¦SPECIAL MS WRITTEN EXAMINATION IN BIOSTATISTICSÈ ¦PART IÈ ¦May 8, 1992: 9:30Û,212:30 p.m.È ‚INSTRUCTIONSƒ: ™a)™This is a ‚closed bookƒ examination. ™b)™Answer any ‚threeƒ questions during the three-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. þèî1.™For iÛV21, 2, Û\5, k, let Y¬i1È, Y¬i2È, Û\5, Y¬inÈ constitute a random sample of size n from a N(Ûm;¬iÈ, Ûs;«2È) þîpopulation, where the population variance Ûs;«2È can be assumed to be known. It is of interest to estimate the parameter ¦Ûh;ÛV2Û!0µkɶi=1Èc¬iÈÛm;¬iÈ ,È ™where the c¬iÈ's are known constants. A statistician suggests using the estimator ¦Ûh;µ^ÈÛV2Û!0µkɶi=1Èc¬iȤYʬiÈ ,È where ¤YʬiÈÛV2n«-1È Û!0µnɶj=1ÈY¬ijÈ . þî13 pts.™™a)™ Using the fact that Ûh;µ^È is a linear combination of independent normal variates, construct þî an appropriate 100(1Û,2Ûa;)% confidence interval for Ûh;. Compute a 95% confidence interval for (2Ûm;¬1ÈÛ,2Ûm;¬2ÈÛ,2Ûm;¬3È) when nÛV225 , ¤Yʬ1ÈÛV27.0 , ¤Yʬ2ÈÛV24.0 , ¤Yʬ3ÈÛV26.0, and Ûs;«2ÈÛV24 . þî 12 pts.™™b)™ If Ûs;«2ÈÛV24 , what is the smallest sample size n required so that a 95% confidence interval þî for (2Ûm;¬1ÈÛ,2Ûm;¬2ÈÛ,2Ûm;¬3È) will have a width no wider than 0.50? þî 2.™Suppose that the time to failure T of a certain type of electronic component follows the þìîdistribution ¦f¬TÈ(t; Ûl;)ÛV2º1ÂÛl;Ë e«-(tÛ,2a)/Ûl;È, tÛ>8a , Ûl;Û>80 ;È þìhere, a is a known positive constant. Further, suppose that a random sample of n such electronic components is selected for testing, and the corresponding failure times T¬1È, T¬2È, Û\5, T¬nÈ are considered; in other words, T¬1È, T¬2È, Û\5, T¬nÈ constitute a random sample from f¬TÈ(t; Ûl;). þî4 pts.™™a)™ Show that the most powerful test of H¬0È: Ûl;ÛV2Ûl;¬0È versus H¬AÈ: Ûl;ÛV2Ûl;¬1È (Û>8Ûl;¬0È) can be þìî based on a rejection region of the form ¦Ûr>ÛV2{¤TÊ: ¤TÊÛ;2k} ,È þìwhere TÛV2n«-1È Û!0µnɶi=1ÈT¬iÈ . þî 2 pts.™™b)™ Is the test developed in part (a) a uniformly most powerful test for all Ûl;Û>8Ûl;¬0È? Provide þî a valid argument to support your answer. þî 7 pts.™™c)™ Assume that n is large. Develop an approximate test involving ¤TÊ of H¬0È: Ûl;ÛV2Ûl;¬0È versus þî H¬AÈ: Ûl;ÛV2Ûl;¬1È based on the standard normal distribution. Construct your test so that Ûa;ÛZ2.025. þî 12 pts.™™d)™ Again assume that Ûa;ÛV2.025 , aÛV21 , Ûl;¬0ÈÛV21 , and Ûl;¬1ÈÛV21.2. Using the test developed þî in part (c), what is the smallest sample size n«*È required to ensure that the power of the test will be at least equal to 0.84 in value? þî 3.™Jacob Cohen's book ‚Statistical Power Analysis for the Behavioral Sciencesƒ is a standard reference þîfor finding the sample size required for a study. Cohen expresses the difference between the null and alternative hypotheses in terms of effect size", where, for example, in making a normal-theory test of the null hypothesis that a correlation is zero, the effect size is the actual correlation under the alternative hypothesis, while in making a standard t-test to compare the means of two normal populations (with samples not necessarily of the same size), the effect size is the difference between the population means divided by their common standard deviation. For experimenters who are unable or unwilling to specify effect sizes numerically, Cohen defines conventional small", medium", and large" effect sizes: these are .1, .3, and .5, respectively, for the correlation test, and .2, .5, and .8 for the t-test. þî15 pts.™a)™ Express the effect size of the t-test, as Cohen defines it, as a function of the correlation þî between the response and a dummy variable distinguishing the two samples. þî  10 pts.™b)™ Hence show that Cohen's definitions of small", medium", and large" effect sizes for þî the correlation test are not consistent with his definitions for the t-test. þî Note: For uniformity of notation among answers, let the dummy variable X be the sample number (1 or 2), and let P be the proportion of the total observations which are in Sample 1, with QÛV2(1Û,2P) the proportion in Sample 2. þî4.™Consider stratified simple random sampling, where a sample of size n is chosen from a population þîof size N that has been divided into H strata, each with N¬hÈ elements. þî10 pts.™™a)™ Show that for the estimated mean difference, d¬hh«Ûm2ÈÈÛV2¤yʬhÈÛ,2¤yʬh«Ûm2ÈÈ , between any two of the þìî sampling strata (h and h«Ûm2È), ¦Var(d¬hh«Ûm2ÈÈ)ÛV2(1Û,2f¬hÈ)S«2É°hÈ/n¬hÈÛ+2(1Û,2f¬h«Ûm2ÈÈ)S«2É°h«Ûm2ÈÈ/n¬h«Ûm2ÈÈ ,È þìwhere ¤yʬhÈ is the sample mean per element, n¬hÈ is the sample size, f¬hÈÛV2n¬hÈ/N¬hÈ is the sampling rate, and S«2É°hÈ is the element variance for the h-th stratum. Comparable notation is defined for the h«Ûm2È-th stratum. [NOTE: You need not derive this from scratch. You can cite any results for SRS.] þî 8 pts.™™b)™ If NÛV210000, S«2É°hÈÛV2S«2É°h«Ûm2ÈÈÛV21000 , W¬hÈÛV2N¬hÈ/NÛV20.20 , W¬h«Ûm2ÈÈÛV20.05 , and the stratified þî sample is to be ‚proportionately allocatedƒ among strata, how large must n be to yield a standard error of 5 for d¬hh«Ûm2ÈÈ? þî 7 pts.™™c)™ Will a stratified sample of n elements following the proportionate allocation in part (b) þî be an equal probability sample? Briefly explain.