Problems in quantitative reasoning

Jeanine Meyer

Mathematics/Computer Science

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.

Discussion

• Comments?

• [again] Intersection of – mathematics/quantitative reasoning– psychology/cognition– economics

intriguing area of study.