Lecture 2
Value proposition for a University: what value do you get and how does that fit into this course?
1.) Learn critical/analytical thinking.
2.) Learn content; how to construct a professional argument that data support a scientific model.
3.) Personal development; “The responsibility for us as teachers is to help our students understand that they must be active participants in the creation of their own character and to insist they do so even in the face of discomfort or adversity.” President of University of Portland. This is a bit like the value proposition of insurance being peace of mind (the value is not obvious). The value proposition of an education is character development (responsibility, civility, leadership, tolerance, . .). Here you practice this in group work and class discussions.
http://www.stonybrook.edu/commcms/cas/liberalartseducation.html
I’ll give examples of each point throughout the semester. First let's consider models, which are the foundation of these goals.
Modeling is intrinsic to the way we think.
Besides scientific models this includes heuristics [wikipedia: are simple, efficient rules, learned or hard-coded by evolutionary processes, that have been proposed to explain how people make decisions, come to judgments, and solve problems typically when facing complex problems or incomplete information], use triangles to build a structure. Show examples from (particularly of heuristics in artificial intelligence) http://c2.com/cgi/wiki?HeuristicRule Modeling also involves ways to interpret literature and speech.
Does your model of the way the world works match the data? Does sacrificing people lead to good weather (Mayan model)? Say you collected data which supported this model and tried to publish the data. Why would it be difficult to get it accepted? There is no scientific model of that explains the results. Therefore people would have trouble believing it (they see correlation but no model to cause the effect). We need a model to justify how the world behaves. For example, the blood vapor contains particulates upon which water condenses to form raindrops.
A current controversy involves a propulsion method that doesn’t emit anything. See http://emdrive.com/ There used to be “fringe theory” warnings on this site https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster Why would this idea have trouble being accepted? It helps if there is a model which explains why something happens rather than simply a correlation without a cause. However, the data may support motion but it seems to violate conservation of momentum since the net force on a body is the time rate of change of the total momentum. Since there is no net force there can be no change in total momentum. How do you make an argument that the model is valid? Does that sound like what you would hear in an english class where you are asked to make an argument (both orally and in writing) about an interpretation of a novel. This type of study is the foundation for critical thinking in a liberal arts college.
Example; Value Proposition A (critical/analytical thinking): You collect data which is presented in this graph http://wwwold.mpiwg-berlin.mpg.de/projects/DeptII_Sepkoski_Extinction/Extinction.jpg Here is the source of the data https://www.mpiwg-berlin.mpg.de/en/research/projects/deptii_sepkoski_extinction
What do you think? What questions do you have? By the end of this class I hope that the first question you have is what is the error or mistakes in the data? Maybe it's just noise. How do you check if it’s noise? REPEAT the expt. However, you can’t do that in this case. How do determine the error? Use multiple ways of measuring the data to check it. How well you know the assumptions in each measurement method and their validity will determine your estimation of the error.
Assume that these extinction events are valid. What model is supported by the data? Periodic volcanic eruptions, asteroids that intersects the Earth's orbit periodically due to a Nemesis star or planet X orbiting every 27 million years, dark matter, etc. Would these models be accepted compared with the sacrifice model?
Some people think the data is valid but that no model is appropriate because the random events lead to this distribution. Then if you were to repeat the expt it would not look the same. This again focuses on our nature to come up with models of what we observe even if the data is random.
Example; Value Proposition A (critical/analytical thinking): RC decay part (a)
-When asked to set up the circuit you don’t start plugging components into the circuit board. You construct a circuit diagram and critically analyze it first. Someone observing you would conclude that you’re an analytic thinker. -In processing your RC data to eliminate 60 Hz noise you integrate the voltage across the capacitor. What complication does that generate?
Example; Value Proposition A (critical/analytical thinking): RC lab part (b)
The scientific method involves testing if the data support a model. There are two ways to test the model: (1) The model in the RC lab is exponential decay. How would you test only if the decay is exponential without concern for the values of R and C? Semi-log plot. (2) How would errors in R can C affect the determination of the time constant? Error in slope = Sqrt[ (partial slope/particle R error R)^2 + (particle slope/particle C error C)^2]. Make a table of 1/RC, error 1/RC, R, error R, C, error C. Make a table of slope calculated (plug in R and C), error in slope calculated (from quadrature), slope from data, R, error in R, C, error in C. Is the slope calculated the same as the data slope within error? You will learn how to get an estimate of the error in the slope just from the data from a least squares procedure.
Example; Value Proposition C (Personal development): RC lab
You don’t notice that your lab partner sticks a wire on the proto-board shorting the power supply. In diagnosing your circuit the lab instructor find the short. This mistake makes you look like you don’t know much about electronics. How do you react? Does that reaction help the two of you to work together on the final report?
Error analysis (see section 3.7 of Baird) Value Proposition B, Content:
-Do error of the product of two quantities. Z = xy. Use percent error. -Do error of the sum Z = x + y -What’s the error in the mean value of n voltage measurements? Vmean =Sum[Vi,{i,1,n}]/n. This is a sum. How does error in this sum depend on errors in the individual terms of the sum. Use the heuristic that to understand need to simplify the notation: use (V1+V2+V3+ )/n rather than sum notation. Assume dV1 = dV2 = dV3 . . . and the error in dV3/n is just due to dV3 and not n since n is an integer with no error.
delVmean=Sum[dVmean/dVi delVi, {i,1,n}]/n where dVmean/dV1 = 1/n, dVmean/dV2=1/n , etc. So, delVmean=Sum[delVi/n, {i,1,n}] . Assume delVi = delV is constant, that is every measurement of Vi has the same error. Adding the errors in quadrature we get delV=Sqrt[Sum[(delV/n)^2, {i,1,n}] but summing identical terms n times give a factor of n so delV=Sqrt[(delV)^2/n] = delV/Sqrt[n].
The error in the mean value is the error in an individual measurement divided by the square root of the number of measurements.
Mean and stnd dev (see section 2.4 Baird) How do you determine errors in a measurement? REPEAT the measurement! Voltmeter may give 6 sig figs but only 3 repeat. Don’t take only the first measurement.
I repeat the measurement of the lifetime of a light bulb 1000 times and get a Gaussian distribution with mean T and stnd dev Tsigma. What happens to the distribution if I measure 10^6 bulbs? Show the definition of standard deviation here https://en.wikipedia.org/wiki/Standard_deviation Note that it depends on sqrt[N-1]. Does this mean the standard deviation gets smaller with N? Run the animation to show how the stnd dev does not change with samples. http://onlinestatbook.com/stat_sim/sampling_dist/index.html Critical thinking comments? (do all measurements behave this way? If there is time dependence due to temperature changes or something else then there is NOT a Gaussian distribution.) The derivation for the error in the mean does not use a Gaussian distribution nor does it mention the standard deviation of the mean. However the two ideas are related.
