AP STATISTICS AP Set: Probability Name:
AP STATISTICS AP Set: Probability Name: Every year, each student in a nationally representative sample is given tests in various subjects. Recently, a random sample of 9.600 twelfth-grade students from the United States were administered a multiple-choice US History exam. One of the multiple choice questions is below. (The correct answer is C.) Of the 9,600 students, 28 percent answered the multiple-choice question correctly. Assume that students who actually know the correct asnwer have a 100 percent chance of answering the question correctly, and students who do not know the correct answer to the question guess completely at random from among the four options. Let k represent the proportion of all United States twelfth-grade students who actually know the correct asnwer to the question. a) A tree diagram of the possible outcomes for a randomly selected twelfth-grade student is provided below. Write the correct probability in each of the five empty boxes. Some of the probabilities may be expressions in terms of k. b) Based on the completed tree diagram, express the probability, in terms of k, that a randomly selected twelfth-grade student would correctly answer the history question. The Probability of Penalizing the Innocent Due to Bad Test Results In modern society two-outcome tests are everywhere. They include drug tests, sobriety tests, disease tests, genetic tests, etc.. The outcome of these tests are either positive or negative, yes or no. We like to think these tests are at least 99% accurate, and yet, horror stories of spurious results seem to abound. Take company-wide drug testing, opponents may claim that at least a third of those identified as drug users will actually be innocent. If we assume the test is 99% accurate, this claim sounds ridiculous. But is it? To analyze the claim we will "grow" a decision tree. Decision trees are a wonderful little device for analyzing anything with two possible outcomes. Every time we reach the end of a branch and have two possibilities we simply create a set of two new branches. For our analysis, we will assume that 2% of all employees actually use drugs. This is lower than the general population but keep in mind that a lot of drug users are unemployed. Also, a company with a clearly stated anti-drug policy will probably have a low proportion of users. The tree's trunk represents the population of all employees. The first set of branches represent the two possible conditions: drug user, not drug user. Next, we add two sets of branches representing the drug test. One set of branches is attached to each of the original two branches. There is a 1% chance of getting a wrong or incorrect result from the test. Finally we add the tree's leaves. Each leaf represents a possible final outcome of the entire process. Note that there are four possibilities. Two of the four possibilities are correct: drug users and drug free individuals are both correctly identified. However, two of the four possibilities are spurious: drug users and drug free individuals are not correctly identified. We are unlikely to hear complaints from a drug user who is incorrectly identified as being drug free. The drug free person identified as a user is another matter. This would be a very upsetting situation. To find the probability of each final outcome as represented by the four leaves simply multiply the probabilities of each branch one must "climb" on the way to reaching the leaf. Note that all the leaf probabilities have to add up to 100%. The percentage of people identified as drug users who are actually innocent can be calculated. Your task: Draw the tree of the above situation. Calculate the probability that if a person fails the drug test they are actually clean.