Section 3EHow Numbers Deceive: Polygraphs, Mammograms and More

Pages 199-212

Ex1/200- 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%

Simpson’s Paradox

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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!

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Men Women

Applied Admitted Percent Applied Admitted Percent

Total 2691 1198 44.5% 1835 557 30.4%

Simpson’s Paradox

Pg201 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!

About Mammograms (pg202)

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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

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?

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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 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?

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.)

Ex3/203- 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?

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About Polygraphs (pp 203-4)

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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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(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

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.

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-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

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?

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%).

Homework

Pages 207-212# 22, 25,27, 28, 30, 31

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