LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 49

FOURTH SEMESTER – APRIL 2006

ST 4804 – STATISTICS FOR COMPETITIVE EXAMINATIONS

Date & Time : 25-04-2006/9.00-12.00 Dept. No. Max. : 100 Marks

SECTION A

Answer ALL the Questions (40 ´1 = 40 Marks)

If the difference between two numbers is 1.2, then the variance of them is

(A) 0.18 (B) 1.44 (C) 0.72 (D) 0.36

To test the hypothesis that the variance of a normal distribution is 2, the test procedure used is

(A) Normal test (B) Chi-square test (C) F-test (D) t-test

To test which one of the following hypothesis, F-test is used?

(A) Goodness of fit (B) equality of means of two normal populations

If X has Poisson distribution with 3P[X = 2] = 2P[X ≤ 1], then the expected value of X is

(A) 3 (B) -2/3 (C) 2 (D) 3/2

X₁, X₂ and X₃ are independent observations on a normal random variable with

mean μ and variance σ².What is the efficiency of (3X₁+2X₂+X₃) / 6 as an estimator of μ ?

(A) 6/7 (B) 1 (C) 1/3 (D) 1/12

If E(Y/X) = α X + β and X has standard normal distribution, then E(Y) is

(A) 0 (B) 1 (C) β (D) α

∞

If P ( A_n) = 1, n=1, 2, 3… then the value of P (A_n) is

n=1

(A) 0 (B) 1 (C) 1/2 (D) ¼

A random variable X has characteristic function

Φ (t) = (sin t)/t, t ≠0

1 otherwise

Then, Var(X) is equal to

(A) 1 (B) 0 (C) 1/6 (D) 1/3

If X₁ and X₂ are independent and identically distributed random variables with p(x) = q^x.p, x = 0, 1, 2, 3… and (p+q) = 1, then the distribution of (X₁+X₂) is

(A) Geometric (B) degenerate (C) Negative Binomial (D) Hyper- geometric

In a random sample of size n from the distribution

dF (x) = e^-x.dx, 0<x<∞,

the mean of the smallest sample value is

(A) 1/n (B) 1/n² (C) 0 (D) 1

If the degrees of freedom for error in the analysis of Variance for a Latin square design is 30, the number of treatments is

(A) 5 (B) 6 (C) 7 (D) not possible to determine

If a population consists of 10 units and the population Variance is 20, the Variance of the sample mean of a simple random sample pf size 4 without replacement is

(A) 5 (B) 2 (C) 20 (D) 3

The number of simple random samples of size 4 that can be drawn without replacement from a population with 12 units is

(A) 12⁴ (B) 495 (C) 11,880 (D) 48

The standard deviation of a symmetric distribution is 4. The value of the fourth moment about the mean in order that the distribution be leptokurtic is

(A) greater than 768 (B) equal to 768 (C) equal to 256 (D) less than 48

Given Maximize subject to

For what values of the above problem will have several optimum solutions?

(A) 2 (B) 3 (C) 6 (D) 1

The objective function in the phase-I (when we use two phase simplex method) is formed by

summing all the variables
summing all the artificial variables

subtracting the sum of artificial variables from the sum of other variables

The following set of constraints require x artificial variables

where x is

(A) 0 (B) 1 (C) 2 (D) 3

Given the following simplex table (associated with a maximization problem)

Basic z x1 x2 x3 x4 Solution

z 1 -4 -2 0 0 8

x3 0 4 3 1 0 1

x4 0 -1 1 0 1 2

The leaving and entering variables are

(A) x1, x3 (B) x1, x4 (C) x2,x3 (D) x2,x4

An LPP has 4 variables and 2 constraints. How many sets of basic variables are possible?

(A) 10 (B) 6 (C) 3 (D) 20

The power function associated with the UMPT for testing against the alternative in is always

(A) Strictly increasing in (B) Strictly decreasing

Which of the following is the form of UMPT for testing against the

alternative in

(A) (B)

Choose the correct statement

(A) Power functions of UMPTs are always monotone

(B) A UMPT is always UMPUT

(É)All similar tests will have Neyman structure

Choose the correct statement

(A) RR methods are not associated with sensitive attributes

(B)Yates Grundy estimator is non-negative under Midzuno scheme

(D)Balanced systematic sampling is not recommended for populations with linear

trend.

Lahiri’s method

(A) is a PPS selection method involving a given number of attempts

(B) is a PPS selection method involving unknown number of attempts

(D) is an equal probability selection method involving unknown number of

attempts

Ratio estimator is

(A) a particular case of regression estimator

(B) an unbiased estimator

(C)more suitable when y and x have high negative correlation

(D) more suitable when y and x have no correlation

Random group method is due to

(A) Desraj (B) Murthy (C) Hartley-Ross (D) Rao-Hartley-Cochran

Randomised response methods are meant for

(A) homogeneous data

(B) heterogeneous data

(D) stratified populations

Which name is associated with shortest route problems

(A) Kuhn-Tucker

(B) Floyd

(D) Karmakar

Which of the following functions is NOT continuous at 0?

(A) |x| (B) e^x (C) x – [x] (D)Sin x

A tosses a fair coin twice and B throws a fair die twice. Let

a = Probability of getting at least two heads

b = Probability that the sum of the numbers that show up is less than 6

Then

(A) a > b (B) a < b (C) a = b (D) a + b > 1

31.The system of equations

2x + 4y – z = 3

x + 2y +2z = 2

x + (m²+1) y + 7z = 4m – 1

has infinitely many solutions if m equals

(A)0 (B) – 1 (C) 1 (D)2

The mean and variance of 8 items are 10 and 100 respectively. An observation 3

is deleted from the data. The variance of the remaining 7 observations is

(A)100 (B)106 (C)112 (D)120

Let T: R³ ® R² be defined as T(x, y, z) = (3x + y – z , x + 5z). The matrix

corresponding to this linear transformation is

(A) (B) (C) (D)

Which of the following is NOT true of a normal variable with mean 0?

(A) E(X²) = 1, E(X³) = 0 (B) E(X²) = 1, E(X⁴) = 2

If X and Y are uncorrelated random variables with equal means and variances,

then

(A) X + Y and X – Y are identically distributed

(B) X + Y and X – Y are independent

(D) X + Y and X – Y have equal variance

In a bivariate dataset {(X_i, Y_i), i =1, 2, …,n}, X assumed only two values namely 0 and – 1 and the correlation coefficient was found to be –0.3. Then , the correlation coefficient for the transformed data {(U_i, V_i), i =1, 2, …,n}, where U_i = 3 – 5 X_i² and V_i = 2Y_i –3 is

(A) –0.3 (B) 0.3 (C) 0 (D) cannot be determined

If X₁, X₂,…, X_n is a random sample from U( 0, q), which of the following is a biased estimator of q?

(A) 2 (B) X_(n) (C)X₁ + X_n (D) (n+1)X_(n) / n

The Cramer-Rao lower bound for estimating the parameter l of a Poisson distribution based on a random sample of size n is

(A) l (B) nl (C) l / n (D) l^1/2

The lower control limit of a c-chart is 4. The upper control limit is

(A)16 (B)20 (C)24 (D) none of these

In a 2⁴ factorial experiment with 4 blocks the degree of freedom for Error Sum of Squares is

(A) 25 (B)35 (C)45 (D)55

SECTION B

Answer any SIX questions (6 X 10 = 60 Marks)

Explain the procedure for solving a game theory problem graphically

Show that family of Uniform densities binomial densities has

MLR in

A sample has two strata with relative sizes and . He believes

that . For a given cost , show that (assuming stratum

sizes are large)

The exponent of a bivariate normal density is given below:

– ⅔(x²+9y²-13x-3xy+60y+103)

Find μ₁, μ₂, σ₁, σ₂ and ρ.

The number of accidents in a town follows a Poisson process with the mean of 2

accidents per day and the number of people involved in i^th accident has the

distribution

P[X₁=k] = 1/ 2^k, k≥1.

Find the mean and variance of the number of people involved in accidents per

week.

If Φ is a characteristic function, show that e ^{λ (Φ -1)}is a characteristic function for

all λ>0.

(a)Let X_{1, …,}X_n, X_n+1 be a random sample from N(m, s²). Let M be the average of the first ‘n’ observations and S² be the unbiased estimator of the population variance based on the first ‘n’ observations. Find the constant ‘k’ so that the statistic k( M – X_n+1) /S follows a t- distribution.

(b) Let X have a Poisson distribution with parameter q. Assume that the

unknown q is a value of a random variable which follows Gamma distribution

with parameters a = a / ( 1- a) and p = r, where ‘r’ is a positive integer. Show

that the marginal distribution of X is Negative Binomial. (5 + 5)

(a) Let X₁,…,X_n be a random sample from Poisson distribution with parameter l. Starting with the initial estimator X₁² – X₁ for l², use Rao-Blackwellization to get an improved estimator by conditioning on the sufficient statistic S X_i. State whether the resulting estimator is UMVUE and justify.

(b) Let X₁, …, X_n be a random sample from N(m, s²). Obtain an unbiased and

consistent estimator of s⁴. (6 + 4)

(a) Derive an expression for E(Mean Treatment Sum of Squares) in LSD.

(b) Consider four quantities T₁, …,T₄ and let T₁ – 2T₂+ T₃ be a contrast. Find

two other contrasts so that all the three are mutually orthogonal. (7 + 3)

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