STAT 496, Spring 2007Homework Assignment #21. The three fundamental principles of a designed experiment are: control of outside variables,randomization, and replication. Consider each of the following scenarios. Does each adhereto all three principles? If so, describe how. If not, indicate how you would change the designto incorporate the missing principle(s). Be specific and be sure to comment on all threeprinciples.(a) An aluminum smelting operation makes its own carbon anodes for use in their aluminumsmelting pots. The density of the anode is an important quality characteristic because itaffects the usable life of the anode. An experiment is conducted to see the effect of baketemperature on anode density. Twenty anodes are made using the same raw materials,methods and people. Five anodes are assigned at random to each of four temperatures,500, 525, 550 and 575oC. A temperature is picked at random and the five anodes arebaked in the oven at that temperature. One of the remaining temperatures is picked atrandom and the five anodes are baked in the oven at that temperature. This randomselection of a temperature and baking 5 anodes continues until all temperatures havebeen used once. The density of each anode is then measured. The order of measurementis completely random.(b) A chemist wants to compare a new and simpler assay method with the standard method.She prepares a batch of solution, divides it into 40 specimens, and then selects 20 atrandom. She asks her technician to analyze these 20 specimens using the new methodand the remaining 20 using the standard method. The technician finishes analyzing the20 specimens using the new method in the morning. He takes a break for lunch anddoes the remaining 20 specimens using the standard method in the afternoon.(c) A fuel economy study of four diesel-powered vehicles was conducted. All four vehicleswere tested under controlled conditions in a laboratory using a dynamometer to controlthe speed and load conditions and to avoid the effects of weather. A single test fuel wasused. Eight tests were run on each vehicle. Fuel economy in miles per gallon (mpg) wasmeasured. The total of 32 tests were run in a completely random order.2. Before the experiment in 1 (c) is run, there is discussion on how much, and what sort of,replication should be involved.(a) If the chances of making errors are set at Alpha=0.05 and Beta=0.10 and a differencein mean fuel economy of 1 standard deviation is to be detected, how many runs arerequired for each vehicle?(b) If resources limit the number of runs to 8 per vehicle, give two combinations of Alphaand Beta and the size of difference that can be detected. Explain briefly the trade-offsbetween the two combinations.(c) An argument develops over what is replication. One person argues that 8 Mercedes, 8Peugeots, 8 Oldsmobiles and 8 Volkswagons must be used. The other person argues thatone vehicle of each type is sufficient provided each vehicle is tested 8 times. Discuss theadvantages and disadvantages of each person’s view of replication. Be sure to commenton the scope of the experiment, that is what sort of generalizations can be made, foreach person’s view.13. Below are the data for the experiment described in 1 (c). The numbers have been orderedfrom smallest to largest for each vehicle. Refer to the JMP output that accompanies thishomework.Mercedes Peugeot Oldsmobile Volkswagon23.8 23.2 18.7 40.124.5 23.3 18.9 40.124.7 23.9 19.2 40.324.7 24.0 19.2 40.424.8 24.1 19.3 40.525.0 24.7 19.7 40.925.1 25.0 19.8 41.825.4 25.0 20.0 41.9Yi24.75 24.15 19.35 40.75si0.475 0.703 0.450 0.725(a) Comment on the side-by-side box plots for the four vehicles. Be sure to discuss bothcenter and spread.(b) Calculate, by hand, the value for the Least Significant Difference (LSD) using t=2. Indi-cate which pairs of vehicles have differences in mean fuel economies that are statisticallysignificant.(c) How does your analysis in (b) compare to JMP using Comparisons - Each pair, Student’st and an individual error rate of α = 0.05?(d) Calculate, by hand, the value for the Least Significant Difference (LSD) using t=3. Indi-cate which pairs of vehicles have differences in mean fuel economies that are statisticallysignificant.(e) How does your analysis in (d) compare to JMP using Comparisons - Each pair, Student’st and an individual error rate of α = 0.01?(f) If you want the best fuel economy, which vehicle would you recommend? Explain yourchoice briefly.(g) If you want to avoid the vehicle with the worst fuel economy, which vehicle should youavoid? Explain your choice