A medical test manufacturer did a trial of a new virus screening test. 100,000 people took the test, and they already knew from other testing whether or not each person was infected with the virus.
Infected (I) Not Infected (NI)
Test Positive (+) 17,600 1,800
Test Negative (–) 5,400 75,200
The manufacturer wants the false negative rate, P(- | I) to be less than 0.03. If the false positive rate is greater than 0.03, they will create a new test. Will they need to create a new test?
To determine if the manufacturer needs to create a new test, we need to compare the false negative rate (P(- | I)) to the threshold of 0.03, and the false positive rate (P(+ | NI)) to the threshold of 0.03.
First, let's calculate the false negative rate:
P(- | I) = (Number of infected people who test negative) / (Total number of infected people)
P(- | I) = 5,400 / (17,600 + 5,400)
P(- | I) ≈ 0.235
The false negative rate is approximately 0.235, which is greater than the threshold of 0.03. Therefore, the manufacturer needs to create a new test.
Next, let's calculate the false positive rate:
P(+ | NI) = (Number of non-infected people who test positive) / (Total number of non-infected people)
P(+ | NI) = 1,800 / (1,800 + 75,200)
P(+ | NI) ≈ 0.0235
The false positive rate is approximately 0.0235, which is less than the threshold of 0.03. Therefore, the false positive rate requirement is met and does not necessitate the creation of a new test.
In conclusion, the manufacturer needs to create a new test because the false negative rate is greater than the threshold of 0.03, but they do not need to create a new test based on the false positive rate.