Answer
The events are not independent.
$P(Republican~|~30-44)\ne P(Republican)$ and $P(30-44~|~Republican)\ne P(30-44).$
Work Step by Step
The sample space: 4000 Iowa Voters. So, N(S) = 4000.
N(Republican) = 2200, N(30-44) = 724 and N(Republican and 30-44) = 340. So:
$P(Republican)=\frac{N(Republican)}{N(S)}=\frac{2200}{4000}=0.55$. (Classical Method, page 259.)
$P(30-44)=\frac{N(30-44)}{N(S)}=\frac{724}{4000}=0.181$. (Classical Method, page 259.)
Now:
$P(Republican~|~30-44)=\frac{N(Republican~and~30-44)}{N(30-44)}=\frac{340}{724}\approx0.4696$ (Conditional Rule, page 288.)
$P(30-44~|~Republican)=\frac{N(Republican~and~30-44)}{N(Republican)}=\frac{340}{2200}=0.1545$ (Conditional Rule, page 288.)
$P(Republican~|~30-44)\ne P(Republican)$ and $P(30-44~|~Republican)\ne P(30-44).$
The events are not independent. See definition, page 292.
