LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 54

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2008

    ST 4808 – STATISTICAL COMPUTING – III

 

 

 

Date : 25/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer ALL the Questions. Each question carries 33 marks

1. (a.)

i.) Draw the OC curve of a single sampling with n = 100 , c =2.Also draw the AOQ

and ATI  curves.

ii.) Draw the tabular CUSUM for the following data.∆ = 0.5, α = 0.005, β = 0.10, σ = 1, n = 5

 

The sample mean values are given below.

34.5, 34.2, 31.6, 31.5, 35, 34.1, 32.6, 33.8, 34.8, 33.6, 31.9, 39.6, 35.4, 34, 37.1, 34.9, 33.5, 31.7, 34, 35.1

 

                                                     OR

(b)

(i) The data below represents the results of inspecting all units of a personal computer produced for the last 10 days.

Obtain the control limits .

 

Day                      1     2          3          4          5          6          7          8         9          10       

 

Units inspected    80   110      90        75        130      120      70        125      105      95

 

No of defectives   4    7          5          8          6          6          4          5          8          7

 

(ii.) The following fraction non confirming control chart with n = 100 is used to control a process.

UCL = 0.075    CL   = 0.04    LCL = 0.005

• Find probability of type I error.
• Find probability of type II error when p = 0.06.
• Draw the OC curve for the control chart.
• Find the ARL when p= 0.06.

 

2. ( a)

The following data were collected from a 2⁵ factorial experiment in two replicates with blocks of size 8 by completely confounding the effects ABC, ADE and BCDE. Analyse the data and identify the significant effects.

 

 

 

 

 

 

Treatment Combination	Yields (Rep I)	Yields (Rep II)
00000 01100 10110 11010 11001 10101 00011 01111	Block 1 56 68 70 73 71 81 69 86	Block5 60 48 77 81 55 51 43 56
11000 10100 00010 01110 00001 01101 11011 10111	Block 2 82 68 59 83 72 88 84 76	Block 6 81 76 56 40 70 56 46 72
10000 11100 01010 01001 00101 10011 11111 00110	Block 3 81 61 56 57 75 72 72 84	Block7 57 37 77 52 51 64 62 70
00111 01011 11101 10001 11110 10010 00100 01000	Block 4 74 69 60 49 46 74 54 72	Block 8 68 46 59 89 50 42 98 62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(OR)

(b) The yield of a chemical process was believed to be dependent mainly on standing time of the process. However, other factors also come into play. The chemical engineers who wanted to compare the effects of various standing times planned to

account for three other factors. So, they conducted an experiment using five types of

 

raw materials, five acid concentrations, five standing times (A, B ,C, D, E) and five catalyst concentrations ( α,β,γ,δ,ε). The following Graeco-Latin square design was

used. Analyse the data and draw your conclusions. Would you recommend a particular standing time over the others to maximize the yield? If so, which standing time is that?

 

Acid Concentrations

Raw

Material type          1                      2                       3                     4                     5

1                (Aα)16             (Bβ) 6                (Cγ) 9           (Dδ) 6             (Eε) 3

 

2                (Bγ) 8              (Cδ) 11               (Dε) 8           (Eα) 1             (Aβ) 11

 

3                (Cε) 10             (Dα) 2                (Eβ) 6           (Aγ) 15            (Bδ) 3

 

4                (Dβ) 5               (Eγ) 5                (Aδ) 12         (Bε) 4               (Cα) 7

 

5                (Eδ) 1               (Aε) 14               (Bα) 7          (Cβ) 7               (Dγ) 4

 

 

(3) (a)

(i) A business man is engaged in buying and selling identical items. He operates from a warehouse having a capacity of 500 items. Each month he can sell any quantity that he chooses up to the stock at the beginning of the month. Each month, he can buy as much as he wishes for delivery at the end of the month so long as his stock does not exceed 500 items. For the next four months he ahs the following error-free forecasts of cost and sales price:

 

Month:                  1       2       3       4

Cost C_n:               27     24     26     28

Sales Price p_n:     28     25     25     27

 

If he has a stock of 220 unit, what quantities should he sell and buy in the next four months. Find the solution using dynamic programming.

 

(ii) Use the Kuhn-Tucker conditions to solve the following non-linear programming problem:

 

Minimize  Z = 2 x₁² + 12 x₁ x₂ – 7 x₂²

Subject to the constraints

2 x₁ + 5 x₂ ≤ 98,       x₁, x₂ ≥ 0

(OR)

(b)

(i) Use integer programming to solve the LPP

 

Maximize Z = x₁ – 2x₂

Subject to the constraints

4 x₁ + 2 x₂ ≤ 15,    x₁, x₂ ≥ 0 and integers

 

(ii) Use Wolfe’s Method to  solve the QPP

 

Maximize Z = 2 x₁ + 3 x₂ – 2 x₁²

Subject to the constraints

x₁ + 4 x₂ ≤ 4

x₁ + x₂ ≤ 2 ,    x₁, x₂ ≥ 0

 

 

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