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An insurance company believes that claim amounts in a certain portfolio of policies follow a normal distribution. An analyst chose 61 policies at random which gave a sample mean of f.523 and a sample standard deviation of f$1 (i) Determine a 95% confidence interval for the mean claim amount in the portfolio. (ii) Determine a 95% confidence interval for the variance of claim amounts in the portfolio. [3] The company assumes that the true mean and standard deviation of claim amounts are the same as those in the sample. The number of claims per month for the portfolio follows a Poisson process with mean 250. (iii) Determine the approximate probability that the number of claims in a particular month exceeds 270, justifying any assumptions you use. [5] (iv) Determine the mean and standard deviation for the total annual amount of claims in the portfolio. The company has changed its loss assessment processes in order to reduce claim sizes on average, targeting a reduction of 120 compared to the current mean. It does not expect a change to the variability of claim amounts. The company intends to verify whether the target has been met by using a sample of claims to test the null hypothesis that there is no change, against a one-sided alternative hypothesis. Company policy is to perform statistical tests at a significance level of 5%. (v) Determine the smallest number of claims that would need to be sampled under the new processes for a E20 reduction to be statistically significant in the test. [5] [Total 20]Draw a separate Venn diagram for each of the following sample spaces: (1) A = (1). B = (1. 2.3) from S = {1. 2, 3, 4, 5, 6} (ii) A = (1, 4, 7), B = (3, 7, 8, 9), C= (4, 7, 8, 9) from S = (1, 2, 3, 4, 5, 6, 7, 8, 9)