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MIE364H1S

Methods of Quality Control and Improvement

Instructor: Prof. V. Makis

Handout #4

Factorial Design 2 Factors

A company wishes to consider changing the way in which one of the materials that it usesin its fabrication process is made. Two subcontractors are invited to make five samplesprepared by the old method, five samples prepared by the new method I and five samplesprepared by the new method II. The response is the tensile strength of the material and ahigher value is desirable.

1. Is there a significant difference between the methods?2. Is there a significant difference between the suppliers?

3. Is there evidence of interaction?

Data Display

Row Y Method Supplier ResI1 FitS1

1 130 1 1 4.2 125.8

2 124 1 1 -1.8 125.8

3 128 1 1 2.2 125.8

4 125 1 1 -0.8 125.8

5 122 1 1 -3.8 125.8

6 125 1 2 -1.8 126.8

7 129 1 2 2.2 126.8

8 127 1 2 0.2 126.89 127 1 2 0.2 126.8

10 126 1 2 -0.8 126.8

11 128 2 1 -0.6 128.6

12 127 2 1 -1.6 128.6

13 133 2 1 4.4 128.6

14 126 2 1 -2.6 128.6

15 129 2 1 0.4 128.6

16 135 2 2 1 134

17 137 2 2 3 134

18 134 2 2 0 134

19 136 2 2 2 134

20 128 2 2 -6 13421 121 3 1 -2.4 123.4

22 129 3 1 5.6 123.4

23 117 3 1 -6.4 123.4

24 127 3 1 3.6 123.4

25 123 3 1 -0.4 123.4

26 126 3 2 1.4 124.6

27 119 3 2 -5.6 124.6

28 130 3 2 5.4 124.6

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29 123 3 2 -1.6 124.6

30 125 3 2 0.4 124.6

Two-way ANOVA: Y versus Method, Suppl ier

Source DF SS MS F P

Met hod 2 278. 600 139. 300 11. 72 0. 000Suppl i er 1 48. 133 48. 133 4. 05 0. 056I nt eract i on 2 30. 867 15. 433 1. 30 0. 291Err or 24 285. 200 11. 883Tot al 29 642. 800

MeanofY

21

132

131

130

129

128

127

126

125

124

321

Supplier Method

Main Effects Plot (data means) for Y

Method

Mean

321

135.0

132.5

130.0

127.5

125.0

1

2

Supplier

Interaction Plot (data means) for Y

Residual

Percent

86420-2-4-6-8

99

95

90

80

70

60

50

40

30

20

10

5

1

Normal Probability Plot of the Residuals(response is Y)

Method

Residual

3.02.52.01.51.0

5.0

2.5

0.0

-2.5

-5.0

-7.5

Residuals Versus Method(response is Y)

FitS1

Residual

135.0132.5130.0127.5125.0

5.0

2.5

0.0

-2.5

-5.0

-7.5

Residuals Versus FitS1(response is Y)

Supplier

Residual

2.01.81.61.41.21.0

5.0

2.5

0.0

-2.5

-5.0

-7.5

Residuals Versus Supplier(response is Y)

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Estimation of parameters:

...

.. ...

1.. 2.. 3..

1 2 3

. . ...

.1. .2.

1 2

. .. . . ...

11

21

31

127.2

126.3, 131.3, 124 0.9, 4.1, 3.2

125.93, 128.47

1.27, 1.27

( )

( ) .767

( ) 1.433

( ) 0.66

i i

j j

ij ij i j

Y

Y Y

Y Y Y

Y Y

Y Y

Y Y Y Y

= =

=

= = =

= = =

=

= =

= =

= +

=

=

=

12

22

32

11. 12.

21. 22.

31. 32.

7

( ) 0.767

( ) 1.433

( ) 0.667

125.8, 126.8

128.6, 134

123.4, 124.6

Y Y

Y Y

Y Y

=

=

=

= =

= =

= =

Degrees of freedom:

( ) 1, ( ) 1 2, ( ) 1 1,

( ) ( 1)( 1) 2, ( ) 24

df df a df b

df a b df e

= = = = =

= = =

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Sums of squares:

3 2 5 3 2 52 2

1 1 1 1 1 1

3 2 52

1 1 1

*

0.05,2,24

( ) ( ) 278.6, ( ) ( ) 48.387,

( ) ( ) 30.867,

642.8

284.946

( ) 139.3

( ) 48.387

( ) 15.4335

11.87

11.74 3.4, R

i j

i j r i j r

ij

i j r

A

SS SS

SS

TCSS

SSE

MS

MS

MS

MSE

F F

= = = = = =

= = =

= = = =

= =

=

=

=

=

=

=

= > =

*

0.05,1,24

*

e

4.08 4.26,

1.3 3.4,

A

B B

AB AB

ject H

F F FTR H

F FTR H

= < =

=