1. False Positives Sensitive Surveys Lesson 6.3.2

2. Starter A bag contains 5 red marbles and 4 blue marbles. Two marbles are drawn without replacement. What is the probability they are both red? –Answer using the conditional probability formula we saw yesterday. –Answer by a different approach involving combination theory.

3. Objectives Students will analyze a disease testing model to determine the probability that a positive test result really means a positive finding. Students will participate in a survey of a personally sensitive issue designed to correct for evasive responses.

4. False Positives Disease testing is usually quite accurate, but occasionally a mistake is made. Suppose you take a test for cancer detection that is known to be 98% accurate, and the result comes back positive. What is the probability that you have cancer? Most people would say the probability is 98%, but it turns out not to be so simple.

5. Taking Disease Incidence Into Account Suppose further that it is known that only about 1% of the general population actually has the disease. How does that affect the question of probability? To answer, draw a branching diagram with two outcomes and two levels (like flipping a coin twice). –Let the first level be whether or not a person has the disease. Start with 10,000 people and show how many go to each branch. –At the second level, assume all people took the test and the test is correct 98% of the time. Show how many people are at the end of each of the branches. Now answer the main question: If you are told the test was positive, what is the probability that you really DO have the disease?

6. Sensitive Issues Surveys What proportion of my students have ever cheated on a test? (Any test, not just mine!) If I ask that question, I am likely to get at least some untruthful answers, so how can I estimate the TRUE proportion? Flip a coin and note the heads / tails outcome. –Don’t show anyone else what you got. When I ask you if you have ever cheated, answer as follows: –If you flipped heads, answer YES regardless of the truth. –If you flipped tails, answer YES or NO, whichever is true

7. Analyzing Responses (in general) Assume (just for a moment) that the true proportion of those who have cheated is 40%. Assume further that the coins came up about 50% heads. If 40 people participate in this survey, how many would say “yes”? –Draw another two-stage branching diagram –Let the coin be first and the response be second You should have 28 “yes” and 12 “no” responses –How could you manipulate those results to find the true 40%? –Subtract the 20 “yes” answers that came from heads, then calculate based on what’s left. –In general, if there are n responses, subtract n/2 “yes” answers and calculate proportion remaining.

8. Analyzing Your Responses I will remind you of the number of “yes” responses in this class and the total number of responses of any kind. Based on those numbers, calculate the estimated proportion who have cheated.

9. Problems with this Method? With a large enough group, this method should give a reasonable estimate of the true proportion. What factors might cause the estimate to be wrong? –People might still answer untruthfully –The coin flips don’t have to come out exactly 50% heads But it’s a lot better than nothing!

10. Objectives Students will analyze a disease testing model to determine the probability that a positive test result really means a positive finding. Students will participate in a survey of a personally sensitive issue designed to correct for evasive responses.

11. Homework Complete the worksheet.worksheet.