https://file.aiquickdraw.com/mathbot/file/17303991247933w1zvr07.jpg

Let's go through each part of the question step-by-step.

### Part (a)
The question asks us to choose the correct Normal model for the tree diameters.

Given:
- **Mean diameter (μ)** = 10.7 inches
- **Standard deviation (σ)** = 4.7 inches

Using the 68-95-99.7 rule:
- **68%** of data falls within one standard deviation from the mean (μ ± σ).
- **95%** of data falls within two standard deviations from the mean (μ ± 2σ).
- **99.7%** of data falls within three standard deviations from the mean (μ ± 3σ).

#### Calculations:
1. **Mean ± σ (10.7 ± 4.7)** = 6.0 to 15.4
2. **Mean ± 2σ (10.7 ± 2×4.7)** = 1.3 to 20.1
3. **Mean ± 3σ (10.7 ± 3×4.7)** = -3.4 to 24.8

Looking at the images provided:
- **Model B** is the correct one, as it has these ranges: 
  - 6.0 to 15.4 for 68%
  - 1.3 to 20.1 for 95%
  - -3.4 to 24.8 for 99.7%.

### Part (b)
We need to find the range for the central 95% of tree diameters using the 68-95-99.7 rule.

For the central 95%, we consider **μ ± 2σ**:
- **Mean ± 2σ = 10.7 ± 2×4.7**
- **Calculation**: 
  - Lower bound: 10.7 - 9.4 = 1.3
  - Upper bound: 10.7 + 9.4 = 20.1

So, the central 95% of tree diameters are expected to be between **1.3 inches and 20.1 inches**.

### Final Answers
- **(a)**: Model **B**.
- **(b)**: The central 95% of the tree diameters are between **1.3 inches** and **20.1 inches**.

Would you like further clarification on any part of this solution, or additional details?

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Here are 5 related questions that expand on this topic:
1. What is the 68-95-99.7 rule, and how does it apply to Normal distributions?
2. How would we calculate the probability of a tree diameter being less than 1 inch?
3. How can we determine the percentage of tree diameters that are greater than 15 inches?
4. What is the impact of a larger standard deviation on the shape of a Normal distribution?
5. If the mean diameter were to increase by 2 inches, how would that change the distribution?

**Tip:** When using the 68-95-99.7 rule, always confirm your calculations of ±σ, ±2σ, and ±3σ ranges carefully to match the context.

A forester measured 35 of the trees in a large woods that is up for sale. He found a mean diameter of 10.7 inches and a standard deviation of 4.7 inches. Suppose that these trees provide an accurate description of the whole forest and that a Normal model applies.

(a) Choose the correct Normal model for tree diameters.
(b) What size would you expect the central 95% of all tree diameters to be? Using the 68-95-99.7 rule, the central 95% of the tree diameters are between ___ inches and ___ inches.

Explore how to apply the Normal distribution model to find the range of tree diameters. Given a mean diameter of 10.7 inches and standard deviation of 4.7 inches, learn how to use the 68-95-99.7 rule to find the central 95% range of diameters in a forest. Suitable for high school statistics students.

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