1. Copyright © 2009 Pearson Education, Inc. 4.4 Statistical Paradoxes LEARNING GOAL Investigate a few common paradoxes that arise in statistics, such as how it is possible that most people who fail a “90% accurate” polygraph test may actually be telling the truth.

2. Slide 4.4- 2 Copyright © 2009 Pearson Education, Inc. Better in Each Case, But Worse Overall It is possible for something to appear better in each of two or more group comparisons but actually be worse overall. This occurs because of the way in which the overall results are divided into unequally sized groups.

3. Slide 4.4- 3 Copyright © 2009 Pearson Education, Inc. Table 4.7 gives the shooting performance of two players in each half of a basketball game. Shaq had a higher shooting percentage in both the first half (40% to 25%) and the second half (75% to 70%). Can Shaq claim that he had the better game? Solution: No, and we can see why by looking at the overall game statistics. Shaq 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%. Vince made a total of 8 baskets on 14 shots, for an overall shooting percentage of 8/14 = 57.1%. Surprisingly, even though Shaq had a higher shooting percentage in both halves, Vince had a better overall shooting percentage for the game. EXAMPLE 1 Who Played Better?

4. Slide 4.4- 4 Copyright © 2009 Pearson Education, Inc. Does a Positive Mammogram Mean Cancer? We often associate tumors with cancers, but most tumors are not cancers. Medically, any kind of abnormal swelling or tissue growth is considered a tumor. A tumor caused by cancer is said to be malignant (or cancerous); all others are said to be benign. About 1 in 100 breast tumors turns out to be malignant.

5. Slide 4.4- 5 Copyright © 2009 Pearson Education, Inc. Suppose a patient’s mammogram comes back positive. Mammograms are not perfect, so the positive result does not necessarily mean that she has breast cancer. Let’s assume that the mammogram screening is 85% accurate: It will correctly identify 85% of malignant tumors as malignant and 85% of benign tumors as benign. Because the mammogram screening is 85% accurate, most people guess that the positive result means that the patient probably has cancer. Consider a study in which mammograms are given to 10,000 women with breast tumors. Assuming that 1% of tumors are malignant, 1% × 10,000 = 100 of the women actually have cancer; the remaining 9,900 women have benign tumors.

6. Slide 4.4- 6 Copyright © 2009 Pearson Education, Inc. Table 4.8 summarizes the mammogram results. The mammogram screening correctly identifies 85% of the 100 malignant tumors as malignant. Thus, it gives positive (malignant) results for 85 of the malignant tumors; these cases are called true positives. In the other 15 malignant cases, the result is negative, even though the women actually have cancer; these cases are false negatives. The mammogram screening correctly identifies 85% of the 9,900 benign tumors as benign. Thus, it gives negative (benign) results for 85% × 9,900 = 8,415 of the benign tumors; these cases are true negatives. The remaining 9,900 – 8,415 = 1,485 women get positive results in which the mammogram incorrectly identifies their tumors as malignant; these cases are false positives.

7. Slide 4.4- 7 Copyright © 2009 Pearson Education, Inc. Overall, the mammogram screening gives positive results to 85 women who actually have cancer and to 1,485 women who do not have cancer. The total number of positive results is 85 + 1,485 = 1,570. Because only 85 of these are true positives (the rest are false positives), the chance that a positive result really means cancer is only 85/1,570 = 0.054, or 5.4%. Therefore, when a patient’s mammogram comes back positive, there’s still only a small chance that she has cancer.

8. Slide 4.4- 8 Copyright © 2009 Pearson Education, Inc. By the Way... The accuracy of breast cancer screening is rapidly improving; newer technologies, including digital mammograms and ultrasounds, appear to achieve accuracies near 98%. The most definitive test for cancer is a biopsy, though even biopsies can miss cancers if they are not taken with sufficient care. If you have negative tests but are still concerned about an abnormality, ask for a second opinion. It may save your life.

9. Slide 4.4- 9 Copyright © 2009 Pearson Education, Inc. Suppose you are a doctor seeing a patient with a breast tumor. Her mammogram comes back negative. Based on the numbers in Table 4.8, what is the chance that she has cancer? Solution: For the 10,000 cases summarized in Table 4.8, the mammograms are negative for 15 women with cancer and for 8,415 women with benign tumors. The total number of negative results is 15 + 8,415 = 8,430. Thus, the fraction of women with cancer who have false negatives is 15/8,430 = 0.0018, or slightly less than 2 in 1,000. In other words, the chance that a woman with a negative mammogram has cancer is only about 2 in 1,000. EXAMPLE 2 False Negatives

10. Slide 4.4- 10 Copyright © 2009 Pearson Education, Inc. TIME OUT TO THINK While the chance of cancer with a negative mammogram is small, it is not zero. Therefore, it might seem like a good idea to biopsy all tumors, just to be sure. However, biopsies involve surgery, which means they can be painful and expensive, among other things. Given these facts, do you think that biopsies should be routine for all tumors? Should they be routine for cases of positive mammograms? Defend your opinion.

11. Slide 4.4- 11 Copyright © 2009 Pearson Education, Inc. Polygraphs and Drug Tests Suppose the government gives the polygraph test to 1,000 applicants for sensitive security jobs. Further suppose that 990 of these 1,000 people tell the truth on their polygraph test, while only 10 people lie. For a test that is 90% accurate, we find the following results: Of the 10 people who lie, the polygraph correctly identifies 90%, meaning that 9 fail the test (they are identified as liars) and 1 passes. Of the 990 people who tell the truth, the polygraph correctly identifies 90%, meaning that 90% × 990 = 891 truthful people pass the test and the other 10% × 990 = 99 truthful people fail the test.

12. Slide 4.4- 12 Copyright © 2009 Pearson Education, Inc. Figure 4.16 A tree diagram summarizes results of a 90% accurate polygraph test for 1,000 people, of whom only 10 are lying. The total number of people who fail the test is 9 + 99 = 108. Of these, only 9 were actually liars; the other 99 were falsely accused of lying.

13. Slide 4.4- 13 Copyright © 2009 Pearson Education, Inc. That is, 99 out of 108, or 99/108 = 91.7%, of the people who fail the test were actually telling the truth. Assuming the government rejects applicants who fail the polygraph test, then almost 92% of the rejected applicants were actually being truthful and may have been highly qualified for the jobs.

14. Slide 4.4- 14 Copyright © 2009 Pearson Education, Inc. TIME OUT TO THINK Imagine that you are falsely accused of a crime. The police suggest that, if you are truly innocent, you should agree to take a polygraph test. Would you do it? Why or why not?

15. Slide 4.4- 15 Copyright © 2009 Pearson Education, Inc. All athletes participating in a regional high school track and field championship must provide a urine sample for a drug test. Those who fail 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? Solution: The easiest way to answer this question is by using some sample numbers. Suppose there are 1,000 athletes in the meet. Then 4%, or 40 athletes, actually use drugs; the remaining 960 athletes do not use drugs. EXAMPLE 3 High School Drug Testing

16. Slide 4.4- 16 Copyright © 2009 Pearson Education, Inc. Solution: (cont.) 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, fail the test. The other 2 athletes who use drugs pass the test. 95% of the 960 athletes who do not use drugs pass the test, but 5% of these 960, or 0.05 × 960 = 48 athletes, fail. The total number of athletes who fail the test is 38 + 48 = 86. But 48 of these athletes who fail the test, 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. EXAMPLE 3 High School Drug Testing

17. Slide 4.4- 17 Copyright © 2009 Pearson Education, Inc. The End