(a) In an agricultural expreiment, the wheat yield, X,, from field i is believed to be normally distributed with mean z, where z is the known quantity of fertiliser spread on the field. Assuming that the yields in the different fields are independent, and that the variance is known to be 1, so that Xi ~ N(uzi, 1) for i = 1, ..., n, (i) Write down the log-likelihood for this model. (1 mark) (ii) Find the score and expected information functions for this model. (Re- member that the z; values are constants). (2 mark) (iii) Show that the MLE A = Sinn zizi. (1 mark) (iv) Show that f is an unbiased estimator. (1 mark) (v) Obtain an expression for an approximate 95% confidence interval for based on the asymptotic distribution of p. (1 mark) (vi) Obtain an expression for the deviance D(#) (in terms of ris, zis and #) under this model, and explain how you would calculate a 95% confidence interval for / based on the asymptotic distribution of D(). (Note: - you don't need to attempt to simplify beyond substituting for the MLE, stop once you have an expression that only depends on the specified) (2 marks) (b) Now consider the following alternative model, where Xi ~ N(uzi, = ) for i = 1, ..., n, so that the variance of the yield also increases with the known value of z;. Draw plots (sketches are sufficient) of how the data (z;, r;), i = 1, . ..,n are likely to appear in the models in each the two models. You can assume any positive value for / but your plot should illustrate the behaviour for all z; > 0. Explain the key difference between the behaviour under the two models. (2 marks)