Solutions to Homework 2, Stat 402, Spring 2004 1a. The experimental units are the litters. b. The observational units are animals. c. The number of replications per treatment is 2. d. An alternative approach would be to do a RCBD by assigning one animal per litter at random to treatment A and then another randomly selected animal per litter to treatment B. Below is a possible method of assigning treatments: Litter# ID# R.N. Treatment 1 1 8 B 1 2 7 - 1 3 0 A 2 4 2 A 2 5 3 B 3 6 6 - 3 7 5 - 3 8 7 B 3 9 5,0 A 4 10 4 - 4 11 1 A 4 12 4,9 B The assignments of treatments were based on the rule that the smallest random number in the litter group received treatment A and the largest random number received treatment B. If a random number was duplicated within a litter group, a second digit was drawn to replace the duplicate digit. 2a. c = 7. b. c = 8. c. c is the mean. d. c is the median. 3a. Calculate the overall mean = [(72*8)+(68*8)+(55*7)]/23 = 65.43. Calculate SSB = (8*(72-65.43)^2)+(8*(68-65.43)^2)+(7*(55-65.43)^2) = 1159.65. Calculate SSE = ((8-1)*312)+((8-1)*310)+((7-1)*321) = 6280. The rest of the ANOVA calculations are now straight forward: Source df SS MS F p-value Treatment 2 1159.65 579.83 1.85 > 0.1 Error 20 6280 314 Total 22 7439.65 b. There is no evidence of differences among the phosphorous content means of the three varieties. (F = 1.85, w/2 and 20 df, p-value > 0.1) c. No, this is not a violation of the assumption of equal variances. The sample variances are estimates of the true variances. The observed difference could be easily explained by chance. As a rough rule, we generally don't become too concerned about non-constant variance unless one variance is 4 times larger than another. d. t(0.975,20) = 2.086 SE = (314/7)^0.5 = 6.7 CI: 55+/-2.086*6.7 (41 to 69) The 95% confidence interval for the mean phosphorous content of leaves from variety C trees is 41 to 69. 4. 143 replicates would be needed in each treatment group to have a test with a power of 80%. Below is the input necessary to do the calculations. data one; do r=2 to 150; alpha=0.05; t=2; variance=9; sumtausq=0.5; The sumtausq is the only one not given directly in the problem. Recall that tau represents a treatment effect. A treatment effect is simply a treatment mean minus the average of all treatment means. It is fairly easy to see, that in the case of two treatments, the treatment effects must be + and - 1/2 the distance between the two treatments. More formally, suppose the mean for the control group is X. We want to be able to detect a difference of 1 unit, so the mean of the new treatment is either X + 1 or X - 1. Thus, the overall mean of these two groups would be either (X+X+1)/2=X+1/2 or (X+X-1)/2=X- 1/2. Either way, the two treatment effects are 1/2 and -1/2. Squaring and summing these results in 0.25 + 0.25 = 0.5 which is the sumtausq.