KEY-Sample Problem-Child Abuse-Bayes Theorem + PDF

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ANSWER KEY for Sample Problem on Child Abuse It is estimated that about 3 percent of school-aged children in a typical American city are physically abused by their parents. It is possible for the schools to screen the children for evidence of abuse (e.g., scars), with the intent of follow-up by contacting the suspected parents. While the potential damage caused by letting an abused child go undetected is great, the costs of falsely suspecting a parent are also great. It is therefore important that school and health officials be very confident of their suspicions before approaching the parents. School health officials assert that the physical examination they use is very reliable. They claim that 95 percent of abused children will be detected by a positive physical, while only 10 percent of non-abused children will be falsely positive. Remember that 3 out of 100 children are abused. A. Given these data, and without using pencil and paper, estimate the probability that a child is abused if the physical exam is positive. (Just give a guess.) P(Abused/Exam Positive) = answers will vary

B. Assume 10,000 children are to be tested. Fill in the following raw data table:

Physical Exam Positive (Says Abused) Physical Exam Negative (Says Not Abused)

Abused 285 15 300

Not Abused 970 8730 9700

1255 8745 10,000 Children

C. For parts D through H use the raw data table, the probability tree or Bayes’ Theorem. Explain how you get your answer in each case, either in words or by writing down calculations. Find the probability an abused child has a positive exam. P(Positive/Abused) = from table is 285/300 = 0.95, given in the problem statement Look just at the column of 300 Abused children, then find the specific case of those 285 that are positive. D. Find the probability a non-abused child has a positive exam. P(Positive/Not Abused) = from table is 970/9700 = 0.1, given in the problem statement Look just at the column of 9700 Not Abused children, then find the specific case of those 970 that are positive.

E. What fraction of exams come out positive? From raw data table: P(Positive) = (285+970)/10,000=1255/10,000 Look at the row of 1255 Positive exam results, out of the total of 10,000 in the table. From tree: 285/10000 + 970/10000 = 1255/10000 From Bayes’: P(POS) = P(A)*P(POS/A) + P(NA)*P(POS/NA) = .03*0.95 + 0.97* 0.1 = 0.1255 F. Find the probability a child is abused if the exam is positive.

From raw data table: P(Abused/Positive)= 285/1255 Look just at the row of 1255 Positive exam results, then find the specific case of 285 abused children From tree: (285/10000) / (285/10000 + 970/10000) Bayes' Theorem: P(A|POS) = [P(A)*P(POS/A)] / P(POS) = [(0.03)(0.95)]/0.1255 = 0.2271 G. Find the probability a child is abused if the exam is negative.

From raw data table: P(Abused/Negative) = 15/8745 Look just at the row of 8745 Negative exam results, then find the specific case of 15 abused children From tree: (15/10000) / (15/10000 + 8730/10000) From Bayes': P(A/NEG) = [P(A)*P(NEG/A)] / P(NEG) = [(0.03)(0.05)]/[(0.03)(0.05) + (0.97)(0.9)] = 0.0015/0.8745 = 0.00172 H. What is the posterior probability of abuse given “Exam Negative”?

A posterior probability is one that is computed after (i.e., post) new information is received. The new (“given”) information is that the exam is negative. P(Abused/NEG) = 15/8745 = 0.00172 (Same as Problem G.) I. Compare the answer to F with your answer to A. Which is higher? answers will vary, usually F is lower If your answer to (A) was too high, you’re typical of most people in being too eager to reject prior knowledge in favor of new evidence....