d) Now assume that the uncertainty in each value of f grows with f: o(f) = 0.13 + 0.05 * f (MHz). Determine the slope and the intercept of the best-fit line using the least-squares method with unequal weights (weighted least-squares fit). Comment on any differences in the fit parameters and uncertainties.
d) Now assume that the uncertainty in each value of f grows with f: o(f) = 0.13 + 0.05 * f (MHz). Determine the slope and the intercept of the best-fit line using the least-squares method with unequal weights (weighted least-squares fit). Comment on any differences in the fit parameters and uncertainties.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Please answer part d
Transcribed Image Text:2.0.5 Problem 5: Optical Pumping
In the optical pumping experiment (OPT), we measure the resonant frequency of a Zeeman transi-
tion as a function of the applied current (proportional to the applied magnetic field). This exercise
is identical to the analysis that you will need to perform for OPT. Consider a mock data set:
Current I (Amps)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
Frequency f (MHz) #
0.13
0.70
1.48
2.60
3.04
3.86
4.98
5.68
6.52
6.85
7.72
8.85
4
Transcribed Image Text:a) Plot a graph of the pairs of values. Assuming a linear relationship between I and f, determine
the slope and the intercept of the best-fit line using the least-squares method with equal
weights, and draw the best-fit line through the data points on the graph.
b) From what they know about the equipment used to measure the resonant frequency, your
lab partner hastily estimates the uncertainty in the measurement of f to be σ(f)
= 0.13
MHz. Estimate the probability that the straight line you found is an adequate description
of the observed data if it is distributed about this line with the uncertainty guessed by your
lab partner. (Hint: use scipy.stats.chi2 class to compute the quantile of the x² distribution).
What can you conclude from these results?
c) Assume that the best-fit line found in part (a) is a good fit to the data. Estimate the
uncertainty in measurement of y from the scatter of the observed data about this line. Again,
assume that all the data points have equal weight. Use this to estimate the uncertainty in
both the slope and the intercept of the best-fit line. This is the technique you will use in
the Optical Pumping lab to determine the uncertainties in the fit parameters. The resulting
uncertainties are determined by the scatter of the data about the model and do not depend
on your estimate of the uncertainty.
d) Now assume that the uncertainty in each value of f grows with f: σ(f) : = 0.13 + 0.05 * f
(MHz). Determine the slope and the intercept of the best-fit line using the least-squares
method with unequal weights (weighted least-squares fit). Comment on any differences in the
fit parameters and uncertainties.
Expert Solution
Step 1
Given the data as
| Current I(Amps) | Frequency f (MHz) |
| 0 | 0.13 |
| 0.2 | 0.7 |
| 0.4 | 1.48 |
| 0.6 | 2.6 |
| 0.8 | 3.04 |
| 1 | 3.86 |
| 1.2 | 4.98 |
| 1.4 | 5.68 |
| 1.6 | 6.52 |
| 1.8 | 6.85 |
| 2 | 7.72 |
| 2.2 | 8.85 |
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