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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