LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION – STATISTICS
THIRD SEMESTER – NOVEMBER 2004
ST 3803 – COMPUTATIONAL STATISTICS – III
02.11.2004 Max:100 marks
1.00 – 4.00 p.m.
SECTION – A
Answer any THREE questions without omitting any section . (3 ´ 34 = 102 marks)
- 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. | Departure | Arrival | Flight No. | Departure | Arrival |
| 1 | 7.00 a.m. | 8.00 a.m | 101 | 8.00 a.m. | 9.15 a.m |
| 2 | 8.00 a.m. | 9.00 a.m | 102 | 8.30 a.m. | 9.45 a.m |
| 3 | 1.30 p.m. | 2.30 p.m | 103 | 12.00 noon | 1.15 p.m |
| 4 | 6.30 p.m | 7.30 p.m | 104 | 5.30 p.m | 6.45 p.m |
(17 marks)
- a) Solve the following unbalanced transportation problem:
To
1 2 3 Supply
From
Demand 75 20 50 (17 marks)
- b) Consider the inventory problem with three items. The parameters of the problem are
shown in the table.
| Item | Ki | bI | hi | ai |
| 1 | Rs.500/- | 2 units | Rs.150/- | 1 ft2 |
| 2 | Rs.250/- | 4 units | Rs. 50/- | 1 ft2 |
| 3 | Rs.750/- | 4 units | Rs.100/- | 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
- 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).
.
(17 marks)
- For a three state Markov chain with states {0,1,2} and transition probability matrix
Find the mean recurrence times of states 0, 1, 2. (17 marks)
- 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)
- 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 m2 = l0. Using the forward equation, find p0y (t), y = 0,1,2. (17 marks)
SECTION – C
- 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 |
X2
Charge rate in (amps) |
X3
Depth of discharge |
| 101 | 0.375 | 60.0 |
| 141 | 1.000 | 76.8 |
| 96 | 1.000 | 60.0 |
| 125 | 1.000 | 43.2 |
| 43 | 1.625 | 60.0 |
| 16 | 1.625 | 76.8 |
| 188 | 1.00 | 100.0 |
| 10 | 0.375 | 76.8 |
| 386 | 1.00 | 43.2 |
| 160 | 1.625 | 76.8 |
| 216 | 1.00 | 70.0 |
| 170 | 0.375 | 60.0 |
(20 marks)
- For the above data given in 5a Test for the significance population partial correlation
coefficient between X1 and X2. (14 marks)
- The stiffness and bending strengths of two grades of Lumber are given below:
| I grade | II grade | ||
| Stiffness | Bending strength | Stiffness | Bending strength |
| 1,232 | 4,175 | 1,712 | 7,749 |
| 1,115 | 6,652 | 1,932 | 6,818 |
| 2,205 | 7,612 | 1,820 | 9,307 |
| 1,897 | 10,914 | 1,900 | 6,457 |
| 1,932 | 10,850 | 2,426 | 10,102 |
| 1,612 | 7,625 | 1,558 | 7,414 |
| 1,598 | 6,954 | 1,470 | 7,556 |
| 1,804 | 8,365 | 1,858 | 7,833 |
| 1,752 | 9,469 | 1,587 | 8,309 |
| 2,067 | 6,410 | 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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