Question

A recent study has shown that 10% of computer science concentrators suffer from symptoms of Carpal Tunnel Syndrome. Researchers have developed a diagnostic test for this condition and in a recent trial, the test gave a positive result in 96% of the patients who were known to truly have Carpal Tunnel Syndrome.

a) For an initial version of the diagnostic test, the false positive rate (the percentage of people without Carpal Tunnel who incorrectly test positive) was 44%. If a student received a positive result from this test, what would be the posterior probability that they truly have Carpal Tunnel Syndrome?
b) The researchers believe their test will be practically useful if a positive result implies the student has Carpal Tunnel with at least 75% probability. What is the largest possible false positive rate that would achieve this accuracy target?

Answer 1

1. 0.1951

2. 3.56%

Explanation:

Probability of disease = 0.10

Probability of disease and positive = 0.96

1.

Probability of no disease and positive = 0.44

Probability of disease and positive result = 0.10x0.96/0.10x0.96+0.90x0.44

= 0.096/0.096+0.396

= 0.096/0.492

= 0.1951

2.

Probability of disease false positive result

P(disease|positive result)

= (0.10x0.96/0.10x0.96+0.90xP) >= 0.75

= 0.096/0.096+0.90p >= 0.75

We cross multiply

0.096 >= 0.75(0.096+0.90p)

0.096 >= 0.072 + 0.675p

0.096-0.072 >= 0.675p

0.024 >= 0.675p

P <= 0.03555

Therefore the largest possible false positive rate = 0.0356

= 3.56%