JMP Output for Drilling ExperimentSTAT 496, Spring 2011 Homework Assignment #6, Due by Thursday, March 24 1. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Because the hardness of the raw material being tooled may vary, heats of raw material are used as blocks. The eight treatment combinations are run in a random order within each block (heat). The levels of each of the 3 factors are given below along with the data. Low (–1) High (+1) A: Cutting Speed 50 rpm 100 rpm B: Tool Geometry Type 1 Type 2 C: Cutting Angle 5o 10o XA XB XC Heat I Heat II Heat III –1 –1 –1 22 31 25 +1 –1 –1 32 43 30 –1 +1 –1 35 50 35 +1 +1 –1 45 55 47 –1 –1 +1 43 45 38 +1 –1 +1 36 38 40 –1 +1 +1 50 61 54 +1 +1 +1 47 43 39 a) Compute means for the low and high levels of each of the three factors. Display these graphically and comment on the apparent effects of each of the factors. b) Construct an interaction plot for factors A and C using the data from Type 1 tool geometry. Construct a second interaction lot for factors A and C using the data from the Type 2 tool geometry. Comment on the plots. What do they indicate about the possible interaction between factors A and C? What do they indicate about the 3-way interaction? c) Compute the overall sample mean and estimate the full effects for each factor, 2-way and 3-way interaction. d) Compute a value of the SSrepError for this experiment. e) Compute means for each heat (block). Use these to compute a value of SSBlock. How many degrees of freedom are associated with this sum of squares? f) Compute the SSError, associated degrees of freedom and the critical effect size (use t=3). g) According to the critical effect size in f), what effects are statistically significant? What effects are not statistically significant? h) Give the reduced model prediction equation that includes only those effects that are found to be statistically significant in g). Be sure to define explicitly any variables you use in the equation. i) Based on the prediction equation in h), if long tool life is the goal, what levels of the three factors would you choose? What is the predicted life and a 95% prediction interval for you optimal choice? 1j) Your prediction in i) will be appropriate for the average hardness of heats of raw material. How would you change the prediction equation, and your prediction for optimal choice, so that it would predict more accurately the tool wear for heats with average hardness similar to Heat II? 2. An experiment is performed on a new piece of drilling equipment. The response variable is the rate of advance of the drill. The four factors (each with a low (–1) and high (+1) level) are A: Load on the drill, B: Flow Rate through the drill, C: Speed of Rotation, D: Amount of Mud used in drilling. The experiment is run as a single replicate full factorial with five center points. The data are given below. Refer to the JMP output for drilling experiment. Trmt. Comb. A B C D Rate 1 –1 –1 –1 –1 1.68 a +1 –1 –1 –1 1.98 b –1 +1 –1 –1 3.28 ab +1 +1 –1 –1 3.44 c –1 –1 +1 –1 4.98 ac +1 –1 +1 –1 5.70 bc –1 +1 +1 –1 9.93 abc +1 +1 +1 –1 9.07 d –1 –1 –1 +1 2.07 ad +1 –1 –1 +1 2.44 bd –1 +1 –1 +1 4.09 abd +1 +1 –1 +1 4.53 cd –1 –1 +1 +1 7.77 acd +1 –1 +1 +1 8.43 bcd –1 +1 +1 +1 11.75 abcd +1 +1 +1 +1 12.30 0 0 0 0 4.54 0 0 0 0 5.18 0 0 0 0 5.53 0 0 0 0 4.49 0 0 0 0 5.01 a) Give the overall sample mean and the estimated full effects for the factorial portion of the experiment. b) Give the mean at the center points and the sample variance at the center points. c) Use the sample variance at the center points to determine the critical effect size for a factor or interaction term. Use t=3. d) Use the critical effect size in c) to determine which factors and/or interactions are statistically significant. e) Is there significant curvature? Support your answer by performing the appropriate test of hypothesis. f) According to JMP, which factors and/or interactions are statistically significant? Use t=3. g) Why are the results in d) different from the results in f)? 2JMP Output for Drilling Experiment Parameter Estimates Term Estimated Half Effect Intercept 5.84 A 0.14625 B 1.45875 A*B –0.11000 C 2.90125 A*C –0.01250 B*C 0.56250 A*B*C –0.10125 D 0.83250 A*D 0.10625 B*D 0.03625 A*B*D 0.10500 C*D 0.48875 A*C*D 0.06250 B*C*D –0.09500 A*B*C*D 0.07875 Center Points Mean 4.95Std Dev 0.4394883N 5 Response: Rate of Advance of the Drill Summary of Fit: Full Factorial in A, B, C, and D Rsquare 0.980451RSquare Adj 0.921802Root Mean Square Error 0.870646Mean of Response 5.628095Observations (or Sum Wgts) 21 Analysis of Variance Source DF Sum of Squares Mean Square F Ratio Model 15 190.08320 12.6722 16.7174 Error 5 3.79012 0.7580 Prob > F C. Total 20 193.87332 0.0028 34Lack Of Fit Source DF Sum of Squares Mean Square F Ratio Lack Of Fit 1 3.0175238 3.01752 15.6227 Pure Error 4 0.7726000 0.19315 Prob > F Total Error 5 3.7901238 0.0168 Max RSq 0.9960 Parameter Estimates Term Estimate Std Error t Ratio Prob>|t| Intercept 5.6280952 0.189991 29.62 <.0001 A 0.14625 0.217662 0.67 0.5314 B 1.45875 0.217662 6.70 0.0011 A*B –0.11 0.217662 -0.51 0.6348 C 2.90125 0.217662 13.33 <.0001 A*C –0.0125 0.217662 -0.06 0.9564 B*C 0.5625 0.217662 2.58 0.0492 A*B*C –0.10125 0.217662 -0.47 0.6614 D 0.8325 0.217662 3.82 0.0123 A*D 0.10625 0.217662 0.49 0.6461 B*D 0.03625 0.217662 0.17 0.8743 A*B*D 0.105 0.217662 0.48 0.6499 C*D 0.48875 0.217662 2.25 0.0747 A*C*D 0.0625 0.217662 0.29 0.7855 B*C*D –0.095 0.217662 -0.44 0.6807 A*B*C*D 0.07875 0.217662 0.36 0.7323 Effect Tests Source Nparm DF Sum of Squares F Ratio Prob > F A 1 1 0.34222 0.4515 0.5314 B 1 1 34.04722 44.9157 0.0011 A*B 1 1 0.19360 0.2554 0.6348 C 1 1 134.67602 177.6671 <.0001 A*C 1 1 0.00250 0.0033 0.9564 B*C 1 1 5.06250 6.6785 0.0492 A*B*C 1 1 0.16403 0.2164 0.6614 D 1 1 11.08890 14.6287 0.0123 A*D 1 1 0.18063 0.2383 0.6461 B*D 1 1 0.02103 0.0277 0.8743 A*B*D 1 1 0.17640 0.2327 0.6499 C*D 1 1 3.82203 5.0421 0.0747 A*C*D 1 1 0.06250 0.0825 0.7855 B*C*D 1 1 0.14440 0.1905 0.6807 A*B*C*D 1 1