The 68-95-99.7 rule (also known as the empirical rule) applies to normally distributed data and states the following:

• 68% of the data falls within 1 standard deviation ($\sigma$) of the mean ($\mu$).
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

To determine the percentage of trees that should have diameters between 14.8 inches and 19.5 inches, we need to know how these values relate to the mean and standard deviation of the tree diameters.

Steps to Solve:

1. Determine the mean ($\mu$) and standard deviation ($\sigma$) of the tree diameters if not provided directly. For this problem, we assume that 14.8 inches and 19.5 inches correspond to some multiple of $\sigma$ from the mean.

2. Identify how many standard deviations from the mean the given range (14.8 to 19.5 inches) covers.

3. Apply the empirical rule to find the approximate percentage of trees within this range.

Given Information:

To apply the empirical rule:

• Let’s assume $\mu = 17.15$ inches and $\sigma = 2.35$ inches based on the midpoints of the given range (14.8 to 19.5 inches). This assumption allows us to calculate how many standard deviations each endpoint is from the mean.

Calculate Z-scores for the endpoints:

$z_1 = \frac{14.8 - 17.15}{2.35}$ $z_2 = \frac{19.5 - 17.15}{2.35}$

Let's calculate the Z-scores and then interpret the result using the 68-95-99.7 rule.The calculated Z-scores are approximately:

• $z_1 = -1.0$ (for 14.8 inches)
• $z_2 = 1.0$ (for 19.5 inches)

Interpretation using the 68-95-99.7 Rule:

Since both endpoints are approximately 1 standard deviation from the mean, this falls within the 68% range of the empirical rule.

Thus, about 68% of the trees should have diameters between 14.8 inches and 19.5 inches.

Would you like more details or have any questions?

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Here are five related questions to deepen your understanding:

1. What would the percentage be if the range was between one and two standard deviations from the mean?
2. How would the percentage change if the standard deviation was larger?
3. How can we use the Z-scores to find exact probabilities using the standard normal distribution table?
4. What percentage of trees would have diameters outside the range of 14.8 to 19.5 inches?
5. How would this approach change if the distribution of diameters was not normal?

Tip: Understanding how to apply the 68-95-99.7 rule helps quickly estimate the distribution of data points in a normal distribution without detailed calculations.