(Same five points.) Using the line y = 0.9x + 1.3, what is the… Step by Step

Step by step

The question

Question: find the residual for the line of best fit y = 0.9x + 1.3 and the point (4, 4).

The point is the real data.

y = 0.9x + 1.3; (4, 4)

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.

residual = actual − predicted

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 = 4.9.

predicted = 0.9(4) + 1.3 = 3.6 + 1.3 = 4.9

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.

Careful: not 4.9 − 4 = 0.9 ✗

The actual y is 4.

4 minus 4.9 is −0.9.

residual = 4 − 4.9 = −0.9

So the residual is −0.9.

It is negative: the point is 0.9 below the line.

3 Check

Let's check with a picture.

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: 4.

So the answer is right!

4.9 − 0.9 = 4

4 Pick the choice

Our answer is residual = −0.9.

That is choice D.

D) −0.9

5 Why not the others

A is 0.9.

Wrong: the residual is actual minus predicted: 4 − 4.9.

That is the Careful mistake from before.

A) 0.9 ✗ the residual is actual minus predicted: 4 − 4.9

B is 0.

Wrong: (4, 4) is not on the line.

B) 0 ✗ (4, 4) is not on the line

C is 4.9.

Wrong: that is the predicted value.

C) 4.9 ✗ that is the predicted value

Let's look back.

First: predict.

Then: actual minus predicted.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer