Serial No. 0995)

STATISTICS
Paper-I

(Time Allowed: Three Hours) (Maximum ,Marks: 200)
INSTRUCTIONS
Candidates should attempt Question Nos. 1 an.d 5
which are compu'lsory. and THREE ofthe
remaining questions, selecting at least
ONE question from each Section.
The number ofmarks carried by each question
is indicated at the end of the question.
Answers must be written in ENGLISH.
Assume suitable data, Ifconsidered-necessary, and
indicate the same clearly.
(Notations and symbols are as usual)

SECTION-A
1. Answer any FOUR parts x 10=40]
5 coins with probability of head equal to each
and 4 coins with probability of head equal to each are tossed simultaneously. Use the probability generating function ofbinomial distribution to find the probability ofgetting 3 heads altogether.
(Coutd.)
State Weak Law ofLarge Numbers (WLLN). Examine ifWLLN and Central Limit Theorem hold for the sequence of random variables where
1
P{X
k k
P{X. K .


Show that X2 is consistent for J.l2 where X is the mean of a random sample of size n from N ell.


Design a randomised sign test of size 0.05, based on a random sample of size for testing the null hypothesis that 500/0 of the new born babies in a hospital are females against the alternative that the percentage is less.


Define a family of Uniformly Most Accurate Unbiased (UMAU) confidence sets oflevel for a parameter e.


Starting from the acceptance region of a suitable 2
,testfor against HI 0 where is the variance ofa nonnal population with Wlknown mean, obtain the UMAU confidence set of level 1-a.
(Contd.)
2. A discrete random variable assuming the values
1,2 ... has aprobabilitymass fimction satisfying
a+J3x f f x . x
where a. and J3 are constants.
Find the arithmetic mean Jl and the mean
deviation about Jl for the distribution.
Hence obtain the mean deviation about Jl for
the binomial distribution with parameters m and p.
Derive the moment generating fimction ofthe negative
binomial distribution, and use this to obtain its mean
and variance.
State and prove the lack of memory property of
exponential distribution. For what type offailure rate
can exponential probability law be used to model
the Wlderlying life distribution?
50% ofthe mangoes of a certain variety in a large
lot have weight 320 grns or mpre, while another
6.68% have weight ofat least 380 grns. Assuming
the underlying distribution to be Donnal, estimate
the percentage ofsuch mangoes weighing between
280 gms and 400 gms.
(You may use the information that for a standard
Donna] variate P{X 0.8413, 0·9332,
and 0,.9772 for K 1.0, 1.5 and 2.0 respectively).

f3} (Contd.)

3. Define Mean Square Error of an estimator T for a parameter 9.
Let X2, ... X be a random sample
n
2
from the population and S2 Ln
i=I 1
X where X Ln X./n. Find a constant C
i=1 I
such that CS2 is the minimum MSE estimator for
A random sample
5·8 is taken from the population with p.d.f.
<e



otherwise
Find the maximum likelihood estimate ofejustifying your ans,ver.
In a large forest there are N number of trees of a certain species haphazardly located throughout the forest and numbered from 1 to N. A stranger, while roaming about in the forest one day, identifies n such trees serial.numbers x..., x
n
Indicate how he would use these observed serial numbers to estimate unknown to him. Give an unbiased estimate of the variance of his estimate, when N is large.
(Contd.)
Use a suitable theorem, to be clearly stated, to obtain
the llllique minimum variance unbiased estimator for
the Poisson parameter based on a random sample
from the population.
4. Random samples from 3 species oftrees in a fores·t
were ,examined for incidence of a certain disease.
The following results emerged

Species Developed the disease?
Yes No Total
A B C 18 13 25 65 91 49 83 104 74
Total 56 205 261

Test·at level ofsignificanceifthespecies behave differently in respect of proneness to the disease.
[You may use the following values of X2.
052
X 3·841 5·991 X2'05,5 11'070]
Describe the median test for comparing the locations oftwo independent populations, clearly mentioning the underlying assumptions, if any, the precise hypothesis to be tested, the test-statistic and the
critical region.
Derive Sequential Probability Ratio Test for Ho: J..l J!o against regarding the parameter Jl in N 1). Indicate how the test can be carned
out graphically.
(Contd.)
Define, using suitable the size-a minimax test from amongst the rival tests w ..., w for
2k
Ho 9 90 against H 9.
I 0SECTION-B
5. Attempt any FOUR parts:­
[4x 10=40]
Use the following set of random numbers, from a
rnndom-startingpoint, to draw asimplerandom sample of 5 trees without replacement from the 324 trees in a forest:
0912 8257 4015 5933 5520 6917 4650 6927 6472 7279 5364 9901 7710 3750 7940
The relation between the dependent variable subject to some observational errors, and the independent variable x is known to be of the fonn
b
y ax . Using a set of observed data ...
It
indicate how you would estimate the above equation by the method of least squares. Derive the normal equations to be solved for estimating the
Construct a Balanced Incomplete Block Design (BIBD) with parameters v 11, r=k 2. Indicate how another BIBD can be constructed
from this design.
(Contd,)
Discuss how you would estimate the percentage of
candidates in a public examination adopting unfair
means, using randomised response technique. Derive
the variance ofthe estimator.
Define Wishart distribution. When does this reduce
to chi-square distribution? State its reproductive
property_
6. The vector variable JS, follows a trivariate
nonna} distribution with mean vector 5:.
and dispersion matrix:
64 28 10
28 49 14
10 14 25
Find:
the nlultiple correlation coefficient and
the conditional mean ofX, given X2 x2 and
X3 x3 ·
Write down the Gauss-Markov linear model. When
is a linear parametric function said to be estimable
under the model State and prove a necessary and
sufficient condition for a linear parametric function
to be linearly estimable.
(Contd.)

Suggest a measure of efficacy of the least squares
bivariate linear regression equation as a prediction
fonnula, and establish its relationship with the product
moment correlation coeffici between the variables

7. Show that, subject to a certain assumption to be
specified by you, V3 V2 V where V2 and
V3 denote the variances ofestimators ofthe population
mean using an umestricted simple random sample, a
stratified random sample with proportional allocation
and a stratified random sample with optimum
(Neyman) allocation, respectively, the total sample
size being identical in case.
What is ratio method of estimation ofthe population
mean Derive the variance ofthe usual ratio estimator
of the population mean to the first order of
approximation and deduce the condition Wlder which
this estimator is more efficient than a comparable
mean per unit estimator.
What is probability proportional to size sampling with
replacement (ppswr) For ppswr sampling, give
the Hansen-Hurwitz estimator ofthe population total.
Show that the estimator is unbiased. Obtain the
variance and the variance-estimator ofthis estimator.

(Contd.)

8. Write down the fixed-effects model for a 2-way
classified data with m observations per cell.
Indicate how the total sum of squares can be
decomposed into various components. What are the
hypotheses that can be tested using these -data?
Mention the test statistic for each ofthese hypotheses.
Obtain the lay-out ofa randomised block design for
testing the treatments D E each to be
replicated 4 times, using the following set ofrandom
numbers and explaining the procedure used
7474 6525 1350 3148 4576
0516 2710 3523 4508 8105
1341 6045 2877 0132 5507
A 32 design is to be laid out in 3 incomplete blocks
ofsize 3 each per replicate. The composition ofone
such incomplete block in a replicate, using standard
notation, is

Identify the factorial effect confounded, and give
the compositions ofthe other two incomplete blocks
of the replicate. If this basic pattern is repeated
r times, write down the first two columns of the
"
ANOVA table to be used for analysing'the data.

(Contd.)
For testing v varieties using a v x v Latin square design, the crop of a comer plot is found to be destroyed by cattle. Discuss how you would proceed to validly analyse the data by appropriately estimating theyield ofthemissingplot.