Answer:
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.
Explanation:
Let X be a random variable which represents the diameters of aspan trees of a park in 1975.
Given that, X ~ N(8, 2.52)
Mean (μ) = 8 inches
Standard deviation (σ) = 2.5 inches
We have to obtain P(X < 5.5 inches)
We know that if X ~ N(μ, σ²) then, X-M Z=1 ~ N(0,1)
P(X< 5.5) = P( X- μ < 5.5 - μ)
σ σ
P(X<5.5) = P(Z < 5.5 – 8 )
2.5
P(X<5.5) = P(Z < -1)
Using "pnorm" function of R we get, P(Z < -1) = 0.1587
:: P(X < 5.5) = 0.1587
The probability that a randomly selected Aspen tree in the park in 1975 would have a diameter less than 5.5 inches is 0.1587.