Topic 5.3 Notes – Linear Regression Models

What a Linear Regression Model Is

You use a linear regression model when a scatterplot of two quantitative variables looks approximately straight-line. The model uses an explanatory variable $x$ to predict a response variable $y$.

$$\hat y = a + bx$$

A few pieces here matter a lot:

• $\hat y$ means the predicted response, not the actual observed response $y$.
• Simple linear regression means one explanatory variable and a straight-line model.
• The regression line is made of predicted points $(x,\hat y)$.
• Your actual data points are observed points $(x,y)$, and they do not all have to lie on the line.

This kind of graph usually shows all of that at once, including the observed point, the predicted point on the line, and the residual between them.

Regression line with predicted value and residual

Two common AP Stats reminders belong here:

• A model that predicts $y$ from $x$ does not automatically prove that $x$ causes $y$.
• You also cannot just reverse the equation to predict $x$ from $y$. Variable roles matter.

Reading the Equation

The equation tells you two things about the line: its slope and intercept.

Slope

The slope $b$ is how much the predicted response changes when $x$ increases by 1 unit.

• If $b>0$, predicted $y$ goes up as $x$ goes up.
• If $b<0$, predicted $y$ goes down as $x$ goes up.
• Units are response units per explanatory-variable unit.

Example. If $\hat y = 12 + 3.5x$, then for each 1-unit increase in $x$, the predicted $y$ increases by 3.5 units.

Intercept

The intercept $a$ is the predicted response when $x=0$.

• It always has a clear math meaning.
• It may have no useful real-world meaning.

If $x=0$ is impossible, unrealistic, or outside the data range, say that. On the exam, don’t force an interpretation that makes no sense.

Calculating and Reporting a Predicted Response

This is the calculation you’ll do most often.

Suppose the model is

$$\widehat{\text{height}}=8.7+0.63(\text{diameter})$$

and diameter is in centimeters.

For a tree with diameter $30$ cm,

$$\widehat{\text{height}}=8.7+0.63(30)=8.7+18.9=27.6$$

So the prediction is 27.6 meters.

Say it like this in context:

• “According to the regression model, the predicted height of a mature red maple tree with diameter 30 cm is 27.6 meters.”

That wording matters. Don’t say “the height is 27.6 meters.” The model gives a prediction, not an observed fact.

Graphically, this means you go to $x=30$, move up to the regression line, and read the corresponding $\hat y$.

Interpolation and Extrapolation

A prediction is only as trustworthy as where it falls relative to the observed $x$-values.

Interpolation

Interpolation means predicting for an $x$-value inside the observed range.

• If the original diameters were 12 to 46 cm, then $x=30$ is interpolation.
• This is usually more defensible because the model is being used where data exist.

Extrapolation

Extrapolation means predicting for an $x$-value outside the observed range.

• If the same data used 12 to 46 cm, then $x=70$ is extrapolation.
• It is less reliable because it assumes the linear pattern continues beyond the data.
• The farther outside the range, the less reliable it usually is.

A predicted value can look reasonable and still be weakly supported. On the AP exam, say that clearly.

When the Model Is Reasonable and Where Students Go Wrong

A linear model makes sense only if the scatterplot looks roughly linear. A strong correlation alone is not enough if the pattern is curved.

Also check whether the prediction makes sense in context:

• impossible values like negative height
• values beyond a fixed scale
• predictions for individuals very different from those in the original data

Big mistakes students make:

• confusing $y$ with $\hat y$
• leaving out units or context
• treating a prediction as an observed value
• ignoring interpolation vs extrapolation
• reversing explanatory and response variables
• using causal language from regression alone