3. (8 marks) Peanut allergy is estimated to affect about 1 in 50 children in Australia. A school has 166 students enrolled. Assume that those enrolled have the same overall propensity for peanut allergy as the population of children in Australia, and let X be the number (among those 166) who have peanut allergy. Use the Poisson distribution to answer the following questions (for the school): a. i. ii. iii. What is the expected value of the number of students who have a peanut allergy? What is the standard deviation of the number of students who have a peanut allergy? What is the probability that there are fewer than 4 students with a peanut allergy? 2a. /1 2b. /1 2c. /2 iv. What is the probability that are more than 4 students with a peanut allergy? 2d. /2 b. A number of diagnostic methods for allergies exist, of which skin-prick testing is considered one of the less risky. It has a sensitivity of about 81% and specificity of about 52%. Find the predictive positive value of this test for a random Australian child. That is, if one tested positive with skin-prick test, what would be the probability that he or she actually had the disease? 3.33% 2d. /2