The regression output, as usual, gives us everything that we need.

All we need to do is to look on the row for the

mother's high school status and take a look at the p-value for that.

Since this is a small p-value, we can determine

that whether or not mom went to high school

is a significant predictor of the cog, cognitive test

scores of children, given all other variables in the model.

Even though we don't need to do any calculations by hand, it's

always a good idea to try to understand how the calculations that

are included in the regression output are actually getting done by the

software that you're using, so that you can understand what they mean.

So, let's go through the mechanics of testing for

the slope within the framework of a multiple linear regression.

As usual with a regression, we use a t-statistic in inference.

The t-statistic looks like point estimate minus

the null value, divided by the standard error.

Our point estimate is simply our slope estimate, and the standard error is the

standard error of this estimate that we

can grab easily from the regression output.

So, the t-statistic for the slope is simply b1

minus 0, divided by the standard error of b1.

How this is different from the single predictor regression case that we

covered in the previous unit is how we calculate the degrees of freedom.

The degrees of freedom here is n minus k minus 1,

where k is the number of predictors included in the model.

Let's take a moment to focus on this new measure of degrees of

freedom and actually highlight that it is not a new measure at all.

We just said that for a multiple linear

regression, the degrees of freedom is n minus

k minus 1, where n is the sample size and k is the number of predictors.

And earlier, in the previous unit, we had said that for a regression with

a single predictor, the degrees of freedom can be calculated as n minus 2.

If you think about it, in a single

predictor regression, the number of predictors is 1.

So, if we were to calculate the degrees of freedom as n minus k minus 1 for that case

as well, we would simply get n minus 1 minus 1, which comes out to be n minus 2.

And remember, the additional minus 1 is because with,

along with every single predictor for which we calculate

a slope estimate, we also calculate an intercept, and

that's where we're losing that one additional degree of freedom.

So, while we've introduced these two formulas slightly

differently, note that they mean exactly the same thing.

You start with your sample size, that is the

total degrees of freedom you have to play with.

And then, you lose the number of degrees of

freedom that goes for however many predictors you have.

And then, you lose one more for the intercept.