MATH 333:Probability & Statistics.  Exam 1 (Spring 2002)                                              Scores         ·

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February 27, 2002          NJIT

Instructors :  S. Balaji,  M. C. Bhattacharjee,  S. Dhar,  H. Fanik  N. Moheb

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1.      Consider the following states from the east coast with their corresponding percentage of the adult population who were obese (1998 data).

 East Coast States CT DE MA MD NJ NY PA VA W.Va Percent (%) obese 14.7 16.6 13.8 19.8 15.2 15.9 19.0 18.2 22.9

(i)  (6 points) What are the average and median percentage obesity values among adults over the  specified east coast states?

(ii)  (7 points) What is the value of sample standard deviation?

(iii)  (7 points) Use the method of box plot to check for  any  (mild or, extreme)  outliers in the

data. Show work.

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2.               In a shipment of 15 transistors;  8 are good (no defects), 4 have minor defects, while the remaining transistors in the shipment have major defects. Suppose we choose 4 transistors at random, from this shipment, without replacement.

Find the probability that:

(a)   (3 points)  none is good,

(b)    (5 points)   at least one is good,

(c)     (3 points)   exactly 2 are good,

(d)    (6 points)   one is good, one has minor defect and 2 have major defects.

3.      (15 points) The Center for Disease Control has determined that 80% of all persons who are given a vaccine, will develop immunity. If eight people are independently given this vaccine , find the probability that:

a)      (5 points) none will develop immunity,

b)      (7 points) at most two  will develop immunity.

c)      (5 points) all will develop immunity.

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4.  Engineers in charge of maintaining a nuclear power plant must check for corrosion inside the pipes that are part of the cooling systems. The inside condition of the pipes cannot be observed directly but a test can give an indication of possible corrosion.  The test has probability 0.7 of detecting corrosion when it is present but it also has probability 0.2 of falsely indicating internal corrosion of falsely indicating corrosion when no internal corrosion is present.  Suppose the probability that any section of pipe has internal corrosion is 0.1.

a)  (6 points) Determine the probability that the test indicates presence of corrosion.

b)   (8 points) Determine the probability that a section of pipe has internal corrosion given that the test indicates its presence.

5.      Seven teams compete for the Gold, Silver and Bronze medals, which correspond to the first, second and third places respectively, in the final round of Figure Skating at the Winter Olympics. An outcome lists the three winning teams along with which medal they won.

a)  (6  points) How many different outcomes are there ?

b)  (8  points)  If  U.S. has three teams among the seven competing in the final round; what is the probability that U.S. wins all three medals ?

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6.         The number of traffic accidents in a certain town during any week is a random variable with the following probability distribution:

 Number of accidents 0 1 2 3 Probability 0.5 0.3 ? 0.05

a)      (2 points)  What is the probability of exactly two accidents in a week ?

b)      (3 points) Find the probability of having at least two accidents in a week.

c)   (8 points) Find the mean and variance of the number of accidents.

d)  (5 points)  If the number of accidents in different weeks are independent, what is the probability that there are no accidents in a consecutive 3 week period ?

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