Answer
Confidence interval: $2.0411\lt β_1\lt3.0209$
We are 95% confident that as the square footage increases by 1 the rent increases between 2.0411 and 3.0209 dollars.
Work Step by Step
From item (d):
$\sqrt {∑(x_i-x ̅)^2}=\sqrt {n-1}s_x=\sqrt {11-1}\times335.19=1059.964$
$n=11$, so:
$d.f.=n-2=9$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.262$
(According to Table VI, for d.f. = 9 and area in right tail = 0.025)
$Lower~bound=b_1-t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=2.531-2.262\times\frac{229.547}{1059.964}=2.0411$
$Upper~bound=b_1+t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=2.531+2.262\times\frac{229.547}{1059.964}=3.0209$
