*Here is the link for Rdata file, incase if you need*: https://drive.google.com/drive/folders/1H_EfjMUd6ehOqhZ43DtHp51Gz7_yg_8-?usp=sharing

Question 2 (13 Marks): use of a software is not recommended

Let X = [X1, X2]' ~ Nz(u, E) be a bivariate normal vector, where A = = 1 2. Let & be d' ( X , / ) = ( X - 1 ) " E- (X -/). (a) (3 Marks) Obtain the distribution of E-/"(X-p). Show your working.(b) (3 Marks) Let ([Xn, Xia] , [X2, Xzz]] , . .., [Xni, Xna]? ) be n inde- pendent copies of X and X = [X1, X2]' be the vector of averages X1 = >Xa and X2 = > Xiz. i=1 1= 1 Let F(r) be the cumulative distribution function (CDF) of nd(X, /). Identify F(x) including its parameter(s). Give your reasoning. (c) (3 Marks) Let the significance level o = 0.1 and F(4.61) =1-o, where F(r) is defined in (b). Suppose that E is known. The sample of size n = 4 is randomly sampled from the distribution of X and the sample mean vector x = [0.3, 0]". Let Ho : # = [0, 0]" and Hi : # * [0, 0]. Determine the interval for all c for which the hypothesis Ho will not be rejected at the significance level o = 0.1 (d) (2 Marks) Now suppose that E is unknown. What is the distribution of nd' (X, /) ? (e) (2 Marks) In this framework, what is the decision rule for rejecting the null hypothesis Ho : / = [0, 0]