportion $\left( \overline{p}=\dfrac{r_1+r_2}{n_1+n_2} \right)$

$n_1$ the sample size of the first sample

$n_2$ the sample size of the second sample

$r_1$ the number of successes of the first sample

$r_2$ the number of successes of the second sample

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