Answer
$X_0^2\lt X_α^2$: null hypothesis is not rejected.
There is not enough evidence to conclude that the machine is “out of control".
Work Step by Step
$x ̅=\frac{64.05+63.99+63.98+63.95+64.05+64.01+64.00+64.05+63.97+64.03+64.01+63.95+64.10+63.97+64.06+64.01+63.98+63.95+63.94+64.08+64.02+64.01}{22}=64.007$
$s^2=\frac{(64.05-64.007)^2+(63.99-64.007)^2+(63.98-64.007)^2+(63.95-64.007)^2+(64.05-64.007)^2+(64.01-64.007)^2+(64.00-64.007)^2+(64.05-64.007)^2+(63.97-64.007)^2+(64.03-64.007)^2+(64.01-64.007)^2+(63.95-64.007)^2+(64.10-64.007)^2+(63.97-64.007)^2+(64.06-64.007)^2+(64.01-64.007)^2+(63.98-64.007)^2+(63.95-64.007)^2+(63.94-64.007)^2+(64.08-64.007)^2+(64.02-64.007)^2+(64.01-64.007)^2+}{22-1}=0.00199$
$H_0:~σ=0.04$ versus $H_1:~σ\gt0.04$
$X_0^2=\frac{(n-1)s^2}{σ_0^2}=\frac{(22-1)0.00199}{0.04^2}=26.119$
Right-tailed test:
$n=22$
$d.f.=n-1=21$
$X_α^2=X_{0.05}^2=32.671$
(According to Table VII, for d.f. = 21 and area to the right of critical value = 0.05)
Since $X_0^2\lt X_α^2$, we do not reject the null hypothesis.
