For this question you will need to download the data cloth.rda and read it into R. Recall this is done using the read.table function.

Description

 The data comprise lengths of cloth samples and the numbers of flaws found in them. 35 observations on 2 variables: x The length of the roll of cloth. y The number of flaws found in the roll. 

(a) Explore the data: produce appropriate graphical summaries to describe the relationship between x and y. (You need to submit (readable) plot(s) to get full marks)

(b) Suppose that we decide to model the Yis as independent, but not identical, Poisson distributed random variables, i.e.

Yi | xi, Pois(xi ) i = 1,...32

Explain why this is a reasonable model.

(c) Derive an expression for the log-likelihood for .

(d) Write a function that finds the log likelihood and plot this function. (Submit a plot to get full marks)

(e) Find the MLE of .

(f) Find a deviance based 95 % CI for (you could do this numerically using

uniroot, or use R to plot the deviance and do by eye).