Chapter 7.2, Problem 28E

Applet Exercise Suppose that Y has an F distribution with ν₁ = 4 numerator degrees of freedom and ν₂ = 6 denominator degrees of freedom.

1. a Use Table 7, Appendix 3, to find F_.025. Also find F_.025 using the applet F-Ratio Probabilities and Quantiles.
2. b Refer to part (a). What quantile of Y does F_.025 correspond to? What percentile?
3. c Refer to parts (a) and (b). Use the applet F-Ratio Probabilities and Quantiles to find F_.975, the .025 quantile (2.5th percentile) of the distribution of Y.
4. d If U has an F distribution with ν₁ = 6 numerator and ν₂ = 4 denominator degrees of freedom, use Table 7, Appendix 3, or the F-Ratio Probabilities and Quantiles applet to find F_.025.
5. e In Exercise 7.29, you will show that if Y is a random variable that has an F distribution with ν₁ numerator and ν₂ denominator degrees of freedom, then U = 1/Y has an F distribution with ν₂ numerator and ν₁ denominator degrees of freedom. Does this result explain the relationship between F_.975 from part (c) (4 numerator and 6 denominator degrees of freedom) and F_.025 from part (d) (6 numerator and 4 denominator degrees of freedom)? What is this relationship?

7.29    If Y is a random variable that has an F distribution with ν₁ numerator and ν₂ denominator degrees of freedom, show that U = 1/Y has an F distribution with ν₂ numerator and ν₁ denominator degrees of freedom.