Stat 502
Homework 6
Assigned 11/8/06
Due 11/15/06
1. Tomatoes: A horticulturist wants to do an experiment on the differences between two varieties of tomatoes, A and B, but only has a small strip of land running North-South, and is worried that the South end may get more sunlight. She decides to divide the land into a grid with 5 rows and 2 columns, the rows running North-South, so there are 10 subplots total. In each row, the scientist randomly assigned variety A to one subplot and B to the other. At harvest time, the tomato yields in kilograms were recorded in each subplot. The results of the experiment were as follows:
A5.4 / B
3.4
B
4.1 / A
4.9
B
4.4 / A
4.3
A
4.6 / B
3.8
B
3.9 / A
4.5
(a) Is this a CRD? Does this design make sense in terms of the researcher’s worry about the potential effect of sunlight? Why or why not?
(b) Use a normal-theory t-test appropriate for this design to evaluate versus , get a p-value, and make a 95% confidence interval for .
(c) List any assumptions you made in (b). Check the assumptions and discuss any evidence that the assumptions are violated.
Now consider using a randomization distribution for testing the hypothesis in (a), using for simplicity the test statistic .
(d) How many possible treatment assignments are there under this randomization scheme?
(e) For any experiment using this same design and test statistic, what is the smallest p-value one could obtain using the randomization test? What is the second smallest p-value?
(f) What would the treatment assignment have to look like in order for the p-value to be a minimum?
(g) Decide if the null hypothesis is rejected at level , using the and the randomization test (you might want to try figuring this out without a computer).
(h) Which do you prefer, a randomization test or a test based on the normal distribution? Why? Do you think there is enough evidence in the data to reject H0?
(i) Think of a simple modification to the researcher’s experimental design that would increase her power.
2. Dick and Jane’s Statistical Consulting : Write a few paragraphs for each of the following scenarios to evaluate Dick and Jane’s statistical consulting service. If they are doing things correctly, describe why. If they are doing things incorrectly, describe why and provide corrections.
(a) Dick and Jane have been hired to analyze data from a paired-sample design, i.e. they have two observations for each person in a random sample of size from a population. They plan on using a paired-sample t-test to compare to , and want to check the assumptions which validate the test. Dick has made two qqnorm plots, one for the A observations and one for the B observations. Jane has compared the variances of the two groups.
(b) Dick and Jane have been hired by some scientists to do a power calculation for a CRD with two treatment groups. The scientists want to be able to detect a mean difference of with a probability of 0.80. They are not sure what the variances in the two groups are, but they think they are approximately equal, and are somewhere between and . To keep the sample size to a minimum, Dick and Jane decide to compute the power of the procedure under . They do this by finding the value of such that the probability that is 0.8, under the condition and that the data are normally distributed.
(c) Dick and Jane run a two factor ANOVA for a RCBD, so that one of the factors is Treatment and the other is Block. The ANOVA suggests there are significant Treatment effects, significant Block effects and significant non-additivity. The scientists want to compare the differences between treatments, averaged across blocks. Since the F-test for Treatment was significant, Dick and Jane decide that it is okay to go ahead and do pairwise comparisons of treatment means. They do this by using the t.test command in R by typing, for example, “t.test(y[trt=A],y[trt=B],var.equal=T).”