Confidence Interval for b_{1}

The second question that we ask when evaluating the regression is: What is the confidence interval for b_{1}?

Like any confidence interval, this one will take the form:

b_{1} ± 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 b_{1} 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 b_{1} 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 PThe S

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%

_{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. b

_{1}itself is the Coef term, in blue. With those two numbers and a t-table, you can construct a confidence interval for b

_{1}. Just remember to use n-2 in the t-table.

With these two values you can also determine the t statistic for hypothesis testing β

_{1}=0 from the previous blog post by dividing b

_{1}/S

_{b1}. But the truth is, you don't have to do that! The t-value for b

_{1}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 b

_{1}. So if that number is less than α/2, then you can reject the hypothesis that β

_{0}is 0.

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