A random person is selected from a large population in which 5% are users of a dangerous illegal drug.A drug test that correctly identifies users 99% of the times and nonusers 95% of the time (this percentage is called "specificity" of the test)is administered to this individual and gives a positive result.What specificity would actually be required to conclude that the probability that this individual is actually a user of this drug is 0.75?
A)0.965
B)0.973
C)0.978
D)0.983
E)0.985

Question

Quizplus · Accepted Answer

To achieve a probability of 0.75 that an individual who tested positive is actually a drug user, we can use Bayes' theorem. The desired probability (posterior probability) can be calculated using the formula:$$ P(\text{User}|\text{Positive}) = \frac{P(\text{Positive}|\text{User}) \times P(\text{User})}{P(\text{Positive})} $$Given:- $ P(\text{User}) = 0.05 $ (prevalence)- $ P(\text{Positive}|\text{User}) = 0.99 $ (sensitivity)- $ P(\text{User}|\text{Positive}) = 0.75 $ (desired posterior probability)We need to find the specificity ($ P(\text{Negative}|\text{Non-User}) $) that would adjust the denominator $ P(\text{Positive}) $ to achieve this posterior probability. The denominator is a combination of true positives and false positives:$$ P(\text{Positive}) = P(\text{Positive}|\text{User}) \times P(\text{User}) + P(\text{Positive}|\text{Non-User}) \times P(\text{Non-User}) $$$$ P(\text{Positive}|\text{Non-User}) = 1 - \text{Specificity} $$Given the desired posterior probability, we can rearrange the formula to solve for specificity. However, without going through the detailed algebra, we know that to increase the probability that a positive test indicates actual drug use to 0.75, the specificity must be very high to reduce the number of false positives. Among the given options, D) 0.983 is the correct specificity required, as it would sufficiently reduce the proportion of false positives, making the probability of being a user given a positive test result closer to 0.75.

A Random Person Is Selected from a Large Population in Which