Answer
Confidence interval: $67.25\lt y\lt68.89$
We are 90% confident that the well-being index composite score of Jane is between 67.25 and 68.89
Work Step by Step
From problem 11 from Section 14.1:
$s_e=0.368$
$∑(x_i-x ̅)^2=85.703^2=7345.00$
$x ̅=\frac{5+15+25+35+50+72+105}{7}=43.857$
$n=7$, so:
$d.f.=n-2=5$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.9=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=2.015$
(According to Table VI, for d.f. = 5 and area in right tail = 0.05)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=68.07-2.015\times0.368\sqrt {1+\frac{1}{7}+\frac{(20-43.857)^2}{7345.00}}=67.25$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=68.07+2.015\times0.368\sqrt {1+\frac{1}{7}+\frac{(20-43.857)^2}{7345.00}}=68.89$
