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. Total Length (cm) 139.0 138.0 139.0 120.5 149.0 141.0 141.0 150.0 166.0 151.5 129.5 150.0 Weight (kg) length 110 60 90 (length, weight). 60 85 100 95 85 155 140 105 110 Weight (kg) 150- 125+ 100- 75- 50- American Black Bears ● 120 130 ● 140 Length (cm) (a) Which variable is the explanatory variable, and which is the response variable based on the goals of the study? The explanatory variable is weight 150 160 170 (length, weight) and the response variable is (b) Use your calculator to compute the linear correlation coefficient r between weight and length.
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. Total Length (cm) 139.0 138.0 139.0 120.5 149.0 141.0 141.0 150.0 166.0 151.5 129.5 150.0 Weight (kg) length 110 60 90 (length, weight). 60 85 100 95 85 155 140 105 110 Weight (kg) 150- 125+ 100- 75- 50- American Black Bears ● 120 130 ● 140 Length (cm) (a) Which variable is the explanatory variable, and which is the response variable based on the goals of the study? The explanatory variable is weight 150 160 170 (length, weight) and the response variable is (b) Use your calculator to compute the linear correlation coefficient r between weight and length.
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Author:Amos Gilat
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Transcribed Image Text:### The American Black Bear: Length and Weight Study
#### Overview
The American black bear (Ursus americanus) is one of eight bear species globally. It is the smallest North American bear and the most common bear species on the planet. In 1969, Dr. Michael R. Pelton from the University of Tennessee initiated a long-term study of the black bear population in the Great Smoky Mountains National Park. The study aimed to develop a model to predict a bear's weight, given the impracticality of weighing bears in the field. One variable related to weight is the length of the bear.
#### Data Presentation
The following data points represent the lengths and weights of 12 American black bears.
**Table: Total Length (cm) and Weight (kg)**
| Total Length (cm) | Weight (kg) |
|-------------------|-------------|
| 139.0 | 110 |
| 138.0 | 60 |
| 139.0 | 90 |
| 120.5 | 60 |
| 149.0 | 85 |
| 141.0 | 100 |
| 141.0 | 95 |
| 150.0 | 85 |
| 166.0 | 155 |
| 151.5 | 140 |
| 129.5 | 105 |
| 150.0 | 110 |
#### Graph: Scatter Plot of Length vs. Weight
To visualize the relationship, a scatter plot plots bear weight (kg) against bear length (cm).
**Explanation of the Scatter Plot:**
- **X-Axis (Horizontal):** Length (cm)
- **Y-Axis (Vertical):** Weight (kg)
- Each point on the graph represents a pair of observations for length and weight of a bear.
- As length increases, there is a general trend of increasing weight, indicating a positive correlation.
#### Analysis
**(a) Identifying Variables:**
- The explanatory variable (independent) is **length** (measured in cm).
- The response variable (dependent) is **weight** (measured in kg).
**(b) Correlation Coefficient:**
- Using a calculator, the linear correlation coefficient \( r \) between weight and length is approximately **0.7039**. This value suggests a moderate positive
![### Linear Regression Analysis for Bear Weight and Length
#### (c) Determining Linear Relationship
**Question:** Does a linear relation exist between the weight of the bear and its length?
**Answer Options:** [yes] [no]
**Observation Inquiry:**
Describe the linear relationship that exists between weight and length by looking at the scatterplot:
**Answer Options:** [positive] [negative]
#### (d) Using a Calculator for Least-Squares Regression Line
**Task:**
Use your calculator to find the least-squares regression line treating total length as the explanatory variable and weight as the response variable.
**Instruction:** Round the slope and y-intercept to three decimal places.
**Equation Format:**
\[ \hat{y} = \_ \_ \_ \_ x + \_ \_ \_ \_ \]
#### (e) Coefficient of Determination
**Question:** What is the coefficient of determination?
**Instruction:** Round your answer to four decimal places.
**Answer Box:** \_ \_ \_ \_
#### (f) Predicting Bear's Weight
**Scenario:**
Suppose a 166.0-cm bear is captured in the field. Use the least-squares regression line you found in part (d) to predict the weight of the bear.
**Instruction:** Round your answer to three decimal places.
**Prediction Answer Box:** \_ \_ \_ \_
**Residual Calculation:**
What is the residual of the 166.0-cm bear?
**Instruction:** Round your answer to three decimal places.
**Residual Answer Box:** \_ \_ \_ \_
**Avg Weight Comparison:**
Is this bear’s weight above or below average for a bear of this length?
**Answer Options:** [above] [below]
---
For sections d, e, and f, use appropriate statistical tools and methods to derive the necessary values. This will aid in the accurate analysis and understanding of the relationship between bear length and weight.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F68f51e6c-f342-4689-afb0-0310aec96525%2Fc8acc2f4-d30e-464e-acc9-98fd9e5e0867%2F6471ard_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Regression Analysis for Bear Weight and Length
#### (c) Determining Linear Relationship
**Question:** Does a linear relation exist between the weight of the bear and its length?
**Answer Options:** [yes] [no]
**Observation Inquiry:**
Describe the linear relationship that exists between weight and length by looking at the scatterplot:
**Answer Options:** [positive] [negative]
#### (d) Using a Calculator for Least-Squares Regression Line
**Task:**
Use your calculator to find the least-squares regression line treating total length as the explanatory variable and weight as the response variable.
**Instruction:** Round the slope and y-intercept to three decimal places.
**Equation Format:**
\[ \hat{y} = \_ \_ \_ \_ x + \_ \_ \_ \_ \]
#### (e) Coefficient of Determination
**Question:** What is the coefficient of determination?
**Instruction:** Round your answer to four decimal places.
**Answer Box:** \_ \_ \_ \_
#### (f) Predicting Bear's Weight
**Scenario:**
Suppose a 166.0-cm bear is captured in the field. Use the least-squares regression line you found in part (d) to predict the weight of the bear.
**Instruction:** Round your answer to three decimal places.
**Prediction Answer Box:** \_ \_ \_ \_
**Residual Calculation:**
What is the residual of the 166.0-cm bear?
**Instruction:** Round your answer to three decimal places.
**Residual Answer Box:** \_ \_ \_ \_
**Avg Weight Comparison:**
Is this bear’s weight above or below average for a bear of this length?
**Answer Options:** [above] [below]
---
For sections d, e, and f, use appropriate statistical tools and methods to derive the necessary values. This will aid in the accurate analysis and understanding of the relationship between bear length and weight.
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