View
219
Download
0
Category
Tags:
Preview:
Citation preview
Outline• Background
– More women murdered on the job– Health Screening (e.g., HIV)
• Cognitive Reflection Test– 3 puzzles– studies by Shane Frederick, MIT
• Discussion
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?
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
What was the problem?
• Mis-use of percentages: – comparing percentages with different bases
• Missing information: – what killed the men?
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
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????
First step
with condition
without condition
Positive test
Negative test
300000*.01=3000
300000*.99=297000
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
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
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
[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
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
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
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?
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?
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?
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.
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
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
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
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
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)
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.
Recommended