https://file.aiquickdraw.com/mathbot/file/1728970720602efuyfdfy.jpg

To solve this problem, we need to use the properties of the normal distribution and the Z-score formula.

### Given:
- Mean (\(\mu\)) = 132.6 feet
- Standard deviation (\(\sigma\)) = 7.4 feet

### Questions:
1. Find \( P(X < 133.3) \)
2. Find \( P(X > 146.7) \)

### Step 1: Calculate the Z-score
The Z-score formula is:
\[
Z = \frac{X - \mu}{\sigma}
\]

### Calculations:

**For \( P(X < 133.3) \):**
\[
Z = \frac{133.3 - 132.6}{7.4} = \frac{0.7}{7.4} \approx 0.0946
\]

Using the Z-table or a calculator, we find the probability associated with \( Z = 0.0946 \). 

**For \( P(X > 146.7) \):**
\[
Z = \frac{146.7 - 132.6}{7.4} = \frac{14.1}{7.4} \approx 1.9054
\]

For this, we need to find the area to the right of \( Z = 1.9054 \), which is \( 1 - P(Z < 1.9054) \).

### Results:
Let's calculate the probabilities:

1. \( P(X < 133.3) \approx 0.5379 \) (using Z = 0.0946)
2. \( P(X > 146.7) \approx 0.0285 \) (using Z = 1.9054)

Would you like to see more details or have any other questions?

Here are 5 related questions:
1. How do you find the Z-score for any given value?
2. What is the relationship between the Z-score and standard normal distribution?
3. How can you use a Z-table to find probabilities?
4. What would the probability be for a tree height less than 120 feet?
5. How would you calculate \( P(X > 130) \) with the same parameters?

**Tip:** When using a standard normal table, remember that it usually gives the area (probability) to the left of a given Z-score. For areas to the right, subtract from 1.

Let X represent the full height of a certain species of tree. Assume that X has a normal probability distribution with a mean of 132.6 ft and a standard deviation of 7.4 ft. A tree in the backyard stands 133.3 feet tall. Find the probability that the height of a randomly selected tree is as tall as mine or shorter. My neighbor's tree is 146.7 feet tall. Find the probability that the full height of a randomly selected tree is at least as tall as hers.

Learn how to calculate probabilities for tree heights using the normal distribution. This problem involves finding the probability that a tree's height is below 133.3 feet and above 146.7 feet using Z-scores. Suitable for students studying probability and statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation