Please address the following questions effectively

An insurance company is investigating past data for two household claims assessors, A and B, used by the company. In particular claims resulting from similar types of water damage were extracted. The following table shows the assessors' initial estimates of the cost (in units of (100) of meeting cach claim. A: 4.6 6.6 2.8 5.8 2.1 5.2 5.9 34 7.8 3.5 1.6 8.6 2.7 B: 5.7 34 4.7 3.6 6.5 3.3 3.8 24 7.0 4.0 4.4 for the A data: " = 13, Ex = 60.6 and Er2 = 340.92 for the A data: no = 11, Er = 48.8 and Er2 = 236.80) (i) Draw a suitable diagram to represent these data so that the initial estimates of the two assessors can be compared. [2] (ii) You are required to perform an appropriate test to compare the means of the assessors' initial estimates for this type of water damage. (a) State your hypotheses clearly. (b) Use your diagram in part (i) to comment briefly on the validity of your test. (c) Calculate your test statistic and specify the resulting P-value. (d) State your conclusion clearly. [10] (tii) You are required to perform an appropriate test to compare the variances of the assessors' initial estimates. (a) State your hypotheses clearly. (b) Use your diagram in part (i) to comment briefly on the validity of your test. () Calculate your test statistic and specify the resulting P-value. (d) State your conclusion clearly and hence comment further on the validity of your test in part (ii). [7] (iv) Use your answers in parts (i) to (iii) to comment on the overall comparison of the two assessors as regards their initial estimates for this type of water damage. [1] [Total 20]Consider the ANOVA model Y, =utt; te;,/ =1,....4, /=1,..., n;, where Y, is the/th sum insured for company , n; is the number of responses for company , e - N(0, ) are independent errors, and )_ nit; = 0. (i) Calculate estimates of the parameters , and t, i =1, 2,3,4. [3] (ii) Test the hypothesis that there are no differences in the means of the sumns insured under such policies by the four companies. [5] [Total 8] The number of claims. X, which arise in a year on each policy of a particular class is to be modelled as a Poisson random variable with mean 2. Let X - (X, X2, ..., X,) bc a random sample of size a from the distribution of Y, and let X =-)X Suppose that it is required to estimate /, the mean number of claims on a policy. (i) Show that 2 , the maximum likelihood estimator of 2, is given by A = Y . [3] (ii) Derive the Cramer-Rao lower bound (CRIb) for the variance of unbiased estimators of A. [4] (iii) (a) Show that 2. is unbiased for A and that it attains the CRIb. (b) Explain clearly why, in the case that " is large, the distribution of I can be approximated by [3] (iv) (a) Show that, in the case n = 100, an approximate 95% confidence interval for ) is given by F=0.196-/5 . (b) Evaluate the confidence interval in (iv)(a) based on a sample with the following composition: observation 0 1 2 3 + frequency 11 28 19 28 [G] [Total 16]