Question

Antibody tests for the Covid-19 virus are not perfect. In New York City, 19% of the population are believed to have been infected with the virus. If the sensitivity of a test is 88.89%, and the false positive rate is 5.86%, what is the probability that a person has the antibodies, given that person got a positive result? Round your answer to the nearest percent. For example, enter 80 for 80 percent.   

Answer 1

We can make 2x2 contingency table which will be easier to solve.

That means that from a total of 100, 19 would be infected. That is total infected percentages.In New York City, 19% of the population are believed to have been infected with the virus. Instead of proportions, we will use absolute values of 100. So total is 100.

If the sensitivity of a test is 88.89%,

Sensitivity = TP / (TP + FN)

This is the probability of testing positive given infected.

Therefore

TP = 16.8891

The proportion of not infected = 1 - 0.19

= 0.81

Number of not infected = 81

the false positive rate is 5.86%,

FPR = FP / (FP + TN)

5.86% =

FP = 4.7466

	Infected	Not infected	Total
test +ve	16.8891 TP	4.7466 FP	21.6357
test -ve	2.1109 FN	76.2534 TN	78.3643
Total	19	81	100

Rest solve the cells by adding and subtracting normally.

what is the probability that a person has the antibodies, given that person got a positive result?

P(Infected | test +ve) = Infected and +ve / Total +ve

= 16.8891 / 21.6357

Ans: 0.78061