Question

A forester studying the effects of fertilization on certain pine forests in the Southeast is interested in estimating the average basal area of pine trees. In studying basal areas of similar trees for many years, he has discovered that these measurements (in square inches) are normally distributed with mean approximately 25 square inches and standard deviation approximately 3 square inches. If the forester samples 16 trees, (a) Find the probability that the sample mean will be within 1 square inch of the population mean. (b) What is the probability that the sample total will exceed 432 square inches? (c) Suppose the forester wants to have the sample mean to be within 1 square inch of the population mean with probability 0.95. How many trees must be measure in order to ensure this degree of accuracy?

Answer 1

Solution:

(a) Find the probability that the sample mean will be within 1 square inch of the population mean.

We have to find:

  

Now using the standard normal table, we have:

Therefore, the probability that the sample mean will be within 1 square inch of the population mean is 0.8164

(b) What is the probability that the sample total will exceed 432 square inches?

We know that:

If X follows N(25,3), then we have:

We have to find:

Now using the standard normal table, we have:

Therefore, the probability that the sample total will exceed 432 square inches is 0.0038

(c) Suppose the forester wants to have the sample mean to be within 1 square inch of the population mean with probability 0.95. How many trees must be measure in order to ensure this degree of accuracy?

Therefore, 35 trees must be measured in order to ensure this degree of accuracy