Lesson 4: Interpreting Linear Models and Predictions

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.

Property

We can use a line of fit (trend line) drawn through a scatter plot to make predictions about data values.

We can estimate values between known data points or predict values beyond the range of our data using the trend line equation.

Property

We can model a linear pattern in data using a line of best fit (also called a regression line).

We can use interpolation to estimate values between known data points or extrapolation to predict values beyond the data range.

Predictions made through extrapolation become less reliable the further we move from the original data.