A symptom validity test (SVT) is sometimes used to confirm diagnosis of
psychiatric disorders. The paper "Developing a Symptom Validity Test for
Posttraumatic Stress Disorder: Application of the Binomial Distribution" (
Journal of Anxiety Disorders [2008]: 1297-1302) investigated the use of SVTs
in the diagnosis of post-traumatic stress disorder. One SVT proposed is a 60
-item test (called the MENT test), where each item has only a correct or
incorrect response. The MENT test is designed so that responses to the
individual questions can be considered independent of one another. For this
reason, the authors of the paper believe that the score on the MENT test can
be viewed as a binomial random variable with \(n=60 .\) The MENT test is
designed to help in distinguishing fictitious claims of post-traumatic stress
disorder. The items on the test are written so that the correct response to an
item should be relatively obvious, even to people suffering from stress
disorders. Researchers have found that a patient with a fictitious claim of
stress disorder will try to "fake" the test, and that the probability of a
correct response to an item for these patients is 0.7 (compared to 0.96 for
other patients). The authors used a normal approximation to the binomial
distribution with \(n=60\) and \(p=0.70\) to calculate various probabilities of
interest, where \(x=\) number of correct responses on the MENT test for a
patient who is trying to fake the test.
a. Verify that it is appropriate to use a normal approximation to the binomial
distribution in this situation.
b. Approximate the following probabilities:
i. \(\quad P(x=42)\)
ii. \(P(x<42)\)
iii. \(P(x \leq 42)\)
c. Explain why the probabilities calculated in Part (b) are not all equal.
d. The authors calculated the exact binomial probability of a score of 42 or
less for someone who is not faking the test. Using \(p=0.96\), they found
$$
P(x \leq 42)=.000000000013
$$
Explain why the authors calculated this probability using the binomial formula
rather than using a normal approximation.
e. The authors propose that someone who scores 42 or less on the MENT exam is
faking the test. Explain why this is reasonable, using some of the
probabilities from Parts (b) and (d) as justification.
