I have a test coming up and these questions were given to prepare for the test! Please help! Thank you!

(a) A survey on 50 claims for a particular class of customers of a motor insurance company found out that the average cost for car damage is £700 with a standard deviation of 400. The insurance manager believes instead that the average cost is £80 bigger.

(i) Does the sample data suggest that the manager is right? Test at a 5% significance level the hypothesis that the average cost is £780. Specify: null hypothesis, alternative hypothesis, critical region and comment on the result of the test.

(b) For the same type of accidents, the number of claims per vehicle insured is known to follow a Poisson distribution with mean 0.03 per year. Using the normal approximation to the Poisson distribution, compute the probability that a group of 400 insured cars produces :

(i) Less than 10 claims per year.

(ii) More than 20 claims per year.

(c) Assuming the cost per claim is 780 with standard deviation of 400, claims’ frequency and portfolio’s size are as specified above:

(i) Compute the risk premium.

(ii) Compute the value at risk at 99.5% level for the Insurer losses.