Tuesday, March 4, 2008

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 ± tα/2, n-2 Sb1

In the last blog post, we found that Sb1 is:

Knowing Sb1, 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 SXY and Sb1 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 Sb1 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/Sb1. 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.

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