Section 3E How Numbers Deceive: Polygraphs, Mammograms and More Polygraph tests are reputed to be 90% accurate. That is, they supposedly catch 90% of the people who are lying and validate 90% of the people who are telling the truth. – So what percent of the people who fail a polygraph test are falsely accused of lying? 10%? No - much higher – as we will see. Numbers may not lie, but they can be deceiving if we do not interpret them carefully. 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. This is a very straight-forward example of Simpson’s Paradox to use as an introduction. 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!

Women were being discriminated against! Simpson’s Paradox Pg201 University of California – Berkeley Graduate Admissions, 1973 Gender Discrimination?? Men Women Applied Admitted Percent Total 2691 1198 44.5% 1835 557 30.4% Women were being discriminated against!

3-E Men Women Department 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 1 in 100 (1%) breast tumors turn out to be malignant. 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. False Negative identify 85% of benign tumors as benign. misidentify 15% of benign tumors as malignant. Tumors = any kind of abnormal swelling or tissue growth. A tumor caused by cancer is malignant (cancerous), while others are benign. Only about 1 in 100 breast tumors turn out to be malignant. False Positive negative mammogram means no cancer – benign. positive mammogram means cancer – malignant.

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) 10,000

Cancer No Cancer Total Mammogram + Test Mammogram - Test 100 9,900 10,000

Cancer No Cancer Total Mammogram + Test .85×100 =85 Mammogram – Test .85×9900 =8415 100 9,900 10,000

Cancer No Cancer Total Mammogram + Test 85 1485 1570 Mammogram – Test 8415 8430 100 9,900 10,000 Use the summary chart to answer the question.

Of those women with positive mammograms, only 85 out of 1570 or Cancer No Cancer Total Mammogram + 85 True + 1485 False + 1570 Mammogram - 15 False - 8415 True - 8430 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.

Cancer No Cancer Total True + False + False - True - 100 Mammogram + 85 True + 1485 False + 1570 Mammogram - 15 False - 8415 True - 8430 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.)

About Polygraphs (pp 203-4) 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) 1,000 Assuming 1% of the population lies (1000x1% = 10).

3-E Lie Tell Truth Total Polygraph + Test (Lie) 9 99 108 Polygraph – Test (Truth) 1 891 892 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.]

Tree Diagram for Polygraphs Note: 1/892 = .11% are permitted to get away with lying – the polygraph thinks they are telling the truth when they aren’t. 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.

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) 1,000

3-E Drugs No drugs Total Drug Test + Test (Drugs) 38 48 86 Drug Test – Test (No Drugs) 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.]

(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