. 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. In 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? (3 points)

a) To find the probability that all 30 subjects will appropriately test as not being infected, we need to calculate the probability of a single subject testing as not infected and then multiply it by itself 30 times.

The probability of a single subject testing as not infected is 1 - 0.02 (since they have a 2% chance of a false positive). So the probability of a single subject appropriately testing as not infected is 0.98.

Now we can calculate the probability of all 30 subjects appropriately testing as not infected:
P(all 30 subjects not infected) = (0.98)^30 ≈ 0.7408

So the probability that all 30 subjects will appropriately test as not being infected is approximately 0.7408 or 74.08%.

b) To find the mean and standard deviation of X, we need to know the distribution of X. In this case, X follows a binomial distribution since each subject can either test positive (success) or not test positive (failure).

The mean of a binomial distribution is given by the formula: mean = n * p
where n is the number of trials (30 subjects) and p is the probability of success (probability of testing positive in this case).

The standard deviation of a binomial distribution is given by the formula: standard deviation = sqrt(n * p * (1 - p))

In this case, since we want to find the mean and standard deviation of X, where X represents the number of subjects who test positive for the disease, we can use the values of n (30) and p (0.02) to calculate them:

Mean = n * p = 30 * 0.02 = 0.6
Standard deviation = sqrt(n * p * (1 - p)) = sqrt(30 * 0.02 * 0.98) ≈ 0.78

So, the mean of X is 0.6 and the standard deviation is approximately 0.78.

c) The viability of a medical test depends on several factors, including its accuracy, reliability, and potential impact on patient care. In this case, the test has a 2% false positive rate, meaning that 2% of healthy subjects will test positive for Hepatitis. While the probability of all 30 healthy subjects testing negative is relatively high (0.7408), there is still a chance of false positive results.

The extent to which this test is viable in the field of medicine would depend on various factors, such as the prevalence of Hepatitis in the population being tested, the consequences of a false positive result (e.g., unnecessary treatment or psychological distress), and the availability of confirmatory tests to verify the results.

It is crucial to consider the overall accuracy and reliability of a test before determining its viability in medical practice. Additionally, the context and specific circumstances should be taken into account when making such judgments.