Page 138 - Modern Analytical Chemistry

1400-CH05 9/8/99 3:59 PM Page 121

Chapter 5 Calibrations, Standardizations, and Blank Corrections 121

indeterminate error. This is called the standard deviation about the regression,
standard deviation about the regression
s r , and is given as The uncertainty in a regression analysis
due to indeterminate error (s r ).
n ( y– y i ) 2
ˆ
s r = i å = 1 i 5.15
n – 2
th
where y i is the i experimental value, and ˆ y i is the corresponding value predicted by
the regression line
ˆ y i = b 0 + b 1 x i
There is an obvious similarity between equation 5.15 and the standard deviation in-
troduced in Chapter 4, except that the sum of squares term for s r is determined rela-
–
tive to ˆ y i instead of y, and the denominator is n – 2 instead of n – 1; n – 2 indicates
that the linear regression analysis has only n – 2 degrees of freedom since two pa-
rameters, the slope and the intercept, are used to calculate the values of ˆ y i.
A more useful representation of uncertainty is to consider the effect of indeter-
minate errors on the predicted slope and intercept. The standard deviation of the
slope and intercept are given as

2 2
ns r s r
= = 5.16
s b 1 2 2 2
nx –( å x i ) å ( x i – x)
å
i
2
2
s å x i 2 s å x i 2
r
r
= = 5.17
s b 0 2 2 2
n å x –( å x i ) n å ( x i – x)
i
These standard deviations can be used to establish confidence intervals for the true
slope and the true y-intercept
5.18
b 1 = b 1 ± ts b 1
5.19
b o = b o ± ts b 0
where t is selected for a significance level of a and for n – 2 degrees of freedom.
do not contain a factor of ( n –1 because the con-
Note that the terms ts b 1 and ts b 0 )
fidence interval is based on a single regression line. Again, many calculators, spread-
and
sheets, and computer software packages can handle the calculation of s b 0 and s b 1
the corresponding confidence intervals for b 0 and b 1. Example 5.11 illustrates the
calculations.
5
EXAMPLE .11
Calculate the 95% confidence intervals for the slope and y-intercept
determined in Example 5.10.
SOLUTION

Again, as you work through this example, remember that x represents the
concentration of analyte in the standards (C S ), and y corresponds to the signal
(S meas ). To begin with, it is necessary to calculate the standard deviation about
the regression. This requires that we first calculate the predicted signals, ˆ y i ,
using the slope and y-intercept determined in Example 5.10. Taking the first
standard as an example, the predicted signal is
ˆ y i = b 0 + b 1 x = 0.209 + (120.706)(0.100) = 12.280