1 HOMEWORK 4 STAT 305B Spring 2016 Due 3/10(R) Name_________________________________________ PROBLEM 1 (35pts) [See book Problem 5-55 on p.183] In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Let X and Y denote (the act of measuring) the milliliters of 120 5 2 XY . , acid and base, respsectively, needed for equivalence. Assume that ( X , Y ) ~ N 2 100 XY 2 (a)(5pts) For XY 0.6 find XY . Solution: (b)(5pts) Find Pr[ X 116 ] . Solution: (c)(10pts) Find Pr[ X 116 | Y 102] . Solution: (d)(5pts) Find Pr[ X 118 Y 96] . To this end, write a Matlab code that uses the Matlab command 'mvncdf'. Include your code HERE. Solution: (e)(10pts) Find Pr[118 X 122 96 Y 104] . To this end, (i) show the event as a shaded area in an x-y plot. Then (ii) write a Matlab code that uses the Matlab command 'mvncdf'. Include your code HERE. Solution: Figure 1(e) Plot showing the event. 2 PROBLEM 2 (40pts) In relation to PROBLEM 1, assume that, in truth, ( X , Y ) has the bivariate normal distribution given in PROBLEM 1. (a)(5pts) Write a Matlab code that will simulate n 100 measurements of ( X , Y ) , and then produce a scatter plot of the resulting data. Solution: [See code under 2(a) in Appendix] Figure 2(a) Scatter plot for n 100 (b)(5pts) From your data in (a) obtain an estimate of XY 0.6 . Then compute the percent error of your estimate. Solution: (c)(10pts) Use your data to obtain a linear prediction model Y mX b . The overlay this model on a scatter plot of the data. Aslo, give the numerical values of your m and b . Solution: [See code under 2(c) in Appendix] m = _______ b =________ Figure 2(c) Scatter plot for n 100 and prediction line. (d)(10pts) Use 104 simulations to arrive at the pdf for the estimator m . Also, give the numerical values of m and m . Solution: [See code under 2(d) in Appendix] Figure 2(d) PDF for m using sample size n 100 . (e)(5pts) Compute the true value of m. Then comment on how it compares to your value for m in (d). Solution: (f)(5pts) Assume that m ~ N (0.24 , 0.07 2 ) . Find Pr[0.23 m 0.25] Solution: 3 PROBLEM 3 (25pts) A press fit refers to fitting two parts together, so as to ensure that they will not slip relative to one another. From https://en.wikipedia.org/wiki/Interference_fit we have the following example: A 10 mm (0.394 in) shaft made of 303 stainless steel will form a tight fit with allowance of 3-10 m (0.0001-0.0003 in). A slip fit can be formed when the bore diameter is 12-20 m (0.0005-0.0008 in) wider than the rod. Suppose that shafts are machined to conform to a shaft diameter distribution S ~ N ( S 10mm , S ) and the bores are machined to conform to a bore diameter distribution B ~ N ( B 10.0065mm , B ) . Denote the clearance between the shaft and bore as C B S (a)(3pts) Use the linearity of E(*) to arrive at C in units of m . Solution: 2 2 (b)(2pts) Suppose we require that C 1 m . From this constraint, obtain an expression for B as a function of S . Solution: (c)(5pts) Suppose that the shaft machining cost as a function of S is T ( S ) 1 / S ($) and the bore machining cost as a function of B is T ( B ) 3 / B ($). Then the total cost of a shaft/bore unit is Ttot T ( S ) T ( B ) (1 / S ) (3 / B ) . From this cost function, along with the constraint you found in (b), one can use the method of Lagrange multipliers [ https://en.wikipedia.org/wiki/Lagrange_multiplier ] to show that the total cost will be minimized for S 0.5700 m and B 0.8218 m . Rather than using this elegant mathematical method, proceed to arrive at these numbers as follows: (i) define the Matlab array S 0.1 : 0.001 : 0.9 (ii) create your B array per your constraint in (b) (iii) compute your Ttot array per the above equation (iv) use the command [Tmin , k 0 ] min(Ttot ) to find the minimum cost and the associated array index, and (v) recover your values of S ( k 0 ) and B ( k 0 ) . Include your code and answers HERE. Solution: min(Ttot ) = ______ ; S = ______ ; B = _______ Code: (d)(10pts) Suppose that one of your colleagues claims that the company has been using S B 1 m for ages, and it is still in business. Hence, to change to the tighter specifications you are suggesting in (c) is simply a waste of money. In this part compute the probability that the clearance will not be in the range 3 10 m for both his specs. and yours. Solution: (e)(5pts) Compute the total cost of a shaft/bore pair using his numbers. Then use your answers in (c) and (d) to compute the anticipated cost of lost units in a run of 10,000 bore/shaft pairs of using your specs. and his. Solution: 4 Matlab Code