Section 3E How numbers deceive

1
Section 3EHow numbers deceive

• Pages 195-202

2
Simpsons Paradox
3-E
Since Shaq has the better shooting percentages in
both the first half and second half of the game,
can he claim that he has the better game
compared to Vince?
3
Simpsons Paradox
3-E
Shaq overall 7 / 14 50 Vince overall
8 / 14 57.1
4
Simpsons Paradox
3-E
Simpsons Paradox 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.
5
Simpsons Paradox A Famous Example
3-E
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
6
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
7
Does Smoking Make You Live Longer?
3-D

• Early 1970s Medical Study in England.
• Involved adult residents from Wickham
• 20 years later, follow-up study looked at
  survival rates of people from the original study.
• Among adult smokers, 24 died during that 20
  year period.
• Among adult non-smokers, 31 died during that
  period.

8
Does Smoking Make You Live Longer?
3-D

• Turns out in the original study, nonsmokers
  were older (on average) than the smokers.
• Thus the higher death rate among the non-smokers
  simply reflected the fact that death rates tend
  to increase with age.
• When the results were broken into age groups,
  they showed that for any given age group,
  non-smokers had a higher survival rate than
  smokers.

9
Conditional Probability
3-E

• Does a positive mammogram always mean cancer?
• Positive test
• True positive really have cancer
• False positive dont really have cancer
• Negative test -
• True negative dont have cancer
• False negative do have cancer

10
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
Cancer No Cancer Total
Mammogram Test
Mammogram - Test
Total 10,000
11
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
Cancer No Cancer Total
Mammogram Test
Mammogram - Test
Total 100 9,900 10,000
12
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
Cancer No Cancer Total
Mammogram Test .85100 85
Mammogram Test .859900 8415
Total 100 9,900 10,000
13
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
Cancer No Cancer Total
Mammogram Test 85 1485 1570
Mammogram Test 15 8415 8430
Total 100 9,900 10,000
14
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
Cancer No Cancer Total
Mammogram Test 85 True 1485 False 1570
Mammogram - Test 15 False - 8415 True - 8430
Total 100 9,900 10,000
15
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
So, chance of actually having cancer if the
mammogram is positive is 85 / 1570 5.4 If
your mammogram is negative what is the chance
its a false negative? (You have cancer in spite
of being told you dont?)
16
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
Cancer No Cancer Total
Mammogram 85 True 1485 False 1570
Mammogram - 15 False - 8415 True - 8430
Total 100 9,900 10,000
17
Mammography 85 accurateresults for 10,000
mammogramsCancer rate 1
3-E
If your mammogram is negative what is the
chance its a false negative? 15 / 8430
.0018 .18
18
Polygraphs Drug Tests
3-E
Suppose a polygraph is 90 accurate 1 of job
applicants lie 1000 applicants so 10 are
lying How many of those applicants who were
accused of lying were actually telling the truth?
19
Tree Diagram for Polygraphs90 accurate
3-E
20
Tree Diagram for Polygraphs
3-E
So 99/108 91.7 of those who are accused of
lying are not actually lying.
21
Athletic Drug Testing
3-E
Drug tests are about 95 accurate Assume 4 of
athletes use drugs Assume 1000 athletes at some
Event. What percentage of the athletes are
falsely suspended from the team?
22
Drug test 95 accurate, 1,000 athletesDrug use
rate 4
3-E
Use Drugs Dont Use Total
Drug Test
Drug Test -
Total 1,000
So 48/86 55.8 of those accused of using drugs
are not actually using them.
23
Drug test 95 accurate, 1,000 athletesDrug use
rate 4
3-E
Use Drugs Dont Use Total
Drug Test 86
Drug Test - 914
Total 40 960 1,000
So 48/86 55.8 of those accused of using drugs
are not actually using them.
24
Drug test 95 accurate, 1,000 athletesDrug use
rate 4
3-E
Use Drugs Dont Use Total
Drug Test .9540 38
Drug Test - .95960 912
Total 40 960 1,000
So 48/86 55.8 of those accused of using drugs
are not actually using them.
25
Drug test 95 accurate, 1,000 athletesDrug use
rate 4
3-E
Use Drugs Dont Use Total
Drug Test 38 48 86
Drug Test - 2 912 914
Total 40 960 1,000
So 48/86 55.8 of those accused of using drugs
are not actually using them.
26
Drug test 95 accurate, 1,000 athletesDrug use
rate 4
3-E
Use Drugs Dont Use Total
Drug Test 38 True 48 False 86
Drug Test - 2 False - 912 True - 914
Total 40 960 1,000
So 48/86 55.8 of those accused of using drugs
are not actually using them.
27
Political Mathematics
3-E
Republicans Tax cut would benefit all families
and the middle class would receive slightly
greater benefits. Democrats Tax cut would send
disproportionate benefits to the rich. Which side
was being more fair?
28
3-D
Republicans calculated the average tax cut that
would be received per family in each group. The
last bar shows that families with incomes over
200,000 would get an average tax cut of
2.9 Someone paying 100,000 in taxes would
reduce their taxes by 2900 while someone paying
1000 in taxes would reduce their taxes by only
29.
29
3-D
Democrats calculated the percentage of total
benefits that would be received by families in
each group. Families with incomes over 200,000
would receive 28.1 of the total benefits from
the tax cut. Because such families pay more than
¼ of the total income taxes collected, they would
see (as a group) more than ¼ of the total benefit
of any across the board cut.
30
3-D
Which side was being more fair? Neither! The
Republicans neglect the fact that most of the
total tax savings would go to the
wealthy. Democrats neglect the fact that the
wealthy already pay most of the taxes.
31
A Cut or an Increase?
3-D

• 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?

32
A Cut or an Increase?
3-D

• 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).

33

• Homework for Wednesday
• Pages 204-207
• 12, 13, 17, 18, 20, 23, 31a,b