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The article “Use of Taguchi Methods and Multiple
A | B | Hardness | |||||
10 | 10 | 875 | 896 | 921 | 686 | 642 | 613 |
10 | 25 | 712 | 719 | 698 | 621 | 632 | 645 |
10 | 50 | 568 | 546 | 559 | 757 | 723 | 734 |
20 | 10 | 876 | 835 | 868 | 812 | 796 | 772 |
20 | 25 | 889 | 876 | 849 | 768 | 706 | 615 |
20 | 50 | 756 | 732 | 723 | 681 | 723 | 712 |
30 | 10 | 901 | 926 | 893 | 856 | 832 | 841 |
30 | 25 | 789 | 801 | 776 | 845 | 827 | 831 |
30 | 50 | 792 | 786 | 775 | 706 | 675 | 568 |
- a. Estimate all main effects and interactions.
- b. Construct an ANOVA table. You may give
ranges for the P-values. - c. Is the additive model plausible? Provide the value of the test statistic and the P-value.
- d. Can the effect of travel speed on the hardness be described by interpreting the main effects of travel speed? If so, interpret the main effects, using multiple comparisons at the 5% level if necessary. If not, explain why not.
- e. Can the effect of accelerating voltage on the hardness be described by interpreting the main effects of accelerating voltage? If so, interpret the main effects, using multiple comparisons at the 5% level if necessary. If not, explain why not.
a.
Find all the main and interaction effects.
Answer to Problem 14E
The interaction effects are:
The main effects are:
Explanation of Solution
Calculation:
The given information is that the experiment involves the response of two factors (A (travel speed) and B (accelerating voltage)).
The first cell refers to travel speed 10 and accelerating voltage 10.
The first cell mean can be obtained as follows:
Similarly the means of remaining cells are given in the below table:
Here, the row means refers to the factor travel speed.
The first row mean can be obtained as follows:
Similarly the means of remaining rows are given in the below table:
Here, the column means refers to the factor accelerating voltage.
The first column mean can be obtained as follows:
Similarly the means of remaining columns are given in the below table:
The remaining row and column mean can be obtained as shown in the table:
10 | 25 | 50 | Row Mean | |
10 | 772.1667 | 671.1667 | 647.8333 | 697.0556 |
20 | 826.5 | 783.8333 | 721.1667 | 777.1667 |
30 | 874.8333 | 811.5 | 717 | 801.1111 |
Column Mean | 824.5 | 755.5 | 695.3333 | 758.4444 |
The row effects can be obtained as follows:
Here,
Substitute
Substitute
Substitute
Thus, the row effects are
The column effects can be obtained as follows:
Here,
Substitute
Substitute
Substitute
Thus, the column effects are
The interaction effects can be obtained as follows:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the interaction effects are
b.
Construct an ANOVA table and the find the ranges for the P-values.
Answer to Problem 14E
The ANOVA table is,
Source | DF | SS | MS | F | P |
A | 2 | 106,912 | 53,456 | 8.74 | 0.001 |
B | 2 | 150,390 | 75,195.2 | 12.29 | 0.000 |
Interaction | 4 | 11,409 | 2,852.2 | 0.47 | 0.760 |
Error | 45 | 275,228 | 6,116.2 | ||
Total | 53 | 543,939 |
For Factor A, the P-value is 0.001.
For Factor B, the P-value is 0.000.
For interaction, the P-value is 0.760.
Explanation of Solution
Calculation:
The factor A is travel speed and factor B is accelerating voltage.
Step-by-step procedure for finding the Two-Way ANOVA table is as follows:
Software procedure:
- Choose Stat > ANOVA > Two-Way.
- In Response, enter the column of Hardness.
- In Row Factor, enter the column of A.
- In Column Factor, enter the column of B.
- Click OK.
Output obtained by MINITAB procedure is as follows:
For Factor A, the F-test statistic is 8.74 and the P-value is 0.001.
For Factor B, the F-test statistic is 12.29and the P-value is 0.000.
For interaction, the F-test statistic is 0.47 and the P-value is 0.760.
c.
Explain whether the additive model is plausible.
Answer to Problem 14E
The additive model is plausible.
Explanation of Solution
Calculation:
Interaction:
Null hypothesis:
Alternative hypothesis:
For interaction, the F-test statistic is 0.47 and the P-value is 0.760.
Decision:
If
If
Conclusion:
Interaction:
Here, the P-value is greater than the level of significance.
That is,
Therefore, the null hypothesis is not rejected.
Thus, the interaction is not significant at
Therefore all the interactions are equal to zero.
Thus, the additive model is plausible.
d.
Check whether the effects of travel speed on the hardness can be described by the main effects of travel speed. If so, interpret the main effects by multiple comparisons at the 5% level. If not explain the reason.
Answer to Problem 14E
Yes, the effects of travel speed on the hardness can be described by the main effects of travel speed.
There is sufficient evidence to conclude that the effect of a travel speed of 10 differs from those of both 20 and 30 at
Explanation of Solution
Calculation:
Factor A is travel speed.
Main effect of factor A:
Null hypothesis:
Alternative hypothesis:
For Factor A, the F-test statistic is 8.74 and P- value is 0.001.
Decision:
If
If
Conclusion:
Factor A:
Here, the P-value is less than the level of significance.
That is,
Therefore, the null hypothesis is rejected.
Thus, some of the main effects of factor A are non-zero.
Hence, it is not plausible that the main effects of travel speed on the hardness are equal to zero at
Since, the main effects of travel speed on the hardness are not all equal to zero, the effects of travel speed on the hardness can be described by the main effects of travel speed.
Thus, the effects of travel speed on the hardness can be described by the main effects of travel speed.
The main effects can be interpret using Tukey’s method.
State the hypotheses:
Null hypothesis:
Alternative hypothesis:
Decision:
By Tukey’s method for multiple comparisons,
If
If
Here
From Appendix A table A.9, the upper 5% point of the
For comparing travel speed in 10 mm/s and 20 mm/s:
The 5% critical value is,
Substitute
From part (a), the row effects are
Which is greater than 63.41.
Thus, reject the null hypothesis
Hence, for travel speed in 10 mm/s and 20 mm/s there is travel speed affect the hardness.
For comparing travel speed in 10 mm/s and 30 mm/s:
Which is greater than 63.41.
Thus, reject the null hypothesis
Hence, for travel speed in 10 mm/s and 30 mm/s there is travel speed affect the hardness.
For comparing travel speed in 20 mm/s and 30 mm/s:
Which is less than 63.41.
Thus, fail to reject the null hypothesis
Hence, for travel speed in 20 mm/s and 30 mm/s there is no travel speed affect the hardness.
Conclusion:
There is sufficient evidence to conclude that the effect of a travel speed of 10 differs from those of both 20 and 30 at
e.
Check whether the effects of accelerating voltage on the hardness can be described by the main effects of accelerating voltage. If so, interpret the main effects by multiple comparisons at the 5% level. If not explain the reason.
Answer to Problem 14E
Yes, the effects of accelerating voltage on the hardness can be described by the main effects of accelerating voltage.
There is sufficient evidence to conclude that the effect of an accelerating voltage in 10 volts differs from those of both 25 volts and 50 volts at
Explanation of Solution
Calculation:
Factor B is accelerating voltage.
Main effect of factor B:
Null hypothesis:
Alternative hypothesis:
For Factor B, the F-test statistic is 12.29 and the P-value is 0.000.
Decision:
If
If
Conclusion:
Factor B:
Here, the P-value is less than the level of significance.
That is,
Therefore, the null hypothesis is rejected.
Thus, some of the main effects of factor B are zero.
Hence, it is not plausible that the main effect of accelerating voltage on the hardness are equal to zero at
Since, the main effects of accelerating voltage on the hardness are not equal to zero, the effects of accelerating voltage on the hardness can be described by the main effects of accelerating voltage.
Thus, the effects of accelerating voltage on the hardness can be described by the main effects of accelerating voltage.
The main effects can be interpret using Tukey’s method.
State the hypotheses:
Null hypothesis:
Alternative hypothesis:
Decision:
By Tukey’s method for multiple comparisons,
If
If
Here
From Appendix A table A.9, the upper 5% point of the
For comparing accelerating in 10 volts and 25 volts:
The 5% critical value is,
Substitute
From part (a), the row effects are
Which is greater than 63.41.
Thus, reject the null hypothesis
Hence, for accelerating in 10 volts and 25 volts there is accelerating voltage affect the hardness.
For comparing accelerating in 10 volts and 50 volts:
Which is greater than 63.41.
Thus, reject the null hypothesis
Hence, for accelerating in 10 volts and 50 volts there is accelerating voltage affect the hardness.
For comparing accelerating in 25 volts and 50 volts:
Which is less than 63.41.
Thus, fail to reject the null hypothesis
Hence, for accelerating in 25 volts and 50 volts there is no accelerating voltage affect the hardness.
Conclusion:
There is sufficient evidence to conclude that the effect of an accelerating voltage in 10 volts differs from those of both 25 volts and 50 volts at
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