A quality engineer is comparing the strength variability of steel rods produced by two different rolling mills. From each mill she takes an independent sample of rods, computes the sample variances $s_1^2$ and $s_2^2$, and forms the $F$ test statistic $F=\frac{s_1^2}{s_2^2}$.

True or False: If $F$ is very close to $1$, then the two samples have essentially the same variability.

A pharmaceutical researcher compares the variability in blood pressure reductions between two independent treatment groups. The sample variances are $s_A^2=6.4$ and $s_B^2=4.9$. She defines the $F$ test statistic as $F=\frac{s_A^2}{s_B^2}$.

True or False: An $F$ statistic calculated this way can ever be a negative number.

Determine the critical $F$-value for a right-tailed test with a value $\alpha =0.025$, numerator degrees of freedom value of $d.f.N=4$, and denominator degrees of freedom value of $d.f.D=10$.

Determine the critical $F$-value for a right-tailed test where $\alpha =0.05$, numerator degrees of freedom $15$, and denominator degrees of freedom $40$.

Two independent samples yield sample variances $s_1^2=4.9$ with $n_1=15$ and $s_2^2=2.5$ with $n_2=12$. Find the value of the $F$ test statistic for testing equality of population variances (use the convention that the test statistic is at least $1$).