Question

A company interested in lumbering rights for a certain tract of slash pine trees is told that the mean diameter of these trees is 14 inches with a standard deviation of 2.8 inches. Assume the distribution of diameters is roughly mound-shaped. a. What fraction of the trees will have diameters between 8.4 and 22.4 inches? b. What fraction of the trees will have diameters greater than 16.8 inches?

Short answer

Answer: The approximate fraction of trees with diameters between 8.4 and 22.4 inches is 4.7%, and the approximate fraction of trees with diameters greater than 16.8 inches is 32%.

Step-by-step solution

Find Z-scores for part a

First, we need to calculate the Z-scores for the given diameters in part a. The Z-score can be calculated using the formula \( Z =\frac{X-\mu}{\sigma} \), where X is the diameter, µ is the mean, and σ is the standard deviation. \n For 8.4 inches: Z-score = \( \frac{8.4-14}{2.8} \) = -2 For 22.4 inches: Z-score = \( \frac{22.4-14}{2.8} \) = 3

Calculate the fraction for part a

Now, using the Empirical Rule (68-95-99.7), we know that approximately 95% of the data falls within 2 standard deviations of the mean, and 99.7% of the data falls within 3 standard deviations of the mean. Since our Z-scores are -2 and 3, the fraction of trees with diameters between 8.4 and 22.4 inches is approximately equal to the fraction of data within 3 standard deviations minus the fraction within 2 standard deviations: \n Fraction = \( \frac{99.7}{100} - \frac{95}{100} \) = \( 0.997 - 0.95 \) = 0.047 or 4.7%

Find Z-score for part b

Now, we will calculate the Z-score for the given diameter in part b. \n For 16.8 inches: Z-score = \( \frac{16.8-14}{2.8} \) = 1

Calculate the fraction for part b

Using the Empirical Rule, we know that approximately 68% of the data falls within 1 standard deviation of the mean. Since our Z-score is 1, we want to find the fraction of trees with diameters greater than 16.8 inches, which means we are looking for the fraction of data outside 68%: \n Fraction = \( 1 - \frac{68}{100} \) = \( 1 - 0.68 \) = 0.32 or 32% So, the fraction of trees that have diameters between 8.4 and 22.4 inches is approximately 4.7%, and the fraction of trees that have diameters greater than 16.8 inches is approximately 32%.

Key concepts

Understanding the Z-score

In statistics, the Z-score is a helpful measure that tells us how many standard deviations a particular data point is from the mean. It helps standardize different data points, making them easier to compare across different distributions. Here's how you calculate it:

• Identify the mean (\( \mu \)) and standard deviation (\( \sigma \)) of the dataset.
• Use the formula: \( Z = \frac{X-\mu}{\sigma} \), where \( X \) is the value for which you're calculating the Z-score.

For example, if you have a tree diameter of 8.4 inches, a mean diameter of 14 inches, and a standard deviation of 2.8 inches, the Z-score is calculated as:

• \( Z = \frac{8.4-14}{2.8} = -2 \).

A Z-score tells us whether the data point is above or below the mean. A positive Z-score means the data point is above the mean, while a negative Z-score indicates it's below. Factors like these make Z-scores fundamental in probability and statistical analysis, as they provide insight into the data's position within its distribution.

Decoding Standard Deviation

Standard deviation is a crucial concept that measures the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out the numbers are in a dataset. Here's how standard deviation works:

• If the standard deviation is low, it means that the values tend to be close to the mean.
• A high standard deviation indicates that the values are spread out over a wider range.

For example, if the diameters of trees in a forest have a mean diameter of 14 inches and a standard deviation of 2.8 inches, we can interpret that the diameters typically vary by 2.8 inches from the mean diameter.
Standard deviation is significant because it helps to understand the extent of variability in data. It's a core part of the Empirical Rule, which uses standard deviation to describe data distribution in normal distributions. In practice, knowing the standard deviation allows us to make educated guesses about where most values lie and assess reliability within a dataset.

Exploring Normal Distribution

Normal distribution, often referred to as a bell curve, is a fundamental concept in statistics that describes how data is distributed in many natural phenomena. When graphed, it forms a symmetrical bell-shaped curve.
Key Characteristics of Normal Distribution include:

• Most data points cluster around the mean, decreasing in frequency as they move away.
• The mean, median, and mode are all located in the center of the graph.
• The curve is symmetric around the mean.

The significance of normal distribution lies in its predictable spread of data, which the Empirical Rule quantifies:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% within 2 standard deviations.
• 99.7% within 3 standard deviations.

Understanding normal distribution allows statisticians to make predictions about data behavior. For example, in the slash pine tree diameter case, with a mean of 14 inches, using normal distribution, we can predict the proportions that fall into particular ranges like 8.4 to 22.4 inches.

Understanding Probability

Probability is a cornerstone of statistics that measures the likelihood of a specific outcome occurring. It ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty.

• Probabilities can tell us how likely it is for certain events to happen.
• They are essential in risk assessment, forecasts, and decision-making processes.

In the context of tree diameters, probability helps estimate the fraction of trees that fall within certain size ranges:

• The probability of a tree diameter between 8.4 and 22.4 inches can be calculated using Z-scores and the Empirical Rule.
• Similarly, probability estimates how many trees exceed a diameter of 16.8 inches.

Real-world applications find probabilities essential for assessing everyday scenarios, from weather predictions to quality control in manufacturing. By understanding and calculating probabilities, we can better plan and make data-driven decisions.