Section 3E How numbers deceive Pages

Simpson’s 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?

Simpson’s Paradox 3-E Shaq overall %: 7 / 14 = 50% Vince overall %: 8 / 14 = 57.1%

Simpson’s Paradox 3-E Simpson’s 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.

3-E MenWomen AppliedAdmittedPercentAppliedAdmittedPercent Total % % Simpson’s Paradox – A Famous Example University of California – Berkeley Graduate Admissions, 1973 Gender Discrimination??

3-E MenWomen DepartmentAppliedAdmitted%AppliedAdmitted% A % % B %251768% C % % D % % E % % F374226%341247% Total % %

Does Smoking Make You Live Longer? Early 1970’s – 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. 3-D

Does Smoking Make You Live Longer? 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. 3-D

Conditional Probability 3-E Does a positive mammogram always mean cancer? Positive test + True positive – identify malignant tumors as malignant False positive – identify benign tumors as malignant Negative test - True negative – identify benign tumors as benign False negative – identify malignant tumors as benign

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E CancerNo Cancer Total + Test (malignant) Mammogram + Test (malignant) – Test (benign) Mammogram – Test (benign) Total10,000

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E CancerNo Cancer Total + Test (malignant) Mammogram + Test (malignant) – Test (benign) Mammogram – Test (benign) Total1009,90010,000

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E CancerNo Cancer Total + Test (malignant) Mammogram + Test (malignant).85 ×100 =85 – Test (benign) Mammogram – Test (benign).85 ×9,900 =8,415 Total1009,90010,000

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E CancerNo Cancer Total + Test (malignant) Mammogram + Test (malignant) ,570 – Test (benign) Mammogram – Test (benign)158,4158,430 Total1009,90010,000

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E CancerNo Cancer Total + Test (malignant) Mammogram + Test (malignant)85 True + 1,485 False + 1,570 – Test (benign) Mammogram – Test (benign)15 False - 8,415 True - 8,430 Total1009,90010,000

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer 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 it’s a false negative? (You have cancer in spite of being told you don’t?)

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E CancerNo Cancer Total + (malignant) Mammogram + (malignant)85 True + 1,485 False + 1,570 - (benign) Mammogram - (benign)15 False - 8,415 True - 8,430 Total1009,90010,000

Mammography 85% accurate results for 10,000 mammograms Cancer rate = Mammography 85% accurate results for 10,000 mammograms Cancer rate = 1% 3-E If your mammogram is negative – what is the chance it’s a false negative? 15 / 8430 =.0018 =.18% 15 / 8430 =.0018 =.18%

Polygraphs & Drug Tests 3-E Suppose a polygraph is 90% accurate 1%lie 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?

3-E LieTell TruthTotal + (Lie) Polygraph Test + (Lie) - (Truth) Polygraph Test - (Truth) Total1,000

3-E LieTell TruthTotal + (Lie) Polygraph Test + (Lie) - (Truth) Polygraph Test - (Truth) Total ,000

3-E LieTell TruthTotal + (Lie) Polygraph Test + (Lie) 9 - (Truth) Polygraph Test - (Truth) 891 Total ,000

LieTell TruthTotal (Lie) Polygraph Test +(Lie) (Truth) Polygraph Test - (Truth) Total ,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.] 3-E

Tree Diagram for Polygraphs 90% accurate 3-E

Tree Diagram for Polygraphs 3-E So 99/108 = 91.7% of those who are accused of lying are not actually lying.

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?

Drug test 95% accurate, 1,000 athletes Drug use rate = 4% 3-E Drug UserCleanTotal + (drugs) Drug Test + (drugs) – (no drugs) Drug Test – (no drugs) Total1,000

Drug test 95% accurate, 1,000 athletes Drug use rate = 4% 3-E Drug UserCleanTotal + (drugs) Drug Test + (drugs)86 – (no drugs) Drug Test – (no drugs)914 Total409601,000

Drug test 95% accurate, 1,000 athletes Drug use rate = 4% 3-E Drug UserCleanTotal + Drug Test +.95 ×40 = 38 - Drug Test -.95 ×960 = 912 Total409601,000

Drug test 95% accurate, 1,000 athletes Drug use rate = 4% 3-E Drug UserCleanTotal + drugs Drug Test + drugs – no drugs Drug Test – no drugs Total409601,000

Drug test 95% accurate, 1,000 athletes Drug use rate = 4% 3-E Drug UserCleanTotal + drugs Drug Test + drugs38 True + 48 False + 86 – no drugs Drug Test – no drugs2 False True Total409601,000 So 48/86 = 55.8% of those accused of using drugs were actually clean and falsely suspended.

Drug test 95% accurate, 1,000 athletes Drug use rate = 4% 3-E Drug UserCleanTotal + drugs Drug Test + drugs38 True + 48 False + 86 – no drugs Drug Test – no drugs2 False True Total409601,000 Of those athletes that passed the drug test, 2 out of 914 or 2/914 =.0021 =.21% were drug users.

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?

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.

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.

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.

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? 3-D

A Cut or an Increase? 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-D

Homework for Wednesday: Pages # 14, 15, 17, 18, 20, 23, 26