Question AC.1: Finding the x-Intercept For the line y = mx + b, the x-inter...
Finding the x-Intercept
For the line y = mx + b, the x-intercept occurs when y = 0, or x = −b/m.
Let’s designate the x-intercept as the function F = −b/m. Find the
x-intercept and its uncertainty for the least-squares line in Figure 4-11.
The following quantities are computed in Section 4-7:
m = 0.615 38 s^{2}_{m} = 0.002 958 6 s^{2}_{y} = 0.038 462 ∑(x_{i}) = 14
b = 1.346 15 s^{2}_{b} = 0.045 859 D = 52
The covariance in Equation C-3 is therefore
s_{mb} = \frac{−s^{2}_{y}∑(x_{i})}{D} = \frac{−(0.038 462)(14)}{52} = −0.010 355
e_{F} = \sqrt{\underbrace{(\frac{\delta F}{\delta m})² s^{2}_{m} + (\frac{\delta F}{\delta b})² s^{2}_{b}}_{\begin{matrix} Variance terms from \\ Equation C-1 \end{matrix} } + \underbrace{2(\frac{\delta F}{\delta m})(\frac{\delta F}{\delta b})s_{mb}}_{\begin{matrix} Covariance accounts for \\ correlation of m and b \end{matrix} } } (C_2)
The x-intercept is just F = −b/m = −(1.346 15)/(0.615 38) = −2.187 5.
To find the uncertainty in F, we use Equation C-2. The derivatives in
C-2 are
\frac{\delta F}{\delta m} = \frac{\delta(−b/m)}{\delta m} = \frac{b}{m²} = \frac{1.346 15}{0.615 38²} = 3.554 7
\frac{\delta F}{\delta b} = \frac{\delta(−b/m)}{\delta b} = \frac{−1}{m} = \frac{−1}{0.615 38} = − 1.625 0
Now we can evaluate the uncertainty with Equation C-2:
e_{F} = \sqrt{(\frac{\delta F}{\delta m})² s^{2}_{m} + (\frac{\delta F}{\delta b})² s^{2}_{b} + 2(\frac{\delta F}{\delta m})(\frac{\delta F}{\delta b})s_{mb}}
= \sqrt{(3.554 7)²(0.002 958 6) + (−1.625 0)²(0.045 859) + 2(3.554 7)(−1.625 0)(−0.010 355)}
=0.527 36
The final answer can now be written with a reasonable number of digits:
F = −2.187 5 ± 0.527 36 = −2.1_{9} ± 0.5_{3}
e_{F} = \sqrt{(\frac{\delta F}{\delta x})² e^{2}_{x} + (\frac{\delta F}{\delta y})² e^{2}_{y} + (\frac{\delta F}{\delta \mathcal{z}})² e^{2}_{\mathcal{z}} + . . .} (C-1)
If we had used Equation C-1 and ignored the covariance term in Equation C-2, we would have computed an uncertainty of ±0.4_{0}.
To learn how to compute variance and covariance and to see how to
include weighting factors in least-squares curve fitting, see J. Tellinghuisen,
“Understanding Least Squares Through Monte Carlo Calculations,” J. Chem. Ed. 2005, 82, 157.