IroY ICJ
00,0862
IB·JGT-K-TUAI
STATISTICS

Paper'!
,ITime Allowed Hours] !Maximum M.arks 200 1

INSTRUCTIONS
Candidates should attempt questions 1 and 5
_which are compulsory, and any THREE ofthe
remaining questions,· selecting at'least ONE
fr(JTn each Section'.
All questil!ns carry equal marks.

Marks allotted to parts ofa are each. Answers must be written in ENGLISH only. Assume suitable data, necessary,
and indicate the same clearly.

(Notations and symbols are as usual)
SECTION A
1. Answer any four partS of the following 4xlO=40
Let X and Y be two independent variables with
2 .2
N(1l cr and 16)
distributions respectively. If .
144 Z 16X2 9y2-64X-54Y 145, find
and State (without prooO any theorems you

have used.
B-JGT-K-TUA 1 [Contd .J

A· random sample of n units is Clivided into k categories such that the'ith category has nj units. multinomial probabilities, derive and cOIT.(ni, n). .
j
Find the constant K such that
-00 x 0 is a density function. Obtain the maxjmum likelihood estimator of Also suggest an estimator for e based on the method of moments.
With an illustrative example discuss how non-p·arametric tests are useful in drawing inferences about the parameters, giving all the necessary mathematical details.
t
Let X have a distribut.ion under Ho and a .
Cauchy distribution 1 x 00 . 1t l+x
under the alternative hypothesis HI" Find a most powerful size a test of Ho against HI and derive its power.
2. Let.til k be independent unbiased estimators of 0 with yeti) Obtain the best . linear combination of t.1 which is unbiased for e and has minimum variance among such combinations. Further show that kL k(k is an unbiased estimator of i=l k V(t )where L tiIk. 1 10
B-JGT-K-TUA 2 [Contd.J
State Chebyshev's Weak Law of Large Numbers (WLLN). For the sequence of r.v.'s IX} such that
n
.
Prob,(Xk ± a. verify whether WLLN holds. 10
Let X ..., x be n independent observations
n
on a r.v. X with density function G). Further,
let x2, 0 be an estimating
o equation for O. Show that, if E(D then j where
dg(x1, x2, ..., x and j is the total
n
information in the sample. What happens when
.. d Log LidO, where L is the likelihood
8) ... 10
n
Explain the concept of 'sufficient statistics' and briefly indicate their in obtaining minimum variance unbiased estimators. stating clearly "-the theorems you refer to. 10
3. Let X be the mean of a random sample of size n from a.2 known. Suppose that the

parameter IJ has a prior distribution which is
distributed as Show that the posterior
distribution' of III X=x is again normaL. Obtain
the posterior mean" and posterior variance. 12
" Describe the chi-square test for testing the homogeneity' of two multinomial populations and discuss its application with reference to an example. 12
B-JGT-K-TUA 3 fContd.]
For testing independence in a i x 2 cl;mtingency table, derive the test statistic. how you would apply the continuity correction. Further if cell frequencies are very small, indicate briefly how you would treat the problem. 12
Give an example of anyone variance stabilizing transformation, explaining how it is used. 4

4. Let ..., ... be a sequence of i.i.d.
'random variables with known dl. Construct a Sequential Probability Ratio Test for testing the hypothesis Ho against the alternative
J.l 10
Let Prob.ex a 8x x ... where
x ax may be zero for· some x. Let X2 ..., xbe n
n independent observations from this distribution. Show that the m.l.e. of 8 is a root of the equation x which is also the same obtained from the method of moments. Calculate the information content in a single obs.ervation. 10
Indep'endent random samples of size n from two
normal populations with known variances
and are used to test the null hypothesis
Ho: III -Jl2 11 against the alternative HI For specified values of a and the probabilities of type I and type II errors respectively, show that the required sample size
IS n ZIl/2)2 (.11
where Z is the standard normal point. 10
B-JGT-K-TUA 4 (Contd.l
Iflog X is distributed as then X is said to have a log nonnal distribution. Write down the density function of X and show that the ·coefficient of variation IS defined by
.1)112
eO"2_


5 [Contd.l
SECTION B

5. Answer any four parts of the following: 4xlO=40

Describe Yates' method of computing factorial effect totals in-a 23 exper1mental design.


What are the sources of non-sampling errors during the field operations stage in a large scale sample survey and how do you control them?


Let y be the response variable and X2, ..., X


be p predictor variables. Let ..., Xp) be
predictor of y. Show that E(y Xz,
minimum when- X21 ..., X is the
conditional expectation
X2l ..., Xp) IXl' X2, Further

prove that Ifor any function g. When is the lower bound attained?
A field researcher has to select one out of the five plantations, in a forest at :andom for further study. Having no access to random numbers, how does she make the required selection using a fair coin? 5
(iir A statistics student was asked to select a ra.ndom sample of size two from a normal population with mean =50 and variance 9. How should he proceed assuming that he was given random number tables? 5
B-JGT-K-TUA 6 (Contd.J
Write down the density of a r.v. distributed as with mean 'vector and variance covariance. matrix L. Specify the distributions along with parameters of Cp Y where I isa p x 1 vector ..., lp) of constants and Z ex, where C isa qxpmatrix representing q 'linear' functions. (No proofs required).
6. Describe the ratio method of estimation for estimating the population total Y of a study variable y when auxiliary information on a related variable x is available. Denote this
A
estimator bYYand derive its Bias and Mean
R
•
Squared Error. Find a condition under which
A 1
YR would, fare better than the unbiased estimator obtained without using information on
x. 15
Obtain the variance expression for the Horvitz
1
Thompson estimator YHT or the population total
Y of a study variable y for fixed sample size given by Horvitz and Thompson. Also cast this in the form given by Yates and Grundy, namely
• ,I



y.)2

V a.. and hence write
bLJ 1J 1t
1(. i ;to j 1 j.
A
down an unbiased estimator V of V [uij has to be clearly specified by you]. Comment on the non
A
negativity of V . 15

B-JGT-K-TUA 7 [Contd.]
Discuss how uniformity trials In experimental designs and pilot surveys in large scale sample surveys are conducted mentioning their usefulness. 10
7. Describe the technique of Analysis of Dispersion as a generalization of ANaVA in the univariate case wherein the decomposition IS gIven by 'Residual Sum of Squares' and 'Deviation from the hypothesis'. Give a practical illustration -where this technique could be applied. 12 ..
Define Mahalanobis-n2 statistic. How IS it related to Hotelling's-T2 statistic Explain clearly how n2 IS treated as a measure of distance mentioning an application. 12 •
If the matrix S rv Wishart and ILl show that lSI is distributed as the product of p independent central X2 variables. What are their degrees of freedom (State clearly any resp.lts that you have assumed). . 9
What you understand by the term 'multiple correlation coefficient' . of Y on the variables<. Xz, ..., Xp denoted by PO'12...p What does this measure? What range of values can it take? 7
B-JGT-K-TUA 8 (Contd.]

B. Explain what is meant by a split-plot design.
Suppose that factor A has p levels which are
arranged in a Randomized Block Design having
r blocks. Let factor B have q level,S which are
applied to the plots of a block after subdividing
each plot jnto q sub plots. Write ·down the model
clearly explaining the notations and present a
blank ANaVA table describing sources of
variation and the corresponding degrees of
freedom. 10
How is the F-test in ANaVA based on normality
assumption When this assumption does not
hold good even after transformation of the
variable, explain how a non parametric test
based on ranks could be used for testing that the
distribution functions of the continuous
populations under study are the same. . 10
Describe the analysis for a Latin Square Design
giving the model, Blank ANOVA table and test
statistics. Also mention the advantages of LSD.
What are 'Orthogonal Latin Squares' 10
What do you understand by 'randomized response
Illustrate this technique with
reference to Warner's unrelated question model.
How efficient is this method (w.r.t, variances)
compared to Direct Response Conventional
method? 10
...
.
B-JGT-K-TUA 9

•

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