Answer
$z_0\gt z_{\frac{α}{2}}$: null hypothesis is rejected
There is enough evidence to conclude that the proportion of adult Americans who totally abstain from alcohol has changed.
Work Step by Step
$N_1,n_1~and~p_1$ refer to 1947 and $N_2,n_2~and~p_2$ refer to 2010.
$H_0:~p̂ _1=p̂ _2$ versus $H_1:~p̂ _1\ne p̂ _2$
$p̂ _1=\frac{x_1}{n_1}=\frac{407}{1100}=0.37$ versus $p̂ _2=\frac{x_2}{n_2}=\frac{333}{1100}=0.303$
Requirements:
$n_1p̂ _1(1-p̂ _1)=1100\times0.37(1-0.37)=256.41\geq10$
$n_2p̂ _2(1-p̂ _2)=1100\times0.303(1-0.303)=232.3101\geq10$
$n_1\leq0.05N_1$
$n_2\leq0.05N_2$
$p̂ =\frac{x_1+x_2}{n_1+n_2}=\frac{407+333}{1100+1100}=0.336$
$z_0=\frac{p̂_1-p̂ _2}{\sqrt {p̂ (1-p̂ )}\sqrt {\frac{1}{n_1}+\frac{1}{n_2}}}=\frac{0.37-0.303}{\sqrt {0.336(1-0.336)}\sqrt {\frac{1}{1100}+\frac{1}{1100}}}=3.33$
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.
Since $z_0\gt z_{\frac{α}{2}}$, we reject the null hypothesis.
