Question

a forester measured 29 of the trees in a large woods that is up for sale. he found a mean diameter of 10.2 inches and a standard deviation of 4.8 inches. suppose that these trees provide an accurate description of the whole forest and that a normal model applies.
a) choose the correct normal model for tree diameters.

b) what size would you expect the central 95% of all tree diameters to be?
using the 68 - 95 - 99.7 rule, the central 95% of the tree diameters are between 6 inches and 19.8 inches.

c) about what percent of the trees should have diameters below 5.4 inches?
using the 68 - 95 - 99.7 rule, about 16% of the trees should have diameters below 5.4 inches.

(do not round. type an integer or a decimal.)

d) about what percent of the trees should have diameters between 15.0 and 19.8 inches?
using the 68 - 95 - 99.7 rule, about % of the trees should have diameters between 15.0 and 19.8 inches.

(do not round. type an integer or a decimal.)

Explanation:

Step1: Recall the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule for a normal distribution states that about 95% of the data lies within \(\mu\pm2\sigma\), about 68% lies within \(\mu\pm\sigma\), and about 99.7% lies within \(\mu\pm3\sigma\).
Given \(\mu = 10.2\) and \(\sigma=4.8\)

Step2: Calculate \(\mu - 2\sigma\) and \(\mu + 2\sigma\)

\(\mu - 2\sigma=10.2-2\times4.8=10.2 - 9.6 = 0.6\)
\(\mu + 2\sigma=10.2+2\times4.8=10.2 + 9.6 = 19.8\)

Step3: Calculate \(\mu - 3\sigma\)

\(\mu - 3\sigma=10.2-3\times4.8=10.2-14.4=- 4.2\) (not relevant for part c as we are looking for values below \(5.4\))
For part c:
We know that \(\mu-\sigma=10.2 - 4.8=5.4\)
The percentage of data below \(\mu-\sigma\) is \(\frac{100 - 68}{2}=16\%\)
For part d:
We want to find the percentage between \(15.0\) and \(19.8\)
First, \(\mu+\sigma=10.2 + 4.8 = 15.0\)
The percentage between \(\mu+\sigma\) and \(\mu + 2\sigma\) is \(\frac{95 - 68}{2}=13.5\%\)

Answer:

b) The central 95% of all tree diameters is between \(0.6\) inches and \(19.8\) inches.
c) About \(16\%\) of the trees should have diameters below \(5.4\) inches.
d) About \(13.5\%\) of the trees should have diameters between \(15.0\) and \(19.8\) inches.