Please address these statistical questions effectively. Show clear derivations

Consider a random sample X], ..., X, from a normal distribution with expectation and variance ". The sample size is n =9. We define the statistics X = - ) X; and $2 =. Six; - x). n - i=1 Determine the following probabilities: [2] (ii) PIX> ul s' > 62] [2] (iii) P[X-[1>0] [2] (iv) P[X-H >S] [2] [Total 8] A random sample of size 49 from a normal distribution gives a 99% confidence interval for the population mean as (30, 50). Determine a 90% confidence interval for the population mean based on this information [5] A sample is drawn from the normal distribution with mean u and variance o. Denote the sampling variance by S'. (i) Show that the expected value of $2 is of, using the sampling distribution of $2 [2] Due to a spreadsheet error a scientist accidentally uses the following to estimate the variance G' = =(Ex; - RT) (ii) (a) Determine the bias of G2 as an estimator of o2. [2] (b) Comment on how the bias behaves as a gets large. [1] [Total 5]We consider a random sample X, ..., X,, from a normal distribution with expectation F[ X; ]= =13 and variance V( X,) = 0- for all i (1) Derive an estimator 0 for the unknown parameter O using the method of moments. (ii) Estimate d for a sample of size 200 for which > x; =900 using the estimator derived in part (i). [1] (iii) Comment on the suitability of the estimator in part (i). In particular, consider the two cases \\ X,