Answer
Confidence interval: $2343.05\lt y\lt2799.75$
We are 95% confident that the mean price of 52-inch 3D televisions is between 2343.05 and 2799.75 dollars.
Work Step by Step
$s_e=269.049$ (item (i))
$∑(x_i-x ̅)^2=24.5358^2=602.0055$ (item (m))
$x ̅=\frac{40+40+46+46+46+52+55+60+65}{9}=50$
$n=9$, so:
$d.f.=n-2=7$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.365$
(According to Table VI, for d.f. = 7 and area in right tail = 0.025)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=2561.4-2.365\times269.049\sqrt {\frac{1}{9}+\frac{(52-50)^2}{602.0055}}=2343.05$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=2561.4+2.365\times269.049\sqrt {\frac{1}{9}+\frac{(52-50)^2}{602.0055}}=2799.75$
