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American Black Bears The American black bear (Ursus americanus) is one of eight bear species in the world. It is the smallest North American bear and the most common bear species on the planet. In 1969 , Dr. Michael R. Pelton of the University of Tennessee initiated a long-term study of the population in the Great Smoky Mountains National Park. One aspect of the study was to develop a model that could be used to predict a bear's weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The following data represent the lengths and weights of 12 . American black bears. (a) Which variable is the explanatory variable based on the goals of the research? (b) Draw a scatter diagram of the data. (c) Determine the linear correlation coefficient between weight and length. (d) Does a linear relation exist between the weight of the bear and its length?

Step by step solution

01

Identify the Explanatory Variable

The explanatory variable is the one that explains or predicts changes in the response variable. Based on the goals of the research, the length of the bear is used to predict its weight. Therefore, the explanatory variable is 'length'.

02

Draw a Scatter Diagram

To draw a scatter diagram, plot each pair of data points on a graph where the x-axis represents the length of the bears and the y-axis represents their weights. This will help visually inspect the relationship between the two variables.

03

Calculate the Linear Correlation Coefficient

The linear correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. Use the formula:

04

Determine if a Linear Relation Exists

A linear relation exists if the absolute value of the correlation coefficient r is close to 1. Typically, a value of r greater than 0.7 or less than -0.7 indicates a strong linear relation. Compare the calculated r to these benchmarks to determine if a significant linear relation exists between the weight of the bear and its length.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Explanatory Variable

In linear regression, the explanatory variable is crucial because it helps predict or explain changes in another variable known as the response variable. Think of it like a cause-and-effect relationship. For the American black bear study, the goal was to use a measurable characteristic to predict the bear's weight.
In this exercise, **length** is chosen as the explanatory variable. This means we observe the bear's length to estimate its weight accurately. Understanding the explanatory variable is vital because it helps researchers focus on the right data to make meaningful predictions.

In simple terms:

• **Explanatory Variable (Length):** The factor you use to predict another variable.

Scatter Diagram

A scatter diagram, also known as a scatter plot, is a graphical representation that shows the relationship between two variables. In the scatter diagram for this research, the x-axis represents the length of the bear, while the y-axis represents its weight.

Creating a scatter diagram involves the following steps:

• Plot each pair of data points on the graph.
• Look for a pattern in the points to see if there's a visible relationship.

This visual inspection can help us quickly understand whether the length and weight of the bears move together in any specific pattern. If the points trend upward (positive correlation) or downward (negative correlation), it suggests a possible linear relationship.
Think of it like plotting a bunch of dots on a graph to see if they align neatly along a line!

Linear Correlation Coefficient

The linear correlation coefficient, denoted as **r**, quantifies the strength and direction of the linear relationship between two variables. It ranges from -1 to 1:

• **r = 1:** Perfect positive linear relationship.
• **r = -1:** Perfect negative linear relationship.
• **r = 0:** No linear relationship.

A larger absolute value of **r** indicates a stronger linear relationship. The correlation coefficient is calculated using the formula:
\[ r = \frac{n(\text{sum}(XY)) - (\text{sum}(X)\text{sum}(Y))}{\text{sqrt}([n(\text{sum}(X^2)) - (\text{sum}(X))^2][n(\text{sum}(Y^2)) - (\text{sum}(Y))^2])} \]
Where:

• **X** and **Y** are the variables (bear length and weight).
• **n** is the number of data points.
• The sum terms represent summation operations over the dataset.

By calculating **r**, we determine if a strong linear connection exists between the bear's length and its weight.

Linear Relationship

A linear relationship exists when two variables are proportionally related to each other. For weight and length in bears, this would mean that as the length increases, the weight also increases or decreases consistently.
To affirm if there's a linear relationship, the correlation coefficient **r** calculated earlier is examined:

• If **r** > 0.7 or **r** < -0.7, it implies a strong linear relationship.
• If **r** is close to 0, the relationship is weak or nonexistent.

The strength of **r** indicates the predictive accuracy. When a significant linear relationship is found, we can confidently use the explanatory variable (length) to predict the response variable (weight).
This valuable insight helps in understanding and managing wildlife like the American black bears effectively.