Question 1 (20%) i. Prove that the sample mean X is an unbiased estimator of /. ii. True or false: if we generate 30 times of a 95% confidence interval for an unknown parameter 0 based on repeated random sampling (with sufficiently large sample size) of the same population that follows a normal distribution, then it is impossible that all the confidence intervals we have generated contain the true value of 0. Explain your reasoning.Question 2 (40%) Consider a random sample of n = 12 from a normally distributed population of large size has X = 25 and 82 = 8. i. Assume that you know the population has a population variance of = 16. Form a 99% confidence interval for /. ii. Assume that the population variance is unknown. Form a 99% confidence interval for u. Now consider a random sample of n = 30 from a population with unknown distribution has X = 25 and 82 = 8. iii. Assume that you know the population has a population variance of = 16. Form a 99% confidence interval for /. iv. Assume that the population variance is unknown. Form a 99% confidence interval for /.Question 3 (40%) Consider a random sample from a normally distributed population of large size. i. If the population variance o' = 35, what sample size is needed to estimate the mean within 12 with 99% confidence? ii. If instead we would like to estimate some true proportion, what sample size is needed to estimate the true proportion within 12% with 99% confidence? Now consider a random sample from a population of large size with unknown distribution. iii. If the population variance o = 50, what sample size is needed to estimate the mean within +1 with 95% confidence (using the z2.5% value)? iv. Why is it the case that such estimating process is still legitimate