Problems 17–22 use the results from Problems 27–32 in Section 4.1.
[DATA] American Black Bears (Refer to Problem 30, Section 4.1.) 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 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 data below represent the lengths of 12 American black bears.
b. Explain the significance of the slope and y-intercept in this situation, if applicable.

Verified step by step guidance

1. Step 1: Understand the context of the problem. The goal is to interpret the slope and y-intercept of a linear regression model that predicts the weight (y) of American black bears based on their total length (x).
2. Step 2: Recall the general form of a linear regression equation: \(y = \beta_0 + \beta_1 x\), where \(\beta_0\) is the y-intercept and \(\beta_1\) is the slope.
3. Step 3: Interpret the slope (\(\beta_1\)). The slope represents the average change in the bear's weight (kg) for each one centimeter increase in the bear's length. In other words, it tells us how much the weight is expected to increase or decrease as the length increases by 1 cm.
4. Step 4: Interpret the y-intercept (\(\beta_0\)). The y-intercept represents the predicted weight of a bear when the length is zero centimeters. While this may not be practically meaningful (since a bear cannot have zero length), it is a necessary part of the linear model and provides a baseline for the prediction.
5. Step 5: Summarize the significance: The slope gives insight into the relationship between length and weight, indicating how weight changes with length, while the y-intercept anchors the regression line on the weight axis when length is zero.

Key Concepts

Linear Regression Model

A linear regression model describes the relationship between a dependent variable (weight) and an independent variable (length) using a straight line. The model predicts the dependent variable based on the independent variable, typically expressed as y = mx + b, where m is the slope and b is the y-intercept.

Slope Interpretation

The slope in a regression model represents the average change in the dependent variable (weight) for each one-unit increase in the independent variable (length). In this context, it indicates how much the bear's weight is expected to increase or decrease with each additional centimeter of length.

Y-Intercept Interpretation

The y-intercept is the predicted value of the dependent variable when the independent variable is zero. In this case, it represents the estimated weight of a bear with zero length, which may not be meaningful biologically but is necessary for defining the regression line mathematically.