Answer
$F_0\lt F_{1-α,n_1-1,n_2-1}$: null hypothesis is rejected.
There is enough evidence to conclude that the variability in the new machine is less than that of the old machine.
Work Step by Step
$s_1,n_1~and~d.f._1$ refer to the new machine and $s_2,n_2~and~d.f._2$ refer to the old machine.
$H_0:~σ_1=σ_2$ versus $H_1:σ_1\ltσ_2$
$s_1=\sqrt {\frac{∑(x_{1_i}-x ̅_1)^2}{n_1-1}}=0.0459$
$s_2=\sqrt {\frac{∑(x_{2_i}-x ̅_2)^2}{n_2-1}}=0.0759$
$F_0=\frac{s_1^2}{s_2^2}=\frac{0.0459^2}{0.0759^2}=0.37$
$d.f_1=n_1-1=15-1=14$
$d.f_2=n_2-1=15-1=14$
Left-tailed test:
$F_{α,n_1-1,n_2-1}=F_{0.05,14,14}=2.40$
(According to table VIII, for $d.f._1=15$, the closest value to 14, $d.f._2=15$, the closest value to 14, and area in the right tail = 0.05)
$F_{1-α,n_1-1,n_2-1}=F_{0.95,19,19}=\frac{1}{F_{0.05,19,19}}=\frac{1}{2.40}=0.42$
Since $F_0\lt F_{1-α,n_1-1,n_2-1}$, we reject the null hypothesis.
