MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS FINAL EXAM - STATISTICS 350 - FALL 1999 Instructor: A. Oyet Date: December 16, 1999 Name(Surname First): Student Number INSTRUCTIONS 1. Don t forget to write your name and number.. Make sure that your examination paper has 6 QUESTIONS. 3. No books or notes are allowed. You may use your calculator. 4. Clearly outline your answers. Show your work and answer each question carefully. 5. Tables can be found on the last few pages. Question # Mark Score 1 5 15 3 15 4 0 5 5 6 0 Total 10
1. (5 points) An experiment was conducted to test the effects of nitrogen fertilizer on lettuce production. Five rates of ammonium nitrate were applied to four replicate plots in a completely randomized design. The data, Y ij, are the number of heads of lettuce harvested from the plot. Treatment (lb N/acre) Heads of lettuce/plot Total 0 104 114 90 140 448 50 134 130 144 174 58 100 146 14 15 156 596 150 147 160 160 163 630 00 131 148 154 163 596 PP Y ij = 41508 85 Source: Dr. B. Gardner, Department of Soil and Water Science, University of Arizona. (a) This experiment was conducted in a completely randomized design with the field plots in a 4 5 rectangular array of plots. Show a random allocation of the five treatments to the 0 plots using a random permutation of the numbers 1 through 0. (b) Write the linear statistical model for this study, and explain the model components. State the assumptions necessary for an analysis of variance of the data. (c) Construct the ANOVA table for this experiment. (d) Compute the standard errors of the least squares estimate of the treatment effects for each nitrogen level. Compute the residual corresponding to observations 1 and under treatment 50. (e) Order the treatment means - ascending order - based on the SNK multiple-range test at the 0.05 significant level.. (15 points) Use Figure 1 and the SAS output on the lettuce yield experiment to answer the questions that follow. The SAS output represent two distinct response curves. POLYNOMIAL RESPONSE FUNCTION Model: MODEL1 Dependent Variable: YIELD Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 4741.5486 370.77143 11.3 0.0008 Error 17 3591.5714 11.504 C Total 19 833.80000 Root MSE 14.53446 R-square 0.5690 Dep Mean 14.60000 Adj R-sq 0.5183 C.V. 10.1947
Figure 1: Model Diagonistics (a) Yield vs Treatment (b) Residual vs Predicted (c) Sequence Plot (d) Normal probability plot of residuals. Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP 1 114.11486 6.8393645 16.685 0.0001 N1 1 0.6349 0.160351 3.847 0.0013 N 1-0.0057 0.00077690 -.905 0.0099 Model: MODEL Dependent Variable: YIELD Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 4809.1486 1603.0476 7.79 0.007 Error 16 353.65714 0.857 C Total 19 833.80000 Root MSE 14.84010 R-square 0.5771 Dep Mean 14.60000 Adj R-sq 0.4978 C.V. 10.40680 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > T INTERCEP 1 11.81486 7.36685905 15.314 0.0001 N1 1 0.80976 0.37481076.160 0.0463 N 1-0.004857 0.0047594-1.01 0.37 N3 1 0.000008667 0.00001564 0.554 0.587 3
(a) Derive the normal equations for the lettuce data. (b) Based on the plots and the SAS output, write down the best polynomial response function that describes the relationship between lettuce yield and nitrogen. Justify your choice. (c) Using your choice of response in part b, predict the average lettuce yield at a rate of 55.7 lb N/acre. (d) Comment on the adequacy of the completely randomized design for the lettuce yield experiment. 3. (15 points) The coefficients shown to you by a colleague for a set of contrasts among treatment means in a balanced design follow. He wants you to check them out. Treatment A B C D E C1 1-1 -1-1 C 1-1 0-1 1 C3-1 1-1 1-1 C4 0 0-1 -1 (a) Does each of the proposed set of coefficients constitute a contrast? Justify your answer. (b) Are C1 and C4 orthogonal? C and C3? Justify your answer. (c) Outline the Scheffé test procedure at 0.05 level of significance to test the null hypothesis about the orthogonal contrasts in part b. Compute those terms in your expressions that can be evaluated. 4. (0 points) A manufacturer of direct ignition systems (DISs) uses one of two probe cards and one of four hot testers to test the circuits of a DIS. The probe cards provide the interface between the test and the DIS circuits. The tester measures the current, in milliamperes, under specified conditions. A completely randomized experiment using 16 DISs produced the following data. Assume that the cards and testers used in the experiment were randomly obtained from large universes of cards and testers. Tester Card 1 3 4 Total 1 40.498 40.35 40.5 40.458 40.513 40.53 40.77 40.503 3.989 40.38 40.158 40.185 40.460 40.375 40.131 40.15 40.507 3.413 Total 161.768 160.777 160.99 161.98 645.40 PP Y ij = 6034.71 (a) Construct the ANOVA table for this study. (b) What percentage of the total variation in current is due to (i) cards (ii) testers. (c) Is there evidence in favour of interaction effect? Test at 0.05 level of significance. 4
5. (5 points) Consider a four-factor factorial experiment where factors A and B are at 3 levels each and factors C and D are at levels each and there are 4 replicates. Suppose that A and B are fixed and C and D are random. (a) Write down the degrees of freedom and the expected mean squares for the effects in the model describing the experiment. Also write down the sums of squares for main effects only. (b) Obtain expressions for estimating the variance components in the model. (c) Do exact tests exist for all effects? If not, propose test statistics for those effects that cannot be directly tested. (d) Now suppose that only three factors A, B, and C are of interest, so that we have a three-factor factorial model defined by y ijkl = µ + A i + B j + AB ij + C k + AC ik + BC jk + ABC ijk + ε ijkl, i =1,, 3; j =1,, 3; k =1, ; l =1,,,n. If all the factors are random and the experimenter was able to replicate the experiment only once (i.e. n = 1), can any effects be tested? If the three factor interaction ABC ijk and the AB ij interaction are dropped from the model, can all the remaining effects be tested? Justify your answer. 6. (0 points) A horticulturalist studied the germination of tomato seed with four different temperatures (5 o C,30 o C,35 o C, and 40 o C) in a balanced incomplete block design because there were only two growth chambers available for the study. Each run of the experiment was an incomplete block consisting of the two growth chambers as the experimental units. Two experimental temperatures were randomly assigned to the chambers for each run. The data that follow are germination rates of the tomato seed. Run 5 o C 30 o C 35 o C 40 o C 1 4.65 - - 1.34-4.38 -.4 3 9.17 1.5 - - 4 - - 5.90 1.83 5 8.90-18.7-6 - 5.53 8.4 - Source: Dr. J. Coons, Department of Botany, Eastern Illinois University. (a) Write a model for the experiment including assumptions - explain the terms. (b) Compute the least squares estimates of the temperature effects, say τ j, and their standard errors. (c) Construct a 98% interval estimate for the difference in means of germination rates at 5 o C and 30 o C. 5