Chapter 10.4, Problem 21BB

Confidence Intervals for β₀ and β₁ Confidence intervals for the y-intercept β₀ and slope β₁, for a regression line (y = β₀ + β₁x) can be found by evaluating the limits in the intervals below.

$b_0\text{ - }E\text{ < }{\beta }_{\text{0}}<b_0+E$

where

$\begin{array}{l}E=t_{\alpha /2^{s_l}}\sqrt{\frac{1}{n}+\frac{x^{-2}}{\sum x^2-\frac{{(\sum x)}^2}{n}}}\\ \\ b_1\text{ - }E\text{ < }{\beta }_{\text{1}}<b_1+E\end{array}$

where

$E=t_{\alpha /2}\cdot \frac{s_l}{\sqrt{\sum x^2-\frac{{(\sum x)}^2}{n}}}$

The y-intercept b₀ and the slope b₁ are found from the sample data and t_α/2 is found from Table A-3 by using n − 2 degrees of freedom. Using the 40 pairs of shoe print lengths (x) and heights (y) from Data Set 2 in Appendix B, find the 95% confidence interval estimates of β₀ and β₁.