Discussion: Reasoning with Data Statistics Questions

Discussion: Reasoning with Data Statistics Questions ORDER NOW FOR CUSTOMIZED AND ORIGINAL ESSAY PAPERS ON Discussion: Reasoning with Data Statistics Questions Homework 5 36-200/247 Fall 2020 Due: On Gradescope, Wednesday, October 21, by 8:00PM (Pittsburgh time) Instructions: 1. Write up your solutions on separate paper. 2. Please put your name and Andrew ID neatly at the top of your homework. 3. Upload a pdf of your solutions to Gradescope. 4. In Gradescope, remember to indicate what page(s) each answer is on. 5. The work and words that you submit must be your own. 1. [A discrete random variable as an example of Actuarial Science] Actuaries are statisticians who assess risk and cost, for example in the insurance and finance industries. Much of actuarial science is involved with probabilities. The beginning of modern actuarial science was the investigation of ‘life tables’ by the British scientist Edmond Halley, better known for the comet named after him. Actuaries also deal with probabilities of accidents and other rare events; here is a simplified example. Suppose you work for State Farm homeowners insurance. Your company sells a fire insurance policy at a yearly ‘premium’ of $3000 (i.e., your company gets the initial $3000 each year from the homeowner, regardless of what else happens). If the homeowner’s house is completely destroyed by fire, your company must pay the homeowner $250,000; suppose the probability of a house being completely destroyed in a given year is 0.001.Discussion: Reasoning with Data Statistics Questions If the homeowner’s house is only partly damaged by fire, your company must pay the homeowner $125,000; suppose the probability of a house being only partly damaged by fire in a given year is 0.003. For the simplicity of the example, assume no other damage is possible. 1 (a) Let X represent the annual net gain (income minus payout) for your company [note this could be negative if your company pays out more than it gets]. The following partially completed table represents the probability distribution of the discrete random variable X. Give the completed probability table. [12pts] Hints: • Discrete probabilities must always sum to 1; • Remember that your company initially gets the $3000 from the homeowner, regardless of what else happens. If the company pays out nothing then it makes $3000 net profit; and if the company pays out something, then the $3000 minus the payout is the resulting net gain or loss. X [net earnings (or loss) by company] P(X) 1 (b) 0.001 0.003 [8pts] Compute the expected (i.e., average) annual net gain for the company from the policy. [Hint, see the example on p. 7 of lecture 13.] [homework continues] 36-200/247 Fall 2020 Homework 5 Page 2 of 3 2. [A Real-World Scientific Application of a Binomial distribution] Dowsing rods are a nonsensical folk belief, which claimed that unseen underground water sources could be found by holding a twig. The water source supposedly would be magically ‘sensed’ by the twig, which was supposed to swivel in the hand when water was nearby. The notion is not true (and there’s no physical reason why it should be). Discussion: Reasoning with Data Statistics Questions Nevertheless, untrue beliefs are often repackaged by the unscrupulous and sold to the desperate. A particularly shocking example involved a fraudulent device marketed to the military during the Iraq war which was claimed to ‘sense’ hidden roadside bombs by some unknown means. The device, named the MOLE, was produced by Global Technical Ltd., UK. It had an antenna which was free to swivel, and it was claimed that the antenna would swivel when explosives were nearby. There were elementary electronic components inside the device, but they were not connected in any meaningful way, there was no battery, and there is no known technology that can detect explosives at such distances. The MOLE device with swiveling aerial (photo from Sandia National Labs publication, reference below) So the device was essentially nothing more than a modern dowsing rod, a scam device being marketed to soldiers who undoubtedly would have needlessly died if they had trusted the device. Many similar devices were marketed around the same time, and the Iraqi government actually spent 85 million dollars on one such worthless device. [You are encouraged to learn about this at the following link: http://news.bbc.co.uk/2/hi/programmes/newsnight/8471187.stm] Luckily, the U.S. Sandia National Labs were skeptical about the MOLE device, so they conducted a test. [“DoubleBlind Field Evaluation of the MOLE Programmable Detection System,” Dale Murray, Sandia National Laboratories.] In the test, 20 grams of C4 explosive were placed in one of four possible boxes at the test facility. Discussion: Reasoning with Data Statistics Questions A representative from the Global Technical Ltd company, who claimed that the device really worked, would then use the detector to attempt to find which box contained the hidden explosive. The test was double-blind. 2 (a) If the operator of the device during the test was essentially just guessing completely at random regarding which of the four boxes contained the explosive, what is his probability p of getting the right box? [4pts] The test was conducted a total of 20 times (with the explosive placed in a different box each time). Consider the number of times X the operator of the device could have correctly found the explosive, out of the 20 trials. Then X is a Binomial random variable, with constants n and p. 2 (b) [8pts] If the device didn’t really work, and if for each of the 20 ‘runs’ of the study the operator of the device was therefore essentially just guessing completely at random regarding which of the four boxes contained the explosive, state n and p for this scenario (be sure to say which is which). 2 (c) [20pts] 2 (d) [4pts] Does the probability in part c represent something that would be relatively surprising to happen just by random, or is it so relatively large that it represents something that is not surprising to happen just by random? [If you’re unsure, look back at what was said on page 2 of lecture 9.] Use the Binomial Formula (hint, see pp. 4-5 of lecture 14) to compute P(X=6), the likelihood of guessing right in exactly 6 of the 20 trials. Show your work. Postscript: 36-200/247 Fall 2020 Homework 5 Page 3 of 3 In the actual study, the operator of the device was in fact correct in only 6 of the 20 trials. Since such an outcome was likely to happen just by random guessing, Sandia National Labs properly concluded, “Based on statistical analysis of the double-blind test results, the MOLE performs no better than a random selection process,” and on May 2, 2013, the managing director of the company selling the device was sentenced to 10 years in prison for fraud (see http://www.bbc.com/news/uk-22380368 ). Discussion: Reasoning with Data Statistics Questions This is a compelling example of the vital importance of scientific thinking and statistical literacy. Unfortunately, unscrupulous fraudsters continue to prey on the desperate; and similar devices, when actually used by soldiers, undoubtedly were responsible for unnecessary deaths. You are encouraged to read the Wikipedia page on the GT200, a renamed version of the MOLE: http://en.wikipedia.org/wiki/GT200 3. For each of the following, use the standardization formula, z= X ?µ ? , and the Standard Normal probability table as appropriate, like shown in lecture 16: 3 (a) [20pts (8 pts for z score, 8 pts for probability, 4 pts for sketch)] Among the developers of statistics was the Belgian astronomer, mathematician, and social scientist Adolphe Quetelet* (1796 – 1874), who was the first person to use the Normal Distribution as a model of real data. * See http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Quetelet.html For instance, based on information provided by the Scottish Army, Quetelet found that the chest measurements of male soldiers closely followed a Normal distribution, with mean µ = 40 inches, and standard deviation ? = 2 inches (according to An Introduction to Mathematical Statistics and its Applications, 2nd ed, by Larsen and Marx, p. 213). Based on this distribution, what proportion of male soldiers had chest measurement above 43.5 inches? For the purpose of the exercise, find the z score and then use the normal table as shown in lecture 16, since this is how you will do it on the exams. Also, sketch a picture of the Normal distribution, with the mean labeled, and the area in question shaded. 3 (b) Suppose you are working as a quality control investigator, testing circuit boards. Discussion: Reasoning with Data Statistics Questions Suppose circuit boards thinner than 7 mm are at risk of breaking. Suppose the distribution of thickness of circuit boards produced by Intel is supposed to be Normal, with mean µ = 10 mm, and standard deviation ? = 0.94 mm, if the manufacturing process is working correctly. (i) [20pts (8 pts for z score, 8 pts for probability, 4 pts for sketch)] Intel claims to you that the process is working correctly. To test this claim, suppose you select one circuit board at random. If the claimed distribution is true, how likely should it be for that one randomly-selected circuit board to be thinner than 7 mm? For the purpose of the exercise, find the z score and then use the normal table as shown in lecture 16, since this is how you will do it on the exams. Also, sketch a picture of the Normal distribution, with the mean labeled, and the area in question shaded. (ii) [Critical Thinking] If, in fact, the board you select does turn out to be thinner than 7 mm, do you think the manufacturing process is working correctly, as claimed? [Say yes or no.] [4pts] [end of homework 5] … Get a 10 % discount on an order above $ 100 Use the following coupon code : NURSING10