4-4 Linear Regression

Property

Linear regression is a statistical method for finding the line of best fit for a set of data points.
The method calculates the values for $a$ and $b$ in the equation $y = ax + b$ that minimize the sum of the squared residuals.

Examples

• A real estate agent uses linear regression to model the relationship between a house''s square footage and its selling price.
• A biologist might use linear regression to analyze the connection between daily temperature and the growth rate of a certain plant.

Explanation

Linear regression is the mathematical process used to determine the "line of best fit" for a scatter plot. This line is the one that comes closest to all of the data points simultaneously. By finding this line, we can model the linear relationship between two variables and make predictions. The method works by minimizing the overall error, specifically the sum of the squared vertical distances from each point to the line.

Property

Linear regression on a graphing calculator finds the line of best fit $y = ax + b$ and correlation coefficient $r$ by entering data into lists and using the LinReg function to calculate optimal parameters.

Examples

Property

The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It always satisfies $-1 \leq r \leq 1$.

Direction:

Property

A linear model such as $y = mx + b$ can be used to represent a trend line and describe the relationship between two variables. In this form, the model provides a simple way to interpret the overall pattern shown in a scatter plot.

• The slope ($m$) represents the rate of change. It is the predicted change in the dependent variable ($y$) for each one-unit increase in the independent variable ($x$).
• The y-intercept ($b$) represents the starting value. It is the predicted value of the dependent variable ($y$) when the independent variable ($x$) is zero.

Examples

• A linear model represents the relationship between hours studied ($x$) and test score ($y$) as $y = 5x + 60$. This equation represents a trend line for the data. The slope, $m=5$, means the score is predicted to increase by 5 points for each additional hour of studying. The y-intercept, $b=60$, is the predicted score for a student who studies for 0 hours.
• The value of a car ($V$) in dollars, $t$ years after it was purchased, is modeled by $V = -2000t + 25000$. The slope, $m=-2000$, means the car''s value decreases by 2000 dollars each year. The y-intercept, $b=25000$, represents the car''s initial purchase price of 25000 dollars.

Explanation

Interpreting a linear model means understanding what the slope and y-intercept mean in a real-world context. The slope describes how quickly the dependent variable is changing relative to the independent variable. The y-intercept gives the predicted starting point or initial condition of the dependent variable. Analyzing these two values provides a complete description of the linear relationship between the two variables.