1) Consider an experiment like the barley salinity experiment, where containers are randomly assigned to one of three treatments with C containers per treatment. Measurements are taken on P plants per container. Previous studies suggest that for this experiment, the variance components are sigma^2(containers) = 0.49 and sigma^2(plants) = 6.76. 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 one treatment mean? b) If you measure P=5 plants per container and C=8 containers per treatment, what is the s.e. of one treatment mean? c) Does either design (a or b) satisfy the precision requirement (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). Since this is calculated for 1 container, it is the variance between container averages (i.e., averaging over plants). Is this the same as the variance component between containers given at the beginning of the problem (sigma^2(containers) = 0.49)? Explain why it should or should not be similar. h) Repeat f and g using P=1000 plants and C=1 container. Is this squared s.e. of the mean similar to the variance component between containers (0.49)? Explain why it should or should not be similar.

i) Calculate and report the correlation between two plants in the same container. If the correlation is less than 0.10, you are willing to consider plants as independent and ignore container effects in your analyses. Is it appropriate to consider plants as independent when you analyze data collected in this experiment? Briefly explain your answer.