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STATISTICS 402 - Assignment 6 Due March 30, 2012 1. For each of the following problem statements, identify the response, conditions and experimental material. Indicate the design (completely randomized, randomized complete block, Latin square) that is used to collect the data and indicate if factorial crossing is used. Construct a partial ANOVA table by identifying all sources of variability and associated degrees of freedom. a. Industrial psychologists wish to investigate the effect of music in the factory on the productivity of workers. Four distinct music programs (Country, Soft Rock, Hard Rock, and Classical) and no music (a control) make up the five treatments. In order to account for variability from day to day and week to week, a different program is used on each day (M, T, W, Th, and F). The programs are rotated from week to week, so that over 5 randomly chosen weeks, each program appears once on any specific working day of the week. b. An experiment is conducted to see the effect of fishing rod length and sinker weight on the length of the cast. There are three fishing rod lengths: 6 feet, 6.5 feet and 7 feet. There are four sinker weights: 2 ounces, 3 ounces, 4 ounces, and 5 ounces. Each of the twelve combinations of fishing rod length and sinker weight will be replicated four times. The order in which the 48 casts will be made is completely randomized. c. A sports physiologist is conducting an experiment on eye focus time. She is interested in the distance of the object from the eye on the focus time. Five different distances are of interest. She chooses 10 players at random from a college softball team. Each player will have her focus time measured at each of the five distances. The order of the distances will be randomized for each player. d. An experiment is performed to see whether different operators obtain different results in the routine analysis to determine the amount of nitrogen in soil. 50 soil samples are chosen at random and divided at random into 5 groups of 10. Each operator is assigned a group of 10 soil samples at random and asked to determine the amount of nitrogen in each sample. 2. A grain refiner is a chemical added to a molten metal or alloy to restrict grain growth that would otherwise lead to softening of the material at high temperatures. A master alloy manufacturer produces grain refiners in four furnaces. Each furnace has unique operating characteristics and so must be considered a nuisance factor for any experiment run in the foundry. Process engineers want to investigate the effect of stirring rate on the grain size of the final product. Each furnace can be run at four different stirring rates; 5 rpm, 10 rpm, 15 rpm and 20 rpm. All four stirring rates are used with each furnace and the order in which the stirring rates are used is randomized for each furnace. The data on grain size appear on the next page. a. What is the response? b. What are the conditions? c. Why is this block design? How are blocks made? 1 Furnace 1 Furnace 2 Furnace 3 Furnace 4 5 rpm 8 4 4 6 Stirring Rate 10 rpm 15 rpm 14 14 5 6 6 8 9 8 20 rpm 18 9 12 11 d. Compute the sample means for the four stirring rates and an overall grand sample mean. e. Compute the estimated effect for each stirring rate. Which stirring rate(s) appear to produce the smallest grain size? Explain briefly. f. Construct a complete ANOVA table including all appropriate sources of variation. g. Are there statistically significant differences among the stirring rates in terms of average grain size? Report the appropriate F- and P-values and explain why these support your answer. h. If there are statistically significant differences among the stirring rates, indicate where those differences lie. Give the value of HSD. Clearly state which of the positions for the feet have statistically significant differences in sample means and which do not. i. If one wishes to have the smallest grain size, what statistically valid recommendations can you make for the choice of stirring rate? Explain your reasoning. j. Is there a statistically significant linear relationship between stirring rate and grain size? Support your answer with an appropriate test of hypothesis. k. Construct appropriate plots of the residuals and comment on whether the equal variance condition and the normally distributed errors condition are satisfied. 3. Traffic engineers attempt to design traffic flow systems so that the maximum number of vehicles can go through the system safely in the least amount of time with the fewest stops at red lights. A study was done to evaluate five different traffic-light signal sequences (A, B, C, D and E). The traffic engineers were interested in the unused red light time, which is the amount of waiting time for traffic facing a red light when there is no traffic in the direction(s) of the green light(s). The experiment was conducted with five randomly selected intersections and five time periods during the day. A lower unused red light time indicates a better outcome in terms of traffic flow. The data appear on the next page. 2 Intersection 1 Intersection 2 Intersection 3 Intersection 4 Intersection 5 a. b. c. d. e. f. g. h. i. 7 to 9 am C 12.1 B 16.5 A 15.2 E 14.6 D 10.7 10 am to 12 D 31.4 C 26.5 B 33.8 A 31.7 E 34.2 1 to 3 pm B 30.2 A 22.7 E 29.1 D 23.8 C 21.6 4 to 6 pm E 17.0 D 19.2 C 13.5 B 16.7 A 19.5 7 to 9 pm A 31.5 E 25.8 D 27.4 C 26.3 B 27.2 What is the response? What are the conditions? Why is this experiment a Latin square design? Compute the sample means for the five traffic-light signal sequences and an overall grand sample mean. Compute the estimated effect for the five traffic-light signal sequences. Which signal sequence appears to produce lowest average unused red light time? Construct a complete ANOVA table including all appropriate sources of variation. Are there statistically significant differences among the traffic-light signal sequences in terms of average unused red light time? Report the appropriate Fand P-values and explain why these support your answer. If there are statistically significant differences among the signal sequences, indicate where those differences lie. Give the value of HSD. Clearly state which of the signal sequences have statistically significant differences in sample means and which do not. What statistically valid recommendations can you make for minimizing the unused red light time? Is there a best or worst traffic-light signal sequence? Explain your reasoning. 3