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'''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; 

'''Comment; Value Proposition A (critical/analytical thinking)'''
I often have students collect data then assume both that the model is valid and the data MUST support the model. After all the model is on wikipedia. That’s not science! I understand your whole education often has been a game of getting good grades when you say you understand the model that is being presented and bad grades when you question it. Many of the labs you do here are designed to kick you of that habit. Don’t assume that the data you get will support the model. Let the data drive that conclusion. Try not to get frustrated because this is the type of data that you will see when you get out of school.


'''Example; Value Proposition A (critical/analytical thinking)''' 

A company makes mechanical pencils. The lead cylinders have a mean diameter of 1.00 mm and standard deviation of 0.2 mm. 

Part (a) What diameter should I design the metal opening into which the lead fits so that 97 percent of the lead cylinders fit my pencil?

Sketch a Gaussian and label the axes as probability density function, PDF, and “led” diameter. Ask what impression does the misspelling give you? Talk about charge density function for a charge uniformly distributed on a line or surface or volume. What is similar and different from the PDF? Talk about quantum mechanics and the PDF associated with finding an atom along the x-axis. Review the integration of the Gaussian here
https://en.wikipedia.org/wiki/Standard_deviation

Ans: 1.0 mm + 0.4 = 1.4 mm or 2 std dev away from the mean of the population.

Part (b) We sell a rectangular box into which 100 lead pieces fit together on the bottom of the box (it is an automatic dispenser which loads the lead into the mechanical pencil) dispenser which loads the lead into the mechanical pencil). If 98 percent (integrate from 0 to 2 sigma to right of mean) of the boxes can be cleanly (fit flat on the box bottom) loaded with pencil leads, what box length do I need? 2 percent of the boxes will then have some cylinders ajar in the box bottom.

Here are two answers:
-One answer is 100 (1.00 + 0.2) = 100 + 20 mm. To have this be the correct answer then the length distribution of 100 cylinders next to each other would have to consist of randomly taking one cylinder from the population and putting 100 cylinders which are of IDENTICAL diameter next to each other. Next take  a different cylinder from the population and putting 100 cylinders which are of IDENTICAL diameter next to each other. In this way you build up a distribution of lengths which has mean value 100 mm and standard deviation 20 mm.

-Another answer is to take 100 random samples and line them up. However, when you fill the box, the 100 cylinders are randomly drawn from the population and are NOT identical. In this case the distribution you get is the distribution of means for a random sample of 100 cylinders placed in a line. The mean diameter for these 100 is 1.00 + 0.2/Sqrt[100]. Now when you line up 100 of cylinders with this mean and standard deviation you get a distribution of total lengths given by 100 x( 1.0 +0.2/Sqrt[100]) = 102 mm for 1 standard deviation and 104 mm for 2 standard deviations.
Why isn’t it 100 x(1.0) + 0.2/Sqrt[100]? 

Remember the derivation of error in an average value using derivatives? delVmean=Sqrt[(delV)^2/n] = delV/Sqrt[n]. This is an error in the average value NOT the error in the sum value. The sum value is the total length which is about 100 mm while the average value of 100 readings has a mean of 1 mm. Label the plots of the distributions for individual lead diameters (mean 1.0 mm), the length of 100 next to each other (mean 100 mm), and avg. values (mean 1.0 mm)

https://en.wikipedia.org/wiki/Standard_deviation
This formula looks like the population stnd dev depends on 1/sqrt[N]. Why doesn’t it?

Show applet. 
http://onlinestatbook.com/stat_sim/sampling_dist/index.html

-Note the normalized distribution (increase number of samples but peak height stays same)
-Note also the horizontal axes don’t have units. What is the physical significance of this math exercise? Say the distribution is of the failure time of light bulbs. You get the population distribution of number vs failure time. Next, take a group of 25 light bulbs and ask what is the average failure time for this group. What does this mean physically? The horizontal axis is the average failure time per light bulb in a group of 25 whereas the axis of the population distribution is the failure time per bulb. Not much physical significance to this information.
The horizontal axes for the means is probability vs mean value (mean length of 100 cylinders). Without the labels the population and mean plots are confusing.

'''Example; Value Proposition C (Personal development): Writing'''

I submitted a manuscript paper and got the following reviewer comment:

“The authors consider a bipartite system consisting of a particle and a mirror, whose respective positions are measured with a time-delay. A joint probability density is calculated, which exhibits a superposition of the mirror state.
The paper is somewhat lengthy, not without redundancy, rather economical in the use of commas and now and then, one is taken by surprise by the syntax.”

Note that the first sentence reviews what the manuscript is about (like sending me your section and title), the reviewer is acknowledging having read it and that the review has been sent to the correct author.
What is most important on the reviewers mind? The writing and not the content!

Ask the students: How do you respond to such criticism? Do you see it as the reviewers problem not to focus on the science? Do you go online and troll the journal? Do you get over your disgust and fix the manuscript up? Do you realize that it is not the journal's job to help you fix up your manuscript. They need only accept or reject manuscripts they think are of value to their readership.  

How you respond is an illustration of your character. The comments the TAs and I provide about your lab reports are intended to help you develop into a professional writer. 

'''Example; Value Proposition C (Personal development): Creativity'''

What is art? I like the definition that art makes you think about the world in new ways. How does an artist know they have created a very special work of art? How do you know you’ve done a special job of conveying your data? When you do know it’s art.


Data visualization Part (a).

Go over the population pyramid example found here
http://www.randalolson.com/2015/07/14/rethinking-the-population-pyramid/