Problems in quantitative reasoning Jeanine Meyer Mathematics/Computer Science Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Outline Background More women murdered on the job Health Screening (e.g., HIV) Cognitive Reflection Test 3 puzzles studies by Shane Frederick, MIT Discussion Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Background Communicating Quantitative Information 'gen-ed' course using news stories to teach mathematics piloted Spring, 2005. 2 sections this semester newmedia.purchase.edu/~Jeanine/charts.html How do you all do in applying / using mathematics to understand issues of the day? When considering decision making, are the choices made by mathematically-able people the correct choices? Mathematics/Computer Science Senior Seminar

More women murdered on the job Headline for actual news stories a dozen years ago in the New York Times 93% of people who die 'on the job' are men 14% of the men are murdered; 40% of the women Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar What was the problem? Mis-use of percentages: comparing percentages with different bases Missing information: what killed the men? Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Observation Wrong, or, more typically incomplete information is common. This is good for pedagogy! Other topics for course include false positives in health screening polling lottery map projections trends in sports records Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Health Screening Consider: HIV (or other) screening 300,000 people tested 1% have condition Test is 99% accurate at returning positive result when patient has condition Test is 98% accurate at returning negative result when patient does not have condition The test result is positive: what is the probability of it being correct???? Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar First step with condition without condition Positive test Negative test 300000*.01=3000 300000*.99=297000 Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Second step with condition without condition Totals Positive test 3000*.99 = 2970 297000*.02=5940 2970+5940=8910 Negative test 30 297000*.98=291060 30+291060=291090 300000*.01=3000 300000*.99=297000 8910+ 291090=300000 Mathematics/Computer Science Senior Seminar

How many false positives? 5940 out of 8910!!!! probabilistic, not guaranteed, but surprising anyway: what is expected with a test 99%/98% accurate Lesson: screening of generally healthy population can produce false…alarms. This can be okay. Complex public health issue Mathematics/Computer Science Senior Seminar

Panning for Terrorists John Allen Paulos: applies same methodology to automatic/semi-automatic systems for monitoring phone calls http://www.nytimes.com/2006/02/12/weekinreview/12read2.html Mathematics/Computer Science Senior Seminar

[Economic] Decision making Why do people make the decisions they do? especially, relating to investments, buying and selling, 'life decisions' Intersection of mathematics psychology: cognition, emotion (affective) economics Mathematics/Computer Science Senior Seminar

Cognitive Reflection & Decision making Shane Frederick, Sloan School, MIT mit.edu/people/shanefre/publications.htm studies (questionnaires) relating performance on a test consisting of 3 puzzles with other tests (e.g., SAT, SAT-math) stated choices Mathematics/Computer Science Senior Seminar

New York Times news story by Virginia Postrel http://www.nytimes.com/2006/01/26/business/26scene.html?_r=1 Headline: Would you choose $1000 or 75% chance at $4000 Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar CRT 1) A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost? 2) If it takes five machines five minutes to make five widgets, how long would it take 100 machines to make 100 widgets? 3) In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half the lake? Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Frederick claim These problems all have a intuitive answer that is wrong: 10 cents 100 minutes 24 days So people who get the correct answer are more reflective… Comments? Mathematics/Computer Science Senior Seminar

My claim: problems different The ball and bat problem: yes, but this is also the easiest one: just use algebra. The workers? Another problem: if a chicken and a half takes a day and half to lay an egg and a half, how many eggs do 3 chickens lay in 3 days? The lake? Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Studies college students (population of choice for most such studies…), plus others CRT correlates well with other, more extensive tests high score (3/3) on CRT correlates with making choices requirement patience, [some] knowledge of expectations AND willingness to take risk high score CRT also correlates with some decisions involving [real] risk, expectation lower. Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Recall Expectation (aka expected value of a bet) is probability of win * value of win If the stake is $1000 and the chance of getting it is 1/100 then expected value is .01 * 1000 = $10 so this bet is worth $10 Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar 50-50 raffle Common fund raising device Collect money: say $1 per chance. Split the take half to winner and half to organization Expectation is 50%. Value of bet is .50 Why do people pay $1 for something worth 50 cents? Want to support the organization AND like betting Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Return to study Analyzed how people scoring well (3 out of 3 correct) on the 3 question test (CRT) vs people doing badly (0 or 1 correct) answered on questions of choice Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Examples Percentage choosing riskier option: Low CRT High CRT $1000 vs 90% chance of $5000 52% 74% $100 vs 25% chance of $200 7% 10% Lose $100 for sure vs 75% chance to lose $200 54% 31% Willingness to pay for overnight shipping vs 2 week wait $4.54 mean $2.18 mean Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar More Gender difference high scoring females were more patient whereas high scoring males were more risk takers (which may or may not have required more patience) Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar from Frederick Are the decisions by high-scoring people the right decisions? (paraphrase): Following the model of smart/analytic people in choice of mortgage may be correct, but choosing apples over oranges because Einstein liked apples may not be warranted. Mathematics/Computer Science Senior Seminar

Mathematics/Computer Science Senior Seminar Discussion Comments? [again] Intersection of mathematics/quantitative reasoning psychology/cognition economics intriguing area of study. Mathematics/Computer Science Senior Seminar