(a)Let 1,2,...,and 1,2,...,be two independent random samples from the normal distributions (,5/2)and (,5/2), respectively. (i)Construct a 100(1)%confidence interval for 3.(6marks)(ii)A researcherclaims that she is 98% confident that the reaction time from chemicalis three timesthat of chemical. Two random samples, each of size 16, were collected as follows.Chemical: 10, 9, 6,5,11,7,9,5,8,12, 6,7,5,12,4,8Chemical: 28, 24, 17, 19, 30,21,28,22,36,18,20,33,16,23,24,29Using the results in part (a)(i), determine whether the claim by the researcheris justifiable.(6marks)(b)Let 1,2,...,be a random sample from a distribution with pdfgiven by()={1232/,0

(a) Let2'5f1,}1'2,...,X,,l and Y1, Y2, Y\" be two independent random samples om the normal distributions N 02X, EXZ) and N (Jay, 5(2), respectively. (i) Construct a 100(1 a}% condence interval for 311),- try. (6 marks) (ii) A researcher claims that she is 93% confident that the reaction time from chemical Y is three times that of chemical X. Two random samples, each of size 16, were collected as follows. Chemical X: 10, 9, s, s, 11, r, 9, s, a, 12, s, 3, s, 12, 4, a Chemical Y: 23, 24, 11', 19, 30, 21, 2s, 22, 3s, 1s, 20, 33, 15, 23, 24, 29 Using the results in part {a)[i), determine whether the claim by the researcher is justiable. (6 marks) (13) Let X1, X2, , X71 be a random sample from a distribution with pdf given by D, elsewhere. {i} Find the method of moments estimator for 6. (T marks) (ii) Given that ELY?) = 1232, determine whether this estimator in part (b){i) is a consistent estimator for 6. (6 marks) [Total : 25 marks]