Solve the below questions with precision

The variables X. X2. .... Xin give the size (in units of $100) of each of 40 claims in a random sample of claims arising from damage to cars by vandals. The size of each claim is assumed to follow a gamma distribution with parameters a = 4 and 2. -0.5 and each is independent of all others, Let X= = ) X; he the random variable 40 giving the mean size of such a sample. (i) State the approximate sampling distribution of X and determine its parameters. 121 (ii) Determine approximately the median of X . [Total 3] A survey is undertaken to investigate the frequency of motor accidents at a certain intersection. It is assumed that, independently for each week, the number of accidents follows a Poisson distribution with mean 2 In a single week of observation two accidents occur. Determine a 95% confidence interval for A, using tables of "Probabilities for the Poisson distribution". [3] (Wi) In an observation period of 30 weeks on average of 2.4 accidents is recorded. Determine a 95% confidence interval for 2., using a normal approximation. [3] (iii) Comment on your answers in parts (i) and (ii) above. [1] [Total 7] A random sample of 25 recent claim amounts in a general insurance context is taken from a population that you may assume is normally distributed, In units of $1,000, the sample mean is X =9.416 and the sample standard deviation is $ = 2.105. Calculate a 95% one-sided upper confidence limit (that is, the upper limit k of a confidence interval of the form (0,k)) for the standard deviation of the claim amounts in the population. 151In a genetic plant breeding experiment a total of 1,500 plants were categorised into one of four classes (labelled A. B. C' and D)) with the following results: class: A C D frequency: 1071 62 68 299 A genetic model specifies that the probability that an individual plant belongs to each class is given by: class: A B C probability: 1 (2+0 ) 4 where 0 is an unknown parameter such that 0