Question

Consider a diagnostic test for a hypothetical disease based on measuring the amount of a

certain biomarker present in blood. High levels of the biomarker are often found in individuals

with the disease, but a number of non-disease conditions can also cause high levels

of the biomarker. Individuals without the disease have biomarker levels that are normally

distributed with mean 1.6 ng/mL (nanograms per milliliter of blood), and standard deviation

0.50 ng/mL. Individuals with the disease have biomarker levels that are normally

distributed with mean 5 ng/mL and standard deviation 1.2 ng/mL. Values of 2.5 ng/mL

and higher constitute a positive test result. Compute the accuracy of the test for those who have the disease and the accuracy of

the test for those who do not have the disease.

Answer 1

For those without the disease, we are given the distribution here as:

The accuracy here is determined by finding the probability of a negative result. Therefore the accuracy for non disease subjects is computed here as:
P( X < 2.5)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

therefore for those who do not have the disease, the accuracy of the test is 96.41%.

Now for those with the disease the distribution is given here as:

The probability accuracy or the probability of correct result here is computed as:

P(X > 2.5)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

therefore 98.14% is the required accuracy here.