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!