False PositivesSensitive Surveys

Lesson 6.3.2

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.

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.

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.

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?

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

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.

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.

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!

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.

Homework

• Complete the worksheet.