1. Describe stratified random sampling. Suppose we want to estimate the population proportion P of units in the population having acharacter by a stratified random sampling. Let Pn denote population proportion of units having A in stratum h and Ph denote sample proportion of units with A in stratum h.
Obtain unbaised estimator of P. Also obtain its variance under proportinal allocation.
Assuming that the estimated population proportion should not differ from the true population proportion by more than 10% with a probability (1.. calculate the sample size for proportional allocation.

The following table gives the Ph and Nh values for different strata Assuming that the estimated population proporation should not differ from the true population by more than 10% with a proba bility of .95, calcukate the
sample size for proportional allocation
h Ph Nh
1 0.3 100
2 0.2 200
3 0.4 250
4 0.2 50
2. State and prove Holder's inequality.
Let Xi i ..... 10 be independent random variables each being uniformly
distributed over calculate P
3. Describe cumularive total method of drawing a ppswr sample.
In ppswr sampling, give an unbised estimator of population total. Also obtain
its variance.
A population consists of 5 units. The values of response variable Yi, size of unit
Xi are given below
Unit No. 1 2 3 4 5
Size Xi 5 10 15 12 8
Yi 32 41 25 30 35
Draw a ppswr sample of size 2 using random numbers 0.40 and 0.26 from
estimate the population total based on yair sample. Is yaer estimate unbiased
B Linear Models and Economics Statistics
4. Define the follows and state the limitations of each
Simple aggregative price index.
Index of price relatives.
Weighted aggregative price index.
Laspeyre's price index.
Paasche's price index.
What is Fisher's ideal index number. Show tha it satisfies time reversal test and
factor reversal test.
Acer 2
. 7.
10
i 1
i X . . ..

CON 324—3 9
Compute Laspeyre's, Paasche's and Finsher's price index numbers for the following
data using 1981 as base period. The table gives prices in Rs. per
Tonne, quantities in million tonnes. Figures in bracket are quantities.
Commodities 1981 1985
Wheat 554 (9.67) 673 (10.77)
Rice 427 (31.95) 622 (36.32)
5. Consider the two-way classification model—
yij j µ xi ßj rij eijk
i ......p, j ..... q k ..... r
Explain the terms involved in the model. State the assumptions required and
derive test for the hypotheses
H :rjj 0 ...... for i ......p, j ..... q
A menu factuter wanted to study production rates for different combinations of
reagents and catalists. The data given in the following table shows coaded
valuesof produciton rates. Obtain ANOVA and write your conclusions. You
are given reagents X catalysts interaction S. S. 84.
Catalyst
Regent 1 2 3
A 6 11, 7 9
B 4 13, 15 7
C 13, 15 15, 9 13, 13
D 12, 12 12, 14 9
6. Write down multiple regression model stating all the assumptions, Describe
various tests of hypotheses associated with such a model. Indiate method of
testing in each case.
Consider a linear regressiun model with six regressors having cuefficients
ß1, .... ß6 and intercept ß0 Data is available on 30 observations and it is given
that regression S. S. 3147.97, residual S. S. 1149.00.
Test the hypothesis ß1 ß2 . ß6 0 at 1 level of significance with
appropriate d.f. 3.71).
Find the value of multiple correlation coefficient.
C Statistical Inference
7. Prove that empirical distribution function is an unbiesed estimator of F df of X.
Let x1, x2, ..... xn be iid vvs from the following pdf.
f ... .
Q M a c x c .
O otherwise
where M is non- negative and obsolutaly continuous over is diffeentiable
everywhere. Find UMVUE of g where g is also differentiable every where.

CON 324 10
8. Let the VV X has the pdf f ... Assume that f belomgs to one
parameter exponential family then prove that the test ..if U is continuous.
. (x.. .
1 if u . C1 w u . c2
0 otherwise
is UMPU of size x for testing
HU . .U against H1 .... .....where U
Let X be a random variable with pdf .. .
f ....
2 0 . x1 . .
. 2
0 otherwise
Obtain a MP test of sine x to test
Ho . .o against H1 . .1 .o
Ho . .o against H1 . .1 .o
9. Lef x1, x2, ...... x4 be a random sample of sine n from a normal distribution with
with mean u and vanance o2. Obtain the likelined ration test for testing
Ho . .. .o against H1 . .o, where .2 is unknown.
In a singing competition, the judges acreed that 7 exhibits were outstanding
and these were numbered from 1 to 7. Three judges have given the following
raankings
Judge A 7 2 4 3 1 5 6
Judge B 1 3 5 2 4 7 6
Judge C 4 1 2 3 5 6 7
Compute Kandall's sample tau coefficient T from the three possible pairs of
rankings.
D Stochastic Processes
10. Consider a time homogeneous Markov Chain xn with finite state space and the
transition mature obtain the probablity distribution of xr, xr .....
xr n in terms of transition probablities and initial distribution of xr, Also derive
chapman — kolmogorov equation for computing n step transition probablities
A company asesses credit worlhiners of various firmsevery quanter, the ratings
are in order of decreasing merit C and D (default). Historical data
support the view that the credit rating of a tyrical firm evoles as a Markov
Chain with following transition probability matrix—
1.a.a2 a a2 0
P a 1.2a.a2 a a2
a2 a 1.2a.a2 a
0 0 0 1
for some panameter a.
Determine the range of values of a for which matrix P is valia transition
matrix.
Whether chain is inducible and apeniodic
Derive the station any probability distrubution of the chain.
For a 0.2 calculate the probabilities
P [x3 D/x1 P [x3 D/x1 P [x3 D/x1
T
n
1 1 ..
Paper II Descriptive type on Statistics

CON 324 11
11. Consider time homogeneous Markor Chain xn with state place S .1, 3..
and transition matrix.
p11 p12 p13
P p21 p22 p23
p31 p32 p33
Given the startig value and transition matrix p explain the method of
simulating path of the Markor Chain.
For the simulated path obtained in above give the estimate of transition
matrix p of the Markor Chain.
A gambler begins with Rs. 500. Each game he may win Rs. 100 with probability
0.3 and lose with probability 0.7. He willplay until he doubles his money
or losses it all use the simulation method to determine the path of the game
using random numbers given below
0.77, 0. 75, 0.14, 0.26, 0.20, 0.51, 0.72, 0.76, 0.44, 0.20, 0.67, 0.84, 0.27,
0.22, 0.07, 0.89, 0.18, 0.69, 0.10, 0.04. using the path obtained calculated
the estimate of transition matrix p.
12. A time series model is specified bym yt x1 yt 1 x2 yt 2 et where et is
a wite noise process with variance s
Determine whether the process is stationary.
Obtain yule walker equations.
Hence obtain autocorrelation of order one and two also obtain partial
autocorrelations.
Give procedure to estimate and x2.
The time series Yt is assumed to be stationary and to follow an ARMA
process defined by
yt 1 8 yt 1 1 yt 2 et 1 et 1
15 15 7
where et are independent N random variables.
Determine the roots of the characteristic polynominal and explain how their
values relate to the stationanity of the process. Find autocorrelation for lags
2.
E Multivariate Analysis
13. Define population principle components show that
........ V
where yi denotes the ith principle component of p-variates population.
Obtain th first principal component y1 of the following corrolation matrix also
find its variance and proportion of variation explained by y1
. . .
.
.
. . .
.
.
. . 0.631.00
1.000.63
14. What is cluster analysis What is distance and similarity cofficient for a pair
of item. Give one example of eachof them.

The vocabulary "richness" of a text can be quantitatively described by counting
the words used once, the words used twice and so forth. Based on these
coutns, a linguist proposed the following distances between chapters of the
Old Testament book Lamentations.
Lamentations chapter
1 2 3 4 5
1 0
Lamentations 2 0.76 0
chapter 3 2.97 0.98 0
4 4.88 4.17 0.21 0
5 3.86 1.92 1.51 0.51 0
Cluster the chapters of lamentations using the single linkage hierarchical mehod.
Draw the dendrogram.
15. Define Mahalanobis distance a measure of the distance between the two
normal populations and its estimate D2; based on two random samples of
sizes n1 and n2 from the two multivariate normal population ... or .
Suppose the n1 11 and n2 12 observations are made on two random vectors
X1 and X2 which are assumed to have bivariate normal distribution with a
common covariance matrix but possibly different mean vectots M1 and M2.
The sample mean vectors and pooled covariance matrix are
obtained the mahalabis sample distance D2 and obtain the linear discriminant
function. Assign the observation Xo
1 to either population .1
or .2. Assume equal cost and equal prior probabilities.
F Numerical Analysis and Basic Computer Techiques.
CON 324 12
.2
..
.
..
.
. . ..
.
. ..
.
. . . ..
.
. ..
.
.
1.1 4.8
7.3 1.1
pooled
1
2
x
1
1
x1 2 S
16.

CON 324 13 CON 324 13
17.
18.