LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 55

FOURTH SEMESTER – APRIL 2007

ST 4808 – STATISTICAL COMPUTING – III

 

 

 

Date & Time: 23/04/2007 / 9:00 – 12:00 Dept. No.                                            Max. : 100 Marks

 

 

 

Answer Any three Questions:

 

 

• Analyse the following 2³  Confounded factorial design

 

 

Replication -1

 

Block-1	N    120	P  121	K 141	Npk 151
Block-2	(1) 121	Nk 145	Np 167	Kp  211

 

 

 

 

 

Replication -2

 

Block-1	Knp  131	K 101	Np  141	(1)  51
Block-2	Nk  62	N 83	P 43	Pk 32

 

 

 

 

Replication -3

 

Block-1	Knp 142	Pk  123	N  195	(1) 143
Block-2	Np  143	Nk 105	P 165	K 212

 

 

 

 

 

 

 

 

• 2) Analyse the following Repeated Latin Square design, stating all the Hypotheses, Anova and Inferences. The data represent the Production in Millions of five different soft drinks in five seasons at five different Companies( for the first two weeks).

 

WEEK-1

 

Company/Season	1	2	3	4	5
S1	A125	B338	C345	D563	E233
S2	B635	C453	D634	E784	A345
S3	C455	D901	E344	A124	B466
S4	D781	E443	A235	B 948	C452
S4	E245	A378	B565	C712	D344

 

 

 

WEEK-2

 

Company/Season	1	2	3	4	5
S1	A255	B385	C455	D156	E273
S2	B165	C454	D645	E748	A734
S3	C475	D903	E354	A124	B456
S4	D078	E432	A253	B498	C455
S5	E485	A782	B556	C142	D534

 

3 a). The data given below are temperature readings from a chemical process in a

 degrees centigrade, taken every two minutes.

853    985    949    937    959

945    973    941    946    939

972    955    966    954    948

945    950    966    935    958

975    948    934    941    963

 

The target value for the mean is m₀ = 950

 

i). Estimate the process standard deviation.

 

ii). Set up and apply a tabular CUSUM for this process, using standardized values

h = 5 and k = 0.5. Interpret this chart.

Reconsider the above data. Set up and apply an EWMA control chart to these

data using l = 0.1 and L =2.7.

 

b). Find a single sampling plan for which p₁ = 0.05, a = 0.05, p₂ =0.15 and

b = 0.10.

 

 

1. a). A paper mill uses a control chart to monitor the imperfections in finished rolls of paper. Production output is inspected for 20 days, and the resulting data are shown below. Use these data to set up a control chart for nonconformities per roll of paper. Does the process appear to be in statistical control? What center line and control limits would you recommend for controlling current production?

 

Day:                Number of rolls produced       Total number of imperfection

 

1                                18                                            12

2                                18                                            14

3                                24                                            20

4                                22                                            18

5                                22                                            15

6                                22                                            12

7                                20                                            11

8                                20                                            15

9                                20                                            12

10                               20                                            10

11                               18                                            18

12                               18                                            14

13                               18                                            9

14                               20                                            10

15                               20                                            14

16                               20                                            13

17                               24                                            16

18                               24                                            18

19                               22                                            20

20                               21                                            17

1. (b) Solve the following IPP  :

Maximize

Subject to

 

are nonnegative integers

 

1. Consider the design of an electronic device consisting of three main components. The three components are arranged in series so that the failure of one component will result in the failure of the entire device. The reliability of the device can be enhanced by installing standby units in each component. The design calls for using one or more standby units, which means that each main component may include upto three units in parallel. The total capital available for the design of the device is $10,000. The data for the reliability, cost for various components for given number of parallel units are summarized below. Determine the number of parallel units for each component that will maximize the reliability of the device without exceeding the allocated capital. You should use dynamic programming technique to solve the given problem.

 

1	0.6	1	0.7	3	0.5	3
2	0.8	2	0.8	5	0.7	4
3	0.9	3	0.9	6	0.9	5

 

 

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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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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 50

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – April 2009

ST 4808 – STATISTICAL COMPUTING – III

 

 

 

Date & Time: 25/04/2009 / 9:00 – 12:00  Dept. No.                                                    Max. : 100 Marks

 

 

Answer any three questions            

                                                                                               

1. a) The data shown here are  and R values for 24 samples of size n=5 taken from a process producing bearings.  The measurements are made on the inside diameter of the bearing, with only the last three digits recorded.

 

Sample number	1	2	3	4	5	6	7	8	9	10	11
	34.5	34.2	31.6	31.5	35.0	34.1	32.6	33.8	34.8	33.6	31.9
R	3	4	4	4	5	6	4	3	7	8	3
Sample number	12	13	14	15	16	17	18	19	20	21	22
	38.6	35.4	34	37.1	34.9	33.5	31.7	34	35.1	33.7	32.8
R	9	8	6	5	7	4	3	8	4	2	1
Sample number	23	24
	33.5	34.2
R	3	2

 

(i). Sep up  and R charts on this process.  Does the process seem to be in statistical           control?  If necessary, revise the trial control limits.

 

(ii). If specifications on this diameter are 0.50300.0010, find the percentage of nonconforming bearings produced by this process.  Assume that diameter is normally distributed.

 

b). In the semiconductor industry, the production of microcircuits involves many steps.  The wafer fabrication process typically builds these microcircuits on silicon wafers and there are many microcircuits per wafer.  Each production lot consists of between 16 and 48 wafers.  Some processing steps treat each wafer separately, so that the batch size for that step is one wafer.  It is usually necessary to estimate several components of variation: within-wafer, between-wafer, between-lot and the total variation. A critical dimension (measured in mm) is of interest to the process engineer. Suppose that five fixed position are used on each wafer (position 1 is the center) and that two consecutive wafers are selected of each batch. The data that results several batches are shown below.

 

(i) What can you say about over all process capability?

 

(ii)  Can you construct control charts that allow within- wafer variability to be evaluated?

 

(iii) What control charts would you establish to evaluate variability between wafers? Set

up these charts and use them to draw conclusions about the process.

 

 

Lot No.	Wafer No.	Position
		1	2	3		4		5
1	1	2.15	2.13	2.08	2.12		2.10
	2	2.13	2.10	2.04	2.08		2.05
2	1	2.02	2.01	2.06	2.05		2.08
	2	2.03	2.09	2.07	2.06		2.04
3	1	2.13	2.12	2.10	2.11		2.08
	2	2.03	2.08	2.03	2.09		2.07
4	1	2.04	2.01	2.10	2.11		2.09
	2	2.07	2.14	2.12	2.08		2.09
5	1	2.16	2.17	2.13	2.18		2.10
	2	2.17	2.13	2.10	2.09		2.13
6	1	2.04	2.06	2.00	2.10		2.08
	2	2.03	2.10	2.05	2.07		2.04
7	1	2.04	2.02	2.01	2.00		2.05
	2	2.06	2.04	2.03	2.08		2.10
8	1	2.13	2.10	2.10	2.15		2.13
	2	2.10	2.09	2.13	2.14		2.11
9	1	2.00	2.03	2.08	2.07		2.08
	2	2.01	2.03	2.06	2.05		2.04
10	1	2.04	2.08	2.09	2.10		2.01
	2	2.06	2.04	2.07	2.04		2.01

(17 +17)

 

2. a). Bath concentrations are measured hourly in a chemical process. Data (in PPM) for the

last 32 hours are shown below (read down from left).

160	186	190	206
158	195	189	210
150	179	185	216
151	184	182	212
153	175	181	211
154	192	180	202
158	186	183	205
162	197	186	197

The process target is =175 PPM.

(i). Estimate the process standard deviation.

 

(ii). Construct a tabular cusum for this process using standardized values of h = 5 and

k =  .

 

b). A product is shipped in lots of size N = 2000.  Find a Dodge-Romig single-sampling plan for which the LTPD = 1%, assuming that the process average is 0.25% defective.  Draw the OC curve and ATI curve for this plan.  What is the AOQL for this sampling plan?                                                                                                                                 (20+14)

 

 

 

 

 

 

3)    (a)  Analyze the following 3² factorial design                                                                 (24)

Replicate I                                            Replicate II

 

a0b0 20	a1b0 32	a0b2 40
a1b1 60	a0b1 48	a2b0 55
a2b1 60	a1b2 31	a2b2 51

a1b1 42	a1b2 60	a0b1 40
a2b0 25	a0b0 62	a1b0 45
a2b2 61	a2b1 31	a0b2 42

 

(b) Construct BIBD using the following :

V = 7, b =7, r = 3, k = 3, λ=1                                                                                      (10)

 

4)  (a) Analyze the following 2³ factorial experiment in blocks of 4 plots, involving three fertilizers N,

P and K each at two levels.                                                                                     (17)

Replicate I                                                  Replicate II

Block 1	np 88	npk 90	(1) 115	k 75
Block 2	p 101	n 111	pk 75	nk 55

Block 3	(1) 115	npk 95	nk 90	p 80
Block 4	np 125	k 95	pk 80	n 100

 

Replicate III

Block 5	pk 53	nk 76	(1) 65	np 82
Block 6	n 75	npk 100	P 55	k 92

 

(b) Use the Kuhn-Tucker conditions to solve the following Non-Linear Programming Problem:

Maximize z =  2x₁ + x₂ -x₁²

Subject to the constraints:

2x₁+ 3x₂ ≤ 6,

5x₁+ 2x₂ ≤ 10

x₁, x₂ ≥ 0                                                                                (17)

 

5)  (a)  Use Penalty method to solve the following L.P.P:

Minimize = 9x₁ + 10x₂

Subject to the constraints:

2x₁ + 4x₂  ≥ 50,

4x₁ + 3x₂  ≥ 24,

3x₁ + 2x₂   ≥ 60

x₁, x₂ ≥ 0                                                                                                      (17)

(b)   Use Beale’s method to solve the following Q.P.P:

Minimize z = 6- 6x₁ + 2x₁² – 2x₁x₂ + 2x₂²

Subject to x₁ + x₂ ≤ 2

x₁, x₂ ≥ 0                                                                                 (17)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2012

ST 4812 – STATISTICAL COMPUTING – III

 

 

Date : 23-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

Answer any THREE questions

WEEK-II

• Develop the ANOVA for R.L.S.D from the given data: (34)

WEEK-I

	VIVEK	VASANTH	CHELLA MANI	VGP
MON	A12	B 22	C 18	D 18
TUE	B 15	C 30	D 22	A 22
WED	C 18	D 40	A 14	B 15
THU	D 20	A 50	B 17	C 20

	VIVEK	VASANTH	CHELLA MANI	VGP
MON	A14	B 24	C 18	D 21
TUE	B 12	C 31	D 23	A 25
WED	C 19	D 39	A 14	B 17
THU	D 24	A 56	B 18	C 23

 

WEEK-III

 

 

 

 

 

	VIVEK	VASANTH	CHELLA MANI	VGP
MON	A13	B 25	C 17	D 23
TUE	B 14	C 33	D 23	A 26
WED	C 20	D 38	A 15	B 16
THU	D 20	A 57	B 19	C 24

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. (a) Solve the following LPP by solving its Dual problem :                                                (16+18)

Max Z = 6X +8Y

 

Subject to, 5X +2Y ≤ 20

                                                X +2Y ≤ 10

and X, Y ≥0

(b) Solve the following LPP by Big-M Method:

Max Z = 5X₁ -4 X₂+ 3X₃

Subject to,      2X₁ +X₂ -6 X₃= 20

                                                              6X₁ +5X₂+10 X₃ ≤ 76

8X₁ -3X₂+6 X₃ ≤ 50

and X₁, X₂ ,X₃≥0

3. (a) At public telephone booth in a post office, arrivals are considered to be Poisson,        with an average inter-arrival time of 12 minutes. The length of a phone call may        be assumed to be distributed exponentially with an average of 4 minutes.               Calculate the following.

(i) What is the probability that a fresh arrival will have to wait for the phone?

(ii) Find the average number of units in the system.

(iii) What is the average length of the queue that forms from time to time?

 

(b) Surface defects on 20 steel plates were counted and the data are reported below:

1       4        3        1        2        5        0        2        1        8

2        1        3        4        6       5        3         1       4        2

Construct the relevant control chart for the process. Compute the OC function when the      average number of defects increases to: 4.5,   5.0,   5.5,    6.0,    6.                   (16+18)

 

4. In a study carried by agronomist to determine if major differences in yield response to N fertilizer exist among different variables/ varieties of jowar. The main plot treatments were 3 varieties of jowar ( V₁: CO-18, V₂: CO-19 and V₃: CO-22) and the sub- plot treatment were N rates of 0, 30 and 60 kg/ ha. The study was replicated 4 times, and the  data gathered for the experiment are shown in table

 

Replication	variety	N rate, kg/ha              0                     30                         60
I	V1	14.5	16.5	19.8
	V2	19.5	23.5	29.2
	V3	14.6	17.2	17.5
II	V1	17.9	19.2	23.5
	V2	14	19.5	17.9
	V3	15	14.8	17.3
III	V1	11.9	13.5	12.5
	V2	19.2	17.5	24.5
	V3	14.9	19.5	21.5
IV	V1	11.9	12.5	17.5
	V2	12.5	16.5	13.9
	V3	11.5	10.9	9.5

 

Analysis the above data by using split plot design.                                                                            (34)

 

5. (a) Construct an Exponentially-Weighted Moving Average Control Chart for the following data on temperatures of a chemical process (in degrees centigrade) with the latest data point getting weight 0.3:

953,   949,   937,   958,   952,   946,   939,   955,   931, 954,   963,   927,   941,   938,   957

 

(b) Compute the OC function and ASN of the Double Sampling Plan (n₁ = 25, c₁ = 2,

n₂ = 10, c₂ = 4) corresponding to the lot fraction defective values p = 0.02, 0.04, 0.06,                  0.08, 0.10, 0.12.

(15 + 19)

 

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