Answer
Confidence interval: $0.2021\lt β_1\lt1.1499$
We are 95% confident that as the 7-day strength increases by 1 psi, the 28-day strength increases between 0.2021 and 1.1499 psi.
Work Step by Step
From item (d):
$∑(x_i-x ̅)^2=\sqrt {n-1}s_x=\sqrt {10-1}\times439.591=1318.773$
$n=10$, so:
$d.f.=n-2=8$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.306$
(According to Table VI, for d.f. = 8 and area in right tail = 0.025)
$Lower~bound=b_1-t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=0.676-2.306\times\frac{271.042}{1318.773}=0.2021$
$Upper~bound=b_1+t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=0.676+2.306\times\frac{271.042}{1318.773}=1.1499$
