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Bank robberies in various countries are assumed to occur according to Poisson processes with rates that vary from year to year. It was reported that the number of robberies in a particular country in a specific year was 123. The number of robberies in a different country in the same year was 111. It can be assumed that each robbery is an independent event and that robberies occur independently in the two countries. Determine an approximate 90% confidence interval for the difference between the true yearly robbery rates in the two countries. [6] A survey is undertaken to investigate the proportion p of an adult population that support a certain government policy. A random sample of 100 adults is taken and contains 30 who support the policy Calculate an approximate 95% confidence interval for p. (2] (ii) Comment on the validity of the interval obtained in part (i). [1] A different sample of 1,000 adults is taken and it contains 300 who support the policy. (iii) Explain how the width of a 95% confidence interval for p in this case will compare to the width of the interval in part (i). without performing any calculations. [1] [Total 4] A regulator wishes to inspect a sample of an insurer's claims. The insurer estimates that 10% of policies have had one claim in the last year and no policies had more than one claim. All policies are assumed to he independent (1) Determine the number of policies that the regulator would expect to examine before finding 5 claims. [1] On inspecting the sample claims, the regulator finds that actual payments exceeded initial estimates by the following amounts: (35 $120 $48 (200 676 (ii) Find the mean and variance of these extra amounts. [3] It is assumed that these amounts follow a gamma distribution with parameters a and A. (iii) Estimate these parameters using the method of moments. [3] [Total 7]