Chapter 5 - Analytical Sciences Digital Library

176 Analytical Chemistry 2.0
where y i is the i th experimental value, and ŷ i
is the corresponding value predicted
by the regression line in equation 5.15. Note that the denominator
of equation 5.19 indicates that our regression analysis has n–2 degrees of
freedom—we lose two degree of freedom because we use two parameters,
the slope and the y-intercept, to calculate ŷ i
.
A more useful representation of the uncertainty in our regression is
to consider the effect of indeterminate errors on the slope, b 1 , and the y-
intercept, b 0 , which we express as standard deviations.
s
b1
=
∑
ns
2
n x − x
i
i
2
r
∑
i
i
2
=
2
sr
2
∑( x − x
i ) 5.20
i
s
b0
=
∑
s
x
2 2
r i
i
2
n x − x
i
i
∑
∑
i
i
2
=
s
r∑
x
2 2
i
i
∑( −
i )
n x x
i
2 5.21
We use these standard deviations to establish confidence intervals for the
expected slope, b 1 , and the expected y-intercept, b 0
β 1
= b 1
± ts b 5.22
1
β 0
= b 0
± ts b 5.23
0
You might contrast this with equation
4.12 for the confidence interval around a
sample’s mean value.
As you work through this example, remember
that x corresponds to C std , and
that y corresponds to S std .
where we select t for a significance level of a and for n–2 degrees of freedom.
Note that equation 5.22 and equation 5.23 do not contain a factor of
( n)
−1 because the confidence interval is based on a single regression line.
Again, many calculators, spreadsheets, and computer software packages
provide the standard deviations and confidence intervals for the slope and
y-intercept. Example 5.10 illustrates the calculations.
Example 5.10
Calculate the 95% confidence intervals for the slope and y-intercept from
Example 5.9.
So l u t i o n
We begin by calculating the standard deviation about the regression. To do
this we must calculate the predicted signals, ŷ i
, using the slope and y‐intercept
from Example 5.9, and the squares of the residual error, ( y − yˆ )
2 .
i i
Using the last standard as an example, we find that the predicted signal is
yˆ = b + bx = 0. 209 + 120. 706×
0. 500 60.
562
6 0 1 6 ( )=
and that the square of the residual error is