# Applied Sciences

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Problem 1:

The diameter of the casting is an important quality characteristic. A coordinate measuring machine is used to measure the diameter of each casting at five different locations. Data for 20 castings are shown below. Perform a process capability study using x-bar and R charts. The specification limits are: Lower specification limit: 11.7; Upper specification limit: 11.8

Measurements

Casting

1

2

3

4

5

1

11.7629

11.7403

11.7511

11.7474

11.7374

2

11.8122

11.7506

11.7787

11.7736

11.8412

3

11.7742

11.7114

11.753

11.7532

11.7773

4

11.7833

11.7311

11.7777

11.8108

11.7804

5

11.7134

11.687

11.7305

11.7419

11.6642

6

11.7925

11.7611

11.7588

11.7012

11.7611

7

11.6916

11.7205

11.6958

11.744

11.7062

8

11.7109

11.7832

11.7496

11.7496

11.7318

9

11.7984

11.8887

11.7729

11.8485

11.8416

10

11.7914

11.7613

11.7356

11.7628

11.707

11

11.726

11.7329

11.7424

11.7645

11.7571

12

11.7202

11.7537

11.7328

11.7582

11.7265

13

11.8356

11.7971

11.8023

11.7802

11.7903

14

11.7069

11.7112

11.7492

11.7329

11.7289

15

11.7116

11.7978

11.7982

11.7429

11.7154

16

11.7165

11.7284

11.7571

11.7597

11.7317

17

11.8022

11.8127

11.7864

11.7917

11.8167

18

11.7775

11.7372

11.7241

11.7773

11.7543

19

11.7753

11.787

11.7574

11.762

11.7673

20

11.7572

11.7626

11.7523

11.7395

11.7884

a) Set up the x-bar and r-charts for this process, assuming the measurements on each casting form a rational subgroup. Show the xbar and r-charts for the stable process. Ensure you have a stable process.

b) Calculate the Cp and Cpk indices based on the stable process.

c) Do your capability analysis using Minitab and show Minitab output.

d) Is the process capable?

e) Is the process centered?

Problem 2:

A paper mill uses a control chart to monitor the imperfections in finished rolls of paper. Production output is inspected for twenty days, and the resulting data are shown below. Set up a control chart for nonconformities per roll of paper. Remove any out of control points, and re-calculate the control limits and center line. Provide the final control chart center line and limits below. Show your equations

Day

Number of Rolls Produced

Total Number of Imperfections

Day

Number of Rolls Produced

Total Number of Imperfections

1

18

12

11

18

18

2

18

14

12

18

14

3

24

20

13

19

8

4

22

18

14

20

10

5

22

15

15

20

14

6

22

12

16

20

13

7

20

11

17

24

16

8

20

13

18

24

18

9

20

12

19

22

20

10

20

10

20

21

17

Problem 3:

The number of nonconforming switches in samples of size 150 is shown below. Create an np-chart with the following data and then remove the out of control points and provide the revised control charts. What are the final center lines, upper and lower control limits? Show your equations

Sample

1

8

2

1

3

3

4

0

5

2

6

4

7

0

8

1

9

10

10

6

11

6

12

0

13

4

14

0

15

3

16

1

17

15

18

2

19

3

20

0

Problem 4

Bath concentrations are measured hourly in a chemical process. Data (in ppm) for the last 32 hours are shown in the table below. (Read down from left)

The process target uo=175 ppm

a. Construct a tabular Cusum for this process using h=5 and k=1/2

b. Provide your interpretation based on the chart

Sample#

Conc.

Sample#

Conc.

Sample#

Conc.

Sample#

Conc.

1

160

9

186

17

190

25

206

2

158

10

195

18

189

26

210

3

150

11

179

19

185

27

216

4

151

12

184

20

182

28

212

5

153

13

175

21

181

29

211

6

154

14

192

22

180

30

202

7

158

15

186

23

183

31

205

8

162

16

197

24

186

32

197

Problem 5:

The following data are temperature readings from a chemical process in degrees Celsius, taken every two minutes. Read the observations down from the left. The target value for the mean is 950.

a) Set up an EWMA control chart with lambda = 0.2 for this process. Show the EWMA chart.

b) Interpret the results

953

985

949

937

959

948

958

952

945

972

941

946

939

937

955

931

972

955

960

954

948

955

947

928

945

950

966

935

958

927

941

937

975

948

934

941

963

940

938

950

970

957

937

933

973

962

945

970

957

940

946

960

949

963

963

933

973

930

952

968

942

943

967

960

940

965

935

959

965

950

969

934

936

973

941

956

962

938

981

927

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