SPECIAL MS WRITTEN EXAMINATION PART I MAY 12, 1994: 9:00 AM - 1:00 PM INSTRUCTIONS: a)This is a closed book examination. b)Answer any three questions during the four-hour time period. c)Put the answers to different questions on separate sets of paper. d)Put your code letter, not your name, on each page. e)Return the examination with a signed statement of the honor pledge on a page separate from your answers. f)You are required to answer only what is asked in the questions, not to tell all you know about the topics. 1.Let X1, X2, ..., Xn be a random sample from a N(m;, 1) population. Suppose that it  is desired to test H0: m; = 0 versus H1: m; = m;1 > 0, where m;1 is a known positive  number. 4 pts. a) Show that the most powerful test of size a; for testing H0 versus H1   involves a rejection region R of the form  R = {X_ : X_ ;2 k1,2a;} ,   where X_ = 1n S;nȶi=1 Xi and where k1,2a; is chosen so that  pr(X_ ;2 k1,2a; |8 H0) = a;. 4 pts. b) For a; = 0.025, what is the value of k.975 when n = 100 ? 2 pts. c) Is the test developed in part (a) a uniformly most powerful (UMP) test  of size a; ? 7 pts. d)Suppose that we decide to reject H0 when X_ ;2 0.80.  What is the power of this test when m;1 = 1 and n = 25 ? 8 pts. e)As in part (d), again suppose that we decide to reject H0 when X_ ;2  0.80. What is the minimum sample size required to insure that we  reject H0: m; = 0 with a probability of at least 0.975 when,  in fact, m;1 = 1 ? 2.Suppose that the height X (in inches) for adult males in the United States is assumed to be a N(70, 16) random variable, and that the height Y (in inches) for United States adult females is assumed to have a N(65, 9) distribution. Assume that males and females are paired randomly without regard to height. 5 pts. a)In what proportion of pairs will there be a height difference of at least  six inches ? 5 pts. b)What is the probability that the male in a pair will be taller than the  female ? 5 pts. c)If the male and female heights for a pair differ by more than six inches,  what is the probability that the male is taller? 10 pts. d)Now, suppose that X [2N(m;m, 16) and that Y [2N(m;f, 9),  where m;m and m;f are unknown parameters. It is desired to make a  statistical inference about the unknown population mean height  difference (m;m ,2 m;f). Let X1, X2, ..., Xn and Y1, Y2, ..., Yn  constitute random samples of size n from each of these two normal  populations. Let X_ = n-1 S;nȶi=1Xi and Y_ = S;nȶi=1Yi.    Find two random variables    A and B, which are to be functions of (X_ ,2 Y_) and V(X_ ,2 Y_),    such that  pr{A < (m;m ,2 m;f) < B} = 0.90 . 3. A manufacturer must choose between two processes, Process 1 and Process 2,  for producing bolts. Suppose that X, the length of a bolt in centimeters, is a   continuous random variable with process-specific densities as follows:  PROCESS 1: fX(x) = 3x-4, 1 < x < r2;  PROCESS 2: fX(x) = 4x-5, 1 < x < r2.  Only bolts with lengths between 1.5 and 2.0 centimeters are acceptable. 5 pts a)Which of these two processes produces the greater proportion   of acceptable bolts? 5 pts b) What is the average length of all acceptable bolts produced by  Process 1? 8 pts c) Suppose that Process 1 produces 106 bolts per day, and that Process 2  produces 2 x 106 bolts per day. Further, at the end of each day,  suppose that all 3 x 106 bolts are put into a large container and mixed  together, thus making it impossible to tell which process produced any  particular bolt. If one bolt is selected at random from this large  container and is found to be acceptable, what is the probability that  it was produced by Process 2? 7 pts  d)If bolts are selected randomly one-at-a-time, with replacement, from  the container described in part (c), provide an exact expression for the  probability distribution of X, the number of bolts selected until exactly  10 bolts are found to be unacceptable? 4.Consider stratified simple random sampling, where a sample of size n is chosen from a population of size N that has been divided in H strata, each with Nh elements. 10 pts. a)Show that for the estimated mean difference, dhh = Y_Ȭh ,2 Y_Ȭh,  between any two of the sampling strata (h and h ),  Var(dhh) = (1 ,2fh)S2ɬh/nh + (1 ,2fh)S2ɬh/nh,  where Y,2Ȭh is the sample mean per element, nh is the sample size,  fh = nh/Nh is the sampling rate, and S2ɬh is the element variance for  the h-th stratum.  Comparable notation is defined for the h-th stratum.  [Note: You need not derive this from scratch. You can cite any results  for SRS.] 8 pts. b)If N = 10,000, S2ɬh = S2ɬh = 1,000, Wh = Nh/N = 0.20,  Wh = 0.05, and the stratified sample is to be proportionately  allocated among strata, how large must n be to yield a standard error  of 5 for dhh ? 7 pts. c)Will a stratified sample of n elements following the proportionate   allocation in part (b) be an equal probability sample? Briefly explain.