Union Public Service Commission (UPSC) take ISS (Indian Statistical Service) exam for the recruitment in the department. Every year approx 1 Lakh students applies for this prestigious exam. A very few among them have got selected in ISS test paper.

Below we are going to share ISS 2012 exam paper to our readers. You can read and download full test exam paper of ISS 2012, Indian Statistical Service question paper with answers. You can download this ISS previous year pape along with answer key and solved paper.

STATISTICS-I

Time Allowed : Three Hours                                                       Maximum   Marks : 200

INSTRUCTIONS

Candidates should attempt FIVE questions in ALL including Question Nos. 1 and 5, which are compulsory. The remaining THREE questions should be answered by choosing at least ONE question each from Section-A and Section-B. The number of marks carried by each question is indicated against each. Answers must be written only in ENGLISH. ( Symbols and abbreviations are as usual, unless otherwise indicated. ) Any essential data assumed by candidates for answering questions must be clearly stated. A Normal Distribution Table and a ‘t ‘ Table are attached with this question paper. All parts and sub-parts of a question being attempted must be completed before moving on to the next question.

Section-A

1. (A) You arc given the following information :

1. In random testing, you test positive for a di sea se.
2. In 5% of cases, the test shows positive even when the subject does not have the disease.
3. In the population at large, one person in 1000 has the disease.
• (B)  What is the conditional probability that you have the disease given that you have been tested positive, assuming that if someone has the disease, he will test positive with probability I ?
• (C)  Items from a large lot are examined one by one until r items with a rare manufacturing defect are found . The proportion of items with this type of defect in the lot is known to be p. Let X denote the number of items needed to be examined. Derive the probability distribution of X, and find E (X).
• (D)  A 4-digit number is formed by selecting 4 digits from the set (0, 1, 2, ……, 9) at random, 0 at the left-most position(s) being permissible. Find the probabilities that-
(i) all 4 digits will be alike;
(ii) 3 will be alike, 1 different;
(iii) there will be 2 pairs of identical digits;
(iv) 2 will be alike , 2 different;
(v) all 4 will be different.
• (E)  12·3% of the candidates in a public examination score at least 70%, while
another 6-3% s core at most 30%. Assuming the underlying distribution to be normal, estimate the percentage of candidates scoring 80% or more.
Let { X n } be a sequence of random variables with
P{Xn = -2^n} = P{Xn = +2^n} = 1/2 n = 1, 2, ···
Examine if the sequence obeys Weak Law of large numbers and Central Limit Theorem.

2. (a) Of three independent events A, Band C,

1. A only happens with probability 18/4·
2. only happens with probability 1/8 and
3. only happens with probability 1/2Find the probability that at least one of these three events happens.

(b) For the Cauchy distribution given by

where k is a constant to be suitably chosen , derive the expression for the distribution function. Hence obtain in a measure of central tendency and a measure of dispersion. What are the points of inflexion of the distribution?

(C) The ith box contains 2i white balls and 6 -2i black balls, i == 1 (1)3. A fair die is cast once. 3 balls are taken at random from box 1, box 2 or box 3 according as the die shows up face 1 , any of 2 and 3, or any of 4, 5 and 6, respectively. Let X denote the number of white balls drawn. Find E(X)

(d) Write down the probability mass function of geometric distribution. State and prove its ‘lack of memory property’. Find also the mean and the variance of the distribution.

3. (a) Two persons Amal and Bimal come to the club at random points of time between 6 p.m. and 7 p.m., and each stays for 10 minutes. What is the chance that they will meet?

(b) Show that m a sequence Bernoullian trials with probability p =1/2, the most
number of successes is s, corresponding probability is

(c) Find E(X) and V(X) for the random variable X having the probability density function

where c is a constant to be suitably chosen.

(d) Let X be a random variable with F (x) as the distribution function. m is a suitable measure of central tendency and M is the mean square error about m. Show that

4. (a) An urn contains a white and b black balls. After a ball is drawn at random, it is to be returned to the urn if it is white; if it is black, it is to be replaced by a white ball from another urn. What is the probability of drawing a white ball from the urn after the foregoing operation has been repeated n times?

(b) For a discrete random variable X assuming the values 0, 1, 2 ,……, the probability mass function satisfies the recursion relation

Find the mean and the mean deviation about the mean of X.

(c)

(d) The continuous random variable X has the distribution function F(x). X ^1 and X ^2 are two randomly selected values of X. Show that

Section-B

5. (a) The three sides x₁, x₂ and x₃ (in decimetre) of a solid rectangular parallelepiped satisfy the relation x₁ + 2x₂+ 4x₃ = 12. Exploiting the inequality relation between two well-known measures of central tendency, determine the maximum possible volume of this parallelepiped.

(b) In respect of the following observations, arranged in a non-decreasing sequence
8, 10, 10, x, 12, 14, 14, 16
where x is unknown, indicate, with a brief justification, if the following statements are correct :

• x is necessarily larger than the standard deviation of the entire set including x.
• The mean deviation about the median is necessarily smaller than that about the mean.

(c)

(d) (i) In a ticket counter, at some point of time the sequence of males (M)
and females (F) was found as

MMFMFFFMFMMFFFFMFMMMM

Use runs test to examine if the sequence is random (5% critical value of the number r of runs with n ₁ = 11, n ₂ = 10 is 6).

(ii) Name two non-parametric tests for comparing locations of two correlated populations.

(e) 125 out of 285 college-going male students in 1995 from a city were found to be smokers. Another sample of 325 such students from the same city in 2012 included 95 smokers. Examine at 5% level of significance if smoking habit among college-going students is on the decrease in this city.

6. (a) Show, by proving all intermediate results, that for a set of n unsorted data, the measure of skewness based on mean, median and standard deviation, necessarily lies between -3 and +3.

(b) Scores in a UPSC examination in Statistics Paper-I and Paper-II awarded to 10 candidates were as under

Calculate Spear man’s rank correlation coefficient between the scores in the two papers.

(c) Solve the equation f (x) = 0 by using a suitable interpolation formula on the following values :

(d) Let (X, Y) follow the bivariate normal distribution N ₂ (0, 0 , 1, 1, p). Show that
the expected value of the absolute difference between X and Y is

7. (a) Consider a 3-point symmetrical distribution having the values x₀ – h, x₀ ,x₀ + h with the corresponding relative frequencies f, 1 – 2/ and f Calculate the coefficient of kurtosis b2 and find its limiting values as f → 0 and f →1/2Hence comment on the suitability of b2 as a measure of uni-modality versus bi modality.

(b) For a set of 10 pairs of observations (xi, yi ), i = 1(1)10, the following calculations are available

Examine at 5% level of significance if the two variables arc uncorrelated in the population.

(c) An unknown function u x has been tabulated below for some selected values of x. Use Newton’s divided difference formula on these to find an approximate value of u3 :

(d) Define partial correlation coefficient r^12.3 between x₁ and x₂ eliminating
from each the effect of x₃. Derive the express ion for r^12.3 and comment on its usefulness in multivariate data analysis.

8. (a) Mention two measures-one of skewness and the other of Kurtis-based on central moments. State and prove an inequality relation involving these two measures and a constant term. Give an example of a distribution for which equality holds in this relation.

(b) The speed y (in km/hr) of a car at different points of time x between 10:00 a.m. and 10:40 a.m. on some day was recorded as follows :

Calculate the approximate distance covered by the car between 10:00 a.m. and 10:40 a.m. on that day using Simpson’s one-third formula for numerical integration.

(c) Indicate how you would test the hypothesis that the means of k independent
normal populations a re identical, clearly mentioning the null and the alternative hypotheses, the assumptions made, the test statistic used , and the critical region.

(d) Ten s h ort-distance runners were put to a rigorous training for two months. Times taken by them to clear 100 metres before and after the training were as follows :

Use Wilcoxn’s paired sample signed rank test to examine at 1 % level if the training was at all effective. [The critical value of Wilcoxon’s statistic at 1 % level of significance for n = 10 is 5]