Download Applied Statistics Final Exam Problems | STAT 740 and more Study notes Statistics in PDF only on Docsity!
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF MARYLAND
GRADUATE WRITTEN EXAMINATION
JANUARY, 2003
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.
- Let Yij = μ+ai +eij , i = 1,... , I, j = 1,... , J, be data from a one-way random effects ANOVA, where the ai are i.i.d. N (0, σ a^2 ) and the eij are i.i.d. N (0, σ^2 e ).
(a) Write out the usual ANOVA table and compute the expected mean squares, E(M SA) and E(M SE ).
(b) Find the distribution of the statistic F = M SA/M SE under general conditions.
(c) Find a 1 − α confidence interval for the intraclass correlation coefficient
ρ =
σ^2 a σ^2 a + σ^2 e
- A questionnaire is to be sent to a sample of high schools to find out which schools provide certain facilities, such as a computer laboratory or a course in Russian. The ith school has an enrollment of Mi students and the total number of students is K =
∑N i=1 Mi. For a certain facility, it is desired to estimate the proportion of students attending a school with the facility:
pU =
∑ w Mi ∑N i=1 Mi
where
∑ w is a sum over the schools^ with^ the facility. A sample of n schools is selected with replacement and with probability proportional to Mi. For one facility of interest, it was found from the sample that a schools had the facility.
(a) Show that ˆp = a/n is an unbiased estimator of pU and that
Var (ˆp) =
pU (1 − pU ) n
(b) Show that an unbiased estimator of Var (ˆp) is
Vˆ (ˆp) = pˆ(1^ −^ pˆ) n
[Hint: Let ti = Mi if the ith school has the facility and 0 otherwise.]
- Consider the linear model Y = Xβ + e, where X is an n × p matrix with rank p ≤ n, E(e) = 0 and Var-Cov(e) = σ^2 I. Let ξi denote the ith column of X. Suppose { βˆ 1 ,... , βˆp} is a set of least squares estimates under
the general model. Show that { βˆ 1 ,... , βˆm}, m < p are also least squares estimates under the null hypothesis H 0 : βm+1 =... = βp = 0 if and only if
ξi ⊥
∑p j=m+1 βˆj^ ξj ,^ i^ = 1,... , m.
- Independent observations Yij , i = 1, 2, j = 1, 2, were modeled as a two factor ANOVA: Yij = μ + αi + βj + eij ,
where the eij are independent random variables with a common N (0, σ^2 ) dis- tribution. Representing the data in vector form, the following decomposition was calculated:
Y 11
Y 12
Y 21
Y 22
(a) Compute the ANOVA table for the data.
(b) Compute statistics for testing the hypotheses HA: no Factor A effect and HB : no Factor B effect. What are the the distributions of the test statistics under the null hypothesis?
(c) Is there some test of whether this additive model fits this data? Would there exist a test if there had been three levels of Factor A and two levels of Factor B?
