Question

please provide minitab outputs and instructions

Use MINITAB to do the calculations and include the outputs in the report Conclusions and interpretations must be hand written and included when possible. 1. Two engineers describe an experiment to investigate the warping of copper plates The two factors studied were the temperature and the copper content of the plates The response variable was a measure of the amount of warping. The data were as follows: Copper Content (%) Temperature (C) 40 60 80 100 50 75 100 125 17, 20 12, 13 16, 17 21,19 17, 21 24, 22 28, 27 15,1417, 12 28, 31 18, 21 25, 2330, 25 23, 21 23, 2428, 31 a. Fit a full quadratic regression model to predict the warping as a function of the copper content and temperature. Use a0.05 i. State the hypotheses ii. What is the value of the test statistic? ii What is the p-value? iv. What is the decision? v. Make conclusions b. Test the hypothesis for the quadratic coefficient individually for the temperature. i. State the hypotheses ii. What is the value of the test statistic? ii. What is the p-value? iv. What is the decision? v. Make conclusions c. Fit the best regression model to predict the warping as a function of the copper content and temperature. Use a 0.05. Write down the resulting model d. Analyze the residuals from the model in part c). Are the basic analysis of variance assumptions satisfied? e. Find the best operating conditions to minimize warping

Answer 1

Anwser a) Two-way ANOVA Warping versus Temperature, Copper Source DF MS Temperature3 155.344 51.781 13.47 0.000 Copper Interaction 3 623.344 207.781 54.06 0.000 97.031 10.781 2.80 0.035 16 61.5003.844 Error Total 31 937.219 S-1961, R-Sq 93.44% R-Sq (adj) 87.29% The estimated p-value for interaction is 0.035 and lessthan 0.05 significance level. Hence, we can conclude that there is interaction between the factors The Estimated p-values for factors temperature and cop- per are 0.000 and 0.000 respectively and lessthan 0.05 level of significance. Hence, we can conclude that both the factor are a significant effect on the amount of warping at the 0.05 significance level

b)i Scatterplot of RESII vs Temperature, Copper 40 60 80 100 Temperature 40 60 80 100 120 From the above plots, the minimum standard deviation of the residulas for a level of a factor is lessthan twice of the maximum standard deivation of a level of the same factor (for both temperature and copper). Hence thebasic assump- tion of variance for the analysis is satisfied

Multiple Comparisons Warping8 Tukey HSD 95% Confidence Interval LowerUpper Bound Bound Mean Std Difference Sig Temperature Temperature Error 125 50 75 50 75 75 1.8750.98027 0.1250 .98027 98027 1.7500 98027 6.0000*.98027 .98027 262 -4.6796 9296 .999 2.9296 2.6796 .003 1.3204 6.9296 .316 -1.0546 4.5546 000 3.1954 8.8046 .003 1.4454 7.0546 100 4.1250* 125 50 4.2500*

Multiple Comparisons Warping8 Tukey HSD Mean Difference (l-J) 95% Confidence Interval LowerUpper Bound Bound Std Sig Copper Copper Error 000 8.8204 14.4296 000 6.9454 12.5546 000 4.4454 10.0546 .2624.6796 0.9296 0027.17961.5704 089 5.3046 0.3046 If low wrapping is desirable than 40 level of copper and 75 40 60 80 60 80 80 11.6250* 98027 9.7500* .98027 .98027 1.8750.98027 -4.3750 98027 2.5000 98027 100 7.2500* 40 60 level of temperature will be selected

d) Multiple Comparisons Warping Bonferroni Mean Difference (1-J) 95% Confidence Interval LowerUpper BoundBound Std Error Si lg Copper Copper 40 60 80 11.6250 98027 000 8.6760 14.5740 6.8010 12.6990 4.3010 10.1990 100 -11.6250 98027 .00014.5740 8.6760 1.8750 98027 443-4.8240 1.0740 7.3240 1.4260 9.750098027 000 12.6990 6.8010 1.0740 4.8240 5.4490 04490 7.2500*.98027 000 10.1990 4.3010 1.4260 7.3240 -4490 54490 100 9.750098027 7.2500*98027 .000 40 60 80 100 40 80 100 40 60 4.3750 98027 002 60 1.875098027 443 2.500098027 128 80 4.3750*.98027 002 2.500098027 128

Multiple Comparisons Warping Fisher Mean Difference (1-J) 95% Confidence Interval Lower Upper Bound Bound Std Sig Temp. Temp. Error 1.8750 98027 4434.8240 1.0740 0.1250 980271.003.0740 2.8240 4.1250* 98027 004 1.1760 7.0740 1.8750 .98027 4431.0740 4.8240 1.7500 980275591.1990 4.6990 6.0000*98027000 3.0510 8.9490 1250 98027 1.0002.8240 3.0740 1.7500 98027 5594.6990 1.1990 4.2500* 98027 003 1.3010 7.1990 100 -4.1250* 98027 0047.0740 1.1760 125 -6.0000* 98027 000 8.9490 -3.0510 50 -4.2500 98027 003 7.1990 1.3010 125 100 50 75 100 125 50 75 100 50 125 75 75

Similar to c) if low wrapping is desirable than 40 level of copper and 75 level of temperature will be selected e) The copper plant 40 is to be used. It does not change the answer from c) and d).