Transcript Section 3B Putting Numbers in Perspective

Section 3E
How Numbers Deceive:
Polygraphs, Mammograms
and More
Pages 199-212
3-E
Ex1/200- Who Played Better?
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%
3-E
Simpson’s Paradox
Simpson’s 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: Don’t average percentages!
3-E
Simpson’s Paradox
Pg201 University of California – Berkeley
Graduate Admissions, 1973
Gender Discrimination??
Men
Applied
Total
2691
Admitted
1198
Women
Percent
44.5%
Applied
1835
Admitted
Percent
557
Women were being discriminated against!
30.4%
3-E
Men
Department
Applied
Admitted
Women
%
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!
3-E
About Mammograms
(pg202)
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.
-identify 85% of benign tumors as benign.
-misidentify 15% of benign tumors as malignant.
negative mammogram means no cancer – benign.
positive mammogram means cancer – malignant.
False
Negative
False
Positive
3-E
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?
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
3-E
Cancer
No Cancer
Total
100
9,900
10,000
Mammogram +
Test
Mammogram Test
Total
3-E
Cancer
Mammogram +
Test
Total
.85×100
=85
Mammogram –
Test
Total
No Cancer
.85×9900
=8415
100
9,900
10,000
3-E
Cancer
No Cancer
Total
Mammogram +
Test
85
1485
1570
Mammogram –
Test
15
8415
8430
100
9,900
10,000
Total
Use the summary chart to answer the question.
3-E
Cancer
No Cancer
Total
Mammogram +
85
True +
1485
False +
1570
Mammogram -
15
False -
8415
True -
8430
100
9,900
10,000
Total
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.
3-E
Cancer
No Cancer
Total
Mammogram +
85
True +
1485
False +
1570
Mammogram -
15
False -
8415
True -
8430
100
9,900
10,000
Total
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.)
3-E
About Polygraphs
(pp 203-4)
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.
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
3-E
Lie
Tell Truth
Total
Polygraph +
Test (Lie)
9
99
108
Polygraph –
Test (Truth)
1
891
892
10
990
1,000
Total
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.]
3-E
Tree Diagram for Polygraphs
So 99/108 = 91.7% of those who are accused of lying are not
actually lying.
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.
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
3-E
Drug Test +
Test (Drugs)
Drug Test –
Test (No Drugs)
Total
Drugs
No drugs
Total
38
48
86
2
912
914
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.]
3-E
(ex5/206) A Cut or an Increase?
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%).
3-E
Homework
Pages 207-212
# 22, 25,27, 28, 30, 31