[DATA] American Black Bears In 1969, Dr. Michael R. Pelton of the University of Tennessee initiated a long-term study of the American black bear (Ursus americanus) population in 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 that is 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. Use the results from Problem 20 in Section 4.2 to answer the following questions:

d. What is the mean weight of American black bears of length 146.0 cm?

Verified step by step guidance

1. Identify that the problem requires predicting the mean weight of American black bears given a specific length (146.0 cm) using a linear regression model.
2. Recall that the linear regression equation has the form: \(\hat{y} = b_0 + b_1 x\), where \(\hat{y}\) is the predicted weight, \(b_0\) is the y-intercept, \(b_1\) is the slope, and \(x\) is the length of the bear.
3. Use the results from Problem 20 in Section 4.2 (which should provide the slope \(b_1\) and intercept \(b_0\) of the regression line) to substitute the values into the regression equation.
4. Substitute \(x = 146.0\) cm into the regression equation to calculate the predicted mean weight \(\hat{y}\) for a bear of that length.
5. Interpret the predicted value \(\hat{y}\) as the estimated mean weight of American black bears with a length of 146.0 cm.

Key Concepts

Simple Linear Regression

Simple linear regression models the relationship between two variables by fitting a linear equation to observed data. One variable is independent (length), and the other is dependent (weight). The model predicts the mean value of the dependent variable for a given independent variable.

Regression Equation and Prediction

The regression equation is typically written as ŷ = b0 + b1x, where ŷ is the predicted mean response, b0 is the intercept, and b1 is the slope. To find the mean weight for a specific length, substitute the length value into the equation to calculate the predicted mean weight.

Interpretation of Predicted Mean Response

The predicted mean response from a regression model represents the average value of the dependent variable for all observations with the given independent variable value. It is not the weight of a single bear but the expected average weight for bears of that length.