CSS past paper

CSS Statistics Past Paper 2026

CSS Statistics 2026 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)
  Define Type I and Type II errors in hypothesis testing. Explain the trade-off between these errors.
  (10)
- (b)
  The first four moments of a frequency distribution about the origin are
  (10)
  $\mu'_{1} = -0.55, \quad \mu'_{2} = 4.46, \quad \mu'_{3} = -0.43 \quad \text{and} \quad \mu'_{4} = 68.52.$
  Find skewness and kurtosis. Also comment on the shape of the data distribution.
Q. 3
(20)
- (a)
  Explain the concept of exchangeability in probability. How does it differ from independence?
  (10)
- (b)
  Define skewness of a distribution. What are the different measures of skewness?
  (10)
  - (i)
    In a frequency distribution, the coefficient of skewness based upon quartiles is 0.6. If the sum of the upper and lower quartiles is 100 and median is 38, find the values of $Q_{1}$ and $Q_{3}$ .
  - (ii)
    Karl Pearson’s coefficient of skewness of a distribution is 0.32. If the standard deviation is 6.5 and mean is 29.6, determine the mode and the median of the distribution.
Q. 4
(20)
- (a)
  Explain the difference between Parametric and Non-Parametric tests. Provide examples of each in the context of ANOVA.
  (10)
- (b)
  A secretary makes 2 errors per page, on average. What is the probability that on the next page he or she will make
  (10)
  - (i)
    4 or more errors?
  - (ii)
    No error?
Q. 5
(20)
- (a)
  Define conditional probability with examples
  (10)
- (b)
  In a small town only three newspapers, A, B, and C, are available for the readers. Suppose that 60% of readers subscribe to newspaper A, 40% subscribe to newspaper B, and 30% to newspaper C. Suppose also that 20% of them subscribe to both A and B, 10% subscribe to both A and C, 20% subscribe to both B and C, and 5% subscribe to all three newspapers.
  (10)
  - (i)
    Construct Venn Diagram to present the above situation.
  - (ii)
    What percentage of newspaper readers subscribe at least one of the three newspapers?
  - (iii)
    What percentage of newspaper readers subscribe none of the three newspapers?

SECTION-B

Q. 6
(20)
- (a)
  Critically evaluate the role of probability theory in statistical inference.
  (8)
- (b)
  One of the larger components of the cost of living is the amount spent on housing. Housing cost includes rent (for tenants), mortgage payments (for home owners), heating, electricity and water. An Economist undertook a five years study to determine how housing costs have changed. Five years ago, he took a random sample of 200 households and recorded the percentage of total income spent on housing. This year he took a sample of 100 households.
  (12)
  The statistics from the two samples are:
  $\overline{x_{1}} = 32.42;\ s_{1} = 6.08;\ \overline{x_{2}} = 33.72;\ s_{2} = 6.75.$
  Test the hypothesis at 5% level, to determine whether the Economist can infer that housing cost as a percent of total income has increased over the last five years.
Q. 7
(20)
- (a)
  Suppose you want a confidence interval for a Normal Mean with small sample size. Explain how would you construct it.
  (10)
- (b)
  The population and death of two towns, according to age groups are given below:
  (10)
  Age group Town A Town B
  Population No. of deaths Population No. of deaths
  Under 10 100 40 1000 300
  11---25 1500 150 500 40
  26---50 850 108 400 24
  Above 50 50 37 100 6
  - (i)
    Calculate crude death rate for town A
  - (ii)
    Calculate SDR of town A taking the population of town B as a standard population.
Q. 8
(20)
- (a)
  Critique the use of p-values in statistical hypothesis testing. Discuss alternatives for interpreting statistical significance.
  (10)
- (b)
  Suppose in a field experiment there are 4 treatments (T1, T2, T3, and T4) and 4 blocks (B1, B2, B3, and B4). Consider the field is portioned into 4 plots (blocks), each of these blocks is homogeneous. Assign each treatment randomly to each plot (block) by fulfilling the requirements of randomized completely block design conditions (requirements), that you have to complete the layout of the experiment according to the RCBD. What will be the ANOVA table (a general form of RCBD ANOVA table) for this experiment?
  (10)
  Blacks T 1 T 2 T 3 T 4
  Block I 2 5 4 1
  Block II 2 3 3 1
  Block III 4 6 6 2
  Block IV 1 4 2 3
  Perform the analysis of variance to test at the 0.05 level of significance.

Age group	Town A		Town B
Under 10	100	40	1000	300
11---25	1500	150	500	40
26---50	850	108	400	24
Above 50	50	37	100	6

Blacks	T 1	T 2	T 3	T 4
Block I	2	5	4	1
Block II	2	3	3	1
Block III	4	6	6	2
Block 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.

Age group	Town A		Town B
	Population	No. of deaths	Population	No. of deaths
Under 10	100	40	1000	300
11---25	1500	150	500	40
26---50	850	108	400	24
Above 50	50	37	100	6