How much do the ads explain?

Topic: Linear regression · more questions on this topic
‹ A line through the origin Where is the prediction interval narrowest? ›

For the advertising data, S_{xx}=0.444, S_{xy}=23.34 and S_{yy}=1600.9. Compute R^2, the residual standard error S on 8 degrees of freedom, and a 95% confidence interval for the slope, using t_{8;0.975}=2.306.

Worked answer (click to show)

R^2=\frac{S_{xy}^2}{S_{xx}S_{yy}}=\frac{23.34^2}{0.444\times1600.9}=0.766.

The line accounts for about 77% of the variation in sales. The residual sum of squares is S_{yy}(1-R^2)=374.0, so S^2=374.0/8=46.75 and S=6.84. The standard error of the slope is S/\sqrt{S_{xx}}=6.84/0.666=10.26, and the 95% interval for the slope is

52.6\pm2.306\times10.26=(28.9,\ 76.2).