Lecture 8 - Inferences About the Regression Slope - Part 2

Confidence Interval for b₁
The second question that we ask when evaluating the regression is: What is the confidence interval for b₁?

Like any confidence interval, this one will take the form:
b₁ ± t_{α/2, n-2} S_b1

In the last blog post, we found that S_b1 is:
S_b1 = S_XY/SQRT(SSX)

Knowing S_b1, we can look up t in the t-table and construct the confidence interval relatively easily.

Note that for the confidence interval for b₁ we use n-2 in looking up the t-score.

Using Minitab to evaluate the regression
We won't be expected to calculate S_XY and S_b1 by hand for the final (or so we were told). But we will likely be asked to create a confidence interval for b₁ given a snippet of Minitab output. So it's worthwhile to take a look at it:

We used the site.mtw dataset and ran the standard regression analysis and got this:

Predictor      Coef  SE Coef      T      P
Constant     0.9645   0.5262   1.83  0.092
Square Feet  1.6699   0.1569  10.64  0.000
S = 0.966380   R-Sq = 90.4%   R-Sq(adj) = 89.6%

The S_b1 value is calculated for us, but it's not obvious where it is. It's the SE Coef term that I've highlighted in red. b₁ itself is the Coef term, in blue. With those two numbers and a t-table, you can construct a confidence interval for b₁. Just remember to use n-2 in the t-table.

With these two values you can also determine the t statistic for hypothesis testing β₁=0 from the previous blog post by dividing b₁/S_b1. But the truth is, you don't have to do that! The t-value for b₁ is right there in the Minitab output also. I've highlighted it in green. The number in the p column (highlighted purple) is the p-value for b₁. So if that number is less than α/2, then you can reject the hypothesis that β₀ is 0.