Wiki Page, Math 13 - Perez Sandbox : Module 9 - Linear Regression (3b of 4)

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2026-01-17T21:15:56-08:00

Interpreting the Slope and Intercept

The constants in the equation of a line give us important information about the relationship between the predicted y-values and x. In the next examples, we focus on how to interpret the meaning of the constants in the context of data.

EXAMPLE Body measurements

In the body measurement data collected from 21 female community college students, we found a strong positive correlation between forearm length and height. The least-squares regression line is

Predicted height = 39 + 2.7*Forearm length

The equation tells us that to predict the height of a woman, start with 39 inches and add 2.7 inches for every inch of forearm length.

The 39 is the predicted height when x = 0. Obviously, it does not make sense to predict the height of a woman with a 0-inch forearm length. This is an example of extrapolating outside the range of the data. Regardless, 39 inches is the starting value in the least-squares equation for predicting height based on forearm length.

The value of b is a positive 2.7. This means that a 1-inch increase in forearm length corresponds to a predicted 2.7-inch increase in height.

Another way to say this: there is an average increase of 2.7-inches in predicted height when we compare women with forearm length of x to women with forearm length of x + 1.

Now let's look at how these two constants relate to the graph of the line.

The slope is the rate of change between any two points on the line. For example, in the graph below, we arbitrarily chose two points (0 , 39) and (8, 60.6). For these two points, an 8-inch increase in forearm length ( 8 – 0 = 8) corresponds to a 21.6-inch increase in predicted height (60.6 – 39 = 21.6). The slope is the change in height divided by the change in forearm length: b = 21.6 / 8 = 2.7. If we chose two other points on the line, we will get the same rate of change.

In the graph, an arrow points to the initial value a = 39. This is the predicted y-value when x = 0.

EXAMPLE Highway sign visibility data

Recall that in a sample of 30 drivers, we see a strong negative linear relationship between the age of a driver (x) and the maximum distance (in feet) at which a driver can read a highway sign (y). The least-squares regression line is

Predicted y-value = (starting value) + (rate of change)*x

Predicted distance = 576 + (-3)*Age

Note: This equation is equivalent to what we wrote before

Predicted distance = 576 – 3*Age

because adding (-3*Age) is the same as subtracting 3*Age:

The equation tells us that to predict the maximum visibility distance for a driver, start with a distance of 576 feet and subtract 3 feet for every year of the driver’s age.

The 576 is the predicted value when x = 0. Obviously, it does not make sense to predict a maximum sign visibility distance for a driver who is 0 years old. This is another example of extrapolating outside the range of the data. We obviously would not use this prediction; however, this initial predicted value is an essential part of the least-squares equation for predicting distances based on age.

The value of b is −3. This means that a 1-year increase in age corresponds to a predicted 3-foot decrease in maximum distance at which a driver can read a sign. Another way to say this is that there is an average decrease of 3 feet in predicted sign visibility distance when we compare drivers of age x to drivers of age x + 1.

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