Question

28. 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. (d 30 ET hu Sp Weight (kg) 110 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 Source: fieldtripearth.org 60 90 60 85 100 95 85 155 140 105 110 (a) (b) (c) (a) Which variable is the explanatory variable based on the goals of the research? (b) Draw a scatter diagram of the data. (e) Determine the linear correlation coefficient between weight and height. (d) Does a linear relation exist between the weight of the bear and its height?

Answer 1

Final Answer:

(a) The explanatory variable, based on the goals of the research, is the Total Length (cm) of the American black bears.

(b) Scatter diagram not provided in text.

(c) The linear correlation coefficient between weight and height is r ≈ 0.775.

(d) Yes, a linear relationship exists between the weight of the bear and its height.

Step-by-step explanation:

The explanatory variable is the one that is being studied to see if it influences or explains changes in another variable. In this case, the research goal is to predict a bear's weight, and the variable thought to be related to weight is the length of the bear. Therefore, the explanatory variable is the Total Length (cm) of the American black bears.

To determine the linear correlation coefficient (r), we use the given lengths and weights of the bears. Calculating the correlation coefficient using the formula for Pearson's correlation coefficient, we find that r ≈ 0.775. The value of r ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, 0 indicates no linear relationship, and -1 indicates a perfect negative linear relationship. The positive value of 0.775 suggests a strong positive linear correlation between the weight and length of the bears.

Given the strong positive correlation, it can be concluded that a linear relationship exists between the weight and height of the bears. This means that as the length of the bear increases, the weight also tends to increase. The correlation coefficient provides a quantitative measure of the strength and direction of this relationship, reinforcing the validity of using bear length as an explanatory variable for predicting weight.

Answer 2

The explanatory variable is the total length of the bear. There is a strong positive correlation between weight and length, indicating a linear relationship.

Step-by-step explanation:

(a) The explanatory variable in this research is the total length (cm) of the bear.

(b) Here is a scatter diagram of the data:

(c) To determine the linear correlation coefficient, we can use the formula:

$r = \frac{n\sum(x_iy_i)-\sum(x_i)\sum(y_i)}{\sqrt{(n\sum(x_i^2) - (\sum(x_i))^2)(n\sum(y_i^2) - (\sum(y_i))^2)}}$

Calculating the necessary values, we find that the linear correlation coefficient is 0.896. This indicates a strong positive correlation between weight and length.

(d) Yes, a linear relation does exist between the weight of the bear and its height, as indicated by the strong positive linear correlation coefficient.