The article "Measurements of the Thermal Conductivity and Thermal Diffusivity of Polymer Melts with the Short-Hot-Wire Method" (X. Zhang, W. Hendro, et al., International Journal of Thermophysics, 2002:1077-1090) reports measurements of the thermal conductivity (in W· m-1 . K') and diffusivity of several polymers at several temperatures (in 1000°C). The following table presents results for the thermal conductivity of polycarbonate. Cond. Temp. 0.236 0.028 0.241 0.038 0.244 0.061 0.251 0.083 0.259 0.107 0.257 0.119 0.257 0.130 0.261 0.146 0.254 0.159 0.256 0.169 0.251 0.181 0.249 0.204 0.249 0.215 0.230 0.225 0.230 0.237 0.228 0.248 Denoting conductivity by y and temperature by x, fit the linear model y = Bo + Bix + ɛ. a. For each coefficient, test the hypothesis that the coefficient is equal to 0. b. Fit the quadratic model y = Bo + Bix + Bzx? + ɛ. For each coefficient, test the Page 661 Fit the cubic model y = Bo + Bix + Bx + Bax + ɛ. For each coefficient, test the %3D hypothesis that the coefficient is equal to 0. C. hypothesis that the coefficient is equal to 0. d. Fit the quartic model y = Bo + B1x + Bzx? + Bzx³ + Bx* + ɛ. For each coefficient, test the hypothesis that the coefficient is equal to 0. e. Which of the models in parts (a) through (d) is the most appropriate? Explain. f. Using the most appropriate model, estimate the conductivity at a temperature of 120°C.

(11) Let X, Y be independent random variables with X ~ Exp(A1) and Y - Erp(A2). Find the joint density of Z = X +Y,W = X – Y. The following answers are provided. Read the answers carefully before making your choice. (a) The inverse transformation is I = v (z, w) = (z + w)/2, y = v2(z, w) = (z – w)/2. The jacobian of the transformation is the following determinant J = = The bivariate density of Z, W is g(z, w) = A1A2e 1(=+w)/2e¬A2(z¬w)/2 |J], if 쁘 > 0, > 0. This is equivalent to saying that if – z 0, " > 0. This is equivalent to saying that if – z 0, 를 >0. This is equivalent to saying that if - 00 0," > 0. This is equivalent to saying that if – z 0, > 0. This is equivalent to saying that if – z< w < z, 0 < z g(z, w) = if otherwise, (a) (b) (c) (d) (e) N/A (vii- Select One)

Let X1, .., Xn be a random sample from a population with 0 unknown and given by the density = * itz>0 { e f(x; 8) if x (Ə ln(f(X;0) nE

Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y. (If an answer doesn't exist enter DNE.) Sxe-x(1 + y) x 2 0 and y 2 0 f(x, y) = otherwise Determine E(XY). E(XY) = Calculate Cov(X, Y) and p. Cov(X, Y) = P =