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Posted By: Atul Member Level: Diamond Posted Date: 01 May 2008
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2007 Anna University B.Tech Chemical engineering MA 036 — STATISTICS AND LINEAR PROGRAMMING Question paper
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B.Tech. DEGREE EXAMINATION.
Fourth Semester
Chemical Engineering
MA 036 — STATISTICS AND LINEAR PROGRAMMING
(Common to Textile Technology, Leather Technology and Textile Chemistry)
Time : Three hours Maximum : 100 marks
Answer ALL questions.
PART A — (10 ´ 2 = 20 marks)
The random variable X is exponentially distributed with mean 100. Find the probability that
The probability density function of X is given by
Find the p.d.f. of
Define Covariance. Prove that
State Central limit theorem and explain its significance.
A process of manufacturing a certain engine part has a mean of 50 mm and standard deviation of .01 mm. Groups of five are sampled every hour and the values of sample mean are recorded and plotted. Find the upper and lower control limits for .
Explain what is meant by randomised block design.
Formulate the following problem as a linear programming problem.
An animal feed company must product 200 kg of a mixture consisting of ingredients and daily. costs Rs. 3 per kg and Rs. 8 per kg. Not more than 80 kg of can be used and atleast 60 kg of must be used. Find how much of each ingredient should be used if the company wants to minimise the cost.
How are the alternative optima and unbounded solutions indicated in the simplex procedure for a LPP?
What are the necessary and sufficient conditions for the existence of a feasible solution for a transportation problem? If it has no such solution, how will you solve the problem?
Write down the dual of the following LPP :
Max
subject to
.
PART B — (5 ´ 16 = 80 marks)
(i) The following is the data on the number of defective electronic components in 20 samples each of size 50. Establish the control limits for
p–charts and find whether the process is under control.
Sample : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of defective components : 8 6 5 7 2 5 3 8 4 4 3 1 5 4 4 2 3 5 6 3
(ii) Carry out the analysis of the following Latin square design using 0.05 level of significance for each test :
A48 B38 C42 D53
B39 C43 D50 A54
C42 D50 A47 B44
D46 A48 B46 C52
(a) (i) Obtain the moment generating function of X whose p.d.f. is given by
(ii) Use Chebyshev inequality to show that, if X is the number scored in a throw of a fair die,
(iii) If X has geometric distribution, show that for any two positive integers s and t
Or
(b) (i) In a component manufacturing industry, there is a small probability of for any component to be defective. The components are supplied in packets of 10. Use Poisson distribution to calculate the approximate number of packets containing
(1) atleast one defective (2) atmost one defective.
(ii) The probability that an experiment will succeed is 0.8. If the experiment is repeated until four successful outcomes have occured, what is the expected number of repetitions required?
(iii) are independent, identically distributed exponential random variables each with paramter . Prove that is a Gamma variable with parameters and r.
(a) (i) Let X and Y have joint density function . Check whether X and Y are independent.
(ii) The fraction X of male runners and the fraction Y of female runners who complete marathon races can be described by the joint density function :
Find the covariance of X and Y.
Or
(b) (i) The joint density function of X and Y is
Find the density function of the random variable
(ii) Let (X, Y) be two–dimensional random variable with joint pdf
Find the correlation coefficient between X and Y.
(a) (i) Solve graphically :
Minimise
subject to
(ii) Using artificial variable technique, solve :
Minimise
subject to
Or
(b) (i) Solve graphically :
Maximise
subject to
(ii) Use Simplex method, to solve the following LPP :
Maximise
subject to
(a) (i) A company is sending a certain commodity from its factories to different warehouses. The details of the capacities, requirements and unit shipping cost are given below. Solve the transportation problem, starting with the solution obtained by Vogel's approximation method. If an alternate optimal solution exists, find it also.
Warehouses Capacities
Factories 12 5 23 9 2100
7 10 11 13 1750
8 12 4 14 1150
Requirements 1200 2250 850 700
(ii) Solve the following assignment problem, to minimise the total time needed for all the four jobs.
Job
Operator 12 10 10 8
14 12 15 11
6 10 16 4
8 10 9 7
Or
(b) (i) State duality theorem of a LPP. What can you say about the optimal solution of the dual problem of the primal problem does not have a feasible solution?
(ii) Solve the following LPP, using duality theory :
Minimise
subject to
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