Question

Medical screening tests are used to check for the presence on disease, evidence of illegal drug use, etc. The its sensitivity and its specificity. The sensitivity among those with the condition that will test positive. The specichy proportion among those without the condition that will test neg sensitivity of a test is defined to be the conditional ng those without the condition that will test negative. More formally, the test is defined to be the conditional probability that a person with the condition tests positive, and the specificity of the test is the conditi ne specificity of the test is the conditional probability that a person without the condition tests negative. Consider selecting a negative. Consider selecting a person at random from some population Then used to check for the presence of some condition, such as a Tillegal drug use, etc. The accuracy of such a test is described in terms of specificity. The sensitivity of a screening test is the overall proportion that will test positive. The specificity of the test is the overall Sensitivity = P(+1D) and Specificity = P(-Dº) where the events "D","+", "-" are defined as follows: D = the selected person actually has the condition + = the selected person tests positive for the condition - = the selected person tests negative for the condition The proportion of people in the population of interest that have the condition is denoted by P(D), and is called the prevalence. The prevalence can also be thought of as the probability that a randomly selected person from the population actually has the condition. When assessing the results of a test (positive or negative), a clinician and a patient will be more interested in the positive predictive value (PPV) and the negative predictive value (NPV) of the test than in its sensitivity and specificity. The PPV is defined to be the conditional probability that a person has the condition, given that the test is positive. The NPV is the conditional probability that the person does not have the condition, given that the test is negative. Suppose that a company performs random drug screening on its employees. Suppose further that the screening test used by the company has sensitivity 0.95 and specificity 0.95. An employee is randomly selected for screening and tested. 3. Assume that 5% of the company's employees are users of the drug being tested for. (a) If the screening test for the employee comes back positive for drug use, what is the probability that this employee actually uses the illegal drug? (8 points) (b) If the screening test for the employee comes back negative for drug use, what is the probability that this employee does not use the illegal drug? (8 points)

Answer 1

Given:

P(D) = 0.05, thus P(D') = 1 - P(D) = 0.95

P(+ | D) = 0.95

P(- | D') = 0.95, thus P(+ | D') = 1 - P(- | D') = 0.05

P(+) = P(+ | D)*P(D) + P(+ | D')*P(D')

= 0.95 * 0.05 + 0.05*0.95 = 0.095

Thus, P(-) = 1 - P(+) = 0.905

(a) The required probability = P(D | +)

= P(+ | D)*P(D)/P(+)

= 0.95 * 0.05/0.095 = 0.5

(b) The required probability = P(D' | -)

= P(- | D')*P(D')/P(-)

= 0.95*0.95/0.905 = 361/362 = 0.99724