Symptom validity tests (SVTs) are sometimes used to confirm diagnosis of psychiatric disorders. The paper “Developing a Symptom Validity Test for Post-Traumatic 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 fake claims of post-traumatic stress disorder. The items on the MENT 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 fake 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 = 0.7 to approximate 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. P(x = 42)

ii. P(x , 42)

iii. P(x ≤ 42)c. Explain why the probabilities computed in Part (b) are not all equal.

c. Explain why the probabilities computed in Part (b) are not all equal.