chapter 1

Chapter 13: Nonlinear and Multiple Regression2d. σ = Var(Yˆ) + Var( Y −Yˆ) suggests that we can estimate Var( ˆ )iiiY iY i− by2 2s − s ˆ yand then take the square root to obtain the estimated standard deviation of each2residual. This gives .040 − (.955) =2 1.059, (and similarly for all points) 10.59,1.236, 1.196, 1.196, 1.196, and .233 as the estimated std dev’s of the residuals. The− .327standardized residuals are then computed as = −.31 , (and similarly) 1.10, -1.28,1.059-.76, .16, 1.49, and –1.28, none of which are unusually large. (Note: Minitab regressionoutput can produce these values.) The resulting residual plot is virtually identical to they − yˆ−.327plot of b. = = −.229≠ −.31, so standardizing using just s would nots1.426yield the correct standardized residuals.e. Y ) Var(Y )(ffy f yˆ = 2.638 =ˆVar + is estimated by 2.040 (.777) 2 = 2. 638+ , sos 1.624 . With y ˆ = 31. 81 and t 2. 132+ f( 2.132)( 1.624) ( 28.35,35.27)31 .81=± .. 05,4=, the desired P.I. is32.. 3463 1.2933 x − x + 2.3964 x − x2 − 2. 3968 x − x− .a. ( ) ( ) ( ) 3b. From a, the coefficient of x 3 is -2.3968, so βˆ3 = −2. 3968. There sill be a contributionto x 2 both from 2.3964( x − 4.3456) 2and from 2.3968( − 4.3456) 3− x .Expanding these and adding yields 33.6430 as the coefficient of x 2 , so βˆ2= 33. 6430 .c. x = 4 .5 ⇒ x′= x − x = . 1544 ; substituting into a yields ˆ = . 1949− 2.3968t , which is not significant (0: β3= 02.4590d. = = −.97the inclusion of the cubic term is not justified.y .H cannot be rejected), so413