Mathematics 376 – Probability and Statistics II Problem Set 5 due: February 24, 2006

A) (A follow-up problem to C from Lab Project 3.) Recall that in Lab Project 3, to determine the endpoints of the confidence interval for the ratio of variances σ_{1}^{2}/σ_{2}^{2}, you needed to work with a formula like this

P (\frac{S_{1}^{2}/S_{2}^{2}}{f_{α/2}} ≤ σ_{1}^{2}/σ_{2}^{2} ≤ \frac{S_{1}^{2}/S_{2}^{2}}{f_{1−α/2}} ) = 1 − α.

In the problem from the Lab Project, you were looking for a 95% confidence interval, so you needed f_{.025}, f_{.975}. The F -table in the back of the book gives the values f_{α} for α = .1, .05, .025, .01, .005, but not the complementary values α = .9, .95, .975, .99, .995. As you might have wondered, why aren’t those other entries there?? Fortunately, with the FCDF function from our Maple package, you could compute f_{.975}, etc., by the same method we used for χ^{2} and t distributions in the lab, so this wasn’t a problem. In this question you will show that the other values could be derived from the table entries given with a little cleverness, too!

1) Let the numbers f_{α}(n, m) be defined by

P (Y ≥ f_{α}(n, m)) = α

if Y has an F -distribution with n numerator degrees of freedom and m denominator degrees of freedom. Show that for all 0 < α < 1, these numbers satisfy:

f_{1−α}(n, m) = \frac{1}{f_{α}(m, n)}

(note the reversal in the numbers of degrees of freedom on the right.) Hint: What is the standard way to get a random variable that has an F -distribution?

2) Using the table in the text and part 1, determine the number f_{.99}(7, 10).

Additional problems from the text:

• Chapter 8/74,75,82,86;