CSS past paper

CSS Statistics Past Paper 2025

CSS Statistics 2025 carries 100 marks over 3 hours. It has 7 questions.

Marks: 100
Time: 3 hours
Questions: 7

Years

How to attempt: Attempt FOUR questions in all by selecting TWO Questions each from SECTION. ALL questions carry EQUAL marks.

Questions

Question 1 (Part I) is the paper's MCQs, which are not reproduced here.

SECTION-A

Q. 2
(20)
- (a)
  What is the purpose of frequency distribution and what are its desirable qualities? For a certain frequency distribution, the mean was 40.5 and the median 36. Find the mode approximately using the formula connecting the three.
  (10)
- (b)
  Define Statistics. Discuss the importance of study of statistics by giving examples. How it can help the extension of scientific knowledge?
  (10)
Q. 3
(20)
- (a)
  Explain what is meant by the skewness of the distribution, and define a suitable measure of Skewness. For given p.d.f.
  (10)
  $f(x) = k(x - x^{2}), \qquad 0 \leq x \leq 1$
  Find the Skewness and discuss.
- (b)
  Define the normal distribution and obtain its mean and variance. Show that for the normal distribution, the mean, mode and median are the same.
  (10)
Q. 4
(20)
- (a)
  If X has a binomial distribution, then show that E(X) = np, Var(X) = npq. Derive the m.g.f of the binomial distribution and explain its uses?
  (10)
- (b)
  What are the main characteristics of the Poisson distribution? Explain with the help of examples. Also give its properties, applications and relationship with other distributions.
  (10)
Q. 5
(20)
- (a)
  From the following set of values:
  (10)
  Y 6.5 5.3 8.6 1.2 4.2 2.9 1.1 3.0
  X 3.2 2.7 4.5 1.0 2.0 1.7 0.6 1.9
  - (1)
    Compute the residuals and verify that they add to zero and draw the conclusion about the results.
  - (2)
    Compute the standard error of estimate, $s_{y.x}$ .
- (b)
  Under what conditions the correlation among the random variables exists. Also describe the properties of correlation coefficient. Calculate the correlation co-efficient for a sample of 20 pairs of observations, given that
  (10)
  Mean of X = 2, Mean of Y = 8, $\sum X^{2} = 180$ , $\sum Y^{2} = 1424$ and $\sum XY = 404$
  Also interpret its results.

SECTION-B

Q. 6
(20)
- (a)
  What is finite-correction factor? When is it appropriately used in sampling applications and when can it, without the great undesirable consequences, be ignored?
  (10)
- (b)
  Given the population 2, 4, 8, 8, 10, 10.
  (10)
  - (1)
    How many samples of size n = 2 can be drawn without replacement from this population?
  - (2)
    Compute and tabulate the sampling distribution of the mean for samples of size n = 2.
Q. 7
(20)
- (a)
  Explain what is meant by:
  (10)
  - (i)
    a statistical hypothesis,
  - (ii)
    test-statistic,
  - (iii)
    test of significance,
  - (iv)
    level of significance,
  - (v)
    type-I error and type-II error
- (b)
  The weights of 4 persons before they stopped smoking and 5 weeks after they stopped smoking are as follows:
  (10)
  Person 1 2 3 4
  Before 148 176 153 116
  After 154 176 151 121
  Use the t-test for paired observations to test the hypothesis at the 0.05 level of significance, that giving up smoking has no effect on a person’s weight.
Q. 8
(20)
- (a)
  Explain the procedure of randomization in a completely randomized design where we have 3 varieties of wheat and 18 experimental plots available. Also explain what is explained by significant F value in an experiment.
  (10)
- (b)
  Four varieties of wheat were tried in a randomized complete block design in four replications. Yield in kilogram per plot is shown in the table given below. Test the hypothesis that there is no difference in the means of four varieties. $\alpha$ = 0.05.
  (10)
  Replicates Varieties
  $V_{1}$ $V_{2}$ $V_{3}$ $V_{4}$
  I 2 5 4 1
  II 2 3 3 1
  III 4 6 6 2
  IV 1 4 2 3

Replicates	Varieties
I	2	5	4	1
II	2	3	3	1
III	4	6	6	2
IV	1	4	2	3

Part-II is to be attempted on the separate Answer Book.
Attempt FOUR questions in all by selecting TWO Questions each from SECTION. ALL questions carry EQUAL marks.
All the parts (if any) of each Question must be attempted at one place instead of at different places.
Write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
Extra attempt of any question or any part of the question will not be considered.
Use of Calculator is allowed.

Y	6.5	5.3	8.6	1.2	4.2	2.9	1.1	3.0
X	3.2	2.7	4.5	1.0	2.0	1.7	0.6	1.9

Person	1	2	3	4
Before	148	176	153	116
After	154	176	151	121

Replicates	Varieties
	$V_{1}$	$V_{2}$	$V_{3}$	$V_{4}$
I	2	5	4	1
II	2	3	3	1
III	4	6	6	2
IV	1	4	2	3