The degrees of freedom we had said was 25, and what we want to do first is

to find the critical t-score associated with this

degrees of freedom, and the given confidence level.

To find the critical t-score, let's draw our curve and mark the middle

95% and note that each tail is now left with 2.5%, or 0.025.

So, the cutoff value, or the critical t-score, can be calculated using R

and the qt function, as qt of 0.025 with degrees of freedom of 25.

This is going to yield a negative value, negative roughly 2.06.

But note that for confidence intervals the critical

value that we use always needs to be positive.

So the t-star is going to be simply 2.06.

We know our slope estimate, 0.9014 plus or minus 2.06 is the

critical value times the standard error that also comes from the regression

output, gives us 0.7 to 1.1 as our confidence interval.

And what do these numbers mean?

How do we interpret this confidence interval?

Basically what this means is that we are 95% confident that for each additional

point on the biological twins' IQs, the foster twins' IQs are expected on average

to be higher by 0.7 to 1.1 points.

So, to recap, we said that we could do a hypothesis test for

the slope, doing a t-statistic, where our point estimate is b1, our null, and

then we subtract from that a null value and divide by the standard error,

and the degrees of freedom associated with this test statistic is n minus 2.

To construct a confidence interval for the slope,

we simply take our slope estimate b1, and

add and subtract the margin of error, that's

composed of a critical t-score and a standard error.