Answer
Confidence interval: $3.707\lt y\lt7.487$
We are 90% confident that the mean interest rate of all individuals whose credit score is 730 is between 3.707 and 7.487 percent.
Work Step by Step
From problem 12 from Section 14.1:
$s_e=1.424$
$∑(x_i-x ̅)^2=166.68^2=27782.22$
$x ̅=\frac{545+595+640+675+705+750}{6}=651.67$
$n=6$, so:
$d.f.=n-2=4$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.9=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=2.132$
(According to Table VI, for d.f. = 4 and area in right tail = 0.05)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=5.5967-2.132\times1.424\sqrt {\frac{1}{6}+\frac{(730-651.67)^2}{27782.22}}=3.707$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=5.5967+2.132\times1.424\sqrt {\frac{1}{6}+\frac{(730-651.67)^2}{27782.22}}=7.487$
