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:

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	K_i	b_I	h_i	a_i
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:

p_k0= (k+1) /(k+2) and p_k,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 p_0y (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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