Answer
Confidence interval: $1439.43\lt y\lt2318.37$
We are 90% confident that the rent of a particular 900-square-foot apartment in Queens is between 1439.43 and 2318.37 dollars.
Work Step by Step
$s_e=229.547$ (item (b))
$∑(x_i-x ̅)^2=1059.964^2=1123523.7$ (item (f))
$x ̅=\frac{500+588+1000+688+825+1259+650+560+1073+1452+1305}{11}=900$
$n=11$, so:
$d.f.=n-2=9$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.90=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=1.833$
(According to Table VI, for d.f. = 9 and area in right tail = 0.05)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=1878.9-1.833\times229.547\sqrt {1+\frac{1}{11}+\frac{(900-900)^2}{1123523.7}}=1439.43$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=1878.9+1.833\times229.547\sqrt {1+\frac{1}{11}+\frac{(900-900)^2}{1123523.7}}=2318.37$
