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ISU STAT 496 - Homework assignment 6

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STAT 496, Spring 2007Homework Assignment #61. A foundry produces ingots of an aluminum alloy. The quality characteristic of interest isthe grain size of the alloy. During the production process the molten alloy is stirred. Anexperiment is performed to see the effects of stirring rate on the grain size. The molten alloyis stirred in one of four furnaces. Since each furnace has its own operating characteristics, thefurnaces will be used as blocks. Each stirring rate will be used (in a random order) in eachfurnace. The data from the experiment, run as a randomized complete block design, appearsbelow. A partial analysis of variance table is also given.Stirring Rate (rpm)5 10 15 201 8 10 13 15Furnace 2 3 4 8 93 3 6 7 124 6 4 10 10(a) Compute means and variances for each stirring rate. Comment on the apparent effectsof stirring rate on the mean grain size of aluminum ingots. Also comment on the amountof variation within each stirring rate.(b) Compute a value of the SSrepErrorfor this experiment.(c) Compute means for each furnace (block). Use these to compute a value of SSBlock. Howmany degrees of freedom are associated with this sum of squares?(d) Compute the SSError, associated degrees of freedom and the LSD (use t=3) for com-paring the stirring rate means.(e) According to the LSD in d), what stirring rate means are statistically different fromwhat other stirring rate means? What stirring rate means are statistically similar towhat other stirring rate means?(f) Instead of using the LSD to look at differences among the stirring rates we can also lookat the linear relationship between grain size and stirring rate using regression. Use acoded stirring rate and compute the slope and intercept for the simple linear regressionof grain size on coded stirring rate.(g) What proportion of the total variation in grain size is explained by the linear relationshipwith coded stirring rate?(h) Is the simple linear relationship statistically significant? Report the appropriate valueof the t-test statistic. Be careful, remember that this is an experiment that used a blockdesign.(i) Give the simple linear regression equation that relates grain size directly to stirring rate(no coding).(j) Use the equation in (i) to predict the grain size for ingots using a stirring rate of 17.5.(k) Your prediction in (j) will be appropriate for the average grain size from the four furnaces.How would you change the prediction equation, and your prediction for a stirring rateof 17.5 rpm, so that it would predict more accurately the grain size for alloys producedin Furnace 1?12. An experiment is performed on a new piece of drilling equipment. The response variable isthe 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: Amountof Mud used in drilling. The experiment is run as a single replicate full factorial with fivecenter points. The data are given below. Refer to the JMP output for drilling experiment.Trmt.Comb. A B C D Rate1 −1 −1 −1 −1 1.68a +1 −1 −1 −1 1.98b −1 +1 −1 −1 3.28ab +1 +1 −1 −1 3.44c −1 −1 +1 −1 4.98ac +1 −1 +1 −1 5.70bc −1 +1 +1 −1 9.93abc +1 +1 +1 −1 9.07d −1 −1 −1 +1 2.07ad +1 −1 −1 +1 2.44bd −1 +1 −1 +1 4.09abd +1 +1 −1 +1 4.53cd −1 −1 +1 +1 7.77acd +1 −1 +1 +1 8.43bcd −1 +1 +1 +1 11.75abcd +1 +1 +1 +1 12.300 0 0 0 4.540 0 0 0 5.180 0 0 0 5.530 0 0 0 4.490 0 0 0 5.01(a) Give the overall sample mean and the estimated full effects for the factorial portion ofthe 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 afactor or interaction term. Use t=3.(d) Use the critical effect size in c) to determine which factors and/or interactions are sta-tistically significant.(e) Is there significant curvature? Support your answer by performing the appropriate testof hypothesis.(f) According to JMP, which factors and/or interactions are statistically significant? Uset=3.(g) Why are the results in d) different from the results in


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