STATISTICS 402 - Assignment 4 Due February 22, 2008 1. In an attempt to find an effective treatment for the ingestion of a fast acting deadly poison, chemists have come up with a potential antidote. An experiment is to be performed with three different exposures to the poison (1-Low, 2-Medium, and 3-High). Additionally, four different amounts, relative to body weight, of the antidote will be administered 0.5 mg/g, 1 mg/g, 1.5 mg/g, and 2 mg/g. The experiment will be performed on mice. The mice will all be of the same age and approximately the same size. The mice will be exposed to the poison and given the antidote and there survival time (hours) will be recorded. a) Identify the response, conditions and experimental units. b) Give an example of an outside variable that is controlled in the study. How is it controlled? c) How many treatment combinations are there? List all the treatment combinations. d) If the experimenter wishes to detect a difference in means for the three poison exposure levels as small as 0.9 standard deviations with Alpha=0.05 and Beta=0.05, how many mice are needed? e) With the number of mice you found in c), how small a difference in means for the four amounts of the antidote can be found with Alpha=0.05 and Beta=0.10? f) The experimenter is only able to use 4 mice for each treatment combination. For Alpha=0.05 and Beta=0.10 what is the size of the difference in means ( Δ σ ) for the poison exposure levels? for the antidote amount levels? g) The experimenter comes to you for advice on randomization. She explains she can only use 4 mice per treatment combination. Explain to the experimenter how to randomly assign the mice. Do the randomization you describe. Turn in your randomized assignment of the mice with this assignment. 2. The experiment described in problem 1 is run. The data are available on the course web site in the file Poison.JMP. There is also an SPSS file Poison.sav. Download the data and use a statistical analysis package to help with the analysis of the data. a) Analyze and comment on the distribution of survival time. b) Fit a model that includes Poison, Antidote and Poison*Antidote interaction effects. Give the value of R2 and the value of the estimate of the error standard deviation, σ. c) Test the hypothesis that there are no differences among the population treatment means. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. d) Test the hypothesis that the poison exposure effects are all zero. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. 1 e) Where are the statistically significant differences among the poison exposure means? Use the LSD (Student’s t) multiple comparisons procedure. f) Test the hypothesis that the antidote amount effects are all zero. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. g) Where are the statistically significant differences among the antidote amount means? Use the HSD (Tukey-Kramer) multiple comparisons procedure. h) Is a linear contrast on antidote amount statistically significant? If it is, what does this indicate? i) Test the hypothesis that the interaction effects are all zero. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. j) Construct and comment on the interaction plot. k) Construct plots of residuals versus each of the factors in the experiment. Comment on the plots. l) Analyze the distribution of residuals. Comment on what the analysis of residuals says about the conditions necessary for statistical analysis. 3. An alternative analysis uses 1/(Survival Time) as the response. This response can be thought of loosely as a “rate of death” or deaths per hour. If a mouse survives 1 hour, then there is 1 death in that hour. If a mouse survives 0.5 hours followed by a second mouse surviving 0.5 hours, then there are 2 deaths (1/0.5) in that hour. If a mouse survives 2 hours, then there is an average of 1 death per 2 hours, ½=0.5 deaths per hour. a) Construct a new column in your data table for the new response variable 1/(Survival Time). Fit a model with 1/(Survival Time) as the response and Poison, Antidote and Poison*Antidote as effects. b) Test the hypothesis that the poison exposure effects are all zero. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. c) Where are the statistically significant differences among the poison exposure means? Use the LSD (Student’s t) multiple comparisons procedure. d) Test the hypothesis that the antidote amount effects are all zero. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. e) Where are the statistically significant differences among the antidote amount means? Use the HSD (Tukey-Kramer) multiple comparisons procedure. f) Test the hypothesis that the interaction effects are all zero. Be sure to include the F statistic, P-value, decision, reason for the decision and a conclusion. m) Construct plots of residuals versus each of the factors in the experiment. Comment on the plots. n) Analyze the distribution of residuals. Comment on what the analysis of residuals says about the conditions necessary for statistical analysis. g) Comment on any differences in conclusions between this alternative analysis and the analysis in problem 2. Remember to turn in copies of your computer output with your assignment. 2