Answer
Confidence interval: $0.9191\lt β_1\lt1.2803$
We are 90% confident that as the rate of return of S&P 500 increases by 1 percent, the rate of return in United Technologies increases between 0.9191 and 1.2803 percent.
Work Step by Step
$n=11$, 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)
$Lower~bound=b_1-t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}$
$Upper~bound=b_1+t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}$
Now, see the results obtained in the MINITAB in item (a).
We can find the lower and upper bounds using the results from MINITAB. Use $\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=SE~Coef$
$Lower~bound=b_1-t_{\frac{α}{2}}(SE~Coef)=1.0997-1.833\times0.0985=0.9191$
$Upper~bound=b_1+t_{\frac{α}{2}}(SE~Coef)=1.0997+1.833\times0.0985=1.2803$
