Download Graduate Exam in Applied Statistics at University of Maryland, January 2004 and more Exams Statistics in PDF only on Docsity! DEPARTMENT OF MATHEMATICS UNIVERSITY OF MARYLAND GRADUATE WRITTEN EXAMINATION JANUARY, 2004 Applied Statistics (Ph.D. Version) Instructions to the Student a. Answer all six questions. Each will be graded from 0 to 10. b. Use a different booklet for each question. Write the problem number and your code number (NOT YOUR NAME) on the outside cover. c. Keep scratch work on separate pages in the same booklet. d. If you use a “well known” theorem in your solution to any problem, it is your responsibility to make clear which theorem you are using and to justify its use. e. You may use calculators as needed. 1. Consider the one-way ANOVA model Yij = θi + ij, i = 1, . . . , I, j = 1, . . . , ni where the ij are independent N(0, σ 2) errors. Let a1, ..., ak be fixed con- stants, chosen before the data were observed. (a) At level α, test H0: ∑k i=1 aiθi = 0 versus H1: ∑k i=1 aiθi 6= 0. (b) Specialize your test to compare treatment 1 to the average of treatments 2 and 3. (c) Would your answer to (a) change if the constants a1, . . . , ak had been chosen after looking at the data? If so, how? If not, why not? 1 2. In a study of which of three machines is preferable for an industrial process, six employees were selected at random and each employee operated each machine three times. The result was Yijk, the output of the kth run of employee j on machine i, for i = 1, 2, 3, j = 1, . . . , 6, k = 1, 2, 3. It was assumed that the following linear mixed model described the data: Yijk = µi + bj + cij + eijk, where µ1, µ2, µ3 are fixed but unknown parameters, bj ∼ N(0, σ2b ), cij ∼ N(0, σ2c ), and eijk N(0, σ 2 e). The usual ANOVA table for a balanced two way layout with replication was calculated, yielding the following results. Source d.f. Mean Square E(MS) Machines 877.63 Employees 248.38 Interaction 42.65 Residual 0.93 (a) Find the missing d.f. and E(MS) values, indicating any functions of the fixed parameters implicitly by notation such as Q(µ1, µ2, µ3). (b) How can one test for differences among machines? (c) Find a point estimate µ̂i for µi and find a point estimate for Var µ̂i. 3. A surveyor makes a single measurement on each of the angles β1, β2, β3 of an area that has the shape of a triangle, and obtains unbiased measure- ments Y1, Y2, Y3 (in radians). It is known that the measurements have a common but unknown variance σ2. (a) Find the least squares estimates of the unknown angles and their vari- ances. (b) Is it possible to obtain an unbiased estimate for σ2? If so, find it, and if not, explain why not. (c) Suppose it is known in advance that β1 = β2. Find the least squares estimates of the unknown angles and their variances in this case. 2
