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The following data arose from a randomized block design, where factor B is the blocking variable…

The following data arose from a randomized block design, where factor B is the blocking variable and corresponds to plots of land on which cotton is planted. Each plot was divided into five subplots, and different concentrations of fertilizer were applied to each, with the response being a strength measurement of the cotton harvested. There were three blocks and five different concentrations of fertilizer. Note that there is only one observation for each block and concentration combination. Further discussion of these data can be found in Experimental Design, 2nd ed., by W. G. Cochran and G. M. Cox (John Wiley & Sons, New York, 1957, pp. 107–108). Suppose we assume the normal regression model with two categorical predictors.

The following data arose from a randomized block design, where factor B is the blocking variable...

(a) Construct the ANOVA table for testing for no effect due to fertilizer and which also removes the variation due to the blocking variable.

(b) Beyond the usual assumptions that we are concerned about, what additional assumption is necessary for this analysis?

 (c) Actually, the factor A is a quantitative variable. If we were to take this into account by fitting a model that had the same slope for each block but possibly different intercepts, then what benefit would be gained? (d) Carry out the analysis suggested in part (c) and assess whether or not this model makes sense for these data.