Question

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?

Answer 1

Given that X be the number of subjects who test positive for the disease out of the 30 healthy subjects used for the test.

The probability of success, i.e. the probability that a healthy subject tests positive is given as 2% = 0.02


Part A:

The probability that all 30 subjects will appropriately test as not being infected, that is the probability that none of the healthy subjects will test positive is given by:

$P(X)={ ^nC_xp^x(1-p)^{n-x}} \\ \\ P(0)={ ^{30}C_0(0.02)^0(1-0.02)^{30-0}} \\ \\ =1(1)(0.98)^{30}=0.5455$


Part B:

The mean of a binomial distribution is given by

$\mu=np \\ \\ =30(0.02) \\ \\ =0.6$

The standard deviation is given by:

$\sigma=\sqrt{np(1-p)} \\ \\ =\sqrt{30(0.02)(1-0.02)} \\ \\ =\sqrt{30(0.02)(0.98)}=\sqrt{0.588} \\ \\ =0.7668$


Part C:

This test will not be a trusted test in the field of medicine as it has a standard deviation higher than the mean. The testing method will not be consistent in determining the infection of hepatitis.