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.

              1 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.

1. Find the probability that a randomly chosen non-allergic individual is screened positive by the blood test. [2]
2. Find the allergen-specific antibody concentration that is exceeded by 20% of the non-allergic individuals. [2]
3. Find the probability that a randomly chosen allergic individual is screened positive by the blood test. [2]
4. Find the probability that a randomly chosen individual is screened positive by the blood test. [2]
5. An individual who undergoes the blood test is screened negative. Find the probability that this individual is truly free of the allergy. [2]

Answer 1