How am I calculating this test statistic wrongly?

Question

Question:

Then I use the 2-sample z-test for sample proportions formula: Z = ( (96/260) - (46/145) - ( 96 - 46 ) ) / sqrt( (96/260)(1-(96/260)) / 260 + (46/145)(1-(46/145))/145 ) = -1021.79

What am I doing wrong here?

Please help Thank you

Answer 1

Let

$$H_0 : p_1=p_2$$

$$H_a : p_1\neq p_2$$

We have

$$\frac{\hat{p}_1-\hat{p}_2}{\sqrt{p^*(1-p^*)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}\sim z_{\alpha/2}$$

where $\hat{p}_1=\frac{96}{260}$, $\hat{p}_2=\frac{46}{145}$, $n_1=260$, $n_2=145$, and $p^*=\frac{96+46}{260+145}=\frac{142}{405}$

We have

$$\frac{\frac{96}{260}-\frac{46}{145}}{\sqrt{\frac{142}{405}(1-\frac{142}{405})\left(\frac{1}{260}+\frac{1}{145}\right)}}\approx 1.05$$

Since $1.05\lt1.96$ we fail to reject the null hypothesis at $\alpha=0.05$. We do not have significant evidence that the population proportions differ.