Answer
Confidence interval: $12.7\lt µ_{Republican}-µ_{Democrat}\lt13.3$
We are 95% confident that $µ_{Republican}-µ_{Democrat}$ is between 12.7 and 13.3 cents.
Work Step by Step
$n=513$, so:
$d.f.=n-1=512$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=1.984$
(According to Table VI, for d.f. = 100, the closest value to 512, and area in right tail = 0.025)
$Lower~bound=(x ̅_1-x ̅_2)-t_{\frac{α}{2}}\sqrt {\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}=(54-41)-1.984\sqrt {\frac{2.9^2}{513}+\frac{2.6^2}{513}}=12.7$
$Upper~bound=(x ̅_1-x ̅_2)+t_{\frac{α}{2}}\sqrt {\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}=(54-41)+1.984\sqrt {\frac{2.9^2}{513}+\frac{2.6^2}{513}}=13.3$
