An article by J.J. Pignatiello, Jr. And J.S. Ramberg in the Journal of Quality Technology, (Vol. 17, 1985, pp. 198-206) describes the use of a replicated fractional factorial to investigate the effects of five factors on the free height of leaf springs used in an automotive application. The factors are A = furnace temperature, B = heating time, C = transfer time, D = hold down time, and E = quench oil temperature. The data are shown in Table P81.

Table P8.1 A B C D E Free Height - 7.78 7.78 7.81 + - + - 8.15 8.18 7.88 + + 7.50 7.56 7.50 + + - 7.59 7.56 7.75 7.88 - - + + 7.54 8.00 + - + 7.69 8.09 8.06 7.44 + - 7.56 7.52 7.56 7.81 7.69 + + + + - - - + 7.50 7.25 7.12 + - + + 7.88 7.88 7.44 - + + + 7.50 7.56 7.50 + + - + 7.63 7.75 7.56 - + + + 7.32 7.44 7.44 + - 7.69 + - + 7.56 7.62 + + - + 7.18 7.18 7.25 + + + + + 7.81 7.50 7.59Month Chevy Malibu Ford Fusion Hyundai Sonata Honda Accord Toyota Camry VW Passat January 22,161 21,753 21,456 19,791 21,435 20,121 February 18,724 19,835 20,442 21,322 20,235 19,555 March 18,384 17,007 16,163 17,631 17,339 16,456 April 19,837 17,870 15,504 14,950 14,543 14,533 May 17,547 16,597 15,473 16,250 16,177 17,325 June 16,694 16,623 14,745 15,508 15,686 17,343Let X1 and X2 be independent random variables having the standard normal distribution. Obtain the joint Lebesgue density of (Y1, Yz), where Yi = VX?+ X} and Y2 = X1/X2. Are Y1 and Y2 independent? Note. For this type of problem, we may apply the following result. Let X be a random k-vector with a Lebesgue density fx and let Y = g(X), where g is a Borel function from (R*, B* ) to (R*, B*). Let A1, . .., Am, be disjoint sets in B* such that R* - (Aj U . . . U Am,) has Lebesgue measure 0 and g on A; is one-to-one with a nonvanishing Jacobian, i.e., the determinant Det(Og(x)/Or) # 0 on Aj, j =1. .... m. Then Y has the following Lebesgue density: fy (x) = >Det (Oh, (x)/ar) | fx (h,(r)). j=1 where h, is the inverse function of g on A;, j = 1, ..., m