Answer
(a) $β_0=−2.324$ and $β_1=2.023$.
(b) $s_e=0.5315$
(c) $s_{b_1}=0.1282$
(d) $t_0\gt t_{\frac{α}{2}}$: null hypothesis is rejected.
There is enough evidence to conclude that a linear relation exists between x and y.
Work Step by Step
(a) According to problem 7d in section 4.2, $b_0=−2.324$ and $b_1=2.023$.
$b_0$ is the estimate for $β_0$ and $b_1$ is the estimate for $β_1$.
(b) $ŷ =2.023x−2.324$
$s_e=\sqrt {\frac{Σ(y_i-ŷ_i)^2}{n-2}}=\sqrt {\frac{[4-(2.023\times3−2.324)]^2+[6-(2.023\times4−2.324)]^2+[7-(2.023\times5−2.324)]^2+[12-(2.023\times7−2.324)]^2+[14-(2.023\times8−2.324)]^2}{5-2}}=0.5315$
(c) According to problem 7d in section 4.2, $s_x=2.073$
$s_{b_1}=\frac{s_e}{\sqrt {∑(x_i-x ̅)^2}}=\frac{s_e}{\sqrt {n-1}s_x}$ (see note on page 684)
$s_{b_1}=\frac{0.5315}{\sqrt {5-1}\times2.073}=0.1282$
(d) $H_0: β_1=0$ versus $H_1: β\ne0$
$t_0=\frac{b_1}{s_{b_1}}=\frac{2.032}{0.1282}=15.850$
$n=5$, so:
$d.f.=n-2=3$
$t_{\frac{α}{2}}=t_{0.025}=3.182$
(According to Table VI, for d.f. = 5 and area in right tail = 0.025)
Since $t_0\gt t_{\frac{α}{2}}$, we reject the null hypothesis.
