Question

A diagnostic test is used to detect a disease that occurs in 1% of the population. 8% of the disease free people test positive for the disease. 10% of people with the disease test negative for the disease. What proportion of people with a positive test result actually have the disease?

Answer 1

To find the proportion of people with a positive test result who actually have the disease, we can use Bayes' Theorem. The proportion is approximately 9.35%.

Step-by-step explanation:

To find the proportion of people with a positive test result who actually have the disease, we can use Bayes' Theorem. Let D be the event that a person has the disease, and T be the event that a person tests positive for the disease. The proportion of people with a positive test result who actually have the disease is given by:

P(D|T) = P(T|D) * P(D) / P(T)

From the given information, we have:

P(D) = 0.01 (since the disease occurs in 1% of the population)

P(T|D) = 1 - 0.10 = 0.90 (since 10% of people with the disease test negative)

P(T) = P(T|D) * P(D) + P(T|not D) * P(not D) = 0.90 * 0.01 + 0.08 * 0.99 = 0.0171 + 0.0792 = 0.0963

Substituting the values into Bayes' Theorem, we get:

P(D|T) = 0.90 * 0.01 / 0.0963 = 0.009 / 0.0963 = 0.0935 (approximately)

Therefore, the proportion of people with a positive test result actually having the disease is approximately 0.0935 or 9.35%.