Use lines of best fit to make predictions

Detailed Explanation: Use lines of best fit to make predictions

A line of best fit describes the overall pattern in a scatterplot. Its equation can be used to predict a response value from an explanatory value.

Suppose a scatterplot compares:

• $x =$ hours a student studies
• $y =$ test score, in points

The data include students who studied between $1$ and $8$ hours. A line of best fit is

Here, $\hat{y}$ means the predicted test score.

Step 1: Identify the value you want to use

Predict the test score for a student who studies $5$ hours.

So, substitute $x=5$ into the equation:

Step 2: Calculate the prediction

Step 3: State the prediction with units

A student who studies for $5$ hours is predicted to score 84 points.

The slope, $6$, means that the predicted score increases by about 6 points for each additional hour studied. The intercept, $54$, is the predicted score for $0$ hours of studying, although that value may not be meaningful unless $0$ hours is represented by the data.

Because $5$ hours is within the observed range of $1$ to $8$ hours, this is an interpolation, which is generally more reliable than predicting outside the data range. The prediction is an estimate based on the overall trend; it does not mean every student who studies for $5$ hours will score exactly $84$ points. A line of best fit summarizes the variation in the data and does not prove that studying causes higher scores.