Question: A blood test is developed to screen for a certain type of allergy in human beings. Individuals who have the allergy tend to have elevated serum concentration of the allergen-specific antibody. In particular, among these allergic individuals, their allergen-specific antibody concentrations follow the Normal distribution with an average of 400 units per litre (U/L) and a standard deviation of 80 U/L. For individuals without the allergy, their allergen-specific antibody concentrations also follow the Normal distribution, but with an average of 200 U/L and a standard deviation of 50 U/L. An individual who undergoes the blood test is screened positive if his/her allergen-specific antibody concentration exceeds 300 U/L. Otherwise, the individual is screened negative. (Note that an allergic individual can have low antibody concentration and be screened negative by the blood test. Similarly, a non-allergic individual can have high antibody concentration and be screened positive.) Suppose 10% of the population have the allergy.
a) Find the probability that a randomly chosen non-allergic individual is screened positive by the blood test.
b) Find the allergen-specific antibody concentration that is exceeded by 20% of the non-allergic individuals.
c) Find the probability that a randomly chosen allergic individual is screened positive by the blood test
d) Find the probability that a randomly chosen individual is screened positive by the blood test
e) An individual who undergoes the blood test is screened negative. Find the probability that this individual is truly free of the allergy.