Question. Provide answers to the questions below. There are four questions.

1.The U.S. economic data are thought to be among the best in the world. Given what you learned in the Clear It Up "How do statisticians measure GDP?" does this surprise you? Or does this simply reflect the complexity of a modern economy?

The ordered remission times (in weeks) of 20 leukaemia patients are given in the table: 4 5 8 8 8 11 11 12 12 15 17 22 23 Suppose the remission times can be regarded as a random sample from an exponential distribution with density: f (X;1) = Me-Ax (i) (a) Determine the maximum likelihood estimator 1 of 1. (b) Obtain the large-sample approximate variance of 2. (c) Hence calculate an approximate 95% confidence interval for 1. (ii) Using the fact that 2AnX has a _2,, distribution, obtain an exact 95% confidence interval for 1, and comment briefly on how it compares with your interval in (i)(c).Calculate a 95% confidence interval for the average height of 10-year-old children, assuming that heights have a N(w,o') distribution (where # and o are unknown), based on a random sample of 5 children whose heights are: 124cm, 122cm, 130cm, 125cm and 132cm.Suppose that unbiased estimators X, and X, of a parameter O have been determined by two independent methods, and suppose that var(X ) = o and that var(X, ) =00 where $> 0. Let Y be the combination given by Y=a X] + BX,, where a and / denote non- negative weights. (i) Find the relationship satisfied by o and / so that Y is also an unbiased estimator of 0. (ii) Determine the variance of Y in terms of o and o if, additionally, the weights are chosen such that the variance of Y is a minimum