And what the multiple regression model is doing is no longer fitting just a line

through here, doesn't make sense if we're in higher dimensions.

It puts the best fitting plane through the data.

So the analog of the line in the simple regression,

is a plane in the multiple regression.

It still fit through the method of least squares.

This is the plane that best fits the data in the sense that it minimizes

the sum of the squares of the vertical distance from the point to the plane now.

So there's our least squares plane.

Now, I said that we can use this plane for

doing forecasting, but we still have our one number summaries around.

Those one number summaries are the regression where R2 and RMSE,

if we calculate R2 for this multiple regression, it comes out to be 84%.

In the simple regression model it was 76%, so our R2 has increased.

We've explained more variation.

By adding in this additional variable, and

we've also reduced the value of root- mean-square error.

Root-mean-square error is now only 3.45, so if we wanted to create

an approximate 95% prediction interval, for the fuel economy of a vehicle.

As long as it's weight and horsepower,

we're in the range described within this data set.

So we're interpolating or extrapolating outside the range.

So as long as we are interpolating we can use our 95% prediction interval, rule of

thumb, again, and say up to the plane plus or minus twice the root-mean-squared area.

So this regression model will give us 95% prediction intervals of a width

of about plus or minus seven, twice 3.45.

So that's the precision with which we can predict based on the current model.

So, through the prediction interval we get a sense of the uncertainty

of our forecast.

So root-mean-squared error is really a critical summary of these regression

models.

So just summarizing this slide,

the multiple regression allows us to estimate this least squares plane.

Once we've got this this multiple regression equation, we can use it for

prediction.

So long as we have a root-mean-squared error estimate, which we do and

we're working within the range of the data we can put the two things together.

The forecast and the root-mean-squared error to come up with a 95% prediction.