Questions: Tree heights: Cherry trees in a certain orchard have heights that are normally distributed with mean μ = 117 inches and standard deviation σ = 18 inches. Round the answers to four decimal places. (a) What proportion of trees are more than 120 inches tall?

Solution Steps

To find the Z-score for a tree height of \( X = 120 \) inches, we use the formula:

\[ z = \frac{X - \mu}{\sigma} \]

Substituting the values:

\[ z = \frac{120 - 117}{18} = \frac{3}{18} = 0.1667 \]

Thus, the Z-score for 120 inches is \( z = 0.1667 \).

Next, we need to find the probability that a tree is more than 120 inches tall. This can be expressed as:

\[ P(X > 120) = 1 - P(X \leq 120) = 1 - \Phi(0.1667) \]

Using the cumulative distribution function \( \Phi \), we find:

\[ P(X > 120) = \Phi(\infty) - \Phi(0.1667) = 1 - \Phi(0.1667) \]

From the calculations, we find:

\[ P(X > 120) = 0.4338 \]

Final Answer