Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

546 Inferences About Normal Linear Models

ci; -0.0054678 -0.0032842
So with approximate confidence 95%, we estimate the drop in winning time to
between 0.0032 to 0.0055 minutes per year.

Based on the fit, the predicted winning time for the men’s 1500 meters in the
2020 Olympics is

yˆ=12. 325411 − 0 .004376(2020) = 3. 486. (9.6.12)

Exercise 9.6.8 provides an estimate (predictive interval) of error for this prediction.

1900 1920 1940 1960 1980 2000 2020

3.6

3.8

4.0

Ye a r

Winning time

3.5 3.6 3.7 3.8 3.9 4.0

−0.1

0.1 0.2 0.3

Fitted values

Residuals

Figure 9.6.2:The top panel is the scatterplot of winning times in the men’s 1500

meters versus the year of the Olympics. The least squares fit is overlaid. The

bottom panel is the residual plot of the fit.

9.6.2 ∗GeometryoftheLeastSquaresFit

In the modern literature, linear models are usually expressed in terms of matrices

and vectors, which we briefly introduce in this example. Furthermore, this allows

us to discuss the simple geometry behind the least squares fit. Consider then Model

(9.6.1). Write the vectorsY=(Y 1 ,...,Yn)′,e=(e 1 ,...,en)′,andxc=(x 1 −

x,...,xn−x)′.Let 1 denote then×1 vector whose components are all 1. Then