Question

The reliability of a particular skin test for tuberculosis (TB) is as follows: the sensitivity of the test is 0.9 (the test comes back positive 90% of the time if the subject has TB) and the specificity of the test is 0.95 (if the subject does not have TB, the test comes back negative 95% of the time). In a large population, only 0.3% (0.003) of the people have TB. A person is selected at random and given the test, which comes back positive.

a) create a hypothetical 100,000 table to answer question b.

b) What is the probability that that person actually has TB?

Answer 1

Contingency table based on the information:

No disease	Disease
P(D)=0.997	P(D)=0.003
0.997	0.003
Test positive	Test negative	Test positive	Test negative
5.00%	95.00%	90.00%	10.00%
False Positive	True Negative	True positive	False negative
0.0499	0.9472	0.0027	0.0003

(a) If n = 100000

Population	100000	Total
Test	positive	Negative
Dis	No dis
Postive	270	4985	5255
Negative	30	94715	94745
Total	300	99700	100000

(b) P (has TB / test is positive) = Predictive positive vlaue = True positive/ (True positve + false positive)

P (has TB / test is positive) = 270 / (270 + 4985) = 0.0514

ANS: 0.0514