Answer
Confidence interval: $0.847\lt µ_{community~college}-µ_{no~transfer}\lt1.153$
We are 95% confident that $µ_{community~college}-µ_{no~transfer}$ is between 0.847 and 1.153 years.
Work Step by Step
$n=268$ (use the smaller value of $n$), so:
$d.f.=n-1=267$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=1.984$
(According to Table VI, for d.f. = 100, the closest value to 267, and area in right tail = 0.025)
$Lower~bound=(x ̅_1-x ̅_2)-t_{\frac{α}{2}}\sqrt {\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}=(5.43-4.43)-1.984\sqrt {\frac{1.162^2}{268}+\frac{1.015^2}{1145}}=0.847$
$Upper~bound=(x ̅_1-x ̅_2)+t_{\frac{α}{2}}\sqrt {\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}=(5.43-4.43)+1.984\sqrt {\frac{1.162^2}{268}+\frac{1.015^2}{1145}}=1.153$
