Question

explain how it is possible for a very accurate polygraph or drug test to result in a large proportion of false accusations.
suppose a positive result is that the test results show not lying/no drug use. also, assume the probability of lying/drug use is very low. choose the correct answer below.
a. since the probabilities of both a false negative and a true negative are very high, the proportion of false negatives to total negatives will be low.
b. since the probabilities of both a false positive and a true positive are very low, the proportion of false positives to total positives will be large.
c. since the probabilities of both a false negative and a true negative are very low, the proportion of false negatives to total negatives will be large.

Explanation:

• Let's analyze each option:
• Option A: If false - negative and true - negative probabilities are high, the proportion of false - negatives to total negatives would be low. But this is about false accusations (false positives or false - negatives depending on the test definition). Since the question is about false accusations (assuming positive is non - lying/drug - free), false - negatives (accusing a non - liar/drug - free person of lying/using drugs) are relevant.
• Option B: If false - positive and true - positive probabilities are low, the proportion of false - positives to total positives would be large. But we are interested in false accusations (false - negatives in our assumed positive - result definition).
• Option C: When the probability of lying/drug use is very low, the number of people who are actually lying/drug using (the base rate) is small. A false - negative (saying a non - liar/drug - free person is lying/using drugs) can be a relatively large proportion of the total negative results (because the number of true - negatives is large due to the low base rate of lying/drug use). For example, if only 1% of people are lying/drug using (\(P(\text{lying/drug use})=0.01\)), and the false - negative rate is \(0.05\) (5% of non - liars/drug - free people are wrongly accused) and the true - negative rate is \(0.95\) (95% of non - liars/drug - free people are correctly identified). The number of true - negatives in a sample of \(N\) people is \(0.99\times0.95N = 0.9405N\), and the number of false - negatives is \(0.99\times0.05N=0.0495N\). The proportion of false - negatives to total negatives (\(0.9405N + 0.0495N\)) is \(\frac{0.0495N}{0.99N}=0.05\) (if we assume the test has some accuracy). But when the base rate is low, even a small false - negative rate can lead to a relatively large proportion of false - negatives among total negative results.

Answer:

C. Since the probabilities of both a false negative and a true negative are very low, the proportion of false negatives to total negatives will be large.