section 3e how numbers deceive: polygraphs, mammograms and more
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Section 3E How Numbers Deceive: Polygraphs, Mammograms and More. Pages 195-202. 3-E. Ex1/196- 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. - PowerPoint PPT PresentationTRANSCRIPT
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Section 3EHow Numbers Deceive: Polygraphs, Mammograms and More
Pages 195-202
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Ex1/196- Who Played Better?
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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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Simpson’s Paradox
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Simpson’s Paradox(pg 196) 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!
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Men Women
Applied Admitted Percent Applied Admitted Percent
Total 2691 1198 44.5% 1835 557 30.4%
Simpson’s Paradox
Pg197 University of California – BerkeleyGraduate Admissions, 1973Gender Discrimination??
Women were being discriminated against!
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Men WomenDepartment 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 (pg198)
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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
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pg 198- 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?
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Cancer No Cancer Total
Mammogram + Test(malignant)
Mammogram – Test (benign)
Total 10,000
Build a summary chart (based on percent of) for 10000 mammograms of women with breast tumors.
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Cancer No Cancer Total
Mammogram + Test
Mammogram - Test
Total 100 9,900 10,000
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Cancer No Cancer Total
Mammogram + Test
.85×100=85
Mammogram – Test
.85×9900=8415
Total 100 9,900 10,000
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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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Cancer No Cancer Total
Mammogram + 85True +
1485False +
1570
Mammogram - 15False -
8415True -
8430
Total 100 9,900 10,000
Of those women with positive mammograms, only 85 out of 1570 or85/1570 = .054 = 5.4% actually have cancer.
pg 198- 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?
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Cancer No Cancer Total
Mammogram + 85True +
1485False +
1570
Mammogram - 15False -
8415True -
8430
Total 100 9,900 10,000
Of those women with negative mammograms, 15 out of 8430 or 15 / 8430 = =.0018 = .18% actually have cancer(about 2 women in 1000.)
Ex3/199- 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?
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About Polygraphs (pg 199)
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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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(pg 200) 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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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 Polygraphs3-E
So 99/108 = 91.7% of those who are accused of lying are not actually lying.
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About Drug Tests (ex4/200)
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-Eex4/200 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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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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(ex5/202) 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?
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Absolute change:$102 million - $100 million = $2 millionThis 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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Homework
Pages 204-207# 12, 15,17, 18, 20, 21
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