l. A young child wonders what causes females to have children. For females she knows, she observes whether their hair is gray and whether they have children. The results are as follows: Children? For exercise I, refer to lecture 10 l3 as needed. 1 (a) 1 (b) 1 (c) 1 (d) 1 (e) 1(0 [Gpts1Tabulate the data in a 2x2 contingency table of counts. [6pts] Using the counts from (a), produce the 2x2 joint probability table representing the scenario, and ll in all the probabilities. [You can round probabilities to the nearest percent] [9pts (3 pts value, 3 pts symbol, 3 pts tenninology Based on your probability table, what is the probability that a woman from the whole group has Children? Give the value; represent this probability in the appropriate \"P( )\" notation [using symbols or words as needed]; and also say whether it is a joint, marginal, or conditional probability. [9pts (3 pts value, 3 pts symbol, 3 pts tenninology Based on your probability table, what is the probability that a woman from the whole group is Grey? Give the value; represent this probability in the appropriate \"P( )\" notation [using symbols or words as needed]; and also say whether it is a joint, maiginal, or conditional probability. [9pts (3 pts value, 3 pts symbol, 3 pts tenninology Based on your probability table, what is the probability that a woman om the whole group is Grey and has Children? Give the value; represent this probability in the appropriate \"P( )\" notation [using symbols or words as needed]; and also say whether it is a joint, marginal, or conditional probability. [9pts (3 pts value, 3 pts symbol, 3 pts terminology\" Based on your probability table, what is the probability that a woman from the whole group has Children or is Grey? Give the value, showing your calculation; represent this probability in the appropriate \"P( )\" notation [using symbols or words as needed]. 1(3) 101) 1 (i) 10') 1(k) [9pts (3 pts value, 3 pts symbol, 3 pts tenninology)] Based on your probability table, what is the probability that a Grey haired woman has Children? Give the value, showing any calculation; represent this probability in the appropriate \"P( )\" notation [using symbols or words as needed]; and also say whether it is a joint, marginal, or conditional probability. [Spts] By comparing two of your previous answers, say whether having Children is independent of being Grey, for the group. Indicate which values you are comparing, and briey say what it is about how they compare that determined your answer. [Spts] By referring to a previous answer, say whether having Children is disjoint from being Grey, for the group. Indicate which value you are considering, and briey say what it is about it that determined your answer. [Spts] Is there an association between being Grey and having Children, for the group? Briey say how you know, based on a previous answer. [3pts] Does your analysis indicate that being Grey causes a women to have Children? 2. - ' ' lication of a Binomial distribution] http:/ews.bbc.co.uklZ/hi/programmes/ newsnightl8471187.stm _ _ _ Do olk belief, Whlch 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). 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 The MOLE device not connected in any meaningful way, there was no battery, and there is no with swiveling aerial known technology that can detect explosives at such distances. (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:/ews.bbc.co.uk/2/hi/programmesewsnight/8471 1 87.stm] Luckily, the US. Sandia National Labs were skeptical about the MOLE device, so they conducted a test. [\"Double- Blind Field Evaluation of the MOLE Programmable Detection System,\" Dale Murray, Sandia National Laboratories; a copy is posted on Canvas for interested students] In the test, 20 grams of C4 explosive were placed in one of four possible boxes at the test facility. A representative from the Global Technical Ltd company, who claimed that the device really worked, would then use the detector to attempt to nd which box contained the hidden explosive. The test was double-blind. [exercise 2 continues on next page] 2 (a) [3pts] 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? 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) [lOpts] Use the Binomial Formula (hint, see pp. 45 of lecture 15) and a calculator as needed, to compute P(X=6), the likelihood of guessing right in exactly 6 of the 20 trials. Show your work. 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 10.] Postscript: 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 doubleblind 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.cornews/uk-223 80368 ). 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 [end of homework 4]