Problem 10.5.1) For an F distribution, find the following
The Correct Answer and Explanation is:
Here are the critical values for the F distribution questions provided:
Answers: a. f0.25,5,10≈0.446f_{0.25,5,10} \approx 0.446 b. f0.10,24,9≈2.056f_{0.10,24,9} \approx 2.056 c. f0.05,8,15≈2.657f_{0.05,8,15} \approx 2.657 d. f0.75,5,10≈1.961f_{0.75,5,10} \approx 1.961 e. f0.90,24,9≈0.486f_{0.90,24,9} \approx 0.486 f. f0.95,8,15≈0.404f_{0.95,8,15} \approx 0.404
Explanation:
The F distribution is a continuous probability distribution that arises frequently in the context of comparing two population variances. Its shape is determined by two parameters: the degrees of freedom for the numerator and the degrees of freedom for the denominator.
Each value of the form fα,d1,d2f_{\alpha, d_1, d_2} represents a critical value for the F distribution. Here, α\alpha denotes the upper-tail probability, d1d_1 represents the degrees of freedom for the numerator, and d2d_2 for the denominator. For instance, in part (a), we are finding the value of F for which there is a 25% chance that a randomly chosen F-statistic with 5 numerator and 10 denominator degrees of freedom would exceed it.
The values in parts (d), (e), and (f) involve upper-tail areas greater than 0.5, which means they lie in the lower portion of the distribution and require reflection across the central axis. This relationship is described by the reciprocal identity:
f1−α,d1,d2=1fα,d2,d1f_{1 – \alpha, d_1, d_2} = \frac{1}{f_{\alpha, d_2, d_1}}
This means that to find a lower-tail critical value such as f0.75,5,10f_{0.75,5,10}, you would use the reciprocal of the upper-tail value f0.25,10,5f_{0.25,10,5}.
The F distribution is not symmetrical, which makes these reciprocal identities essential for navigating both tails accurately. These critical values are key in hypothesis testing procedures such as ANOVA, where they determine whether observed differences in variance between groups are statistically significant.
