Question 1170728: A medical test has a 95% accuracy of detecting a Condition Z if the person has it. It also has a 97% chance to indicate that the person does not have the condition if they really don't have it. If the incidence rate of this disease is 10 out of every 100:
What is the probability that a person chosen at random will both test positive and actually have the disease (i.e., get a true positive)?
What is the probability that a person chosen at random will test positive but not have the disease (i.e., get a false positive)?
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Look at 1000
=====D+========D-===Total
T+----95----------27-----122
T- ----5---------873-----878
Total=100======900===1000
Use a round number like 1000
you know that 100 have the disease
95 of them test positive
5 test negative. You fill that in
You also know that 900 don't have the disease
97% of them test negative as well, so that is 873 of the 900
3% test positive, so that is where the 27 comes from.
Check the horizontal and vertical parts to make sure everything adds up.
then for test positive, read across, and 95 have the disease.
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The issue here is choosing one at random. This means you have to look at one of the four middle squares and divide it by 1000. If you look at a true positive, that is when you test positive, you really have the disease. That is different altogether.
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Multiply the Total with disease by the Sensitivity to get the number of True Positives. Total with disease is 100* 0.95=95. That is 95/1000 or 9.5% if chosen at random from the population.Multiply the Total without disease (900) by the Specificity (0.97) to get the number of True Negatives (873). The number of False Positives, where they have a positive test but no disease is 27 out of the entire 1000 population or 2.7%.
What I see as a false Positive rate are of those who test positive (122), 27 of them do not have the disease, so that the false positive rate is 27/122 or 22.1%.
true positive is 9.5%
false positive is 2.7% WHEN CHOSEN AT RANDOM FROM THE WHOLE POPULATION, NOT IF LOOKING AT THE SUBGROUP OF THOSE WHO TESTED POSITIVE. BIG DIFFERENCE.
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