Answer
Confidence interval: $-0.0589\lt β_1\lt-0.0369$
We are 95% confident that as the commute time increases by 1 minute, the Gallup-Healthways Well-Being Index Composite Score decreases between 0.0369 and 0.0589.
Work Step by Step
From item (c):
$∑(x_i-x ̅)^2=\sqrt {n-1}s_x=\sqrt {7-1}\times34.988=85.703$
$n=7$, so:
$d.f.=n-2=5$
$t_{\frac{α}{2}}=t_{0.025}=2.571$
(According to Table VI, for d.f. = 5 and area in right tail = 0.025)
$Lower~bound=b_1-t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=−0.0479-2.571\times\frac{0.368}{85.703}=-0.0589$
$Upper~bound=b_1+t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=−0.0479+2.571\times\frac{0.368}{85.703}=-0.0369$
