Solve all the questions well

1 The time taken to process simple home insurance claims has a mean of 20 mins and a standard deviation of 5 mins. Stating clearly any assumptions that you make, calculate the probability that: (1) the sample mean of the times to process 5 claims is less than 15 mins [2] the sample mean of the times to process 50 claims is greater than 22 mins [2] (iii) the sample variance of the time to process 5 claims is greater than 6.65 mins [2] (iv) the sample standard deviation of the time to process 30 claims is less than 7 mins [2] (v) both (i) and (iii) occur for the same sample of 5 claims. [1] [Total 9] 2 A statistician suggests that, since a f variable with k degrees of freedom is symmetrical with vle mean 0 and variance K K -2 for k > 2, one can approximate the distribution using the normal variable N O k K - 2 Use this to obtain an approximation for the upper 5% percentage points for a t variable with: (a) 4 degrees of freedom, and (b) 40 degrees of freedom. [2] (ii) Compare your answers with the exact values from tables and comment briefly on the result. [2] [Total 4]8 A random sample of eight observations from a distribution are given below: 4.8 7.6 1.2 3.5 2.9 0.8 0.5 2.3 (1) Derive the method of moments estimates for: (a) 1 from an Exp()) distribution (b) from a x, distribution. Derive the method of moments estimators for: (a) k and p from a Type 2 negative binomial distribution (b) # and of from a lognormal distribution. 9 Show that the likelihood that an observation from a Poisson (1) distribution takes an odd value (ie 1, 3, 5,.) is 4(1-e-24). 10 A discrete random variable has a probability function given by: X 2 4 5 P(X = x) 1 + 20 (1) Give the range of possible values for the unknown parameter a . A random sample of 30 observations gave respective frequencies of 7, 6 and 17. Calculate the method of moments estimate of o . (iii) Write down an expression for the likelihood of these data and hence show that the maximum likelihood estimate