1) You will be doing an experiment at a new research farm. You are considering using a plot size of 0.2 ha (hectares, 0.2 ha = 2,000 m^2). The only previous data about plot-plot variability on this farm is from a study using 0.5 ha plots. Those data suggest that the s.d. between 0.5 ha plots is 3.1 kg/ha.
a) What s.d. (between your 0.2 ha plots) would you expect to see?
b) Your study will compare 6 treatments, using a 1 way ANOVA. You are given a choice of using 24 0.2 ha plots (i.e. 4 per treatment) or 12 0.5 ha plots (i.e. 2 per treament). Please consider s.e. of a treatment mean and error d.f. Does the choice of plot size make much difference? Which plot size is better statistically? Explain your choice.
Hint: You do not need to calculate power, but you should support your choice numerically.

2) Consider an experiment like the salinity experiment, where containers are randomly assigned to one of two treatments and measurements are taken on k plants per container. Previous studies suggest that sigma^2(containers) = 0.49 and sigma^2(plants) = 6.76. These are the variance components for containers and plants. You are told to make sure that your experiment provides a s.e. of a treatment mean that is less than 0.5.

a) If you measure P=2 plants per container and C=16 containers per treatment, what is the s.e. of the mean?
b) If you measure P=5 plants per container and C=8 containers per treatment, what is the s.e. of the mean?
c) Does either design satisfy the precision requirement (i.e. s.e. < 0.5).
d) If the major cost in your study was the cost of measuring a plant, which design should you use? Explain briefly.
e) If the major cost in your study was the space for a container (e.g. on a greenhouse bench), which design should you use? Explain briefly.
f) Calculate the se of the mean for P=2 plants and C=1 container.
g) Calculate the variance between containers when two plants are measured per container by squaring your answer from (f). Is this the same as the variance component between containers given at the begining of the problem (sigma^2(containers) = 0.49)? Explain why or why not.

Please do problem 3 by hand.

3) The following data come from a study of rice yields in response to elevated atmospheric CO2. The data considered here are only from the 'ambient CO2' treatment. This treatment is applied to 8 fields. Grain yield is measured on three subsamples per field. The response is log transformed yield (g/m2). There were a total of 24 observations. The goal is to estimate the variance component for fields, sigma^2_F, and the variance components for subsamples, sigma^2_S. Here is part of the ANOVA table:

Source	d.f.	MS	Expected MS
Fields		0.0321	sigma^2_S + 3 sigma^2_F
Error		0.0012	sigma^2_S

a) Fill in the d.f. for fields and for subsamples.
b) Estimate the variance components, sigma^2_F and sigma^2_S.
c) Test the hypothesis that the field variance component (sigma^2_F) is zero.
d) Without doing any calculations, will increasing the number of subsamples per field to 6 be useful, in the sense of substantially decreasing the s.e. of the mean yield? Briefly explain.