Wiki Page, Math 13 - Perez Sandbox : Module 9 - Linear Regression (1a of 4)

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2026-01-17T21:15:55-08:00

More on lines as statistical models

In this section we will focus on the line as a statistical model.

A statistical model is a mathematical equation that summarizes the pattern in data, taking variability into account. As you now know, measuring variability is at the heart of statistics, so statistical models come with assessments of how well the model fits the data and also the average amount of error.

Why do statisticians create mathematical models to summarize data? The primary motivation is to make predictions. So we will start by making predictions using linear models.

Using a Line to Make Predictions

As you work through this example, focus on how we use the graph of the line and the equation of the line to make predictions. Don't worry about how we got the line's equation. We will discuss that soon.

EXAMPLE Using a line's graph and its equation to make predictions

Earlier, we examined the linear relationship between the age of a driver and the maximum distance at which the driver can read a highway sign.

Suppose we want to predict the maximum distance that a 60-year-old driver can read a highway sign. In the original data set, we do not have a 60-year-old driver and there is variability in the data, so how do we make a prediction?

Here again is the scatterplot of driver ages and maximum reading distances. (Note: Sign Legibility Distance = Max distance to read sign.) We marked 60 on the x-axis.

Of course, different 60-year-olds will have different maximum reading distances. We expect variability among individuals. But here our goal is to make a single prediction that follows the general pattern in the data.

Using the graph of the linear model to make a prediction:

Our first step is to model the pattern in the data with a line. In the scatterplot, you see a red line that follows the pattern in the data. (Later we will learn how to find the best line for summarizing the pattern.)

To use this line to make a prediction, we find the point on the line with an x-value of 60. Simply trace from 60 straight up to the line. We use the y-value of this point as the predicted maximum reading distance for a 60-year-old. Trace from this point straight across to the y-axis.

We predict that 60-year-old drivers can see the sign from a maximum distance of just under 400 feet.

Using the equation of the linear model to make a prediction:

We can also use the equation for the line to make a prediction. The equation for the red line is

Predicted distance = 576 - 3*Age (again, at this point, don't worry about where the equation comes from)

To predict the maximum distance for a 60-year-old, substitute Age = 60 into the equation. This prediction is more accurate than the estimate from the graph.

Predicted distance = 576 - 3*(60) = 396 feet

Soon we will learn how to use StatCrunch to find the equation of the line that best fits the data. Later, we develop a method to measure the accuracy of the predictions from this “best” line. For now, just focus on how to use the line to make predictions.

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