A new medical test provides a false positive result for hepatitis 2% of the time that is a perfectly healthy subject being tested for hepatitis will test as being infected 2% of the time. And research, the test is given to 30 healthy (not having hepatitis) subjects. Let X be the number of subjects who test positive for the disease A. What is the probability that all 30 subjects will appropriately test as not being infected? B. What are the mean and standard deviation of X? C. To what extent do you think this is a viable test to use in the field of medicine?

A. To find the probability that all 30 subjects will test as not being infected, we first need to understand the probability of a single healthy subject testing negative for the disease. Since there's a 2% false positive rate, the probability of testing negative (true negative) is 100% - 2% = 98%, or 0.98 in decimal form.

This scenario can be modeled using the binomial distribution, where the number of trials is n = 30 (the number of healthy subjects) and the success probability on each trial is p = 0.98 (the probability of a true negative result).

The probability that all 30 subjects will test negatively, denoted as P(X=0), where X is the number of positive tests, is simply the probability of having 0 successes (positive tests) in 30 trials:

P(X = 0) = (0.98)^30

Using a calculator, we multiply 0.98 by itself 30 times:

P(X = 0) ≈ (0.98)^30 ≈ 0.541169526

B. To find the mean and standard deviation of X, we use the formulas for the mean (μ) and the standard deviation (σ) of a binomial distribution:

Mean (μ) = n * p Standard Deviation (σ) = sqrt(n * p * (1 - p))

where n is the number of trials and p is the probability of success on each trial.

For our case: Mean (μ) = 30 * 0.02 = 0.6 Standard Deviation (σ) = sqrt(30 * 0.02 * 0.98) ≈ sqrt(0.588) ≈ 0.766964988

C. To determine the viability of this test in the field of medicine, we have to consider its false positive rate in relation to the prevalence of the disease and the implications of false positives in diagnosis. A 2% false positive rate can be considered good for a medical test, but it might not be acceptable in circumstances where the disease is very rare, or the consequences of a false positive are severe (e.g., leading to invasive further testing or unnecessary treatment). The viability of the test also depends on its false negative rate (the rate at which it fails to detect the disease when it is present), the seriousness of the disease, the availability and risks associated with alternative tests, and the healthcare context in which the test is used.

Overall, if the false negative rate is also low and the benefits of early detection outweigh the risks associated with false positives, the test can be considered viable. However, it should be used alongside other diagnostic methods to confirm results, especially in cases where the test result could lead to serious implications for the patient.