Analysis of Service Time and Steel Hardness: Experimental Insights

IE 563: Experimental Design Homework #4 Problem 1 An experiment is conducted to determine the service time (in minutes) at two different banking companies. Factors considered are: day of the week (Monday through Saturday), teller [fixed] (1 or 2), and Bank (A or B). The data is shown in the randomized run order below. (Nested Slide 3) Bank Teller Weekday Service Time (min) Bank Teller Weekday Service Time (min) A 1 Monday 2.793 B 1 Monday 4.9266 A 1 Tuesday 0.987 B 1 Tuesday 3.3894 A 1 Wednesday 1.722 B 1 Wednesday 1.9152 A 1 Thursday 3.066 B 1 Thursday 4.0824 A 1 Friday 6.174 B 1 Friday 4.0572 A 1 Saturday 5.754 B 1 Saturday 5.229 A 2 Monday 6.951 B 2 Monday 2.7594 A 2 Tuesday 2.016 B 2 Tuesday 0.63 A 2 Wednesday 1.47 B 2 Wednesday 0.5796 A 2 Thursday 4.851 B 2 Thursday 1.0962 A 2 Friday 3.066 B 2 Friday 1.0836 A 2 Saturday 4.053 B 2 Saturday 1.1592 A 1 Monday 3.36 B 1 Monday 5.985 A 1 Tuesday 4.746 B 1 Tuesday 2.4192 A 1 Wednesday 5.334 B 1 Wednesday 2.9358 A 1 Thursday 1.659 B 1 Thursday 3.402 A 1 Friday 10.227 B 1 Friday 5.3172 A 1 Saturday 9.765 B 1 Saturday 4.3848 A 2 Monday 9.219 B 2 Monday 2.5578 A 2 Tuesday 4.263 B 2 Tuesday 2.583 A 2 Wednesday 4.242 B 2 Wednesday 2.5326 A 2 Thursday 2.268 B 2 Thursday 3.7926 A 2 Friday 6.069 B 2 Friday 3.2004 A 2 Saturday 9.345 B 2 Saturday 4.1076 a) Analyze the data using the appropriate model.

This is the output for the GLM on a nested model. b) Draw conclusions based on your analysis; including selection of levels appropriate for the application. Bank, Weekday have a significant effect on service time, but the tellers nested at each bank do not. (similar service by tellers at the same bank, but not between the banks themselves). Residual graphs look good. Based on the Tukey test to minimize service time I would go to Bank B and a Wednesday.

Problem 2 A researcher is interested in the hardness of steel bars treated with four coatings (1, 2, 3, and 4) at four furnace temperatures 350 o F, 375 o F, 400 o F, and 425 o F . The furnace is set to a temperature and four steel bars (one with each coating type) are put in the furnace and heated. The following data are shown in the actual run order. Temperatur e Coating Corrosion Resistanc e Temperatur e Coating Corrosion Resistanc e 350 2 192 400 4 232 350 3 177 400 3 215 350 1 148 400 2 222 350 4 178 400 1 251 375 1 145 375 4 302 375 3 184 375 1 182 375 4 182 375 3 186 375 2 191 375 2 208 400 3 142 425 3 482 400 1 148 425 2 451 400 2 124 425 4 398 400 4 189 425 1 458 425 1 530 350 1 163 425 4 482 350 4 196 425 3 516 350 2 219 425 2 445 350 3 169 a) Analyze the data using the appropriate model. Since this is not a CRD since 4 bars are put into the furnace at a given temperature at the same time, but factors are not nested within each other we use a split plot model.

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b) Draw conclusions based on your analysis; i.e. do temperature and/or coating type have a significant effect on hardness? Since the residual plots look good we go back to the analysis of variance and see that only Temperature is the only significant factor in determining corrosion resistance. c) Which temperature and coating would you choose to maximize hardness? Since it’s a split plot we can use the means to determine the levels to use. Coat 4 combined with 425-degree furnace temperature produces the best corrosion resistance

Problem 3 An experiment is designed to study pigment reflectance in paint. We are interested in the effect of mixing technique and paint application method on the percentage reflectance of pigment. Three different mixing techniques (A, B, C) and three application methods (brushing, spraying, rolling) of a particular pigment are studied. The procedure consists of preparing the paint using a particular mixing technique and then applying that mix to a panel using the three application methods. The experiment is conducted over 4 days and the data (% reflectance) obtained follow. Day Method Mix Tech. Reflect. Day Method Mix Tech. Reflect. 1 Spray B 43.6 3 Brush C 31.6 1 Roll A 56.1 3 Roll B 32.2 1 Roll C 46 3 Spray A 39.3 1 Spray C 44 3 Spray C 33 1 Brush B 41.2 3 Brush B 29.5 1 Roll B 45.4 3 Spray B 30.6 1 Spray A 54.5 3 Roll A 39.8 1 Brush C 42.3 3 Brush A 38.6 1 Brush A 52 3 Roll C 34.8 2 Brush C 38.5 4 Brush C 38.7 2 Spray C 41.2 4 Spray B 37.5 2 Brush B 38.6 4 Roll B 39.4 2 Spray B 40.5 4 Spray A 48.1 2 Roll C 43.1 4 Brush A 47.3 2 Roll B 41.7 4 Roll A 48.8 2 Spray A 52 4 Spray C 40.5 2 Brush A 48.4 4 Roll C 42.6 2 Roll A 53.1 4 Brush B 36.2 a) Analyze the data using the appropriate model. This model is a split plot with a block on Day.

b) Draw conclusions based on your analysis; i.e. do mixing technique and/or application method have a significant effect on the percentage reflectance of pigment? Method and mixing technique are significant. Day was a block and Day*Mix technique is the WP factor c) Which mixing technique and application method should you use to maximize percent reflectance? Since we cant use a tukey test for split plot designs we use the fitted means. Rolling technique 3 generally produce the best results on their own. However, given the internation effect Roll and technique 1 produced the highest average mean and should be used to maximize reflectiveness. Problem 4

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A specialized alloy is used to make components for an aircraft fuselage. Cracking is a potentially serious problem in the final part, as it can lead to catastrophic failure. A test is run at the parts producer to determine the effect of four factors on cracks. The four factors are pouring temperature (A), alloy content (B), heat treatment method (C), and amount of grain refiner used (D). Two replicates of a design are run, and the length of crack (in mm) induced in a sample coupon subjected to a standard test is measured. The data are shown below: Actual Run Order A B C D Crack Length Actual Run Order A B C D Crack Length 22 -1 -1 -1 -1 3.68 17 -1 -1 -1 -1 4.87 9 1 -1 -1 -1 4.72 21 1 -1 -1 -1 4.18 25 -1 1 -1 -1 3.61 20 -1 1 -1 -1 4.32 18 1 1 -1 -1 3.58 26 1 1 -1 -1 4.12 8 -1 -1 1 -1 4.34 29 -1 -1 1 -1 5.53 27 1 -1 1 -1 3.96 2 1 -1 1 -1 5.44 23 -1 1 1 -1 4.48 19 -1 1 1 -1 4.91 3 1 1 1 -1 4.57 13 1 1 1 -1 4.02 16 -1 -1 -1 1 5.99 32 -1 -1 -1 1 7.40 11 1 -1 -1 1 5.29 6 1 -1 -1 1 6.89 1 -1 1 -1 1 5.04 7 -1 1 -1 1 6.86 14 1 1 -1 1 4.09 24 1 1 -1 1 5.61 12 -1 -1 1 1 6.40 5 -1 -1 1 1 7.32 15 1 -1 1 1 6.12 30 1 -1 1 1 7.51 28 -1 1 1 1 4.91 31 -1 1 1 1 6.79 4 1 1 1 1 5.35 10 1 1 1 1 6.56 (a) Estimate the factor effects. Which effects appear to be significant?

Based on the bottom line showing the sumproduct function in excel based on the absolute value of the sumproduct function you could say A, B, C, D, ACD and BD would be significant. Its hard to determine a cutoff using this method, but A is 6x less than D and seems like an appropriate place to cut off. Run OrderA B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD Crack Leng 22 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 3.68 9 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 4.72 25 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 3.61 18 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 3.58 8 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 4.34 27 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 3.96 23 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 4.48 3 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 4.57 16 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 5.99 11 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 5.29 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 5.04 14 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1 4.09 12 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 6.4 15 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 6.12 28 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 4.91 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5.35 17 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 4.87 21 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 4.18 20 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 4.32 26 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 4.12 29 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 5.53 2 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 5.44 19 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 4.91 13 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 4.02 32 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 7.4 6 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 6.89 7 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 6.86 24 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1 5.61 5 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 7.32 30 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 7.51 31 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 6.79 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6.56 -0.2775 -0.67625 0.4975 1.7375 -0.1 0.13375 -0.13375 0.0475 -0.2875 -0.02375 0.09625 0.01375 0.3075 -0.01875 0.065 (b) Perform an analysis of variance to confirm your conclusions from part (a).

From the analysis it appears that B, D are the only factors that are significant. The difference in “significant effects from part A and part B is due to not having a point to make the statistical significant cutoff in part A that we can do in part B (c) Analyze the residuals from your ANOVA analysis. The graphs show a slight but of spiralling in the top left graph, but otherwise appears to meet all parametric assumptions

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(d) Construct a regression model for predicting crack length. (e) Analyze the residuals from your regression analysis. The residuals look good for the most part. No specific patterns and good spread across the graphs (f) What levels of A, B, C, and D would you recommend using? Based on the comparison I would use A- low setting, B – low setting, C – high setting, D- High setting. Problem 5

In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C), and temperature (D). A single replicate of a design was run and the resulting data are shown in the following table. Actual Run Yield Order A B C D (lbs) 5 -1 -1 -1 -1 147 9 1 -1 -1 -1 162 8 -1 1 -1 -1 105 13 1 1 -1 -1 138 3 -1 -1 1 -1 270 7 1 -1 1 -1 243 14 -1 1 1 -1 279 1 1 1 1 -1 215 6 -1 -1 -1 1 102 11 1 -1 -1 1 249 2 -1 1 -1 1 103 15 1 1 -1 1 156 4 -1 -1 1 1 221 16 1 -1 1 1 341 10 -1 1 1 1 244 12 1 1 1 1 324 (a) Estimate the factor effects by hand (you may use excel) . Which effects appear to be large? A, C and AD are significantly larger than the remaining affects. Run OrderA B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD Yield 5 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 147 9 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 162 8 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 105 13 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 138 3 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 270 7 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 243 14 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 279 1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 215 6 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 102 11 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 249 2 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 103 15 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1 156 4 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 221 16 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 341 10 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 244 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 324 44.625 -21.375 121.875 22.625 -19.125 -17.375 55.375 18.125 -0.125 8.125 -0.125 -14.375 17.375 6.375 13.625 44.625 21.375 121.875 22.625 19.125 17.375 55.375 18.125 0.125 8.125 0.125 14.375 17.375 6.375 13.625 3 5 1 4 6 8 2 7 14 12 14 10 8 13 11 The factor level settings are as follows: Factor Low (-) High (+) A (hr) 2 4 B (%) 10 20 C (psi) 60 90 D (temp) 50 80

(b) Perform an analysis of variance by hand to confirm your conclusions for part (a). What terms are significant at α=0.05? Effect s SS DoF MS F A 44.62 5 7965.562 5 1 7965.5 63 8.6400 24 C 121.8 75 59414.06 25 1 59414. 06 64.444 78 AD 55.37 5 12265.56 25 1 12265. 56 13.304 11 Error 11063.25 12 921.93 75 Total 90708.43 75 15 6047.2 29 =(16/4)*Effects^2 =SS/ Dof =MSa,c,ac/ Mserror F(0.05,1,12) = 4.75 Fad=Msad/Mse Based on the F test statistic of 4.75 A C and AD are significant (c) Confirm your results in Minitab. Construct a normal probability plot of the factor effects, perform an ANOVA and analyze the ANOVA residuals. Results match!

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Normal probability plot for the factor effects looks good. Maybe some clumping around zero, but otherwise no other concerns. (d) Construct a regression model by hand for predicting yield. Use the significant terms you identified in your part (b) ANOVA. Average response of yield = Sum(yield)/N = 3299/16 = 206.1875 Effect A = 44.625/2 = 22.3125 Effect B = 121.875/2 = 60.9375 Effect AD = 55.375/2 = 27.6875 Y = 206.1875 + 22.3125A + 60.9375B + 27.6875AD (e) Confirm you part (d) results using Minitab. Analyze the residuals from your regression analysis.

(f) What levels of A, B, C, and D would you recommend using to maximize the yield? Given the below comparison I would recommend high setting for A, low setting for B, high setting for C and high setting for D to maximize yield

(g) Using the regression model found in part (e), predict the yield at 3.25 h, 14%, 75 psi, and a temperature of 60. Problem 6 Four factors of interest are included in a factorial design used to develop a etch process for a single-layer tungsten etcher. The four factors of interest are anode-cathode gap (A), pressure (B), flow rate (C), and power applied to the cathode (D). The response variable of interest is the etch rate. First, a single replicate of a is run and the data are shown below in standard order (not actual run order). Data were collected in random order. Run Order A B C D Etch Rate 16 -1 -1 -1 -1 683 9 1 -1 -1 -1 1066 4 -1 1 -1 -1 589 10 1 1 -1 -1 817 1 -1 -1 1 -1 649 8 1 -1 1 -1 929 The factor level settings are as follows: Factor Low (-) High (+) A (cm) 0.8 1.2 B (mTorr) 4.5 550 C (SCCM) 125 200 D (W) 275 325

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14 -1 1 1 -1 405 15 1 1 1 -1 392 12 -1 -1 -1 1 1045 3 1 -1 -1 1 957 13 -1 1 -1 1 1125 11 1 1 -1 1 880 7 -1 -1 1 1 1174 5 1 -1 1 1 1099 6 -1 1 1 1 819 2 1 1 1 1 714 (a) Construct a normal probability plot of the factor effects. Which effects appear to be large? Based on the Normal probability plot factors B and D are significant (b) Perform an ANOVA for the above experiment. What are your conclusions? Based on the ANOVA we can believe that both B and D are significant factors. R-sq is not great, but the residuals look ok so parametric assumptions are met

(c) Now suppose we include the following 5 center point observations in our above experiment Run Etch Orde A B C D Rate 17 0 0 0 0 669 18 0 0 0 0 666 19 0 0 0 0 710 20 0 0 0 0 667 21 0 0 0 0 602 Perform an ANOVA. What are your conclusions? (NOTE: Only include significant effects in your final ANOVA model). Still believe that B and D are significant factors.

(d) Is there any evidence of curvature? Do you feel the linearity assumption is valid? Why or why not? No! Curvature is less than 0.05 so it is not significant. This means the linearity assumption is valid

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IE563_HW4

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