Anna University Chennai 2009-4th Sem B.E Information Technology ./B.Tech , /E (ester, )-PROBABILITY AND STATISTICS - Question Paper
B.E./B.Tech. DEGREE EXAMINATION, MAY/JUNE 2009.
Fourth Semester Information Technology MA 1259 PROBABILITY AND STATISTICS Time : Three hours Maximum : 100 marks
Statistical Tables are permitted.
Answer ALL questions.
PART A (10 x 2 = 20 marks)
1. Four persons are chosen at random from a group containing 3 men, 2 women
and 4 children. Prove that the chance that exactly two of them will be children
. 10 is .
21
2. A continuous random variable X follows the probability law
f{x)~ Ax2,0 <x <1, determine A and find the probability that X lies between .2 and .5.
3. If, on an average 9 ships out of 10 arrive safely to a port. Obtain the mean and standard deviation of the number of ships returning safely out of 150 ships.
4. Suppose that during a rainy season in a tropical island the length of the shower has an exponential distribution, with average 2 minutes. Find the probability that the shower will be there for more than three minutes.
5. The joint p.d.f. of the two dimensional random variable (X,Y) is given by
22 l<x <y <2 9 J
0 otherwise Find the marginal densities of X and Y .
6. Two fair dice are tossed 600 times. Let X denote the number of times a total of
7 occurs. Using central limit theorem find (90 < X < 110).
7. A machine runs on an average of 125 hours/year. A random sample of 49 machines has an annual average use of 126.9 hours with standard deviation of 8.4 hours. Does this suggest to believe that machines are used on the average more than 125 hours annually at 0.05 level of significance?
8. In a sample of 90 university professors 28 own a computer. Can we conclude at
.05 level of significance that at most of the professors own a computer?
4
9. Define completely randomized design.
10. What do you mean by Latin square design?
PART B (5 x 16 = 80 marks)
11. (a) (i) A box of fuses contains 20 fuses, of which five are defective. If three of the fuses are selected at random and removed from the box in succession without replacement, what is the probability that all three fuses are defective? (6)
(ii) The chances of three candidates A, B and C to become the manager of a company are in the ratio 3:5:4. The probability of introducing a special bonus scheme by them if selected are .6, .4 and .5 respectively. If the bonus scheme is introduced, what is the probability that B has become the manager? (10)
Or
(b) (i) The density function of a random variable X is given by
2
otherwise
find k , Mean and Variance of the distribution. (8)
(ii) A random variable X has the probability function
* = 1,2,3,.....
Find its M.G.F. and Mean. (8)
12. (a) (i) It has been found that 80% of printers used on home computers operate correctly at the time of installation. A particular dealer sells 10 units during a given month. Find the probability that at least nine printers operate correctly on installation. Consider
5 months, in which 10 units are sold per month. What is the probability that at least nine units operate correctly in each of the
5 months? (8)
(ii) The average number of traffic accidents on a certain section of a highway is two per week. Assume that the number of accidents follow a Poisson distribution. Find the probability of no accident and atmost two accidents in a 2 week period. (8)
Or
2 J 3293
(b) (i) Let X be the service life of a semiconductor having Weibull with a - .025 and y? = .5 as parameters. Find the probability that the semiconductor will be working after 3,000 hours. (8) /
(ii) The life time of an electric component is normally distributed with mean value of 250 hours and standard deviation of a hours. Find the value of a so that the probability of the component to have life between 200 and 300 hours is 0.7. (8)
13. (a) (i) The joint p.d.f. of the two dimensional random variable (X,Y) is
given by = 0<xty, >. Find the marginal density
functions and state whether the random variables are independent.
(ii) Let Xy and X2 have the joint p.d.f
j-j t \ 22 4 n
p{xux2)= 18 , *1
%2 = 2
Find the COV (XUX2). (8)
Or
(b) (i) If the probability density of X is given by
j_ f6rc(1 jc) forOcxcl [ 0 elsewhere
find the probability density of Y = X3. (6)
(ii) A sample of size 100 is taken from a population whose mean is 60 and variance is 400. Using central limit theorem with what probability can we assert that the mean of the sample will not differ from jU 60 by more than 4. (10)
14. (a) (i) Suppose that 100 litres made by a certain manufacturer lasted on
the average 21, 819 miles with a standard deviation of 1,295 miles.
Test the null hypothesis // = 22,000 miles against the alternative hypothesis pL <. 22,000 miles at llits 0.00 level uf significance. (8)
(ii) In a study designed to investigate whether certain detonators used with explosives in coal mining meet the requirement that atleast 90% will ignite the explosive when charged, it is found that 174 of 200 detonators function properly. Test the null hypothesis p = 0.90 against the alternative hypothesis p < -90 at the 0.05 level of significance. (8)
Or
3 J 3293
(b) (i) In the comparison of two kinds of paint, a consumer testing service finds that four 1-gallon cans of one brand cover on the average 546 square feet with a standard deviation of 31 square feet, whereas four 1-gallon cans of another brand cover on the average 492 square feet with a standard deviation of 26 square feet. Assuming that the two populations sampled are normal and have equal variances, test the null hypothesis =0 against the
alternative hypothesis -//2 > 0 at the .05 level of significance. (8)
(ii) An experiment yielded the following results :
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nt =13, =19.2 .2 = 16 and s\ =3.5. Test the null hypothesis n. o{ = against the alternative cri * cr2 significance. |
at the 0.01 level of |
Detergent A : 77, 81, 71, 76, 80 Detergent B : 72, 58, 74, 66, 70 Detergent C : 76, 85, 82, 80, 77
15. (a)
Test at the 0.01 level of significance whether the differences among the means of the whiteness readings are significant. (16)
Or
(b)
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Monday Tuesday Wednesday Thursday Friday | ||||||||||||||||||||||||
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Test at the 0.05 level of significance whether the differences among the means obtained for different routes are significant and also whether the differences among the means obtained for the different days of the week are significant. (16)
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Attachment: |
| Earning: Approval pending. |
