Question

The American Disease X Foundation reports that 6% of the population over 50 years of age has Disease X. You inquire as to the source of their information, and they cite disease population screening data in the literature which reports that 6% of that population was positive when screened. Referring to the literature, you discover that the screening test used had sensitivity of 95% and specificity of 98%. What proportion of the population over 50 years of age do you think really has the disease?

Answer 1

Let the proportion of population having disease be p

P(having disease) = p

P(not having disease) = 1 - p

Sensitivity = 0.95

P(testing positive | having disease) = 0.95

Specificity = 0.98

P(testing positive | not having disease) = 1 - 0.98 = 0.02

P(testing positive) = 0.06

P(testing positive and having disease) + P(testing positive and not having disease) = 0.06

p x 0.98 + (1 - p) x 0.02 = 0.06

0.98 p + 0.02 - 0.02p = 0.06

0.96p = 0.04

p = 0.0417

4.17% of population over 50 years of age do you think really has the disease