Consider the problem of school officials trying to identify cases of child abuse. In one community, officials believe that 3 percent of the area's 10,000 children enrolled in public grade schools are physically abused. Measures can be taken to help these students, but first they must be located and identified. A prelimin program is proposed, whereby officials will contact and interview parents whenever evidence of abuse (such as unusual bruises) is found. Of course, screening is not entirely reliable. Although ninety-five (95) percent of abused children wll be identified, it is estimated that screening will incorrectly label non-abused children as abused in 10 percent of the cases. School officials are anxious to identify cases of abuse, but they must proceed cautiously given the enormous stigma of charging someone with child abuse and the risk of liability to school officials if such an accusation is proven false.

a. Construct the 2 X 2 contingency table corresponding to the child's true condition (abused versus not abused) and the results of the screening test.

b. Compute all of the marginal probabilities associated with this table.

c. Compute all of the conditional probabilities associated with this table.

d. Compute all of the joint probabilities associated with this table.

e. What is the probability that a child enrolled in grade school is abused?

f. What is the probability that an abused child will have a negative screen?

g. If school officials decide to implement the screening program, what is the probability that they will make an error?

h. Draw a reasonable decision tree that represents the school officials' decision Include the basic decision as well as uncertainties, probabilities, outcomes, and values associated with each alternative. If you do not have numeric estimates to complete the tree, describe the desired value in words or mathematical notation. (Note: You do not have enough information to solve the tree, so don't even try!)