Problems in quantitative reasoning

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Problems in quantitative reasoning

• Jeanine Meyer
• Mathematics/Computer Science

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Outline

• Background
• More women murdered on the job
• Health Screening (e.g., HIV)
• Cognitive Reflection Test
• 3 puzzles
• studies by Shane Frederick, MIT
• Discussion

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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?

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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

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What was the problem?

• Mis-use of percentages
• comparing percentages with different bases
• Missing information
• what killed the men?

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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

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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????

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First step
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Second step
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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 falsealarms.
• This can be okay.
• Complex public health issue

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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/12r
  ead2.html

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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

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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

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New York Times news story

• by Virginia Postrel
• http//www.nytimes.com/2006/01/26/business/26scene
  .html?_r1
• Headline Would you choose 1000 or 75 chance at
  4000

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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?

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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?

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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?

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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.

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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

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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

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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

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Examples
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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)

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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.

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Discussion

• Comments?
• again Intersection of
• mathematics/quantitative reasoning
• psychology/cognition
• economics
• intriguing area of study.