Answer
$-z_{\frac{α}{2}}\lt z_0\lt z_{\frac{α}{2}}$: null hypothesis is not rejected.
There is not enough evidence to conclude that the proportions of males and females who are willing to pay higher taxes to reduce the deficit differs.
Work Step by Step
$N_1,n_1~and~p_1$ refer to males and $N_2,n_2~and~p_2$ refer to females.
$H_0:~p̂ _1=p̂ _2$ versus $H_1:~p̂ _1\ne p̂ _2$
$p̂ _1=0.317$ and $p̂ _2=0.216$ (item (a))
Requirements:
$n_1p̂ _1(1-p̂ _1)=82\times0.317(1-0.317)=17.75\geq10$
$n_2p̂ _2(1-p̂ _2)=116\times0.216(1-0.216)=19.64\geq10$
$n_1\leq0.05N_1$
$n_2\leq0.05N_2$
$p̂ =\frac{x_1+x_2}{n_1+n_2}=\frac{26+25}{82+116}=0.258$
$z_0=\frac{p̂_2-p̂ _1}{\sqrt {p̂ (1-p̂ )}\sqrt {\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.317-0.216}{\sqrt {0.258(1-0.258)}\sqrt {\frac{1}{82}+\frac{1}{116}}}=1.60$
Two-tailed test:
$z_{\frac{α}{2}}=z_{0.025}$
If the area of the standard normal curve to the right of $z_{0.025}$ is 0.025, then the area of the standard normal curve to the left of $z_{0.025}$ is $1−0.025=0.975$
According to Table V, the z-score which gives the closest value to 0.975 is 1.96.
Also, $-z_{\frac{α}{2}}=-1.96$
Since $-z_{\frac{α}{2}}\lt z_0\lt z_{\frac{α}{2}}$, we do not reject the null hypothesis.
