2 Comparing two exponential populations We start by constructing an appropriate test to compare the medical costs (charges) for two different groups in the data (e.g. females and males). Assume that X 2 (X1, X2.., X\") and Y 2 (Y1, Y2.., Ym) are independent and Xi qHEXp()\\1) and Y1- Zv-zleXpOQ). Let x = (931, ..mn) and y = (y1,...,ym) be realizations of X and Y respectively. We want to perform the following test: H0 2 )\\1 = A2 VS. H1 2 A1 75 A2. (1) The generalized likelihood ratio (GLR) in this case is supL()\\1,)\\2; x,y) 90 A = . (X, y) sup L()\\1,)\\2;x,y) 9 Where 9 = {(A1,)\\2) 2 A1 > 0, A2 > 0} and 90 ={()\\1,)\\2)1)\\1 = A2 > 0}. Question 1 [2 marks] Show that A(x, y) = i Q . A1 A2 where X1 = +, 32 = mi and 30 = nn'% and state the distribution of T = 210gA(x, y). Ii 92' i=1 $i+ i=1 C911 i=1 i=1 You may assume that the Hessian of the loglikelihood under 9 is negative denite