Answer
$z_0\lt z_α$: null hypothesis is not rejected.
The researcher wants to know whether more than 41% of the registred voters are Republican. But, there is not enough evidence that more than 41% of the registred voters are Republican.
Work Step by Step
Requirement:
$np_0(1-p_0)=320\times0.41(1-0.41)=77.408\gt10$
$p̂ =\frac{x}{n}=\frac{142}{320}=0.44375$
$z_0=\frac{p̂ -p_0}{\sqrt {\frac{p_0(1-p_0)}{n}}}=\frac{0.44375-0.41}{\sqrt {\frac{0.41(1-0.41)}{320}}}=1.23$
Using the classical method:
$z_α=z_{0.05}$
If the area of the standard normal curve to the right of $z_{0.05}$ is 0.05, then the area of the standard normal curve to the left of $z_{0.05}$ is $1−0.05=0.95$
According to Table V, there are 2 z-scores which give the closest value to 0.95: 1.64 and 1.65. So, let's find the mean of these z-scores: $\frac{1.64+1.65}{2}=1.645$
Since $z_0\lt z_α$, we do not reject the null hypothesis.
