Answer
Confidence interval: $1.441\lt ŷ\lt8.037$
We are 90% confident that the rate of return on United Technologies stock if the rate of return on the S&P 500 for a randomly selected month is 4.2% is between 1.441 and 8.037 percent.
Work Step by Step
From problem 17 from Section 14.1:
$s_e=1.70053$ (see 17b: S in the Model Summary)
$∑(x_i-x ̅)^2=(\frac{s_e}{SE~Coef})^2=(\frac{1.70053}{0.0985})^2=298.055$ (see 17b)
$x ̅=\frac{1.48-8.20-5.39+6.88-4.74+8.76+3.69-0.23+6.53+2.26+3.20}{11}=1.295$
$n=11$, so:
$d.f.=n-2=9$
$level~of~confidence=(1-α).100$%
$90$% $=(1-α).100$%
$0.9=1-α$
$α=0.1$
$t_{\frac{α}{2}}=t_{0.05}=1.833$
(According to Table VI, for d.f. = 9 and area in right tail = 0.05)
$Lower~bound=ŷ -t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=4.739-1.833\times1.70053\sqrt {1+\frac{1}{11}+\frac{(4.2-1.295)^2}{298.055}}=1.441$
$Upper~bound=ŷ +t_{\frac{α}{2}}.s_e\sqrt {1+\frac{1}{n}+\frac{(x^*-x ̅)^2}{∑(x_i-x ̅)^2}}=4.739+1.833\times1.70053\sqrt {1+\frac{1}{11}+\frac{(4.2-1.295)^2}{298.055}}=8.037$
