Please do question number 2 using Program R if necessary. Thank you.

THE CHINESE UNIVERSITY OF HONG KONG Department of Statistics STAT3002: Experimental Design Problem Sheet 4 1. (Chapter 10, Exercise 6 in Dean and Voss) Candle Experiment, Tsai, H.-C., Yang M.-C., Wheeler, D., Schultz, T. (1989) An experiment to determine whether dierent colored candles (red, white, blue, yellow) burn at dierent speeds was conducted by Hsing-Chuan Tsai, Mei-Chiao Yang, Derek Wheeler, and Tom Schultz in 1989. Each experimenter collected four observations on each color in a random order, and experimenter was used as a blocking factor. Thus, the design was a general complete block design with v = 4, k = 16, b = 4 and s = 4. The resulting burning times (in seconds) are shown in Table ??. A pilot experiment indicated that treatments and blocks do interact. The candles used in the experiment were cake candles made by a single manufacturer. Block Tom Derek Tsai Yang Red 989 1032 1077 1019 899 912 911 943 898 840 955 1005 993 957 1005 982 Colour White Blue 1044 979 1011 951 987 1031 928 1022 847 880 899 800 879 830 820 812 840 952 909 790 961 915 871 905 987 960 864 925 920 1001 824 790 Yellow 974 998 1033 1041 886 859 901 907 950 992 920 890 949 973 978 938 Table 1: Data for Candle Experiment (a) Complete an analysis of variance table for the data using the blocktreatment interaction model for a general complete block design. (b) Test the null hypotheses of negligible blockXtreatment interaction and, if appropriate, test the null hypothesis of equality of treatment eects. (c) Use an appropriate multiple comparisons procedure to evaluate which color of candle is best. Interpret the results. (d) Discuss whether blocking was important in this experiment. 2. (Chapter 13, Exercises 3 a)-c) of Dean and Voss) Projectile Experiment, Johnson, N.L. and Leone, F.C. 1977 N. L. Johnson and F. C. Leone, in their 1977 book Statistics and Experimental Design in Engineering and the Physical Sciences, described a single-replicate 24 experiment concerning the performance of a new rie under test. Under study were 1 the eects on projectile velocity of the factors charge weight (A), projectile weight (B), propellant web (C), and weapon (D), where two ries were used. The design included two blocks each of size eight, corresponding to the two days on which data were collected, confounding ABCD. The coded velocity data are given in Table ??. Run 1 7 5 3 6 4 2 8 Day 1 TC y1ijkl 0000 97 0011 26 0101 53 0110 15 1001 145 1010 100 1100 150 1111 54 Run 13 11 9 15 10 16 14 12 Day 2 TC y2ijkl 0001 75 0010 39 0100 68 0111 -16 1000 151 1011 97 1101 141 1110 66 Table 2: Data for Projectile Experiment (a) Fit a model including block eects, treatment main eects, and 2factor interactions. Use residual plots to check the standard model assumptions. (b) Conduct the analysis of variance, and discuss the results. (c) Construct simultaneous condence intervals for any interesting treatment contrasts using an appropriate method of multiple comparisons. 3. (Chapter 13, Exercise 5 in Dean and Voss) Suggest a confounding scheme for a 26 experiment in 8 blocks of 8, assuming that all 2-factor interactions are to be estimated, as are the 3-factor interactions involving both A and F . List all eects confounded. List the treatment combinations in the design block by block. 4. (Chapter 13, Exercises 8a), c)- e)) Decontamination experiment Beta particles An experiment was described by M. K. Barnett and F. C. Mead, Jr. in the journal Applied Statistics in 1956 to explore the eect of four factors on the eciency of a decontamination process for the removal of radioactive isotopes from liquid waste. The measurements taken after the decontamination process were the counts per minute per milliliter of alpha and beta particles. We consider here part of the data for the beta particles, shown in Table ??. The four treatment factors were: A: 0.4 g and 2.5 g per liter of aluminum sulphate (coded 0, 1); B: 0.4 g and 2.5 g per liter of barium chloride (coded 0, 1); C: 0.08 g and 0.4 g per liter of carbon (coded 0, 1); D: Final pH of liquid waste (6 and 10, coded 0, 1). The experimenters selected a design in b = 2 blocks of k = 8 that confounded the four-factor interaction contrast ABCD. The tted 2 model included all main-eects and all 2-factor and 3-factor interactions. The treatmentblock interaction was assumed to be negligible. Block I II 1010 (716) 0010 (1024) Treatment 1111 0110 (686) (498) 0001 0111 (1364) (475) Combinations (Response) 0000 1100 0101 0011 (1437) (527) (579) (1433) 1000 1101 0100 1110 (574) (664) (579) (507) 1001 (906) 1011 (1130) Table 3: Data for Contamination Experiment (Beta) (a) Use a normal probability plot to identify the important contrasts. (b) Draw an interaction plot of any interaction that appears to be nonnegligible by either analysis. (c) Looking at the results of your analysis, which settings of the factors would you recommend for reducing the beta particle counts? (d) Suppose that the experimenters had believed before the experiment that the three-factor interactions were all negligible. What would the analysis of variance table have looked like? Would your recommendations have been any dierent? THE END 3