¦BASIC MPH WRITTEN EXAMINATION IN BIOSTATISTICSÈ ¦PART IÈ ¦May 8, 1992: 9:30Û,211:30 a.m.È ‚INSTRUCTIONSƒ: ™a)™This is an open book examination. ™b)™Answer any ‚twoƒ questions during the two-hour time period. ™c)™Put the answers to different questions on separate sets of paper. ™d)™Put your code letter, ‚notƒ your name, on each page. þì™e)™Return the examination with a signed statement of the honor pledge on a page separate from þìîyour answers. þìî™f)™You are required to answer only what is asked in the questions, not to tell all you know about þìîthe topics. þèìî1.™Suppose that the intensity" X of an earthquake is assumed to have the density function ¦f¬XÈ(x)ÛV22k«-2È(kÛ,2x) , 0Û<8xÛ<8k ,È ™where k is a known positive constant. Let ¦YÛV210X/kÈ þì™be the corresponding ‚Richter Scale Valueƒ, 0Û<8YÛ<810 . 8 pts.™a)™ For r a non-negative integer, find a ‚general expressionƒ for Ûm;«Ûm2É°rÈÛV2E(X«rÈ) , and then use this þî result to find E(X) and V(X). þî 8 pts.™b)™ What is the probability of an earthquake recording a value of at least 5 on the Richter þî Scale? þî 9 pts.™c)™ Find E(Y) and V(Y). 2.™Suppose that X¬1È, X¬2È, Û\5, X¬nÈ are n independent random variables with E(X¬iÈ)ÛV2Ûm; , V(X¬iÈ)ÛV2Ûs;«2È, þîiÛV21, 2, Û\5, n. Further, suppose that Y¬1È, Y¬2È, Û\5, Y¬nÈ are a set of n correlated random variables (but independent of the X¬iÈ's), where E(Y¬iÈ)ÛV2Ûm; , V(Y¬iÈ)ÛV2Ûs;«2È, and where corr(Y¬iÈ, Y¬jÈ)ÛV2Ûr; for every iÛW2j. þî 10 pts.™a)™ Prove that the correlation Ûr;Û;2Û,21/(nÛ,21) by examining VÛB0 Û!0µnɶi=1ÈY¬iÈÛK0 . þì15 pts.™b)™ Consider the estimator T of Ûm;, where ¦TÛV2(Ûa;¤XÊÛ+2Ûb;¤YÊ) .È þì™™ Find values of Ûa; and Ûb; such that E(T)ÛV2Ûm; and V(T) is a minimum. 3.™For SRS (n of N) samples, the population proportion (P) with an attribute can be estimated by þîthe sample proportion (p) with the same attribute. Suppose now that we estimate the population percentage with the attribute, P«*ÈÛV2100P, using the corresponding sample percentage, p«*ÈÛV2100p. þèî9 pts.™a)™ Show that the true variance of the sampling distribution for p«*È is ¦Var(p«*È)ÛV2ºNÛ,2nÂNÛ,21Ë ºP«*È(100Û,2P«*È)ÂnË .È 9 pts.™b)™ In a population with NÛV210,000 and P«*ÈÛV230, how large a sample would be needed for þî the standard error of p«*È to be 5? þî 7 pts.™c)™ If cluster sampling with a design effect (Deff) of 1.5 were used to estimate P«*È in the þî same population, how large a sample would be needed to produce a standard error of 5 for p«*È (i.e., the same standard error for p«*È as in part b)? HINT:™Recall that the true variance of the sample mean (¤yÊ) in SRS (n of N) samples þî is þî ¦Var(¤yÊ) ÛV2 º1Û,2fÂnË S«2È ÛV2 º1Û,2fÂnË Û!0µNɶi=1È(Y¬iÈÛ,2¤YÊ)«2ÈÛ-0(NÛ,21) ,È where fÛV2n/N .