Answer
Confidence interval: $-0.1001\lt β_1\lt-0.0527$
We are 95% confident that as the credit score increases by 1, the interest rate of a 36-month auto loan decreases between 0.0527 and 0.1001 percent.
Work Step by Step
From item (c):
$∑(x_i-x ̅)^2=\sqrt {n-1}s_x=\sqrt {6-1}\times74.543=166.68$
$n=6$, so:
$d.f.=n-2=4$
$t_{\frac{α}{2}}=t_{0.025}=2.776$
(According to Table VI, for d.f. = 4 and area in right tail = 0.025)
$Lower~bound=b_1-t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=−0.0764-2.776\times\frac{1.424}{166.68}=-0.1001$
$Upper~bound=b_1+t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=−0.0764+2.776\times\frac{0.368}{166.68}=-0.0527$
