5.5 7.2 7.3 6.3 8.1 6.8 7.0 7.0 6.8 6.5 7.0 6.3 7.9 9.0 8.6 8.7 7.8 9.7 7.4 7.7 9.7 8.0 7.7 11.6 11.3 11.8 10.7 The data below give accompanying strength observations for cylinders. 6.3 5.8 7.8 7.1 7.2 9.2 6.6 8.3 7.0 8.7 7.4 8.1 7.4 8.5 8.9 9.8 9.7 14.1 12.6 11.6 Prior to obtaining data, denote the beam strengths by X,, ..., X and the cylinder strengths by Y, .., Y. Suppose that the X,'s constitute a random sample from a distribution with mean and standard deviation o, and that the Y's form random sample (independent of the X's) from another distribution with mean u, and standard deviation o,. (a) Use rules of expected value to show that X - Y is an unbiased estimator of u, - H,. O E(X - Y) = - E(Y) = H, - H2 O EX - Y) = V E(X) - E(Y) = H1 - #2 O E(X – ) = E(X) – E(Y) H1- #2 nm O E(X - Y) = nm E(X) - E(Y - H1- 42 O EX - ) = E(X) – E(Y) = H1 -2 %3D Calculate the estimate (in MPa) for the given data. (Round your answer to three decimal places.) MPa (b) Use rules of variance to obtain an expression for the variance and standard deviation (standard error) of the estimator in part (a). v(X - ) = v(X) + V(Y) = a + o,2 Identify the next step in this rule from the options below. 02 O vx - ) = n, n2