Page 299 - Statistics for Environmental Engineers

L1592_frame_C34 Page 305 Tuesday, December 18, 2001 2:52 PM

2
The value of σ is typically unknown and must be estimated from the data; replicate measurements will

2
2
provide an estimate. If there is no replication, σ is estimated by the mean residual sum of squares (s )
which has ν = n − 2 degrees of freedom (two degrees of freedom are lost by estimating the two parameters
β 0 and β 1 ):
(
∑ y i y ˆ ) 2
–
2
s = ------------------------ = ------------
S R
–
n 2 n 2
–
The (1 – α)100% confidence intervals for β 0 and β 1 are given by:
2
x
1
b 0 ± t υ,α/2 s --- + ------------------------
(
n ∑ x i – x) 2
1
b 1 ± t υ,α/2 s ------------------------
∑ x i –( x) 2
These interval estimates suggest that the joint confidence region is rectangular, but this is not so. The
joint confidence region is elliptical. The exact solution for the (1 − α)100% joint confidence region for
β 0 and β 1 is enclosed by the ellipse given by:

2
(
x i b 0 β 0 ) b 1 β 1 ) +
nb 0 β 0 ) + 2   ∑   ( – ( –   ∑  ( b 1 β 1 ) = 2s F 2,n−2,α
2
2
2
–
–
x i
where F 2,n−2,α is the tabulated value of the F statistic with 2 and n − 2 degrees of freedom.
The confidence interval for the mean response (η 0 ) at a particular value x 0 is:
2
x)
(
x 0 –
1
( b 0 + b 1 x 0 ) ± t υ,α/2 s --- + ------------------------
n ∑ x i – x) 2
(
The prediction interval for the future single observation ( = b 0 + b 1 x f ) to be recorded at a setting x f is:
y ˆ f
2
x)
(
x f –
--- +
( b 0 + b 1 x f ) ± t υ,α/2 s 1 + 1 ------------------------
n ∑ x i – x) 2
(
Note that this prediction interval is larger than the confidence interval for the mean response (η 0 ) because
the prediction error includes the error in estimating the mean response plus measurement error in y. This
introduces the additional “1” under the square root sign.

Case Study: A Linear Model
Data from calibration of an HPLC instrument and the fitted model are shown in Table 34.1 and in
Figure 34.2. The results of fitting the model y = β 0 + β 1 x + e are shown in Table 34.2. The fitted equation:

y ˆ = b 0 + b 1 x = 0.566 + 139.759x
© 2002 By CRC Press LLC