02.11.2004
1.00 - 4.00 p.m.
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – NOVEMBER 2004
ST 3803 - COMPUTATIONAL STATISTICS - III
Max:100 marks
SECTION - A
(3  34 = 102 marks)
Answer any THREE questions without omitting any section .
1. a) Use two phase method to solve
Max. z = 5x  2y + 3z
Subject .to
2x + 2y  z ≥ 2
3x  4y
 3
y + 3z < 5
x, y, z ≥ 0
(17 marks)
b) An airline that operates seven days a week between Delhi and Jaipur has the time-table
as shown below. Crews must have a minimum layover of 5 hours between flights.
Obtain the pairing of flights that minimizes layover time away from home. Note that
crews flying from A to B and back can be based either at A or at B. For any given
pairing, he crew will be based at the city that results in smaller layover:
Flight No.
1
2
3
4
Departure
7.00 a.m.
8.00 a.m.
1.30 p.m.
6.30 p.m
Arrival
8.00 a.m
9.00 a.m
2.30 p.m
7.30 p.m
Flight No.
101
102
103
104
Departure
8.00 a.m.
8.30 a.m.
12.00 noon
5.30 p.m
Arrival
9.15 a.m
9.45 a.m
1.15 p.m
6.45 p.m
(17 marks)
2. a) Solve the following unbalanced transportation problem:
To
1
1 5

From
2  6
3  3
Demand
75
2
1
3 Supply
7  10

4
6  80
2 5  15
20 50
(17 marks)
b) Consider the inventory problem with three items. The parameters of the problem are
shown in the table.
Item
1
2
3
Ki
Rs.500/Rs.250/Rs.750/-
I
2 units
4 units
4 units
hi
Rs.150/Rs. 50/Rs.100/-
ai
1 ft2
1 ft2
1 ft2
Assume that the total available storage area is given by A = 20ft2. Find the economic
order quantities for each item and determine the optimal inventory cost.
(17 marks)
SECTION - B
3. a) Suppose the one step transition probability matrix is as given below:
Find i) p00(2)
ii) f00(n)
iii) f13(n)
and
iv) f33(n).
 0.4

 0.2
P
0.5

 0.0

0.0 0.6 0.0 

0.4 0.2 0.2 
.
0.0 0.5 0.0 

0.5 0.0 0.5 
(17 marks)
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b) For a three state Markov chain with states {0,1,2} and transition probability matrix
1 / 2 0 1 / 2 


 0 1 / 3 2 / 3
1 / 4 1 / 4 1 / 2 


Find the mean recurrence times of states 0, 1, 2.
(17 marks)
4. a) An infinite Markov chain on the set of non-negative integers has the transition function
as follows:
pk0 = (k+1) /(k+2)
and pk,k+1 1/(k+2)
i)
ii)
Find whether the chain is positive recurrent, null recurrent or transient.
Find the stationary distribution, incase its exists.
(17 marks)
b) Consider a birth and death process three states 0, 1 and 2, birth and death rates such
that 2 = 0. Using the forward equation, find p0y (t), y = 0,1,2.
(17 marks)
SECTION - C
5. a) From the following data test whether the number of cycles to failure of batteries is
significantly related to the charge rate and the depth of discharge using multiple
correlation coefficient at 5% level of significance.
X1
No. of cycles to failure
101
141
96
125
43
16
188
10
386
160
216
170
X2
Charge rate in (amps)
0.375
1.000
1.000
1.000
1.625
1.625
1.00
0.375
1.00
1.625
1.00
0.375
X3
Depth of discharge
60.0
76.8
60.0
43.2
60.0
76.8
100.0
76.8
43.2
76.8
70.0
60.0
(20 marks)
b) For the above data given in 5a Test for the significance population partial correlation
coefficient between X1 and X2.
(14 marks)
6. The stiffness and bending strengths of two grades of Lumber are given below:
I grade
Stiffness
Bending strength
1,232
4,175
1,115
6,652
2,205
7,612
1,897
10,914
1,932
10,850
1,612
7,625
1,598
6,954
1,804
8,365
1,752
9,469
2,067
6,410
II grade
Stiffness Bending strength
1,712
7,749
1,932
6,818
1,820
9,307
1,900
6,457
2,426
10,102
1,558
7,414
1,470
7,556
1,858
7,833
1,587
8,309
2,208
9,559
Test whether there is significant difference between the two grades at 5% level of
significance, by testing the equality of mean vectors. State your assumptions.
(34 marks)
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