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# Instructions: First, solve the problems without using MINITAB (unless otherwise stated). Then develop also a MINITAB solution. Submit both solutions. Marks 1. Twenty observations on the oxide thickness of individual silicon wafers are shown below. Wafer 1 2 3 4 5 6 7 8 9 10 Oxide Thickness 45.4 48.6 49.5 44.0 50.9 55.2 45.5 52.8 45.3 46.3 Wafer 11 12 13 14 15 16 17 18 19 20 Oxide Thickness 53.9 49.8 46.9 49.8 45.1 58.4 51.0 41.2 47.1 45.7 6 a) Use these data to set up a moving range chart and a chart for individual observations with 3 sigma limits. Estimate the process parameters. b) Check normality of the in-control data by plotting on the attached normal paper. Comment on the plot. 6 c) 9 d) Using the in-control data above and considering the specification limits LSL=32, USL=64, calculate the estimates of the capability indexes C p , C pk and C pm . Find the natural tolerance limits for the process, ? ? 0.95 and ? ? 0.01 . Comment. Consider the pooled sample standard deviation as the estimate of ? . Check the claim that C p >1.1. Formulate and test the appropriate hypothesis at significance level ? ? 0.05 and draw conclusion. Estimate the probability of Type II error for this test if the actual value of C p =1.2. Explain the meaning of the Type I and Type II errors for this test. 2. The data below give the number of nonconforming bearing and seal assemblies in samples of size 100. Number of Nonconforming Sample Number Assemblies 1 7 2 4 3 2 4 3 5 6 6 8 7 0 8 5 9 2 10 7 11 6 12 5 13 0 14 9 15 5 16 3 17 4 18 5 19 7 20 12 6 a) 5 b) 7 c) Construct a fraction nonconforming control chart with probability limits ( ? ? 0.004 ) using these data. Is the process in control? If necessary, revise the trial control limits. If a chart with 3 sigma limits is used, what is the ? ? risk? Is the chart with 3 sigma limits appropriate for this process? Find the control limits (probability limits, ? ? 0.004 ) for the number of nonconforming assemblies. Assume that the process mean shifts to p1 ? 0.15 . For the p chart with probability limits ( ? ? 0.004 ), what is the minimum sample size for which the probability of detecting this shift on the next sample following the shift is greater than or equal to 0.85? 3. A paper mill wishes to monitor the imperfections in the finished rolls of paper. Production output is inspected for 20 days and the resulting data are shown below: Day 1 2 3 4 5 6 7 8 9 10 6 9 Number of Rolls Produced 18 18 24 22 22 22 20 20 20 20 Total Number of Imperfections 12 14 20 18 15 12 11 15 12 10 Day 11 12 13 14 15 16 17 18 19 20 Number of Rolls Produced 18 18 18 20 20 20 24 24 22 21 Total Number of Imperfections 11 14 9 10 14 13 16 18 20 17 a. Set up a control chart with 3 sigma limits to control this process. Estimate ? , the expected number of nonconformities per roll of paper. b. Assume that the expected number of nonconformities per roll of paper has shifted to ?1 ? 1 . Design a control chart with probability limits ( ? ? 0.004 ) and a fixed sample size, to detected this shift on the first or second sample following the shift with probability ? 0.5 . For this chart, and ? ? 1.5 , find the probability that the out-of-control run length is ? 5 . 4. A study was carried out to determine the effect of the amount of carbon fiber and sand additions on casting hardness in a moulding process. The data are in the table below. Sand Addition (%) 0 0 15 15 30 30 0 0 15 15 30 30 0 0 15 15 30 30 Carbon Fiber Addition (%) 0 0 0 0 0 0 0.25 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.5 Casting Hardness 61 63 67 69 65 74 69 69 69 74 74 72 67 69 69 74 74 74 7 6 3 5 a. Formulate a model for this experiment and find the estimates of the model parameters. b. Identify significant effects by testing the appropriate hypotheses at significance level ? ? 0.05 . Find the p-values (using MINITAB). c. What combinations of the factor levels would you recommend (maximizing the hardness) based on the interaction plot? d. Check the model assumptions (only MINITAB plots are required for this problem). Comment on the plots. 5. To find a lower-pollution synthetic fuel, researchers are experimenting with three different factors, each controlled at two levels, for the processing of such a fuel. The measured levels of the undesirable emission of the fuel are shown in the table below for two replications of each treatment. Treatment (1 ) a b c ab ac bc abc _ 6 6 7 Degree of Undesirable Emission Level in ppm Y1 Y2 30 26 18 24 30 25 28 22 43 41 54 46 58 50 24 22 a. Find the estimates of the main effects, interaction effects and variance of the measurement error. b. Identify significant effects. Formulate and test the appropriate hypotheses at 5% significance level. Find the estimates of the standard errors of the significant regression coefficients. What combination of factor levels results in the lowest mean undesirable emission level? c. Assume now that the experiment was performed in two blocks, the data in column 1 (Y1) corresponds to block 1 and the data in column 2 (Y2) corresponds to block 2. Is the block effect significant in this experiment?

Instructions: First, solve the problems without using MINITAB (unless otherwise stated). Then develop also a MINITAB solution. Submit both solutions.
Marks
1.

Twenty observations on the oxide thickness of individual silicon wafers are shown below.

Wafer
1
2
3
4
5
6
7
8
9
10

Oxide Thickness
45.4
48.6
49.5
44.0
50.9
55.2
45.5
52.8
45.3
46.3

Wafer
11
12
13
14
15
16
17
18
19
20

Oxide Thickness
53.9
49.8
46.9
49.8
45.1
58.4
51.0
41.2
47.1
45.7

6

a) Use these data to set up a moving range chart and a chart for individual observations with 3 sigma limits. Estimate the process parameters.

b) Check normality of the in-control data by plotting on the attached normal paper. Comment on the plot.

6

c)

9

d)

Using the in-control data above and considering the specification limits LSL=32, USL=64, calculate the estimates of the capability indexes C p , C pk and C pm . Find the natural tolerance limits for the process,

? ? 0.95 and ? ? 0.01 . Comment. Consider the pooled sample standard deviation as the estimate of ? .
Check the claim that C p >1.1. Formulate and test the appropriate hypothesis at significance level ? ? 0.05 and draw conclusion. Estimate the probability of Type II error for this test if the actual value of C p =1.2. Explain the meaning of the Type I and Type II errors for this test.

2. The data below give the number of nonconforming bearing and seal assemblies in samples of size 100.
Number of
Nonconforming
Sample Number Assemblies
1
7
2
4
3
2
4
3
5
6
6
8
7
0
8
5
9
2
10
7
11
6
12
5
13
0
14
9
15
5
16
3
17
4
18
5
19
7
20
12

6

a)

5

b)

7

c)

Construct a fraction nonconforming control chart with probability limits ( ? ? 0.004 ) using these data. Is the process in control? If necessary, revise the trial control limits. If a chart with 3 sigma limits is used, what is the ? ? risk?
Is the chart with 3 sigma limits appropriate for this process?
Find the control limits (probability limits, ? ? 0.004 ) for the number of nonconforming assemblies.
Assume that the process mean shifts to p1 ? 0.15 . For the p chart with probability limits ( ? ? 0.004 ), what is the minimum sample size for which the probability of detecting this shift on the next sample following the shift is greater than or equal to 0.85?

3. A paper mill wishes to monitor the imperfections in the finished rolls of paper. Production output is inspected for 20 days and the resulting data are shown below:

Day
1
2
3
4
5
6
7
8
9
10

6
9

Number of
Rolls
Produced
18
18
24
22
22
22
20
20
20
20

Total Number
of
Imperfections
12
14
20
18
15
12
11
15
12
10

Day
11
12
13
14
15
16
17
18
19
20

Number of
Rolls
Produced
18
18
18
20
20
20
24
24
22
21

Total Number
of
Imperfections
11
14
9
10
14
13
16
18
20
17

a. Set up a control chart with 3 sigma limits to control this process. Estimate ? , the expected number of nonconformities per roll of paper.
b. Assume that the expected number of nonconformities per roll of paper has shifted to ?1 ? 1 . Design a control chart with probability limits ( ? ? 0.004 ) and a fixed sample size, to detected this shift on the first or second sample following the shift with probability ? 0.5 . For this chart, and ? ? 1.5 , find the probability that the out-of-control run length is ? 5 .

4. A study was carried out to determine the effect of the amount of carbon fiber and sand additions on casting hardness in a moulding process. The data are in the table below.
(%)
0
0
15
15
30
30
0
0
15
15
30
30
0
0
15
15
30
30

(%)
0
0
0
0
0
0
0.25
0.25
0.25
0.25
0.25
0.25
0.5
0.5
0.5
0.5
0.5
0.5

Casting
Hardness
61
63
67
69
65
74
69
69
69
74
74
72
67
69
69
74
74
74

7
6
3
5

a. Formulate a model for this experiment and find the estimates of the model parameters.
b. Identify significant effects by testing the appropriate hypotheses at significance level ? ? 0.05 . Find the p-values (using MINITAB).
c. What combinations of the factor levels would you recommend (maximizing the hardness) based on the interaction plot?
d. Check the model assumptions (only MINITAB plots are required for this problem). Comment on the plots.

5. To find a lower-pollution synthetic fuel, researchers are experimenting with three different factors, each controlled at two levels, for the processing of such a fuel. The measured levels of the undesirable emission of the fuel are shown in the table below for two replications of each treatment.

Treatment
(1 )
a
b
c
ab
ac
bc
abc
_

6
6

7

Degree of Undesirable Emission Level in ppm
Y1
Y2
30
26
18
24
30
25
28
22
43
41
54
46
58
50
24
22

a. Find the estimates of the main effects, interaction effects and variance of the measurement error.
b. Identify significant effects. Formulate and test the appropriate hypotheses at 5% significance level. Find the estimates of the standard errors of the significant regression coefficients. What combination of factor levels results in the lowest mean undesirable emission level?
c. Assume now that the experiment was performed in two blocks, the data in column 1 (Y1) corresponds to block 1 and the data in column 2 (Y2) corresponds to block 2. Is the block effect significant in this experiment?

Interested in a PLAGIARISM-FREE paper based on these particular instructions?...with 100% confidentiality?