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?