Question

Testing the test Are false positives too common in some medical tests? Researchers conducted an experiment involving $250$ patients with a medical

condition and $750$ other patients who did not have the medical condition. The medical technicians who were reading the test results were unaware that they

were subjects in an experiment.

(a) Technicians correctly identified $240$ of the $250$patients with the condition. They also identified $50$ of the healthy patients as having the condition. What

were the false positive and false negative rates for the test?

(b) Given that a patient got a positive test result, what is the probability that the patient actually had the medical condition? Show your work.

Short answer

Part (a) The rates are $6.67\%$ and $4\%$ respectively.

Part (b) The probability is $0.8276$

Step-by-step solution

Part (a) Step 1. Given Information

The table is made up of the following components:

Part (a) Step 2. Concept

$Probability\left(p\right)=\frac{Numberoffavourable}{Totalnumberofexhaustive}$

Part (a) Step 3. Calculation

$240$of the $250$patients with the disease were accurately recognised by technicians. They also recognised $50$healthy people as having the disease, therefore the test's false positive (FP) and false negative (FN) rates can be estimated as follows:

$P\left(FP\right)=\frac{50}{750}=0.0667\%=6.67\%$

$P\left(FN\right)=\frac{10}{750}=0.0400=4.00\%$

As a result, the needed rates are 6.67% and 4%, respectively.

Part (b) Step 1. Calculation

Given a positive test result, the likelihood that the selected patient has the medical condition can be computed as follows:

$P\left(positivetest\right)=\frac{240+50}{1000}=0.290$

$P\left(Incorrectandpositive\right)=\frac{240}{1000}=0.240$

$P\left(Incorrectidentifyandpositivetest\right)=\frac{0.240}{0.290}=82.76\%$

As a result, $0.8276$ is the required probability.