Problems 17–22 use the results from Problems 27–32 in Section 4.1.
[DATA] Credit Scores (Refer to Problem 28, Section 4.1.) An economist wants to determine the relation between one’s FICO score, x, and the interest rate of a 36-month auto loan, y. The given data represent the interest rate (in percent) a bank would offer on a 36-month auto loan for various FICO scores.
b. Describe what the slope and y-intercept represent in this context. Note: Credit scores range from 300 to 850.

Verified step by step guidance

1. Step 1: Understand the variables involved. Here, the independent variable \(x\) is the FICO credit score, and the dependent variable \(y\) is the interest rate (in percent) offered on a 36-month auto loan.
2. Step 2: Recall the general form of a linear equation: \(y = m x + b\), where \(m\) is the slope and \(b\) is the y-intercept.
3. Step 3: Interpret the slope \(m\) in this context. The slope represents the rate of change of the interest rate with respect to the credit score. In other words, it tells us how much the interest rate changes for each one-point increase in the FICO score.
4. Step 4: Interpret the y-intercept \(b\). The y-intercept is the predicted interest rate when the credit score \(x\) is zero. Although a credit score of zero is not realistic, the y-intercept provides a starting point for the linear model.
5. Step 5: Summarize the interpretations: The slope indicates how the interest rate decreases or increases as the credit score changes, and the y-intercept represents the theoretical interest rate when the credit score is zero.

Key Concepts

Linear Regression and the Equation of a Line

Linear regression models the relationship between two variables by fitting a line to the data points. The equation y = mx + b represents this line, where m is the slope and b is the y-intercept. This equation helps predict the dependent variable (interest rate) based on the independent variable (credit score).

Slope Interpretation

The slope (m) indicates the rate of change of the dependent variable with respect to the independent variable. In this context, it shows how much the interest rate changes for each one-point increase in the credit score. A negative slope would mean higher credit scores correspond to lower interest rates.

Y-Intercept Interpretation

The y-intercept (b) is the predicted value of the dependent variable when the independent variable is zero. Here, it represents the estimated interest rate when the credit score is zero, which may not be meaningful practically but is essential for defining the regression line.