2. Let X and Y be two independent random samples of sizes nx and ny, respectively, from two normal population distributions that do not necessarily have the same means or variances. The two distributions refer to the miles per gallon achieved by two 1/2-ton pickup trucks produced by two rival Detroit manufacturers. Define s^ as in Theorem 6.19.

(a) Show that the random variable F ¼ ðs^2 X = s2 XÞ=ðs^2 Y = s2 YÞ

has the F-distribution with (nx  1) numerator and

(ny  1) denominator degrees of freedom.

(b) Let nx ¼ 21 and ny ¼ 31. What is the probability that the random interval :49 s^2 Y=s^2 X

 ; 1:93 s^2 Y=s^2 X

  will have an outcome that will contain the value of the ratio of the variances sY 2

/sX 2

? (This random interval is another example of a confidence interval – in this case for the ratio of the population variances sY 2

/sX 2

.)

(c) Suppose that sx 2 ¼ .25 and sy 2 ¼ .04. Define a confidence interval that is designed to have a .98 probability of generating an outcome that contains the value of the ratio of population variances associated with the miles per gallon achieved by the two pickup trucks. Generate a confidence interval outcome for the ratio of variances.