Answer
Confidence interval: $2.528\lt ŷ\lt7.672$
We are 95% confident that the expcted value of $y$ for a $x=8$ is between 2.528 and 7.672.
Work Step by Step
From problem 6 from Section 14.1:
$s_e=0.7303$
$∑(x_i-x ̅)^2=(\sqrt {5-1}\times3.162)^2=39.993$
$x ̅=7$ (Problem 8d, section 4.2)
$n=5$, so:
$d.f.=n-2=3$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=3.182$
(According to Table VI, for d.f. = 3 and area in right tail = 0.025)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=5.1-3.182\times0.7303\sqrt {1+\frac{1}{5}+\frac{(8-7)^2}{39.993}}=2.528$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=5.1+3.182\times0.7303\sqrt {1+\frac{1}{5}+\frac{(8-7)^2}{39.993}}=7.672$
