Step by step
The question
Question: find the residual for the line of best fit y = 3x + 2 and the point (4, 11).
The point is the real data.
A residual tells how far a point is from the line.
Residual = actual minus predicted.
Actual is the real y.
Predicted is the y the line gives.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Predict
We put x = 4 into the line.
It predicts y = 14.
2 Actual minus predicted
Careful!
Not predicted minus actual.
That flips the sign.
Actual comes first, so a point above the line gets a plus.
The actual y is 11.
11 minus 14 is −3.
So the residual is −3.
It is negative: the point is 3 below the line.
3 Check
Let's check with a picture.
Each square up is 2.
The line of best fit.
The point.
The residual is the distance up or down from the line to the point.
And predicted plus residual gives the actual y: 11.
So the answer is right!
4 Pick the choice
Our answer is residual = −3.
That is choice A.
5 Why not the others
B is 3.
Wrong: 14 − 11: predicted minus actual; a residual is actual minus predicted.
That is the Careful mistake from before.
C is 11.
Wrong: that is the actual y, not the residual.
D is 14.
Wrong: that is the predicted y, not the residual.
Let's look back.
First: predict.
Then: actual minus predicted.
Then: check.
Then: pick the choice.
Last: why not the others.