Page 1 of 7Stat 511 M.S. Exam Spring 2004 ISU engineering students Dodd, Raine, Neaves and Ney experimented on the running of a 48-cavity injection molding machine used to make “pre-forms” for plastic bottles. Pre-forms are produced 48 at a time, in a die with cavities arranged in 4 columns of 12, as shown on the diagram below. The students weighed pre-forms produced in the 5 cavities indicated by the darkened circles under a number of different circumstances. (Weights were recorded in grams.) It is desirable that there be no systematic pattern across the die in the weights of pre-forms produced in the different cavities. That is, ideally the mean pre-form weight produced in row r and column c of the die is (),rcµµ= (1) for some constant µ. A simple possible departure from this might be ()01 2,rc r cµββ β=+ + (2) for some constants 01 2, , and βββ. a) Suppose one pre-form is taken from each of 10 consecutive cycles of the machine (under the same machine set-up), all 5 of the test cavities being represented twice. Let measured mass of the th pre-form from row and column rciyirc= and ()311 312 911 912 621 622 141 142 11,41 11,42,,,,,,,, ,yyyyyyyyy y=Y (for example, 11,41y represents the first pre-form from the 11th row and 4th column). Write three sets of assumptions about the mean of Y in the form E=YXβ for different sets of and X β, representing assumptions (1) and (2), and then “no restrictions on the 5 cavity means.” b) Suppose that error sums of squares for the scenario and 3 models in a) are as in the table below Model for Cavity Means SSE (1) Constant Mean .0324(2) Linear Row and Column Effects .01265 Unrestricted Means .0045Page 2 of 7Give values of F statistics appropriate to testing i) 01 2H: 0ββ== under assumption (2) ii) the hypothesis that there is lack of fit to relationship (2) The R printout attached to this question concerns analyses of weights from 5 consecutive machine cycles (all run under the same machine set-up) where pre-forms from all 5 test cavities were gathered and weighed. With measured mass of the pre-form from row and column on cycle rciyrci= the printout enables inferences under the models 01 2rci i rciyrcβββδε=+ + ++ (3) and (),rci i rciyrcµδε=++ (4) where 01 2, and βββ are constants, the 5 values (),rcµ are constants, 12345,,,,δδδδδ are iid ()2N0,δσ independent of the rciε that are iid ()2N0,σ. c) Under model (3) is there a statistically significant systematic linear trend in pre-form weight left-to-right across the die? (Explain.) d) How is it obvious from the printout for model (3) that the size of cycle-to-cycle variability in pre-form weight for a particular cavity is estimated with poor precision? e) According to the analysis provided for model (4), is there a statistically detectable difference between the mean pre-form weights produced in cavities 18 and 47? (Give approximate 95% confidence limits for this difference.) f) According to model (4), what is the expected value of the sample variance of the weights of the 5 pre-forms produced in any particular cavity? Give a formula for exact 90% confidence limits for this quantity based on pooling 5 such sample variances. g) For an (),rc pair in the data set, let ()5.1/5rc rciiyy==∑ and 6rcy be the pre-form weight produced in row r and column c on a sixth cycle of the machine (immediately after the 5 for which we have data). What are the mean and variance of 6.rc rcyy−? What then is a formula for exact 95% prediction limits for 6rcy ? The students did some experimenting with the machine, using as a response variable the average weight of the pre-forms from the 5 cavities indicated on the diagram on page 1 and running various combinations of levels of 3 factors that can be set on the machine, A-Injection Time, B-Hold Time, and C-Hold Pressure. h) What does model (4) say is the variance of an individual response used in the students’ experimentation?Page 3 of 7i) The table below gives results from a single cycle for 8 different machine set-ups making up a complete 222×× factorial arrangement in the Factors A,B, and C. If the mean response at level of A, level of B, and level of Cijkij kµ= a sensible definition of the main effect of C at level k is .. ...kkγµµ=− Based on the data in the table below, give an estimate of 21γγ−. Then, based on model (4) and the R printout, give a standard error for your estimate. Level of A Level of B Level of C Response1 1 1 22.9362 1 1 23.0001 2 1 23.5182 2 1 23.5661 1 2 23.2722 1 2 23.3181 2 2 23.7362 2 2 23.794 j) The data in the table above are balanced. Below is an ANOVA table giving some corresponding “sums of squares.” Source SS A .00583 B .54497 C .15125 AB .00000 AC .00001 BC .00541 ABC .00010 Define the 81× vectors BC BC,,,, and Y1X X X and the 84× matrix X by BCB22.936 1 1 123.000 1 1 123.518 1 1 123.566 1 1 1,, , ,23.272 1 1 123.318 1 1 123.736 1 1 123.794 1 1 1−−        −−        −    −    ====    −    −                    Y1XXX()CBCBC1111, and | | |1111−−==−−X1XXX Let XP be the perpendicular projection matrix onto ()C X, the column space of X. Evaluate the quadratic form ()′−XYI P YPage 4 of 7Stat 511 Printout > CAVITY<-as.factor(cavity) > Data<-data.frame(y,CAVITY,row,col,cycle) > Data y CAVITY row col cycle 1 23.36 3 3 1 1 2 23.37 3 3 1 2 3 23.33 3 3 1 3 4 23.34 3 3 1 4 5 23.35 3 3 1 5 6 23.36 9 9 1 1 7 23.36 9 9 1 2 8 23.33 9 9 1 3 9 23.34 9 9 1 4 10 23.35 9 9 1 5 11 23.19 18 6 2 1 12 23.20 18 6 2 2 13 23.28 18 6 2 3 14 23.29 18 6 2 4 15 23.23 18 6 2 5 16 23.30 37 1 4 1 17 23.32 37 1 4 2 18 23.32 37 1 4 3 19 23.34 37 1 4 4 20 23.32 37 1 4 5 21 23.38 47 11 4 1 22 23.39 47 11 4 2 23