Lecture 4
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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 loaded with pencil leads, how what box length do I need to be so the leads all fit flat on the box bottom? 2 percent of the boxes will then have some cylinders ajar in the box bottom. | 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 loaded with pencil leads, how what box length do I need to be so the leads all fit flat on the box bottom? 2 percent of the boxes will then have some cylinders ajar in the box bottom. | ||
| − | The avg lead diameter for 100 pieces is 1.00 | + | The avg lead diameter for 100 pieces is 1.00 <math>\pm</math> 0.2/Sqrt[100] or <math>\pm</math> 0.02. The length of the box is then 100 times this or 100 mm <math>\pm</math> 2 mm. The box then has to be of length 102 for one std dev and 104 for two std dev. This is NOT 100 times (1.00 <math>\pm</math> 0.2). |
What diameter should I design the metal opening into which the lead fits so that 97 percent of the lead cylinders fit my pencil? | What diameter should I design the metal opening into which the lead fits so that 97 percent of the lead cylinders fit my pencil? | ||
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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 taking a cylinder from the population and putting 100 cylinders which are of IDENTICAL diameter next to each other. Then taking 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. | 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 taking a cylinder from the population and putting 100 cylinders which are of IDENTICAL diameter next to each other. Then taking 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. | ||
| − | 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. The mean diameter for these 100 is 1.00 | + | 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. The mean diameter for these 100 is 1.00 <math>\pm</math> 0.2/Sqrt[100]. Now when you put 100 of cylinders with this mean and standard deviation you get a distribution of lengths given by 100 ( 1.0 <math>\pm</math> 0.2/Sqrt[100]) = 102 mm for 1 standard deviation and 104 mm for 2 standard deviations. |
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Show applet. | Show applet. | ||
Revision as of 19:07, 30 September 2016
Pencil problem My company makes mechanical pencils. The lead cylinders have a mean diameter of 1.00 mm and standard deviation of 0.2 mm.
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 loaded with pencil leads, how what box length do I need to be so the leads all fit flat on the box bottom? 2 percent of the boxes will then have some cylinders ajar in the box bottom.
The avg lead diameter for 100 pieces is 1.00 
0.2/Sqrt[100] or 0.02. The length of the box is then 100 times this or 100 mm 2 mm. The box then has to be of length 102 for one std dev and 104 for two std dev. This is NOT 100 times (1.00 0.2).
What diameter should I design the metal opening into which the lead fits so that 97 percent of the lead cylinders fit my pencil?
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 taking a cylinder from the population and putting 100 cylinders which are of IDENTICAL diameter next to each other. Then taking 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.
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. The mean diameter for these 100 is 1.00 0.2/Sqrt[100]. Now when you put 100 of cylinders with this mean and standard deviation you get a distribution of lengths given by 100 ( 1.0 0.2/Sqrt[100]) = 102 mm for 1 standard deviation and 104 mm for 2 standard deviations.
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.
-critical thinking about the derivation of error in mean Modeling/critical thinking: math is another example of a heuristic or rule we follow without critical thinking (can you explain in words what the Sch eqn means?) The hamburger problem tests your understanding beyond that of system 1 thinking (in this case is a math heuristic where you just need to figure out what to substitute into the equation). Where did the sqrt n come from? Go back through the derivation of the error in an average value. Start with Vtest = n Vavg = Sum[]. Error in Vtest is Vavg times error in n + n Error in Vavg. Since there is no error in n the error in Vtest is n error in Vavg which is also error in Sum[] = Sqrt[ partial Vavg/V1 del V1 + . . .
You want to see if data support a model. You can use a plug your data into a chi square heuristic. This spits out correlation but not causation. For example, some expt yields the results that wine is good for you. A few years later you hear the opposite. These health issues are so complex that often the statistical methods applied to imperfect data (not all relevant variables are represented in the data to support a model) neglect important variables. Some people think that you can’t even make predictions in some systems (eg sensitivity to initial conditions). Some think that some phenomena become emergent (larger entities arise through interactions among smaller or simpler entities such that the larger entities exhibit properties the smaller/simpler entities do not exhibit). If you have no clue of what causes what, you can analyze the data with “principal functional analysis” to find correlations. For example, measurements of the heights of children over a wide range of ages, races, income, culture, etc. This gives hints as to how to build a model but doesn’t explain why things may be correlated (like sacrificing people to improve weather).
-hope when you see data you ask what is the error in this data. More than that I hope you ask questions when you read an article, about who wrote the article, what are the sources of the information (data) presented, what is the justification of the thesis in the article, what are the assumptions of the argument,
How to ask questions: a procedure
Sept. 28 is ask a stupid question day. What examples do you have?
One of my favorite is “why are meteors always found in craters?”
Marilyn vos Savant (smartest person in the world) answers questions in newspaper column. Here is one which she thought appropriate for this day: “I do not understand women. Would the study of quantum mechanics help?”
What do these types of questions have in common?
I’m going to show you a procedure or process to ask productive questions.
Helicopter video https://www.youtube.com/watch?v=RihcJR0zvfM
what questions do you have? what are you curious about?
6 categories of questions based on how we view the world through models/heuristics/patterns
INCONGRUOUS To clarify seeming violations of a model or understanding of how the world functions Does this violate Newton’s third law because there is no reaction? How did they fake this video? CONGRUOUS To gather information on how to apply your understanding Is it possible to use the work-energy theorem to explain any part of the copter dismantling? MODIFYING To probe your understanding as assumptions, parts, applications, or parameters are changed What if the helicopter has only one propeller? What if . . . ? GENERALIZING / ANALOGY To search for similar patterns in one’s understanding Does the helicopter vibrate the way a loose ceiling fan does when it reaches a certain angular velocity? Is this like . . . ? CAUSAL / CREATIVE To generate novel patterns or improve on existing understanding. Would this still work in a zero gravity environment? INFORMATIONAL To find information simply for its intrinsic interest or diagnostic purposes. Is the helicopter attempting to take off, or is it tied to the ground?
The “stupid questions” indicate the person doesn’t understand the expected models/heuristics/patterns. The observer often finds this humorous
Famous quote: Judge a person by their questions not answers..
