Some clinical psychiatrists advocate the use of a test known as the Dissociative Experiences Scale (DES) to diagnose a condition referred to as Dissociative Identity Disorder (DID) (see Bernstein and Putnam, 1986, for the development of the DES). Patients diagnosed with DID are apparently prone to various symptoms, including amnesia, having two or more distinct personalities, and feelings of detachment from their bodies. The DES is a survey that attempts to monitor the frequency of certain associated traits, and on which high scores are consistent with DID.

A sample of $50$ patients who had been diagnosed DID by independent means were asked to complete the DES. The sample mean score of these patients was , the standard deviation being . A psychologist wishes to make inference about the distribution of DES scores in the population of patients with DID. He believes that the distribution of DES scores is Normally distributed, and he wishes to draw inference about the parameters using a Bayesian approach that permits the incorporation of his prior knowledge.

Bernstein, E.M. and Putnam, F.W. (1986): Development, reliability, and validity of a dissociative scale. J. Nerv. Mental Dis. 174 , 727-735.

Provide answers to the following to two decimal places.

Part a) Suppose that inference is required about the population mean DES score, and that the population variance is taken as known to be the sample variance. A prior distribution for the mean is taken to be Normal with mean and variance . Find the posterior distribution in this case, and provide the posterior mean.


Part b) Find the posterior probability that the mean DES score in the population of DID patients is greater than $45$.


Part c) Suppose now that inference is required about the precision of the DES score in DID patients, and that the population mean is taken as known to be the sample mean. The researcher takes a prior distribution for the precision that is the $Ga(, )$ distribution. Find the posterior probability that the precision is less than 0.005.