GenAG, Inc., a genetics engineering laboratory specializing in the production of better seed varieties for commercial agriculture, is analyzing the yield response to fertilizer application for a new variety of overlineley that it has developed. GenAg has planted 40 acres of the new overlineley variety and has applied a different fixed level of fertilizer to each one-acre plot. In all other respects the cultivation of the crop was identical. The GenAg scientists maintain that the relationship between observed levels of yield, in bushels per acre, and the level of fertilizer applied, in pounds per acre, will be a quadratic relationship as \(Y_{j}=\beta_{0}+\beta_{1} f_{j}+\beta_{2} f_{j}^{2}+V_{j}\), where \(f_{j}\) is the level of fertilizer applied to the \(j\) th one-acre plot, the \(\beta^{\prime}\) s are fixed parameters, and the \(V_{j}^{\prime}\) s are iid random variables with some continuous probability density function for which \(\mathrm{E} V_{j}=0\) and \(\operatorname{var}\left(V_{j}ight)=\sigma^{2}\).

(a) Given GenAg's assumptions, is \(\left(Y_{1}, \ldots, Y_{40}ight)\) a random sample from a population distribution or more general random sampling? Explain.

(b) Express the mean and variance of the sample mean \(\bar{Y}_{40}\) as a function of the parameters and \(f_{j}\) variables. If the sample size could be increased without bound, would \(\bar{Y}_{n}\) converge in probability to some constant? Explain.

(c) Is it true that \(\left(\bar{Y}_{n}-\left(\beta_{0}+\beta_{1} n^{-1} \sum_{i=1}^{n} f_{j}+\beta_{2} n^{-1} \sum_{i=1}^{n} f_{j}^{2}ight)ight)\) \(\xrightarrow{\mathrm{p}} 0\) ? If so, interpret the meaning of this result. Based on your analysis to this point, does it appear that an outcome of \(\bar{Y}_{40}\) will produce a meaningful estimate of any characteristic of the yield process?

(d) Suppose that the 40 one-acre plots were all contiguous on a given 40 acre plot of land. Might there be reasons for questioning the assumption that the \(V_{j}\) 's are iid? What would the outcome of \(V_{j}\) represent in GenAg's representation of the yield process? Presuming that the \(V_{j}^{\prime}\) s were not iid, would this change your answer to (a)?