histograma 2

Universidad Tecnológica de Torreón

Victo Hugo franco García

Procesos Industriales Área de Manufactura

Matricula: 1110167

Prof.: Lic. Gerardo Edgar Mata

Se ha llenado de cierto liquido los embases de 500 ml de refresco de cola, cuyas especificaciones fueron de 44 ± 0.35mg de sodio que se le agrego a la producción arriba mencionada. Para ver si el departamento de embotellado está trabajando bajo el estándar indicado de control de calidad se

tomaron 56 muestras de cada turno, cada 30 min con un periodo de 4 horas., obteniendo los siguientes resultados:

Resultado general de las 896 muestras realizadas por ambos 2 trabajadores

43.83 43.68 43.82 43.88 43.86 43.88

43.81 43.92 43.83 44.28 44.12 44.21 44.29

44.15 44.02 43.96 44.05 44.13

43.77 43.95 43.91 43.81 43.86 43.91

43.91 43.86 43.89 44.08 44.17 44.16 44.09

44.20 44.12 44.17 44.10 44.31

43.87 43.77 43.88 43.90 43.85 43.85

43.79 43.80 43.77 44.17 44.32 44.27 44.17

44.24 44.29 44.18 44.17 43.97

43.93 43.81 43.81 43.88 43.77 43.85

43.84 43.89 43.88 44.18 44.11 44.16 44.14

44.23 44.17 44.06 44.09 44.13

43.93 43.90 43.94 43.92 43.87 43.86

43.83 43.88 43.91 44.02 44.12 44.12 44.07

44.24 44.16 44.16 44.20 44.09

43.96 43.88 43.73 43.67 43.86 43.83

43.95 43.90 43.79 44.13 44.07 44.21 44.21

44.14 44.08 44.04 44.11 44.22

43.71 43.91 43.89 43.88 43.90 43.91

43.76 43.79 43.87 44.10 44.15 44.09 44.07

44.30 44.16 44.24 44.09 43.98

43.83 43.89 43.84 43.95 43.80 43.82

43.79 43.89 43.89 44.28 44.14 44.23 44.29

44.22 44.22 44.27 44.12 44.14

43.92 43.76 43.85 43.74 43.89 44.00

43.84 43.96 43.91 44.17 44.06 44.23 44.23

44.29 44.23 44.06 44.22 43.97

43.78 43.78 43.81 43.85 43.86 43.90

43.78 43.87 43.84 44.19 44.17 44.29 44.39

44.18 44.07 44.05 44.08 44.21

43.80 43.90 43.81 43.94 43.86 43.89

43.76 43.82 43.87 44.10 44.31 44.17 44.20

44.31 44.24 44.13 44.02 44.12

43.74 43.87 43.91 43.96 43.91 43.79

43.71 43.87 43.82 43.96 44.31 44.21 44.31

44.08 44.30 44.18 44.23 44.04

43.73 43.79 43.90 43.87 43.97 43.91

43.89 43.97 43.88 44.16 44.23 44.27 44.23

44.28 44.24 44.16 44.15 44.05

43.79 43.83 43.88 43.86 43.76 43.95

43.83 43.69 43.95 44.26 44.20 44.31 44.20

44.27 44.15 44.21 44.25 44.26

43.80 43.86 43.90 43.97 43.77 43.80

43.88 43.85 43.84 44.07 44.16 44.25 44.09

44.26 43.92 44.20 44.09 44.07

43.71 43.89 43.96 43.85 43.89 43.83

43.71 43.84 43.78 44.16 44.23 44.19 44.05

44.37 43.89 44.30 44.07 44.14

43.81 43.86 43.87 43.81 43.93 43.76

43.93 43.92 43.79 44.01 44.09 44.17 44.12

44.14 44.24 44.11 44.01 44.13

43.82 43.79 43.89 43.91 43.79 43.94

43.72 43.93 43.90 44.14 44.34 44.12 44.16

44.13 44.11 44.22 44.19 44.10

43.86 43.97 43.97 43.85 43.71 43.92

43.82 43.96 43.96 44.15 44.29 44.02 44.17

44.25 44.22 44.08 44.04 44.08

43.83 43.88 43.83 43.92 43.79 43.86

43.79 43.73 43.81 44.11 44.17 44.18 44.22

44.21 44.16 44.16 44.17 44.19

43.83 43.85 43.92 43.88 43.86 43.71

43.84 43.90 43.80 44.31 44.13 44.19 44.10

44.23 44.00 44.20 43.90 44.10

43.83 43.90 43.86 43.95 43.88 43.88

43.86 43.87 43.92 44.18 44.08 44.14 44.28

44.08 44.25 44.26 44.13 44.08

43.94 43.91 43.93 43.84 43.78 43.88

43.96 43.90 43.86 44.22 44.10 43.98 44.14

44.15 44.21 44.32 44.32 43.94

43.84 43.81 43.84 43.79 43.79 43.93

43.82 43.88 43.88 44.06 44.12 44.26 44.08

44.33 44.24 44.20 44.03 44.19

43.84 43.79 43.80 43.75 43.88 43.79

43.80 43.84 44.02 44.17 44.28 44.16 44.15

44.27 44.31 44.26 44.40 44.36

43.82 43.92 43.92 43.68 43.86 43.93

43.80 43.96 43.81 44.27 44.18 44.01 44.09

44.14 44.23 44.11 44.09 44.38

43.98 43.77 43.81 43.86 43.81 43.81

43.89 43.88 43.80 44.13 44.35 44.17 44.22

44.17 44.02 44.35 44.24 44.03

43.91 43.75 43.89 43.87 43.86 43.80

43.93 43.85 43.93 44.18 44.06 44.16 44.22

44.06 44.22 44.15 44.17 44.19

44.01 43.90 43.89 43.85 43.79 43.74

43.76 43.85 43.83 44.06 44.27 44.27 44.26

44.09 44.16 44.16 44.07 44.09

43.81 43.89 43.83 43.86 43.79 43.94

43.91 43.85 43.76 44.09 44.27 44.22 44.32

44.22 44.12 44.27 44.02 44.32

43.96 44.00 43.82 43.87 43.92 43.79

43.95 43.74 43.92 44.13 44.09 44.09 44.18

44.15 44.10 44.05 44.23 44.28

43.74 43.95 43.76 43.95 43.96 43.95

43.81 43.88 43.76 44.18 44.17 44.28 44.25

44.13 44.33 44.13 44.19 44.13

43.89 43.94 43.98 43.86 43.81 43.77

43.86 43.86 43.82 44.29 44.20 44.04 44.19

44.06 44.25 44.30 44.10 44.08

43.91 43.86 43.81 43.78 43.84 43.82

43.80 43.89 43.76 44.11 44.20 44.11 44.04

44.23 44.25 44.13 44.14 44.28

43.98 43.85 43.86 43.76 43.86 43.76

43.90 43.84 43.78 44.14 44.22 44.12 44.24

44.06 44.19 44.18 44.17 44.17

43.84 43.88 43.83 43.86 43.97 43.95

43.87 43.90 43.89 44.16 44.20 44.24 44.26

44.05 44.00 44.27 44.20 44.28

43.82 43.85 44.04 43.91 43.89 43.85

43.85 43.94 43.86 44.14 44.30 44.30 44.26

44.26 44.20 44.22 44.21 44.25

43.89 43.78 43.93 43.75 43.96 43.87

43.78 43.98 43.73 44.19 43.95 44.05 44.19

44.17 44.08 44.19 44.30 44.27

43.82 43.73 43.91 43.81 43.86 43.91

43.83 43.76 43.98 44.25 44.09 44.01 44.12

44.31 44.08 44.10 44.11 44.12

43.90 43.90 43.94 43.88 43.95 43.98

43.90 43.89 43.87 44.13 44.12 44.11 44.13

44.04 44.15 44.12 44.05 44.15

43.75 43.88 43.86 43.84 43.81 43.85

43.85 43.84 43.94 44.14 44.02 44.03 44.04

44.22 44.26 44.19 44.15 44.24

43.79 43.89 43.89 43.83 43.79 43.78

43.92 43.80 43.85 44.22 44.03 44.14 44.20

44.14 44.07 44.11 44.22 44.26

43.75 43.86 43.83 43.84 43.93 43.81

43.78 43.90 43.83 44.20 44.35 44.19 44.24

44.14 44.16 44.31 43.99 44.17

43.8 43.78 43.77 43.82 43.91 43.91 43.9 44.00 43.85 44.25 44.14 44.19 44.31 44.2 44.00 44.16 44.13 44.21

3 0 543.8

5 43.93 43.93 43.80 43.74 43.9143.9

3 43.94 43.73 44.22 44.12 44.18 44.0044.1

6 44.06 44.34 44.14 44.1443.8

5 43.83 43.83 43.89 43.93 43.8843.7

7 43.86 43.80 44.22 44.04 44.15 44.2144.2

0 44.19 44.17 44.08 44.1343.8

3 43.79 43.76 43.85 43.84 43.7543.8

8 43.82 43.94 44.13 44.34 44.18 44.1844.1

3 44.19 44.15 43.98 44.0943.9

9 43.80 43.90 43.93 43.78 43.7643.7

3 43.83 43.86 44.10 44.19 44.19 44.1944.1

4 44.07 44.14 44.14 44.15

43.82 43.90 43.87 43.73 43.8443.8

6 43.78 43.82 44.03 44.12 44.39 44.1144.0

9 44.22 44.24 44.21

43.82 43.88 43.85 43.80 43.8143.9

2 43.85 43.81 44.13 44.19 44.10 44.1644.1

1 44.23 44.32 44.16

Resultados de 448 muestras tomadas del trabajador “Alan”

43.83 43.68 43.82 43.877 43.86 43.8843.8

1 43.92 43.833 43.75 43.864 43.8335 43.8443.9

3 43.81 43.78

43.77 43.946 43.91 43.807 43.86 43.9143.9

1 43.86 43.892 43.83 43.782 43.7694 43.8243.9

1 43.91 43.9

43.87 43.766 43.88 43.898 43.85 43.8543.7

9 43.8 43.766 43.85 43.932 43.9336 43.843.7

4 43.91 43.93

43.93 43.807 43.81 43.885 43.77 43.8543.8

4 43.89 43.884 43.85 43.829 43.8309 43.8943.9

3 43.88 43.77

43.93 43.9 43.94 43.921 43.87 43.8643.8

3 43.88 43.914 43.83 43.795 43.7638 43.8543.8

4 43.75 43.88

43.96 43.881 43.73 43.667 43.86 43.8343.9

5 43.9 43.793 43.99 43.796 43.9004 43.9343.7

8 43.76 43.73

43.71 43.908 43.89 43.881 43.9 43.9143.7

6 43.79 43.875 43.82 43.848 44.037 43.9143.8

9 43.85 43.85

43.83 43.889 43.84 43.947 43.8 43.8243.7

9 43.89 43.885 43.89 43.78 43.9326 43.7543.9

6 43.87 43.78

43.92 43.759 43.85 43.736 43.89 4443.8

4 43.96 43.914 43.82 43.733 43.9081 43.8143.8

6 43.91 43.83

43.78 43.776 43.81 43.853 43.86 43.943.7

8 43.87 43.835 43.9 43.905 43.9379 43.8843.9

5 43.98 43.9

43.8 43.896 43.81 43.939 43.86 43.8943.7

6 43.82 43.865 43.75 43.879 43.8624 43.8443.8

1 43.85 43.85

43.74 43.871 43.91 43.961 43.91 43.7943.7

1 43.87 43.822 43.79 43.891 43.8858 43.8343.7

9 43.78 43.92

43.73 43.788 43.9 43.866 43.97 43.9143.8

9 43.97 43.884 43.74 43.945 43.7598 43.9543.9

6 43.95 43.81

43.79 43.834 43.88 43.86 43.76 43.9543.8

3 43.69 43.954 43.89 43.936 43.9815 43.8643.8

1 43.77 43.86

43.8 43.859 43.9 43.972 43.77 43.843.8

8 43.85 43.84 43.91 43.857 43.8126 43.7843.8

4 43.82 43.8

43.71 43.887 43.96 43.847 43.89 43.8343.7

1 43.84 43.782 43.98 43.85 43.8557 43.7643.8

6 43.76 43.9

43.81 43.859 43.87 43.807 43.93 43.7643.9

3 43.92 43.789 43.84 43.88 43.8345 43.8643.9

7 43.95 43.87

43.82 43.79 43.89 43.91 43.79 43.9443.7

2 43.93 43.898 43.88 43.924 43.7635 43.9843.9

4 43.73 43.81

43.86 43.974 43.97 43.849 43.71 43.9243.8

2 43.96 43.955 43.86 43.765 43.8909 43.8743.8

6 43.8 43.8

43.83 43.882 43.83 43.919 43.79 43.8643.7

9 43.73 43.808 43.89 43.824 43.84 43.9443.8

2 43.94 43.93

43.83 43.855 43.92 43.879 43.86 43.7143.8

4 43.9 43.795 43.84 43.755 43.80 43.8543.8

3 43.86 43.83

43.83 43.905 43.86 43.946 43.88 43.8843.8

6 43.87 43.918 43.9 43.778 43.90 43.8343.7

8 43.82 43.76

43.94 43.907 43.93 43.842 43.78 43.8843.9

6 43.9 43.865 43.94 43.89 44.00 43.8543.8

6 43.81 43.88

43.84 43.808 43.84 43.787 43.79 43.9343.8

2 43.88 43.88 43.98 43.858 43.8016 43.7443.7

9 43.93 43.85

43.84 43.789 43.8 43.748 43.88 43.79 43.8 43.84 44.022 43.83 43.735 43.7362 43.9543.7

9 43.76 43.85

43.82 43.923 43.92 43.681 43.86 43.93 43.8 43.96 43.863 43.82 43.867 43.9407 43.9243.7

9 43.91 43.8543.98 43.771 43.81 43.857 43.81 43.81 43.8 43.81 43.892 43.96 43.996 44.0068 43.9 43.8 43.85 43.87

9 9

43.91 43.747 43.89 43.822 43.9 43.8743.7

3 43.84 43.855 43.82 43.878 43.8538 43.843.8

1 43.92 43.85

Resultados de 448 muestras tomadas del trabajador “Juan”

1.382

1.4559

1.432

1.386

1.438 1.304

1.421 1.48

1.475 1.474 1.41 1.446 1.38 1.38

1.456 1.357

1.518

1.343

1.433 1.424

1.413 1.53

1.434 1.473 1.4 1.333 1.45 1.46

1.394

1.3973

1.512 1.37

1.459 1.391

1.404 1.41 1.29 1.275 1.41 1.456 1.52 1.4

1.504

1.4527

1.418

1.428 1.46 1.397

1.589 1.43

1.466 1.449 1.49 1.457 1.41 1.35

1.531.343

31.44

4 1.381.35

7 1.4661.47

6 1.41.35

3 1.544 1.43 1.472 1.45 1.371.47

2 1.343 1.411.28

71.44

4 1.5411.48

5 1.391.39

6 1.495 1.29 1.382 1.36 1.41.26

21.285

61.23

21.44

6 1.39 1.4361.49

9 1.31.40

5 1.49 1.42 1.505 1.44 1.441.41

31.360

81.44

31.43

7 1.45 1.4221.42

3 1.451.44

6 1.33 1.39 1.506 1.41 1.521.31

1 1.3321.42

31.45

8 1.35 1.5261.52

5 1.331.39

7 1.42 1.37 1.402 1.44 1.481.42

81.466

4 1.291.51

61.40

5 1.458 1.55 1.291.47

4 1.494 1.37 1.367 1.51 1.481.40

51.400

6 1.3 1.51.37

7 1.5371.41

9 1.46 1.48 1.394 1.47 1.501 1.4 1.451.47

81.386

31.43

91.31

61.47

3 1.5481.28

9 1.471.40

4 1.538 1.56 1.444 1.5 1.391.38

31.470

71.48

61.39

21.46

5 1.4991.43

5 1.431.46

9 1.576 1.39 1.569 1.39 1.431.42

31.493

41.49

11.36

9 1.53 1.3531.40

8 1.481.33

5 1.421 1.47 1.438 1.44 1.37

1.359

1.4099

1.296

1.299

1.555 1.363 1.49 1.41

1.465 1.396 1.35 1.449 1.35 1.41

1.338

1.3366

1.392 1.48

1.446 1.502

1.504 1.44

1.456 1.373 1.4 1.43 1.43 1.4

1.396 1.373

1.367

1.409

1.358 1.469

1.448 1.5

1.445 1.402 1.36 1.386 1.47 1.33

1.444

1.4279

1.383

1.333

1.415 1.449

1.475 1.49

1.418 1.501 1.42 1.39 1.41 1.39

1.326

1.4723

1.407

1.415

1.479 1.381

1.385 1.31

1.497 1.345 1.49 1.473 1.48 1.32

1.414

1.4936

1.394

1.398

1.394 1.399

1.451 1.49

1.315 1.452 1.36 1.443 1.4 1.5

1.292

1.4272

1.262 1.49

1.384 1.441

1.381 1.35

1.437 1.377 1.45 1.348 1.44 1.4

1.311.424

91.31

81.53

31.39

1 1.4191.38

6 1.37 1.47 1.489 1.46 1.483 1.55 1.411.49

31.402

21.37

81.41

3 1.51 1.3651.32

6 1.361.29

6 1.471 1.48 1.486 1.47 1.411.35

61.490

11.42

61.50

61.48

2 1.371.37

8 1.431.38

1 1.417 1.33 1.347 1.54 1.461.44

61.355

91.43

81.48

91.54

8 1.411.41

5 1.411.43

9 1.339 1.46 1.432 1.43 1.341.45

31.350

21.29

31.39

31.38

1 1.3171.37

1 1.49 1.32 1.406 1.35 1.44 1.46 1.451.41

21.472

81.48

8 1.281.34

5 1.4941.43

1 1.441.29

9 1.439 1.49 1.53 1.38 1.411.40

41.350

2 1.451.30

11.34

2 1.481.29

2 1.361.42

9 1.296 1.42 1.316 1.49 1.411.39

51.345

61.39

41.39

61.19

6 1.4421.48

4 1.45 1.45 1.435 1.34 1.47 1.42 1.351.44

91.475

11.44

11.36

51.50

6 1.5461.50

4 1.371.39

6 1.431 1.25 1.47 1.36 1.451.58

11.428

51.45

11.48

21.34

7 1.4391.46

6 1.431.46

9 1.541 1.47 1.499 1.33 1.361.48

41.383

41.52

5 1.391.49

5 1.531.41

9 1.361.45

2 1.456 1.43 1.401 1.48 1.411.51

91.333

31.43

41.31

41.53

8 1.4851.38

2 1.481.40

7 1.401 1.42 1.463 1.37 1.45

1.351.403

41.47

51.46

1 1.42 1.2461.34

6 1.431.39

1 1.541 1.45 1.43 1.53 1.531.39

51.337

81.48

61.41

91.40

1 1.3341.50

3 1.431.49

8 1.35 1.22 1.344 1.48 1.57

Solución en las siguientes paginas

Para resolver el problema, primero vamos a elaborar una tabla de frecuencias con su respectivo histograma tomando en conjunto la seria de muestras tomadas por ambos trabajadores.

Resultado general de las 896 muestras realizadas por ambos trabajadores

43.83 43.68 43.82 43.88 43.86 43.88

43.81 43.92 43.83 44.28 44.12 44.21 44.29

44.15 44.02 43.96 44.05 44.13

43.77 43.95 43.91 43.81 43.86 43.91

43.91 43.86 43.89 44.08 44.17 44.16 44.09

44.20 44.12 44.17 44.10 44.31

43.87 43.77 43.88 43.90 43.85 43.85

43.79 43.80 43.77 44.17 44.32 44.27 44.17

44.24 44.29 44.18 44.17 43.97

43.93 43.81 43.81 43.88 43.77 43.85

43.84 43.89 43.88 44.18 44.11 44.16 44.14

44.23 44.17 44.06 44.09 44.13

43.93 43.90 43.94 43.92 43.87 43.86

43.83 43.88 43.91 44.02 44.12 44.12 44.07

44.24 44.16 44.16 44.20 44.09

43.96 43.88 43.73 43.67 43.86 43.83

43.95 43.90 43.79 44.13 44.07 44.21 44.21

44.14 44.08 44.04 44.11 44.22

43.71 43.91 43.89 43.88 43.90 43.91

43.76 43.79 43.87 44.10 44.15 44.09 44.07

44.30 44.16 44.24 44.09 43.98

43.83 43.89 43.84 43.95 43.80 43.82

43.79 43.89 43.89 44.28 44.14 44.23 44.29

44.22 44.22 44.27 44.12 44.14

43.92 43.76 43.85 43.74 43.89 44.00

43.84 43.96 43.91 44.17 44.06 44.23 44.23

44.29 44.23 44.06 44.22 43.97

43.78 43.78 43.81 43.85 43.86 43.90

43.78 43.87 43.84 44.19 44.17 44.29 44.39

44.18 44.07 44.05 44.08 44.21

43.80 43.90 43.81 43.94 43.86 43.89

43.76 43.82 43.87 44.10 44.31 44.17 44.20

44.31 44.24 44.13 44.02 44.12

43.74 43.87 43.91 43.96 43.91 43.79

43.71 43.87 43.82 43.96 44.31 44.21 44.31

44.08 44.30 44.18 44.23 44.04

43.73 43.79 43.90 43.87 43.97 43.91

43.89 43.97 43.88 44.16 44.23 44.27 44.23

44.28 44.24 44.16 44.15 44.05

43.79 43.83 43.88 43.86 43.76 43.95

43.83 43.69 43.95 44.26 44.20 44.31 44.20

44.27 44.15 44.21 44.25 44.26

43.80 43.86 43.90 43.97 43.77 43.80

43.88 43.85 43.84 44.07 44.16 44.25 44.09

44.26 43.92 44.20 44.09 44.07

43.71 43.89 43.96 43.85 43.89 43.83

43.71 43.84 43.78 44.16 44.23 44.19 44.05

44.37 43.89 44.30 44.07 44.14

43.81 43.86 43.87 43.81 43.93 43.76

43.93 43.92 43.79 44.01 44.09 44.17 44.12

44.14 44.24 44.11 44.01 44.13

43.82 43.79 43.89 43.91 43.79 43.94

43.72 43.93 43.90 44.14 44.34 44.12 44.16

44.13 44.11 44.22 44.19 44.10

43.86 43.97 43.97 43.85 43.71 43.92

43.82 43.96 43.96 44.15 44.29 44.02 44.17

44.25 44.22 44.08 44.04 44.08

43.83 43.88 43.83 43.92 43.79 43.86

43.79 43.73 43.81 44.11 44.17 44.18 44.22

44.21 44.16 44.16 44.17 44.19

43.83 43.85 43.92 43.88 43.86 43.71

43.84 43.90 43.80 44.31 44.13 44.19 44.10

44.23 44.00 44.20 43.90 44.10

43.83 43.90 43.86 43.95 43.88 43.88

43.86 43.87 43.92 44.18 44.08 44.14 44.28

44.08 44.25 44.26 44.13 44.08

43.94 43.91 43.93 43.84 43.78 43.88

43.96 43.90 43.86 44.22 44.10 43.98 44.14

44.15 44.21 44.32 44.32 43.94

43.84 43.81 43.84 43.79 43.79 43.93

43.82 43.88 43.88 44.06 44.12 44.26 44.08

44.33 44.24 44.20 44.03 44.19

43.84 43.79 43.80 43.75 43.88 43.79

43.80 43.84 44.02 44.17 44.28 44.16 44.15

44.27 44.31 44.26 44.40 44.36

43.82 43.92 43.92 43.68 43.86 43.93

43.80 43.96 43.81 44.27 44.18 44.01 44.09

44.14 44.23 44.11 44.09 44.38

43.98 43.77 43.81 43.86 43.81 43.81

43.89 43.88 43.80 44.13 44.35 44.17 44.22

44.17 44.02 44.35 44.24 44.03

43.91 43.75 43.89 43.87 43.86 43.80

43.93 43.85 43.93 44.18 44.06 44.16 44.22

44.06 44.22 44.15 44.17 44.19

44.01 43.90 43.89 43.85 43.79 43.74

43.76 43.85 43.83 44.06 44.27 44.27 44.26

44.09 44.16 44.16 44.07 44.09

43.81 43.89 43.83 43.86 43.79 43.94

43.91 43.85 43.76 44.09 44.27 44.22 44.32

44.22 44.12 44.27 44.02 44.32

43.96 44.00 43.82 43.87 43.92 43.79

43.95 43.74 43.92 44.13 44.09 44.09 44.18

44.15 44.10 44.05 44.23 44.28

43.74 43.95 43.76 43.95 43.96 43.95

43.81 43.88 43.76 44.18 44.17 44.28 44.25

44.13 44.33 44.13 44.19 44.13

43.89 43.94 43.98 43.86 43.81 43.77

43.86 43.86 43.82 44.29 44.20 44.04 44.19

44.06 44.25 44.30 44.10 44.08

43.91 43.86 43.81 43.78 43.84 43.82

43.80 43.89 43.76 44.11 44.20 44.11 44.04

44.23 44.25 44.13 44.14 44.28

43.98 43.85 43.86 43.76 43.86 43.76

43.90 43.84 43.78 44.14 44.22 44.12 44.24

44.06 44.19 44.18 44.17 44.17

43.84 43.88 43.83 43.86 43.97 43.95

43.87 43.90 43.89 44.16 44.20 44.24 44.26

44.05 44.00 44.27 44.20 44.28

43.82 43.85 44.04 43.91 43.89 43.85

43.85 43.94 43.86 44.14 44.30 44.30 44.26

44.26 44.20 44.22 44.21 44.25

43.89 43.78 43.93 43.75 43.96 43.87

43.78 43.98 43.73 44.19 43.95 44.05 44.19

44.17 44.08 44.19 44.30 44.27

43.82 43.73 43.91 43.81 43.86 43.91

43.83 43.76 43.98 44.25 44.09 44.01 44.12

44.31 44.08 44.10 44.11 44.12

43.90 43.90 43.94 43.88 43.95 43.98

43.90 43.89 43.87 44.13 44.12 44.11 44.13

44.04 44.15 44.12 44.05 44.15

43.75 43.88 43.86 43.84 43.81 43.85

43.85 43.84 43.94 44.14 44.02 44.03 44.04

44.22 44.26 44.19 44.15 44.24

43.79 43.89 43.89 43.83 43.79 43.78

43.92 43.80 43.85 44.22 44.03 44.14 44.20

44.14 44.07 44.11 44.22 44.26

43.75 43.86 43.83 43.84 43.93 43.81

43.78 43.90 43.83 44.20 44.35 44.19 44.24

44.14 44.16 44.31 43.99 44.17

43.83 43.78 43.77 43.82 43.91 43.91

43.90 44.00 43.85 44.25 44.14 44.19 44.31

44.25 44.00 44.16 44.13 44.21

43.8 43.93 43.93 43.80 43.74 43.91 43.9 43.94 43.73 44.22 44.12 44.18 44.00 44.1 44.06 44.34 44.14 44.14

5 3 643.8

5 43.83 43.83 43.89 43.93 43.8843.7

7 43.86 43.80 44.22 44.04 44.15 44.2144.2

0 44.19 44.17 44.08 44.1343.8

3 43.79 43.76 43.85 43.84 43.7543.8

8 43.82 43.94 44.13 44.34 44.18 44.1844.1

3 44.19 44.15 43.98 44.0943.9

9 43.80 43.90 43.93 43.78 43.7643.7

3 43.83 43.86 44.10 44.19 44.19 44.1944.1

4 44.07 44.14 44.14 44.15

43.82 43.90 43.87 43.73 43.8443.8

6 43.78 43.82 44.03 44.12 44.39 44.1144.0

9 44.22 44.24 44.21

43.82 43.88 43.85 43.80 43.8143.9

2 43.85 43.81 44.13 44.19 44.10 44.1644.1

1 44.23 44.32 44.16

Se agrupan en 9 intervalos y se obtiene la siguiente tabla de distribución de frecuencias:

Intervalos aparentes Marcas de clase Medidas de tendencia central y dispersión

Límite inf. Límite sup. Xi fi fai fri frai (fi)(xi) (|xi-~x|)

(fi) (xi-~x)2(f

43.66722 43.748 43.7075 26 26 0.02902 0.029021136.394

13 7.8658415 2.379672

43.7487 43.829 43.7890 134 160 0.14955 0.178575867.721

529.618337

1 6.546611

43.8302 43.911 43.8705 200 360 0.22321 0.401798774.093

327.906473

2 3.893856

43.9117 43.992 43.9520 93 453 0.10379 0.50558 4087.533 5.3970100 0.313201

43.9932 44.074 44.0335 64 517 0.07143 0.577012818.141

85 1.5019286 0.035247

44.0747 44.155 44.1150 134 651 0.14955 0.726565911.405

514.065662

9 1.476439

44.1562 44.237 44.1965 146 797 0.16295 0.889516452.684

127.224274

6 5.076446

44.2377 44.318 44.2780 81 878 0.09040 0.97991 3586.5221.705378

3 5.816339

44.319 44.400 44.3595 18 896 0.02009 1.00000798.4704

0 6.290417 2.198297

896 1 39432.95

9141.57532

4 27.736109

Media Aritmética

~x 44.00999

881

Desviación Media 0.158008

17

Varianza = S2 0.030955

48

Desviación

Estándar = √S 0.175941

69

Mediante esta tabla podemos observar lo siguiente:

1. Mediante un histograma general elaborado por ambos trabajadores, podemos determinar si surge un error, determinar de dónde proviene., y Como mejorarlo para que ambos turnos trabajen al mismo ritmo.

2. Mediante la tabla de frecuencias, podemos observar que las frecuencias no están distribuidas en forma normal, están distribuidas en forma Bimodal la cual suele presentarse cuando un conjunto de valores son obtenidos por la mezcla de 2 muestras como en este caso, que se tomo en conjunto las muestras de los 2 trabajadores.

3. El USL esperado no está dentro de los limites de especiación que fueron establecidos por la empresa, mientras que el LSL, si esta dentro del rango de tolerancia.

Es recomendable, revisar el histograma para su posterior interpretación:

43.4000 43.6000 43.8000 44.0000 44.2000 44.4000 44.60000

50

100

150

200

250

Las observaciones que se isieron a partir de la tabla de distribución, se pueden visualizar mejor en el histograma: se observa que los datos están distribuidos en forma Bimodal y que algunos valores están fuera de los límites de especificación.

Para resolver el problema, primero vamos a elaborar una tabla de frecuencias con su respectivo histograma por cada trabajador.

TV USLLSL

Resultados de 448 muestras tomadas del trabajador “Alan”

43.83 43.68 43.82 43.877 43.86 43.8843.8

1 43.92 43.833 43.75 43.864 43.8335 43.8443.9

3 43.81 43.78

43.77 43.946 43.91 43.807 43.86 43.9143.9

1 43.86 43.892 43.83 43.782 43.7694 43.8243.9

1 43.91 43.9

43.87 43.766 43.88 43.898 43.85 43.8543.7

9 43.8 43.766 43.85 43.932 43.9336 43.843.7

4 43.91 43.93

43.93 43.807 43.81 43.885 43.77 43.8543.8

4 43.89 43.884 43.85 43.829 43.8309 43.8943.9

3 43.88 43.77

43.93 43.9 43.94 43.921 43.87 43.8643.8

3 43.88 43.914 43.83 43.795 43.7638 43.8543.8

4 43.75 43.88

43.96 43.881 43.73 43.667 43.86 43.8343.9

5 43.9 43.793 43.99 43.796 43.9004 43.9343.7

8 43.76 43.73

43.71 43.908 43.89 43.881 43.9 43.9143.7

6 43.79 43.875 43.82 43.848 44.037 43.9143.8

9 43.85 43.85

43.83 43.889 43.84 43.947 43.8 43.8243.7

9 43.89 43.885 43.89 43.78 43.9326 43.7543.9

6 43.87 43.78

43.92 43.759 43.85 43.736 43.89 4443.8

4 43.96 43.914 43.82 43.733 43.9081 43.8143.8

6 43.91 43.83

43.78 43.776 43.81 43.853 43.86 43.943.7

8 43.87 43.835 43.9 43.905 43.9379 43.8843.9

5 43.98 43.9

43.8 43.896 43.81 43.939 43.86 43.8943.7

6 43.82 43.865 43.75 43.879 43.8624 43.8443.8

1 43.85 43.85

43.74 43.871 43.91 43.961 43.91 43.7943.7

1 43.87 43.822 43.79 43.891 43.8858 43.8343.7

9 43.78 43.92

43.73 43.788 43.9 43.866 43.97 43.9143.8

9 43.97 43.884 43.74 43.945 43.7598 43.9543.9

6 43.95 43.81

43.79 43.834 43.88 43.86 43.76 43.9543.8

3 43.69 43.954 43.89 43.936 43.9815 43.8643.8

1 43.77 43.86

43.8 43.859 43.9 43.972 43.77 43.843.8

8 43.85 43.84 43.91 43.857 43.8126 43.7843.8

4 43.82 43.8

43.71 43.887 43.96 43.847 43.89 43.8343.7

1 43.84 43.782 43.98 43.85 43.8557 43.7643.8

6 43.76 43.9

43.81 43.859 43.87 43.807 43.93 43.7643.9

3 43.92 43.789 43.84 43.88 43.8345 43.8643.9

7 43.95 43.87

43.82 43.79 43.89 43.91 43.79 43.9443.7

2 43.93 43.898 43.88 43.924 43.7635 43.9843.9

4 43.73 43.81

43.86 43.974 43.97 43.849 43.71 43.9243.8

2 43.96 43.955 43.86 43.765 43.8909 43.8743.8

6 43.8 43.8

43.83 43.882 43.83 43.919 43.79 43.8643.7

9 43.73 43.808 43.89 43.824 43.84 43.9443.8

2 43.94 43.9343.83 43.855 43.92 43.879 43.86 43.71 43.8 43.9 43.795 43.84 43.755 43.80 43.85 43.8 43.86 43.83

4 3

43.83 43.905 43.86 43.946 43.88 43.8843.8

6 43.87 43.918 43.9 43.778 43.90 43.8343.7

8 43.82 43.76

43.94 43.907 43.93 43.842 43.78 43.8843.9

6 43.9 43.865 43.94 43.89 44.00 43.8543.8

6 43.81 43.88

43.84 43.808 43.84 43.787 43.79 43.9343.8

2 43.88 43.88 43.98 43.858 43.8016 43.7443.7

9 43.93 43.85

43.84 43.789 43.8 43.748 43.88 43.79 43.8 43.84 44.022 43.83 43.735 43.7362 43.9543.7

9 43.76 43.85

43.82 43.923 43.92 43.681 43.86 43.93 43.8 43.96 43.863 43.82 43.867 43.9407 43.9243.7

9 43.91 43.85

43.98 43.771 43.81 43.857 43.81 43.8143.8

9 43.81 43.892 43.96 43.996 44.0068 43.943.8

9 43.85 43.87

43.91 43.747 43.89 43.822 43.9 43.8743.7

3 43.84 43.855 43.82 43.878 43.8538 43.843.8

1 43.92 43.85




(fi) (xi-~x)2(f

43.66722 43.707 43.6873 7 7 0.01563 0.01563305.8112

2 1.1677625 0.194810

43.7084 43.749 43.7285 20 27 0.04464 0.06027 874.5703 2.5124643 0.315624

43.7496 43.790 43.7697 56 83 0.12500 0.185272451.104

1 4.7277000 0.399128

43.7908 43.831 43.8109 82 165 0.18304 0.36830 3592.495 3.5443036 0.153196

43.8320 43.872 43.8521 102 267 0.22768 0.595984472.915

9 0.2063679 0.000418

43.8732 43.913 43.8933 93 360 0.20759 0.803574082.078

4 3.6434411 0.142738

43.9144 43.955 43.9345 58 418 0.12946 0.933042548.202

0 4.6618536 0.374705

43.9556 43.996 43.9757 25 443 0.05580 0.98884 1099.39 3.0394196 0.369523

43.997 44.037 44.0169 5 448 0.01116 1.00000220.0845

8 0.813884 0.132481

448 1 19646.65

5 24.317196 2.082622

Media Aritmética

~x 43.85413

966


46


71

Desviación


46

Mediante esta tabla podemos observar lo siguiente:

Una vez que tenemos los datos, es imposible determinar una posible conclusión, es recomendable que se ordenen los valores en una tabla de distribución de frecuencias. Se observan que los valores obtenidos por el primer trabajador, están distribuidos en forma normal.


43.6000 43.7000 43.8000 43.9000 44.0000 44.1000 44.2000 44.3000 44.40000

20

40

60

80

100

120

140

Las observaciones que se isieron a partir de la tabla de distribución, se pueden visualizar mejor en el histograma: se observa que los datos obtenidos por las muestras del primer trabajador están distribuidos en forma normal y que los valores están dentro de los límites de especificación que fueron establecidos por la misma empresa.

Resultados de 448 muestras tomadas del trabajador “Juan”

USL TVLSL

1.382

1.4559

1.432

1.386

1.438 1.304

1.421 1.48

1.475 1.474 1.41 1.446 1.38 1.38

1.456 1.357

1.518

1.343

1.433 1.424

1.413 1.53

1.434 1.473 1.4 1.333 1.45 1.46

1.394

1.3973

1.512 1.37

1.459 1.391

1.404 1.41 1.29 1.275 1.41 1.456 1.52 1.4

1.504

1.4527

1.418

1.428 1.46 1.397

1.589 1.43

1.466 1.449 1.49 1.457 1.41 1.35

1.531.343

31.44

4 1.381.35

7 1.4661.47

6 1.41.35

3 1.544 1.43 1.472 1.45 1.371.47

2 1.343 1.411.28

71.44

4 1.5411.48

5 1.391.39

6 1.495 1.29 1.382 1.36 1.41.26

21.285

61.23

21.44

6 1.39 1.4361.49

9 1.31.40

5 1.49 1.42 1.505 1.44 1.441.41

31.360

81.44

31.43

7 1.45 1.4221.42

3 1.451.44

6 1.33 1.39 1.506 1.41 1.521.31

1 1.3321.42

31.45

8 1.35 1.5261.52

5 1.331.39

7 1.42 1.37 1.402 1.44 1.481.42

81.466

4 1.291.51

61.40

5 1.458 1.55 1.291.47

4 1.494 1.37 1.367 1.51 1.481.40

51.400

6 1.3 1.51.37

7 1.5371.41

9 1.46 1.48 1.394 1.47 1.501 1.4 1.451.47

81.386

31.43

91.31

61.47

3 1.5481.28

9 1.471.40

4 1.538 1.56 1.444 1.5 1.391.38

31.470

71.48

61.39

21.46

5 1.4991.43

5 1.431.46

9 1.576 1.39 1.569 1.39 1.431.42

31.493

41.49

11.36

9 1.53 1.3531.40

8 1.481.33

5 1.421 1.47 1.438 1.44 1.371.35

91.409

91.29

61.29

91.55

5 1.363 1.49 1.411.46

5 1.396 1.35 1.449 1.35 1.411.33

81.336

61.39

2 1.481.44

6 1.5021.50

4 1.441.45

6 1.373 1.4 1.43 1.43 1.41.39

6 1.3731.36

71.40

91.35

8 1.4691.44

8 1.51.44

5 1.402 1.36 1.386 1.47 1.331.44

41.427

91.38

31.33

31.41

5 1.4491.47

5 1.491.41

8 1.501 1.42 1.39 1.41 1.391.32

61.472

31.40

71.41

51.47

9 1.3811.38

5 1.311.49

7 1.345 1.49 1.473 1.48 1.321.41

41.493

61.39

41.39

81.39

4 1.3991.45

1 1.491.31

5 1.452 1.36 1.443 1.4 1.51.29

21.427

21.26

2 1.491.38

4 1.4411.38

1 1.351.43

7 1.377 1.45 1.348 1.44 1.4

1.311.424

91.31

81.53

31.39

1 1.4191.38

6 1.37 1.47 1.489 1.46 1.483 1.55 1.411.49

31.402

21.37

81.41

3 1.51 1.3651.32

6 1.361.29

6 1.471 1.48 1.486 1.47 1.41

1.356

1.4901

1.426

1.506

1.482 1.37

1.378 1.43

1.381 1.417 1.33 1.347 1.54 1.46

1.446

1.3559

1.438

1.489

1.548 1.41

1.415 1.41

1.439 1.339 1.46 1.432 1.43 1.34

1.453

1.3502

1.293

1.393

1.381 1.317

1.371 1.49 1.32 1.406 1.35 1.44 1.46 1.45

1.412

1.4728

1.488 1.28

1.345 1.494

1.431 1.44

1.299 1.439 1.49 1.53 1.38 1.41

1.404

1.3502 1.45

1.301

1.342 1.48

1.292 1.36

1.429 1.296 1.42 1.316 1.49 1.41

1.395

1.3456

1.394

1.396

1.196 1.442

1.484 1.45 1.45 1.435 1.34 1.47 1.42 1.35

1.449

1.4751

1.441

1.365

1.506 1.546

1.504 1.37

1.396 1.431 1.25 1.47 1.36 1.45

1.581

1.4285

1.451

1.482

1.347 1.439

1.466 1.43

1.469 1.541 1.47 1.499 1.33 1.36

1.484

1.3834

1.525 1.39

1.495 1.53

1.419 1.36

1.452 1.456 1.43 1.401 1.48 1.41

1.519

1.3333

1.434

1.314

1.538 1.485

1.382 1.48

1.407 1.401 1.42 1.463 1.37 1.45

1.351.403

41.47

51.46

1 1.42 1.2461.34

6 1.431.39

1 1.541 1.45 1.43 1.53 1.531.39

51.337

81.48

61.41

91.40

1 1.3341.50

3 1.431.49

8 1.35 1.22 1.344 1.48 1.57




(fi) (xi-~x)2(f

43.88575 43.942 43.9137 4 4 0.00893 0.00893175.6549

9 1.0086964 0.254367

43.9427 43.999 43.9707 10 14 0.02232 0.03125 439.7075 1.9517411 0.380929

43.9997 44.056 44.0277 37 51 0.08259 0.113841629.026

7 5.1124420 0.706407

44.0567 44.113 44.0847 71 122 0.15848 0.27232 3130.017 5.7633616 0.467836

44.1137 44.170 44.1417 113 235 0.25223 0.524554988.017

6 2.7316741 0.066036

44.1707 44.227 44.1987 100 335 0.22321 0.747774419.874

9 3.2825893 0.107754

44.2277 44.284 44.2557 67 402 0.14955 0.897322965.135

2 6.0183348 0.540602

44.2847 44.341 44.3127 37 439 0.08259 0.97991 1639.57 5.4325580 0.797640

44.342 44.398 44.3697 9 448 0.02009 1.00000399.3277

4 1.834433 0.373905

448 1 19786.33

3 33.135830 3.695476

Media Aritmética

~x 44.16592

282


91


83

Desviación


08

En esta tabla se observan lo siguiente: 1. los datos están distribuidos en forma normal 2. La desviación estándar de este trabajador “Alan” (s = 0.09082308) es mayor que la del trabajador “Juan” (s = 0.06818146).


43.6000 43.7000 43.8000 43.9000 44.0000 44.1000 44.2000 44.3000 44.4000 44.50000

20

40

60

80

100

120

140

1. En el histograma se observa que el USL no cumple con los límites de especificación que fueron establecidos por la misma empresa.

Diagrama de Pareto

A continuación mediante un diagrama de Ishikawa, determinaremos los posibles causantes de dicha fallas en las muestras tomadas del turno vespertino.

MdeO MyE MP Mt MA-15

5

25

45

65

85

105

125

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

TV USLLSL

causas fi fr fra M de O 113 0.38 0.38 %M y E 71 0.24 0.62 %MP 67 0.22 0.84 %MT 37 0.12 0.97 %MA 10 0.03 1.00 %

Mediantes este diagrama, podemos determinar los factores que influye en el proceso y a su vez provocan fallas en la calidad del mismo.

Conclusión: El histograma del trabajador “Alan” confirma nuestras primeras observaciones; es un proceso aceptablemente centrado.

Con esta información observada anteriormente, podemos determinar que el trabajador Alan está cumpliendo con los límites de tolerancia establecidos por la

empresa., mientras que Juan NO cumple con las especificaciones que fueron requeridas, lo cual se pide que se le ponga más atención a este trabajador para que

logre mejorar y seguir al mismo ritmo que Alan.

histograma 2

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