Problem 3. (20%) Let X1, .... Xs be iid ~ N(0, 1) and let X = : Ci_, X, be the sample mean. Let Y ~ N(0, 1) be another random variable which is independent from X1, . .., Xs. Find the distributions of the following random variables and justify your answers. (a) W = Zi=1X? (b) U = EM(X, - X)2 (c) U+ Y2 (d) v5Y/ vw (e) 2Y/ VU Problem 4. (Shear strength of beams) (20%) Consider a population which contains concrete beams. To study the shear strength of these beams, we study a random sample of size 15. The shear strength (in kN) of 15 concrete beams are the following. X X2 X3 XA X5 X6 X7 Xq X10 X11 X12 X13 X14 X15 580 400 428 825 850 875 920 550 575 750 636 360 590 735 950 The proposed statistical model is X1, ..., X15 iid ~ N(700, 1002). (a) What is the exact distribution of Z = >15 (X-700 ) 2. 100 )? Moreover, find the observation z = > , (2-700 ) based on the given data for 1, . . ., $15. (b) Compute P(Z Z z). (c) Based on the result of part b, can you tell whether the proposed statistical model X1, ..., X15 iid ~ N(700, 1002) is reasonable