https://file.aiquickdraw.com/mathbot/file/1728436804458kns5y72d.jpg

The problem presented is related to the Normal (Gaussian) distribution and uses the **Empirical Rule**, also known as the **68-95-99.7 rule**. Here's a breakdown of the problem and how to address it:

### Problem Summary:
- A forester measured 22 trees in a large woods, finding a **mean diameter** of **10.9 inches** and a **standard deviation** of **4.6 inches**.
- You're asked to make calculations regarding the tree diameters using the Normal model and the 68-95-99.7 rule.

### Key Points of the Empirical Rule:
1. **68%** of the data falls within **1 standard deviation** of the mean.
2. **95%** of the data falls within **2 standard deviations** of the mean.
3. **99.7%** of the data falls within **3 standard deviations** of the mean.

Let’s walk through the questions in the image:

#### Question (b):
**What size would you expect the central 95% of all tree diameters to be?**

- For 95%, the data lies within **2 standard deviations** of the mean.
- Mean = 10.9 inches, Standard Deviation = 4.6 inches.

To calculate the range:
\[
Lower \ limit = \text{mean} - 2 \times \text{standard deviation} = 10.9 - 2 \times 4.6 = 1.7 \ \text{inches}
\]
\[
Upper \ limit = \text{mean} + 2 \times \text{standard deviation} = 10.9 + 2 \times 4.6 = 20.1 \ \text{inches}
\]
Therefore, the central **95% of tree diameters** will fall between **1.7 inches and 20.1 inches**.

#### Question (c):
**About what percent of the trees should have diameters below 6.3 inches?**

- For this, we need to calculate how far **6.3 inches** is from the mean in terms of standard deviations (z-score).
\[
Z = \frac{6.3 - 10.9}{4.6} \approx -1 \ (\text{1 standard deviation below the mean})
\]
Using the **Empirical Rule**, about **16%** of the data falls **below 1 standard deviation** from the mean.

#### Question (d):
**What percent of the trees should have diameters between 20.1 and 24.7 inches?**

- **20.1 inches** is **2 standard deviations above** the mean, and **24.7 inches** is **3 standard deviations above** the mean.
- The **68-95-99.7 rule** tells us that **99.7%** of data lies within 3 standard deviations, but for **2 to 3 standard deviations**, the percentage is:
\[
99.7\% - 95\% = 4.7\% \ (\text{split across both tails, so } 2.35\% \text{ on each side})
\]
Thus, approximately **2.35%** of the trees will have diameters between **20.1 inches and 24.7 inches**.

Would you like to explore any specific part of this solution in more detail, or should we go through the remaining questions?

### 5 Related Questions:
1. How do you calculate z-scores in a Normal distribution?
2. What does the standard deviation tell us in a data set?
3. How would the range change if the standard deviation was larger?
4. How can we calculate the exact percentages using a z-score table instead of the Empirical Rule?
5. How is the Empirical Rule applied to non-Normal distributions?

### Tip:
Always sketch a quick bell curve when working with Normal distributions to visually track where the percentages fall!

A forester measured 22 of the trees in a large woods that is up for sale. He found a mean diameter of 10.9 inches and a standard deviation of 4.6 inches. Using the Normal distribution and Empirical Rule, answer the following questions:
(b) What size would you expect the central 95% of all tree diameters to be?
(c) About what percent of the trees should have diameters below 6.3 inches?
(d) About what percent of the trees should have diameters between 20.1 and 24.7 inches?

Analyze tree diameters using the Normal distribution and Empirical Rule (68-95-99.7). This problem involves calculating the range for the central 95% of tree diameters, identifying percentages for trees below 6.3 inches, and those between 20.1 and 24.7 inches.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation