Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 1 Numbers in the Real World 3

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 2 Unit 3E How Numbers Deceive: Polygraphs, Mammograms, and More

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 3 Example Kevin had a higher shooting percentage both in the first half (40% to 25%) and in the second half (75% to 70%). Does this mean that Kevin had the better shooting percentage for the game?

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 4 Example (cont) No, Kevin made a total of 7 baskets (4 in the first half and 3 in the second half) on 14 shots (10 in the first half and 4 in the second half), for an overall shooting percentage of 7/14 = 50%. Kobe made a total of 8 baskets on 14 shots, for an overall shooting percentage of 8/14 = 57.1%. Even though Kevin had a higher shooting percentage in both halves, Kobe had a better overall shooting percentage for the game.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 5 Test Results true positive false positive true negative false negative A test correctly reports a positive result A test incorrectly reports a positive result A test correctly reports a negative result A test incorrectly reports a negative result

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 6 Example Based on the numbers in Table 3.7, what is the percentage of women with negative test results who actually have cancer (false negatives)?

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 7 Example (cont) Solution For the 10,000 cases summarized in Table 3.7, the mammograms are negative for 15 women with cancer and for 8415 women with benign tumors. The total number of negative results is = The percentage of women with false negatives is 15/8430 = = 0.18%, or slightly less than 2 in In other words, the chance that a woman with a negative mammogram has cancer is very small.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 8 Polygraphs Suppose that 1000 people take the polygraph test, 10 of whom lie, and the polygraph is 90% accurate. How many of those applicants who were accused of lying were actually telling the truth?

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 9 Example All athletes participating in a regional high school track and field championship must provide a urine sample for a drug test. Those who test positive for drugs are eliminated from the meet and suspended from competition for the following year. Studies show that, at the laboratory selected, the drug tests are 95% accurate. Assume that 4% of the athletes actually use drugs. What fraction of the athletes who fail the test are falsely accused and therefore suspended without cause?

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 10 Example (cont) Solution The easiest way to answer this question is by using some sample numbers. Suppose there are 1000 athletes in the meet. We are told to assume that 4%, or 40 athletes, actually use drugs. The remaining 960 athletes do not use drugs. In that case, the 95% accurate drug test should return the following results: 95% of the 40 athletes who use drugs, or 0.95 × 40 = 38 athletes, test positive. The other 2 athletes who use drugs test negative.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 11 Example (cont) 95% of the 960 athletes who do not use drugs test negative, but 5% of these 960 athletes test positive. The number of athletes who fail despite not using drugs is 0.05 × 960 = 48. The total number of athletes who test positive is = 86. But 48 of these athletes, or 48/86 = 56%, are actually nonusers. Despite the 95% accuracy of the drug test, more than half of the suspended students are innocent of drug use.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 12 Example Consider the two charts shown on the next slide. Both purport to show effects in 2011 of the tax cuts originally enacted under President Bush in The chart in Figure 3.5a, created by supporters of the tax cuts, indicates that the rich ended up paying more under the tax cuts than they would have otherwise. Figure 3.5b, created by opponents of the tax cuts, shows that the rich received far more benefit from the tax cuts than lower income taxpayers. The two charts therefore seem contradictory, because the first seems to indicate that the rich paid more while the second seems to indicate that they paid less.

Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 3, Unit E, Slide 13 Political Mathematics In fact, both of the graphs are accurate and show data from reputable sources.