Section 3E How Numbers Deceive: Polygraphs, Mammograms and More

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Section 3EHow Numbers Deceive Polygraphs,
Mammograms and More

• Pages 199-212

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Ex1/200- Who Played Better?
3-E
Shaq has higher shooting percentages than Vince
in both the first half and second half of the
game.
So, Shaq had the better game.
BUT, Vince has a higher shooting percentage than
Shaq for the entire game. So, Vince had the
better game. Shaq 7/14 .5 50 Vince
8/14 .57 57
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Simpsons Paradox
3-E
Simpsons Paradox(pg 200) occurs when something
appears better in each of two or more comparison
groups, but is actually worse overall. It occurs
because the numbers/counts in each comparison
group are so unequal.
Abuse of Percentages Dont average percentages!
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Simpsons Paradox
3-E
Pg201 University of California
BerkeleyGraduate Admissions, 1973Gender
Discrimination??
Men Men Men Women Women Women
Applied Admitted Percent Applied Admitted Percent
Total 2691 1198 44.5 1835 557 30.4
Women were being discriminated against!
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3-E
Men Men Men Women Women Women
Department Applied Admitted Applied Admitted
A 825 512 62 108 89 82
B 560 353 63 25 17 68
C 325 120 37 593 202 34
D 417 138 33 375 131 35
E 191 53 28 393 94 24
F 374 22 6 341 24 7
Total 2691 1198 44.5 1835 557 30.4
The admission rates for women are actually higher
than those for men in all but Departments C and
E, and the rates were quite close in those
departments.
Women were admitted at a significantly lower rate
overall BUT no individual department was guilty
of this practice. WOW!
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About Mammograms (pg202)
3-E
About 1 in 100 (1) breast tumors turn out to
be malignant.
Mammograms are 85 accurate

• identify 85 of malignant tumors as malignant.
• misidentify 15 of malignant tumors as benign.

False Negative

• identify 85 of benign tumors as benign.
• misidentify 15 of benign tumors as malignant.

False Positive
negative mammogram means no cancer
benign. positive mammogram means cancer
malignant.
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pg 202- When a doctor tells a woman that her
mammogram is positive, what should he also tell
her about her chances that she actually has
cancer?
3-E
Build a summary chart (based on percent of) for
10000 mammograms of women with breast tumors.
Cancer No Cancer Total
Mammogram Test(malignant)
Mammogram Test (benign)
Total 10,000
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3-E
Cancer No Cancer Total
Mammogram Test
Mammogram - Test
Total 100 9,900 10,000
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3-E
Cancer No Cancer Total
Mammogram Test .85100 85
Mammogram Test .859900 8415
Total 100 9,900 10,000
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3-E
Cancer No Cancer Total
Mammogram Test 85 1485 1570
Mammogram Test 15 8415 8430
Total 100 9,900 10,000
Use the summary chart to answer the question.
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3-E
Cancer No Cancer Total
Mammogram 85 True 1485 False 1570
Mammogram - 15 False - 8415 True - 8430
Total 100 9,900 10,000
pg 202- When a doctor tells a woman that her
mammogram is positive, what should he also tell
her about her chances that she actually has
cancer?
Of those women with positive mammograms, only 85
out of 1570 or 85/1570 .054 5.4 actually
have cancer.
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3-E
Cancer No Cancer Total
Mammogram 85 True 1485 False 1570
Mammogram - 15 False - 8415 True - 8430
Total 100 9,900 10,000
Ex3/203- When a doctor tells a woman that her
mammogram is negative, what should he also tell
her about her chances that she actually has
cancer?
Of those women with negative mammograms, 15 out
of 8430 or 15 / 8430 .0018 .18 actually
have cancer(about 2 women in 1000.)
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About Polygraphs (pp 203-4)
3-E

• Suppose 1 of job applicants lie.
• Suppose a polygraph is 90 accurate
• correctly identifies 90 of liars as liars-
  misidentifies 10 of liars as truth tellers
• correctly identifies 90 of truth tellers as
  truth
• tellers.
• misidentifies 10 of truth tellers as liars.
• positive polygraph means lying detected.
• negative polygraph means no lying detected.

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3-E
(pp 203-4) Suppose 1000 applicants take the
polygraph test. How many of those applicants who
were accused of lying (and rejected for the job)
actually told the truth?
Build a summary chart (based on percent of) for
1000 applicants.
Lie Tell Truth Total
Polygraph Test (Lie)
Polygraph Test (Truth)
Total 1,000
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3-E
Lie Tell Truth Total
Polygraph Test (Lie) 9 99 108
Polygraph Test (Truth) 1 891 892
Total 10 990 1,000
Of those applicants that failed the polygraph, 99
out of 108 or 99/108 .917 91.7 were actually
telling the truth.
Of those applicants that passed the polygraph, 1
out of 892 or 1/892 .0011 .11 were actually
lying.
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Tree Diagram for Polygraphs
3-E
So 99/108 91.7 of those who are accused of
lying are not actually lying.
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3-E
About Drug Tests (ex4/204)

• Suppose 4 of athletes take banned drugs.
• Suppose a drug test is 95 accurate
• correctly identifies 95 of drug uses as drug
  users.- misidentifies 5 of drug users as clean.
• correctly identifies 95 of clean athletes as
  clean.
• misidentifies 5 of clean athletes as drug users.
  
• positive drug test means drugs detected.
• negative drug test means no drugs detected.

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3-E
ex4/200 Suppose 1000 athletes at a regional high
school track meet submit urine samples. What
percentage of the athletes who fail the test are
falsely suspended from the team?
Build a summary chart (based on percent of) for
1000 athletes.
drugs no drugs Total
Drug Test Test (drugs)
Drug Test Test (no drugs)
Total 1,000
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3-E
Drugs No drugs Total
Drug Test Test (Drugs) 38 48 86
Drug Test Test (No Drugs) 2 912 914
Total 40 960 1,000
Of those athletes that failed the drug test, 48
out of 86 or 48/86 .56 56 were actually
clean and falsely suspended.
Of those athletes that passed the drug test, 2
out of 914 or 2/914 .0021 .21 were drug
users.
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(ex5/206) A Cut or an Increase?
3-E

• Government spending for a popular education
  program was 100 million last year. When
  Congress prepares its budget for next year,
  spending for the program is slated to rise to
  102 million. The Consumer Price Index is
  expected to rise by 3 over the next year.
• Is spending on this program being increased or
  cut?

Absolute change 102 million - 100 million
2 million This is an increase in
spending. Relative change 2 million / 100
million 2 This is a decrease in spending
relative to the inflation rate (3).
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3-E

• Homework
• Pages 207-212
• 22, 25,27, 28, 30, 31