Problems 17–22 use the results from Problems 27–32 in Section 4.1.
[DATA] Height versus Head Circumference (Refer to Problem 29, Section 4.1.) A pediatrician wants to determine the relation that exists between a child’s height, x, and head circumference, y. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the following data.
b. Explain the meaning of the slope and y-intercept in this context, if applicable.

Verified step by step guidance

1. Step 1: Understand the variables involved. Here, the independent variable \(x\) is the child's height in inches, and the dependent variable \(y\) is the head circumference in inches. The pediatrician wants to find the relationship between these two variables.
2. Step 2: Recall the general form of the linear regression equation: \(y = \beta_0 + \beta_1 x\), where \(\beta_0\) is the y-intercept and \(\beta_1\) is the slope of the line.
3. Step 3: Interpret the slope \(\beta_1\). The slope represents the average change in head circumference (inches) for each one-inch increase in height. In other words, it tells us how much the head circumference is expected to increase or decrease as the child's height increases by one inch.
4. Step 4: Interpret the y-intercept \(\beta_0\). The y-intercept represents the estimated head circumference when the height is zero inches. While this may not be meaningful in a practical sense (since a height of zero is not realistic for a child), it is a necessary component of the linear equation to model the relationship.
5. Step 5: Summarize the interpretations in the context of the problem: The slope indicates how head circumference changes with height, and the y-intercept is the theoretical head circumference when height is zero, which helps define the linear relationship between height and head circumference.

Key Concepts

Linear Regression and the Regression Line

Linear regression models the relationship between two variables by fitting a line that best describes how the dependent variable (head circumference) changes with the independent variable (height). The regression line equation is typically written as y = mx + b, where m is the slope and b is the y-intercept.

Slope Interpretation

The slope in this context represents the average change in head circumference (y) for each one-inch increase in height (x). A positive slope indicates that as height increases, head circumference tends to increase, while a negative slope would indicate the opposite.

Y-Intercept Interpretation

The y-intercept is the predicted head circumference when the height is zero inches. Although a height of zero is not realistic here, the y-intercept provides a baseline value for the regression line and helps position the line correctly on the graph.