STAT 496, Spring 2011 Homework Assignment #4, Due by Thursday, March 3 1. In an experiment on the effectiveness of laundry detergents, 2 brands of detergent and 2 water temperatures are used in a 22 factorial experiment. A standard load of laundry consists of 2 pairs of jeans, 5 sets of cotton underwear, 5 pairs of white crew socks, 5 cotton shirts, and 3 cotton towels. Each load is mixed with 100 grams of top soil, shaken to remove loose dirt, and weighed. Five standard laundry loads are washed for each of the combinations. One washing machine is used. The assignment of detergent brand and water temperature to a laundry load and the order of washing are completely randomized. The amount of dirt removed (difference between starting weight and ending weight after the load is washed and dried) is the response. The data are given below. Detergent Brand X Y X Y Water Temperature Warm Warm Hot Hot Amount of Dirt Removed (g) 13 17 18 21 16 19 16 20 16 20 17 18 14 17 19 22 17 20 20 23 Mean Variance 15.2 18.6 18.0 20.8 2.7 2.3 2.5 3.7 a) Is this a well designed experiment? Comment briefly on control of outside variables, random assignment and replication within the experiment. b) The experiment does not have a control group. How would you construct a control group for this experiment? c) Calculate means for each brand of detergent and each water temperature. d) Display the means for detergent brand and water temperature and comment on the apparent effects of each factor. e) Construct two interaction plots, one with detergent brand on the horizontal axis, one with water temperature on the horizontal axis. Does there appear to be an interaction between the two factors? Explain briefly. f) Compute the MSrepError by pooling the individual sample variances for each treatment combination. g) Calculate the estimated full effect for each factor and the interaction effects. h) Give the standard error of an estimated full effect and use it to determine what estimated effects are statistically significant. Use t = 3 for the critical value. i) Eliminate any non-significant effects and give the reduced model prediction equation. Be sure to define the coded predictors explicitly. j) Use the reduced model prediction equation to predict the values for the four treatment combinations. Construct prediction intervals for each of these values. k) What recommendation would you make if you wanted to get the most dirt out? l) We could also look at this experiment as a “one-factor” experiment, where the “treatments” for the “one-factor” are combinations of detergent brand and water temperature. Calculate the LSD (use t = 2) for such an experiment. m) Based on the LSD, which “treatments” are significantly different? Which “treatments” are not significantly different? Does this analysis agree or disagree with the one done previously? Explain briefly. Note: You may use JMP, or another computer program, to assist you with any of the graphing or calculations needed to answer the questions above. However, since exams will contain some hand calculations it may be worthwhile to do the calculations by hand and verify the hand calculations with the computer.