Question

A park ranger at a large national park wants to estimate the mean diameter of all the aspen trees in the park. The park ranger believes that due to environmental changes, the aspen trees are not growing as large as they were in 1975. (a) Data collected in 1975 indicate that the distribution of diameter for aspen trees in this park was approximately normal with a mean of 8 inches and a standard deviation of 2.5 inches. Find the approximate probability that a randomly selected aspen tree in this park in 1975 would have a diameter less than 5.5 inches.

Answer 1

To find the approximate probability for the diameter of aspen trees in 1975, calculate the Z-score using the historical mean and standard deviation, then refer to the normal distribution table.

Step-by-step explanation:

The park ranger is interested in estimating the mean diameter of aspen trees and comparing it with historical data. To find the approximate probability that a randomly selected aspen tree in 1975 would have a diameter less than 5.5 inches, we need to use the normal distribution. In 1975, the mean diameter was 8 inches with a standard deviation of 2.5 inches. To find this probability, we can use a Z-score formula:

Z = (X - μ) / σ

Where:

• X = the value for which we want to find the probability (5.5 inches)
• μ = mean value (8 inches)
• σ = standard deviation (2.5 inches)

Substituting the values, we get:

Z = (5.5 - 8) / 2.5 = -1

After calculating the Z-score, we use the standard normal distribution table or a calculator to find the probability corresponding to a Z-score of -1. This gives us the probability that a randomly selected aspen tree had a diameter less than 5.5 inches in 1975.