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L1592_frame_C34 Page 307 Tuesday, December 18, 2001 2:52 PM

The appropriate value of the t statistic for estimation of the 95% confidence intervals of the parameters
is t ν=13,α/2=0.025 = 2.16. The individual confidence intervals estimates are:

β 0 = 0.566 ± 1.023 or −0.457 < β 0 < 1.589

β 1 = 139.759 ± 6.242 or 133.52 < β 1 < 146.00

The joint confidence interval for the parameter estimates is given by the shaded area in Figure 34.2.
Notice that it is elliptical and not rectangular, as suggested by the individual interval estimates. It is
bounded by the contour with sum of squares value:


2

S c = 15.523 1 + ------ 3.81( ) = 24.62
 13 
2
The equation of this ellipse, based on n = 15, b 0 = 0.566, b 1 = 139.759, s = 1.194, F 2,13,0.05 = 3.8056,
2
∑ x i = 1.974, ∑ x i = 0.40284 , is:
(
(
(
(
(
(
(
15 0.566 β 0 ) +2 1.974( ) 0.566 β 0 ) 139.759 β 1 )+ 0.40284) 139.759 β 0 ) = 2 1.194) 3.8056)
2
2
–
–
–
–
This simplifies to:
2
2 568.75β 0 + 281.75β 1 + 0.403β 0 + 8176.52 =
15β 0 – 3.95β 0 β 1 – 0
The confidence interval for the mean response η 0 at a single chosen value of x 0 = 0.2 is:
2
0.2 0.1316)
–
1
0.566 + 139.759 0.2) ± 2.16 1.093) ------ + ( ------------------------------------ = 28.518 ± 0.744
(
(
15 0.1431
The interval 27.774 to 29.262 can be said with 95% confidence to contain η when x 0 = 0.2.
The prediction interval for a future single observation recorded at a chosen value (i.e., x f = 0.2) is:
0.2 0.1316)
2
–
1
(
0.566 + 139.759 0.2) ± 2.16 1.093) 1 + ------ + ( ------------------------------------ = 28.518 ± 2.475
(
15 0.1431
It can be stated with 95% confidence that the interval 26.043 to 30.993 will contain the future single
observation recorded at x f = 0.2.
Comments
Exact joint confidence regions can be developed for linear models but they are not produced automatically
by most statistical software. The usual output is interval estimates as shown in Figure 34.3. These do
help interpret the precision of the estimated parameters as long as we remember the ellipse is probably
tilted.
Chapters 35 to 40 have more to say about regression and linear models.

© 2002 By CRC Press LLC