Thread: Reliability of a Test

15%of the U.S. population has math fever. If an individual has math fever, a testaccurately identifies the condition 80% of the time. If an individual does nothave math fever, the test accurately identifies the individual as not havingmath fever 90% of the time. The following list describes theseoutcomes:

MathFever 15%
PosTest 80%
NegTest 20%

NoMath Fever 85%
PosTest 10%
NegTest 90%

What percent of the time is the test inaccurate?

What is the probability of a false positive?

What is the probability that the individual does not have math fever given that thetest is negative?

I know that the answers should be:
0.885, 0.115, 9623
but I'm not sure how to set up any of these

What does it mean to have a false positive? You know that a a test says you have the fever if you actually have the fever 80% of the time. It also says you don't have the fever when you don't actually have the fever 90% of the time. Consider now that you don't actually have the fever but it gives you a positive test anyway. You should have something like

[Not Sick] * [Positive Test] + [Sick] * [Negative Test] = [False Positive]

In other words, the given how many people are sick or not sick and the chances that the test gets it wrong, you should be able to figure out its false positive rate. I got the correct answer, and from the expression above, so should you. The key here is understanding why it is set up that way. Do you see why?

Ahh yes, that certainly makes much more sense. How would you approach the inaccurate question and what would be the difference?

Think about the definition. A false positive is precisely that: how likely is a test to indicate the false of what is actually (positively) happening? Inaccuracy is how off the mark a test is. This is about the positive rate itself. Using the above algorithm, just think how many of the population that are sick will test sick or how many of the population that aren't sick will test not sick? The answer falls right out. As for the last question, it is a conditional probability. What do you know about that so far?

Thanks so much Bryan! I finally got it! (I didn't realize the last one was conditional probability--thanks!)