Answer
Confidence interval: $3.998\lt ŷ\lt6.102$
We are 95% confident that the mean value of $y$ when $x=7$ is between 3.998 and 6.102.
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 7d, 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)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=5.1-3.182\times0.7303\sqrt {\frac{1}{5}+\frac{(8-7)^2}{39.993}}=3.998$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=5.1+3.182\times0.7303\sqrt {\frac{1}{5}+\frac{(8-7)^2}{39.993}}=6.102$
