Answer
$z_0\gt z_{\frac{α}{2}}$: null hypothesis is rejected.
There is enough evidence to conclude that the proportion of individuals in these groups in favor of capital punishment for persons under the age of 18 is different.
Work Step by Step
$N_1,n_1~and~p_1$ refer to Catholics and $N_2,n_2~and~p_2$ refer to seculars.
$H_0:~p̂ _1=p̂ _2$ versus $H_1:~p̂ _1\ne p̂ _2$
$p̂ _1=\frac{x_1}{n_1}=\frac{180}{580}=0.310$ and $p̂ _2=\frac{x_2}{n_2}=\frac{238}{600}=0.397$
Requirements:
$n_1p̂ _1(1-p̂ _1)=580\times0.3101-0.310)=124.062\geq10$
$n_2p̂ _2(1-p̂ _2)=600\times0.303(1-0.303)=126.7146\geq10$
$n_1\leq0.05N_1$
$n_2\leq0.05N_2$
$p̂ =\frac{x_1+x_2}{n_1+n_2}=\frac{180+238}{580+600}=0.354$
$z_0=\frac{p̂_2-p̂ _1}{\sqrt {p̂ (1-p̂ )}\sqrt {\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.397-0.310}{\sqrt {0.354(1-0.354)}\sqrt {\frac{1}{580}+\frac{1}{600}}}=3.12$
Two-tailed test:
$z_{\frac{α}{2}}=z_{0.005}$
If the area of the standard normal curve to the right of $z_{0.005}$ is 0.005, then the area of the standard normal curve to the left of $z_{0.005}$ is $1−0.005=0.995$
According to Table V, there are 2 z-scores which give the closest value to 0.995: 2.57 and 2.58. So, let's find the mean of these z-scores: $\frac{2.57+2.58}{2}=2.575$
Since $z_0\gt z_{\frac{α}{2}}$, we reject the null hypothesis.
