Statistics /Probability-A blood test is developed to screen for a certain type

Subject: Mathematics    / Statistics
Question

19. 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. [3 marks]
b) Find the allergen-specific antibody concentration that is exceeded by 20% of the non-allergic individuals. [2 marks]
c) Find the probability that a randomly chosen allergic individual is screened positive by the blood test. [2 marks]
d) Find the probability that a randomly chosen individual is screened positive by the blood test. [2 marks]
e) An individual who undergoes the blood test is screened negative. Find the probability that this individual is truly free of the allergy. [2 marks]

20. A 4-member relay team (including the order in which they will swim) is to be randomly selected

from the 9 members of a men’s swimming club.

a) How many different teams are possible?

b) Find the probability that the best swimmer is left off the randomly-selected relay team.

c) Find the probability that the best swimmer is picked for the team, and asked to swim the fourth

lap of the relay.

d) Find the probability that the top two swimmers in the club wind up swimming the third and

fourth laps of the race, in either order; i.e., the best could be either the third or fourth into the

pool.

21. Find the probability that a random selection of 5 cards from an ordinary deck of 52 will produce a
hand known in poker as a “flush” — i.e., the cards are all of the same suit, like 3H, 5H, 6H, 10H,
QH.

22. On a certain computer network each user has to have a password consisting of six to eight
characters in length; each character is either a lowercase letter or a digit. A password must contain
at least one digit. How many possible passwords can be issued?

23. A large shipment of items contains 8% defective items. Four items of the shipment are randomly
selected. What is the chance that the selection will include at least one defective item?

24. In 6/49 lottery, you win cash prize when you have three or more of the six winning numbers. In other words, you do not win any cash prize (i.e. you lose $2 for the ticket) when you have two or less winning numbers. What is the probability that you do not win any cash prize when you purchase a ticket of 6/49 lottery?

25. A special committee of five professionals is formed from a group of four lawyers, three doctors and three teachers. At least three lawyers are needed in the special committee. Define X as the number of teachers in the special committee.
a) Construct the probability distribution of the random variable X. [8 marks]
b) Calculate the expected of X and provide an interpretation of it. [2+2 marks]
c) Calculate the standard deviation of X. [2 marks]