From Algorithms to Z-Scores - matloff - University of California, Davis

15.19. CASE STUDIES 317
8. Consider a random pair (X, Y ) for which the linear model E(Y |X) = β0 + β1X holds, and think
about predicting Y , first without X and then with X, minimizing mean squared prediction error
(MSPE) in each case. From Section ??, we know that without X, the best predictor is EY , while
with X it is E(Y |X), which under our assumption here is β0 + β1X. Show that the reduction in
MSPE accrued by using X, i.e.
is equal to ρ 2 (X, Y ).
E (Y − EY ) 2 − E {Y − E(Y |X)} 2
E [(Y − EY ) 2 ]
(15.59)
9. In an analysis published on the Web (Sparks et al, Disease Progress over Time, The Plant Health
Instructor, 2008, the following R output is presented:
> severity.lm summary(severity.lm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.66233 1.10082 2.418 0.04195 *
temperature 0.24168 0.06346 3.808 0.00518 **
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1
Fill in the blanks:
(a) The model here is
mean = β0 + β1
(b) The two null hypotheses being tested here are H0 : and H0 : .
10. In the notation of this chapter, give matrix and/or vector expressions for each of the following
in the linear regression model:
(a) s 2 , our estimator of σ 2
(b) the standard error of the estimated value of the regression function mY ;X(t) at t = c, where
c = (c0, c1, ..., cr)