[DATA] 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.

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?

Verified step by step guidance

1. Step 1: Organize the data by listing the lengths (X) and weights (Y) of the 12 bears as two separate lists or columns.
2. Step 2: Calculate the mean (average) of the lengths, denoted as \(\overline{X}\), and the mean of the weights, denoted as \(\overline{Y}\). Use the formulas: \(\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i\) \(\overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i\) where \(n=12\) is the number of bears.
3. Step 3: Compute the deviations from the mean for each data point: \((X_i - \overline{X})\) and \((Y_i - \overline{Y})\). Then calculate the products of these deviations for each pair: \((X_i - \overline{X})(Y_i - \overline{Y})\).
4. Step 4: Calculate the sums needed for the correlation coefficient formula: \(S_{XY} = \sum (X_i - \overline{X})(Y_i - \overline{Y})\), \(S_{XX} = \sum (X_i - \overline{X})^2\), \(S_{YY} = \sum (Y_i - \overline{Y})^2\).
5. Step 5: Use the formula for the Pearson correlation coefficient \(r\): \(r = \frac{S_{XY}}{\sqrt{S_{XX} \times S_{YY}}}\) This value will indicate the strength and direction of the linear relationship between bear length and weight.

Key Concepts

Linear Correlation Coefficient

The linear correlation coefficient, often denoted as r, measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 indicate little to no linear correlation.

Scatterplot and Visual Analysis

A scatterplot is a graphical representation of paired data points that helps visualize the relationship between two variables. By plotting bear lengths against weights, one can observe patterns or trends that suggest whether a linear relationship exists before calculating the correlation coefficient.

Interpreting Correlation in Context

Interpreting the correlation coefficient requires understanding the context and data variability. Even a strong correlation does not imply causation, and outliers or measurement errors can affect the coefficient. In this case, it helps determine if length is a good predictor of weight for American black bears.