Predictions From Computer Output

From Computer Output to a Prediction

In “Making Predictions Using the Equation,” you practiced substituting a value into a regression equation. Computer output often gives you the coefficients without displaying the equation in the form you have practiced. The essential task is to find the estimate for the intercept and the estimate for the predictor, put them in the correct places, and evaluate the equation at the requested predictor value.

A coefficient table can contain several columns and rows. The row labeled “(Intercept)” gives the intercept \(a\). The row for the predictor gives the slope \(b\), and the “Estimate” column gives the coefficient values to use. Other columns—such as standard errors, test statistics, or p-values—are not coefficients to substitute when making a prediction.

The names shown in output can differ from the variable names you use in an explanation. For example, software might display “distance_km” as a row label. The context tells you whether that variable is the predictor or the response. As covered in “Reading Regression Computer Output” and “Writing the Equation With Variable Names,” check the row label and the coefficient column before writing an equation.

Once you have the equation, use the predictor value specified in the question. Substitute that value for \(x\), multiply the slope by it, and then add the intercept. The result is a predicted response, \(\hat{y}\), not necessarily the response that any particular individual will actually have.

Worked Example: Predicting a Commute Time

Suppose a fictional transportation class collects data on the distance of a commute and the time it takes. Let \(x\) be commute distance in kilometers and \(y\) be commute time in minutes. The following invented computer output reports the coefficient estimates from a simple linear regression of time on distance.

A useful check is to confirm that the coefficient came from the right row and that the requested value was substituted for the predictor. Here, the input is distance in kilometers, not time in minutes. The calculation also has the form intercept plus slope times predictor, matching the equation \(\hat{y}=a+bx\).

Read the Row and Column Before Calculating

The coefficient table may also contain columns such as “Std. Error,” “t value,” or “Pr(>|t|).” These columns have other purposes; they are not the estimated intercept or slope. For a prediction, locate the “Estimate” column and then match each estimate to its row. Do not choose a number merely because it is next to the predictor name.

The response variable may be identified in a heading, a model formula, or the surrounding description rather than in a coefficient row. A simple regression coefficient table normally lists the intercept and predictor terms; it does not list the response as a coefficient. Before calculating, use the situation or output heading to establish which variable is predicted and which value is being supplied as the predictor.

Worked Example: Predicting Basil Growth

A fictional greenhouse class models the dry mass of basil plants using the number of hours of light per day. Let \(x\) be daily light hours and \(y\) be dry mass in grams. The invented output is:

This example highlights a common output-reading trap. The 0.5 in the predictor row is a standard error, not the slope. Substituting it would produce a different equation and an incorrect prediction. When the table has several numerical columns, name the column you are reading before you write down a coefficient.

Worked Example: Predicting a Swim Time

In another invented example, a coach records weekly training minutes and 100-meter swim times for a group of swimmers. Let \(x\) be weekly training minutes and \(y\) be swim time in seconds. The output gives these coefficient estimates:

A negative estimate is still used in the same equation, \(\hat{y}=a+bx\). You can write it with a plus sign followed by a negative slope, or with a minus sign and the slope’s positive magnitude. Either way, retain the sign in the output. This agrees with the direction of the fitted line described in “Positive and Negative Slopes in Context.”

A Reliable Sequence for Computer-Output Predictions

A short, repeatable sequence helps prevent mistakes when the output is unfamiliar. Identify the response and predictor from the context. Find the “Estimate” column, then take the intercept from its row and the slope from the predictor’s row. Write the equation before substituting, so you can see which coefficient multiplies the requested value. Finally, evaluate the expression and describe the result as a predicted response in context.

Common Mistakes and AP Exam Tips

• Using a standard error instead of an estimate. In a table with multiple columns, the estimate column supplies the coefficients. A standard error describes uncertainty about an estimate; it does not replace that coefficient in the prediction equation.
• Taking both coefficients from the wrong row. The intercept comes from the “(Intercept)” row. The slope comes from the row for the predictor. Do not use the predictor’s estimate as the intercept.
• Reversing the predictor and response. A regression line predicts the response from the predictor used to fit it. Check the context and model description rather than assuming the row label tells you which variable is predicted.
• Dropping a negative sign. Copy the sign of the predictor’s estimate exactly. A changed sign changes the direction and numerical value of a prediction.
• Reporting the input as the prediction. The supplied value is \(x\), while the result of evaluating the line is \(\hat{y}\). State the predicted response, not the predictor value again.
• Calling the prediction an actual result or a cause. As discussed in “Prediction Versus Observed Values” and “Regression Does Not Mean Causation in Slope Statements,” a line gives a prediction; individual observations can differ, and an association alone does not establish causation.

For a clear AP-style response, show the coefficient mapping, the substitution, and a sentence interpreting the calculated prediction. For example: “Using the intercept estimate 6.35 and distance coefficient 2.18, the predicted commute time for a distance of 8.5 kilometers is \(6.35+2.18(8.5)=24.88\) minutes.” This makes the source of each number and the meaning of the result clear.