Chapter 9.4, Problem 23E

Eastern Hemlock Ring shake, which is the separation of the wood between growth rings, is a serious problem in hemlock trees. Researchers have developed the following function that estimates the probability $P$ that a given hemlock tree has ring shake:

$P\left(A,B,D\right)=\frac{1}{1+e^{3.68-0.016A-0.77B-0.12D}}$

where $A$ is the age of the tree (yr), $B$ is $1$ if bird pecking is present and $0$ otherwise, and $D$ is the diameter (in.) of the tree at breast height. Source: Forest Products Journal.

a. Estimate the probability that a $150$-year-old tree with bird pecking present and a breast height diameter of $20in.,$ will have ring shake.

b. Estimate the probability that a $150$-year-old tree, with no presence of bird pecking and a breast height diameter of $20in.,$ will have ring shake.

c. Develop a statement about what can be said about the influence have on the probability of ring shake.

d. Using the total differential, estimate the probability if the actual age of the tree was $160$ years and the diameter at breast height was $25in.,$ Assume that no bird pecking was present. Compare your answer to the actual value. (Hint: Assume that $B=0$ and exclude that variable from your calculations)

e. Comment on the practicality of using differentials in part $d.$