Answer
Confidence interval: $-0.5605\lt β_1\lt0.3005$
We are 95% confident that $β_1$ is between -0.5605 and 0.3005
Work Step by Step
$n=17$, so:
$d.f.=n-2=15$
$level~of~confidence=(1-α).100$%
$95$% $=(1-α).100$%
$0.95=1-α$
$α=0.05$
$t_{\frac{α}{2}}=t_{0.025}=2.131$
(According to Table VI, for d.f. = 15 and area in right tail = 0.025)
$Lower~bound=b_1-t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}$
$Upper~bound=b_1+t_{\frac{α}{2}}\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}$
Now, see the results obtained in the MINITAB in item (b).
We can find the lower and upper bounds using the results from MINITAB. Use $\frac{s_e}{\sqrt {Σ(x_i-x ̅)^2}}=SE~Coef$
$Lower~bound=b_1-t_{\frac{α}{2}}(SE~Coef)=-0.130-2.131\times0.202=-0.5605$
$Upper~bound=b_1+t_{\frac{α}{2}}(SE~Coef)=-0.130+2.131\times0.202=0.3005$
