5.3D WS: Calculated Risks

5.3D WS: Calculated Risks 5.3B Many high schools now have drug-testing programs for athletes. The main goal of these programs is to reduce the use of banned substances by students who play sports. It is not practical to test every athlete for drug use regularly. Instead, school administrators give drug tests to randomly selected student athletes at unannounced times during the school year. Students who test positive face serious consequences, including letters to their parents, required counseling, and suspension from athletic participation. Drug tests aren’t perfect. Sometimes the tests say that athletes took a banned substance when they did not. This is known as a false positive. Other times, drug tests say that athletes are “clean” when they did take a banned substance. This is called a false negative. Suppose that 16% of the high school athletes in a large school district have taken a banned substance. The drug test used by this district has a false positive rate of 5% and a false negative rate of 10%. Use what you have learned in this chapter to help answer the following questions about the district’s drugtesting program. Show your method clearly. 1. What’s the probability that a randomly chosen athlete tests positive for banned substances? 2. If two athletes are randomly selected, what’s the probability that at least one of them tests positive? 3. If a randomly chosen athlete tests positive, what’s the probability that the student did not take a banned substance? Based on your answer, do you think that an athlete who tests positive should be suspended from athletic competition for a year? Why or why not? 4. If a randomly chosen athlete tests negative, what’s the probability that the student took a banned substance? Explain why it makes sense for the drug-testing process to be designed so that this probability is less than the one you found in Question 3. 5. The district decides to immediately retest any athlete who tests positive. Assume that the results of an athlete’s two tests are independent. Find the probability that a student who gets a positive result on both tests actually took a banned substance. Based on your answer, do you think that an athlete who tests positive twice should be suspended from athletic competition for a year? Why or why not? Chapter 5 FRAPPY! A statistics teacher has 40 students in his class, 23 females and 17 males. At the beginning of class on a Monday, the teacher planned to spend time reviewing an assignment due that day. Unknown to the teacher, only 19 of the females and 11 of the males had completed the assignment. The teacher plans to randomly select students to do problems from the assignment on the whiteboard. 6. What is the probability that a randomly selected student has completed the assignment? 7. Are the events “selecting a female” and “selecting a student who completed the assignment” independent? Justify your answer. 8. If the teacher randomly chose a person who completed the assignment, what is the probability that the person was female? Suppose a particular rapid COVID-19 test has a false positive rate of 15% and a false negative rate of 30%. A school district uses this test as one of it’s safety measures for students who will be returning to class. Supposed that 1% of this population actually have COVID-19. 9. What is the probability that a randomly chosen student would test positive? 10. If a randomly chosen student tested positive, what is the probability that the student actually had COVID-19? 11. If a randomly chosen student tested negative, what is the probability that the student actually had COVID-19? 12. If there was a “pod” of 36 independently tested students who all tested negative, what is the probability that at least one of them actually has COVID-19?