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

2. 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

3. Mathematics/Computer Science Senior Seminar
Background
Communicating Quantitative Information
'gen-ed' course using news stories to teach mathematics
piloted Spring, 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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

11. Panning for Terrorists
John Allen Paulos: applies same methodology to automatic/semi-automatic systems for monitoring phone calls
Mathematics/Computer Science Senior Seminar

12. [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

13. 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

14. New York Times news story
by Virginia Postrel
Headline: Would you choose $1000 or 75% chance at $4000
Mathematics/Computer Science Senior Seminar

15. 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

16. 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

17. 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

18. 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

19. 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 * = $10
so this bet is worth $10
Mathematics/Computer Science Senior Seminar

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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