Indian Economic Service - Indian Statistical Service Examination
STATISTICS IV
1. Answer all of the following : 10x5=50
1. (a) A paint company produces both interior and exterior paints using two raw materials M₁ and M₂. The following table provides the basic data of the problem :
Tons of raw material per ton of — Exterior paint — Interior paint — Maximum daily availability (tons)
Raw Material M₁ — 6 — 4 — 24
Raw Material M₂ — 1 — 2 — 6
Profit per ton (in thousand Rs.) — 50 — 40
The daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton. Also the maximum daily demand for interior paint is 2 tons. Determine the optimal product mix. 10
1. (b) Neon lights on the campus of a university are replaced at the rate of 100 units per day. It costs Rs. 10,000 to initiate a purchase order. A neon light kept in storage is estimated to cost about Rs. 2/day. The lead time between placing an order and receiving the order is 12 days. Determine optimal inventory policy for ordering the neon lights. Also, calculate cycle length and reorder point of the order. 10
1. (c) There are five jobs to be performed by employees in a Departmental Store. The time (in hours) that each employee takes to perform each job is given in the following effective matrix :
Employees — I — II — III — IV — V
Jobs
A — 10 — 5 — 13 — 15 — 16
B — 3 — 9 — 18 — 13 — 6
C — 10 — 7 — 2 — 2 — 2
D — 7 — 11 — 9 — 7 — 12
E — 7 — 9 — 10 — 4 — 12
How the jobs should be assigned, one per employee, so as to minimize the total manhours ? 10
UPSC Previous year question paper Indian Economic Service - Indian Statistical Service Examination, 2026 exam
1. (d) Two jobs are to be performed on five machines A, B, C, D and E. Processing times (in hours) are given below :
Machines
Job 1 — Sequence A B C D E — Processing Time 3 4 2 6 2
Job 2 — Sequence B C A D E — Processing Time 5 4 3 2 6
Use graphical method to obtain minimum elapsed time and idle times for the two jobs. 10
1. (e) Define the terms : IFR, IFRA, NBU, NBUE, DMRL 10
2. Answer any two from the following : 25 × 2 = 50
(a) Consider the following table summarizing the details of a project involving 14 activities :
Activity — Immediate predecessor(s) — Duration (in months)
A — - — 2
B — - — 6
C — - — 4
D — B — 3
E — A — 6
F — A — 8
G — B — 3
H — C, D — 7
I — C, D — 2
J — E — 5
K — F, G, H — 4
L — F, G, H — 3
M — I — 13
N — J, K — 7
Draw the network diagram and identify the critical path of the project. 25
2. (b) (i) Write the basic characteristics of queueing model (M/M/I): (FCFS/∞/∞). 5
(ii) A company has received a contract to supply gravel to three new construction projects located in three towns A, B, C from 3 gravel pits located in three towns X, Y and Z. The delivery cost from each pit to each project site of truckloads is given below :
Pits — Project Location A B C — Supply
X — 4 8 8 — 76
Y — 16 24 16 — 82
Z — 8 16 24 — 77
Demand — 72 102 41
Find an optimum schedule of transportation so as to minimize the total cost of transportation. 20
2. (c) (i) Discuss the replacement policy for items whose running cost increases with time and value of money remains constant during a period. 10
(ii) Suppose that a sample of 12 items were put on test and the test was terminated at the 8th failure time, and the lifetimes follow an exponential distribution with mean θ > 0. The failure times are, in hours
30, 57, 145, 167, 320, 440, 505, 675
(I) obtain maximum likelihood estimate of mean lifetime
(II) construct 95% confidence interval for θ
[Given χ²(16, 0.975) = 28.845; χ²(16, 0.025) = 6.908] 15
2. (d) (i) Obtain an optimal solution using simplex method to the following linear programming problem:
Maximize z = 3x₁ + 2x₂
Subject to
−x₁ + 2x₂ ≤ 4
3x₁ + 2x₂ ≤ 14
x₁ − x₂ ≤ 3
x₁, x₂ ≥ 0
Also find an alternative optimal solution if it exists. 15
10
Player A — Player B I II III
I — 30 40 −80
II — 0 15 −20
III — 90 20 50
SECTION 'B'
(Demography and Vital Statistics)
3. Answer all of the following :
3. (a) Define Central Mortality Rate (mₓ) and Force of Mortality (μₓ).
Establish the relation
μ_(x + ½) = mₓ
stating the necessary assumptions involved.
3. (b) (i) Define crude death rate with its limitations.
(ii) Explain standardized death rates including the methods of their computation.
3. (c) Ascertain whether Gompertz's or Makeham's formula for graduation is suitable for the following data :
Age x — 35 40 45 50 55 60
l(x) — 897858 623782 377780 187289 566666
3. (d) Derive an algebraic expression relating the probability of a person surviving between age x and x + 1, pₓ to the force of mortality, μₓ.
3. (e) Fill in the blanks in a portion of life table given below :
Age (x) — lₓ — dₓ — pₓ — qₓ — Lₓ — Tₓ — ρₓ°
49 — 90000 — 500 — ? — ? — ? — ? — ?
50 — 60200 — ? — 5? — 300 — ? — ? — ?
4. Answer any two from the following : 25×2=50
(a) What is vital statistics ? State the uses of vital statistics. Explain registration method and census method of obtaining vital statistics. 5+5+15=25
(b) The sex wise distribution of population and number of births with survival rates of a town in 2022 are given below :
Age group — Population Male — Population Female — Births Male — Births Female — Survival rate (nπₓ)
15-19 — 64326 — 21065 — 640 — 92 — 0-90
20-24 — 53185 — 43012 — 110 — 150 — 0-90
25-29 — 44704 — 58011 — 110 — 105 — 0-87
30-34 — 39603 — 97080 — 80 — 78 — 0-86
35-39 — 35803 — 60060 — 65 — 65 — 0-84
40-44 — 31403 — 50151 — 80 — 80 — 0-83
45-49 — 28902 — 70543 — 0-81
Compute GFR, ASFR, TFR, GRR, NRR.
(c) In usual notations as in a life table, given l₈₀ = 16000 and
Age (x) — 80 81 82 83 84 85 86
dₓ — 5000 3000 2500 2000 1000 1000 500
(i) Compute the values of lₓ and qₓ for x = 81, 82, ..., 86.
(ii) The ages of three persons A, B and C are 81, 82 and 83 respectively. Find the probabilities.
(1) that A, B and C will be alive in two years' time 5
(2) that one at least of the three will be alive in two years' time 5
(3) that exactly one of them will be alive in two years' time 5
(d) What is internal and international migration? 10
Discuss various classifications of internal and international migration. 15
5. Answer all of the following : 10x5=50
(a) For gamma lifetime model f(x, b, p) = b^p e^(−bx) x^(p − 1) / √p, b > 0, p > 0, x > 0, show that hazard function is an increasing function for p > 1 and decreasing function for p < 1. 10
(b) One often hears that the death rate of a person that smokes is, at each age, twice that of a non-smoker. Does it mean that a non-smoker has twice the probability of surviving a given number of years as does a smoker of the same age? Justify your answer. 10
(c) Define hazard function, cumulative hazard function and mean residual life function. If T is a continuous non-negative random variable with cumulative hazard function H(T) then show that H(T) follows standard exponential distribution. 10
(d) What is multicenter trial? Why are multicenter trials conducted? 10
(e) Consider two groups of survival data with hazards λ₁(t), λ₂(t) and survivor functions s₁(t), s₂(t) respectively.
(i) One of the assumptions for Cox model is proportional hazard, what is really meant by "proportional hazard"? 5
(ii) Assuming the two hazard functions are not the same, examine the connection between the crossings of the two hazard functions and the crossing of the two survivor functions. 5
6. Answer any two from the following : 2×25=50
(a) The survival times of two groups of breast cancer patients who had surgical treatment are given below :
Survival group (in months) : 3, 7⁺, 9, 9, 11⁺, 16
Chemotherapy group (in months) : 8, 9, 10⁺, 12⁺, 18, 23⁺
Apply logrank test for comparing survival distributions of the two groups at 5% level of significance using exact method.
[Given χ₁²(0.05) = 3.841] 25
(b) (i) Explain time censoring and number censoring (type-II censoring). State the likelihood functions in each censoring scheme. Assume the life time follows exponential distribution with failure rate 1/λ, λ > 0. Derive the maximum likelihood estimate of survival function at time t = 700 based on the following censored data: t₁ = 80, t₂ = 95, t₃ = 105, t₄ = 180, t₅ = 270, t₆ = 330, t₇ = 670, t₈ = 800, tᵢ ≥ 1000, i = 9, 10, ..., 20. Using the (i) actual failure time/survival time 13 (ii) number of failures observed only without considering the actual failure time. 12
(ii) number of failures observed only without considering the actual failure time. 12
(c) A study is made on the impact of regular exercise and gender on the risk of developing heart diseases amongst 55-75 year olds. A sample of people was followed from 0 if female, 1 if male, the exact age of 55 years until they either develop heart diseases or turn 75 years, whichever comes first. The Cox PH model was used for this study with the two covariates being defined as
The model results were as follows :
Model fitted Maximum log likelihood
1. Null model : -1190
2. Gender only : -1177
3. Gender and Exercise : -1170
4. Gender, Exercise and Interaction : -1166 ; (Interaction = Gender * Exercise)
Covariate Parameter fitted
Gender : β₁ = 0.25
Exercise : β₂ = −0.35
Gender * Exercise : β₀ = −0.45
(i) Give two reasons why the Cox PH model is suitable in this data analysis.
(ii) Perform a statistical test to show that the interaction term is significant in the model (Take α = 0.05).
(iii) Give the hazard functions for a male who does not exercise regularly; a female who exercises regularly; a female who rarely exercises and a male who exercises regularly.
(iv) Identify the baseline hazard for this model.
(v) Interpret your results in (iv) with reference to the baseline hazard and the hazard function for males who rarely exercise.
[Given: χ²(1, 0.05) = 3.841] 2+8+8+2+5=25
(d) What is randomization in a clinical trial? How should the randomization code be determined? Discuss some common randomization methods. 5+4+16=25
SECTION 'D'
7. Answer all of the following: 10×5=50
(a) Why do we make use of Statistical Quality Control? Write the advantages when a process is working in a state of statistical control. 10
(b) Explain the statistical reasoning for using 3-σ limits in statistical quality control. 10
(c) A Metropolitan Transit system uses the number of written passenger complaints per day as a measure of its service quality. For 10 days, the number of complaints received are given below:
Day (sample) no. — 1 2 3 4 5 6 7 8 9 10
No. of complaints/day — 4 8 2 0 3 9 10 0 6 4
Obtain the three control limits for the above data. 10
(d) Explain the differences between CUSUM and Shewhart Control Charts in terms of their uses and methodology. 10
(e) Describe Double-Sampling Inspection plan and discuss its advantages. 10
8. Answer any two from the following: 25×2=50
(a) A machine is set to deliver packets of a given weight. Ten samples of size 5 each were recorded and the recorded data is produced below:
Sample No. — 1 2 3 4 5 6 7 8 9 10
Mean (x) — 15 17 15 18 17 14 18 15 17 16
Range (R) — 7 7 4 9 8 7 12 4 11 5
Can the process be regarded under control? (Given conversion factors for n = 5, A₂ = 0.58, D₃ = 0, D₄ = 2.115) 25
(b) Explain the statistical basis and construction of p and np charts. How is the choice between p and np charts made? 25
(c) (i) It has been decided to sample 100 items at random from each large batch and to reject the batch if more than 2 defectives are found. The acceptable quality level is 1% and unacceptable quality level is 5%. Find the producer's and consumers risks. 10
(ii) Define a single sampling plan. Consider a single sampling plan: Lot size (N) = 2000, Sample size (n) = 50 and acceptance number of defectives (c) = 2. Find the probability of accepting the lot. 15
(d) Hourly concentration (Xᵢ) data collected from a chemical process is given below:
Hour — Xᵢ — Hour — Xᵢ
1 — 5-50 — 11 — 6-75
2 — 4-50 — 12 — 3-25
3 — 5-25 — 13 — 5-25
4 — 6-0 — 14 — 5-05
5 — 5-25 — 15 — 4-5
6 — 3-50 — 16 — 6-50
7 — 5-75 — 17 — 7-20
8 — 6-25 — 18 — 6-80
9 — 4-50 — 19 — 6-75
10 — 5-0 — 20 — 6-50
If the target mean (μ₀) = 5-0, n = 1 and σ = 1, use CUSUM Control Chart to detect the shift to μ₁ = 6-0 for K = 0.5. 25
SECTION 'E'
(Multivariate Analysis)
9. Answer all of the following :
10×5 = 50
(a) Given X ~ N₃(μ, Σ), where
(i) find the regression function of X₁ on X₂ and X₃, and
(ii) compute the conditional variance of X₁ given X₂ and X₃. 10
(b) Show that X = (X₁, X₂, …, Xₚ)′ has p-variate normal distribution if and only if every linear combination (l₁X₁ + l₂X₂ + … + lₚXₚ) of X follows a univariate normal distribution. 10
(c) Let Tₚ² = n Y′A⁻¹Y, where Y ~ Nₚ(μ, Σ) and A ~ Wₚ(n, Σ) which is independent of Y. Show that Tₚ² ≥ T_K² for K ≤ p. 10
(d) Let there be two populations π₁ and π₂. It is known that about 30% of all objects belong to π₂ and
C (2|1) : cost incurred when a π₁ observation is incorrectly classified as π₂ observation = 15
C (1|2) : cost incurred when a π₂ observation is incorrectly classified as π₁ observation = 10
Suppose the two density functions f₁(x) and f₂(x) (corresponding to π₁ and π₂) are evaluated at a new observation x₀ and f₁(x₀) = 0.32, f₂(x₀) = 0.56. Can the new observation be classified as coming from π₁ or π₂? 10
(e) Let X = (X₁, X₂, X₃)′ has the correlation matrix R given by
Obtain the first two principal components and the percentage of population variance explained by the first two principal components. 10
10. Answer any two from the following :
25×2=50
(a) (i) If X is a random p-vector distributed as Nₚ(μ, Σ), then obtain the distribution of X′Σ⁻¹X and specify its parameters. 15
(ii) If A ~ Wₚ(n, Σ), then prove that C A C′ ~ W_q(n, CΣC′), where C is a (q×p) matrix of rank q ≤ p. 10
(b) Define sample generalised variance based on a random sample χ_α (α = 1,2,…,N) of size N drawn from Nₚ(μ, Σ) and obtain its distribution. Also find an expression for its hth moment (h = 1,2,…). If χ₁, χ₂ and χ₃ are independently and identically distributed as N₂(g, Σ) with Σ = [[1, 2], [2, 5]], then obtain E|Σ_{α=1}^3 Z_α Z_α′|. 25
(c) Let X₁, X₂, …, Xₚ represent measurements or characteristics on one member of a twin pair and X_(p+1), X_(p+2), …, X_(2p) represent the same measurements on the other member. Assuming that χ = (X₁, X₂, …, X_(2p))′ ~ N_(2p)(μ, Σ), with Σ unknown, develop a suitable test for testing the hypothesis of equality of measurements of the twins. 25
(d) Define canonical correlations and canonical variates and obtain the characteristic equation they satisfy. Hence, or otherwise, show that multiple correlation and simple correlation are special cases of canonical correlation. 25
SECTION 'F'
(Design and Analysis of Experiments)
11. Answer all of the following :
10×5 = 50
(a) In an RBD there are only two blocks. Let K be the number of treatments and x̄_j, j = 1,2 the average yield of jth block. Show that the between block sum of squares can be expressed as (K/2)(x̄₁ − x̄₂)² and write the ANOVA table. 10
(b) Explain the two basic ways of testing hypothesis involving contrasts of K parameters. Explain orthogonal contrasts and their use. 10
(c) Suppose you have υ varieties compared in V² plots. How will you carry out the experiment under each of the following situations ?
(i) there is no fertility difference among the V² plots.
(ii) the fertility changes along two perpendicular direction.
Write the appropriate ANOVA table for each case. 5+5
(d) State the advantages of a factorial experiment over a simple experiment. Explain Yates' method of computing factorial effect totals. 10
(e) Define the linear model for a split-plot design with two factors replicated r times. Write the ANOVA table. Give an example of the design. 10
12. Answer any two from the following :
25×2=50
(a) In the table given below are the yields of 6 varieties in a 4 replicate experiment for which one observation under treatment 2 in block 2 is missing. Estimate the missing observation and analyse the data :
Blocks — Treatments 1 2 3 4 5 6
1 — 18-5 15-7 16-2 14-1 13-0 13-6
2 — 11-7 - 12-9 14-4 16-9 12-5
3 — 15-4 16-6 15-5 20-3 18-4 21-5
4 — 16-5 18-6 12-7 15-7 16-5 18-0
(Given : F_(3,14)(0.05) = 3.34, F_(5,14)(0.05) = 2.96)
(b) Diet affects weight gain. We wish to compare nine diets : these diets are the factor level combinations of protein source (beef, pork and grain) and number of calories (low, medium and high). There are test animals nine in number that were randomly assigned to the nine diets one animal per diet. The responses (weight gain) are :
Source — Calories Low Medium High
Beef — 76-0 86-8 101-8
Pork — 83-3 89-5 98-2
Grain — 83-8 83-5 86-2
Using an appropriate linear model analyse the data and give the ANOVA table. (Given : F_(2,4)(0.05) = 6.44) 25
(c) Define the analysis of covariance model of CRD with one concomitant variable and explain its statistical analysis. 25
(d) In designing a battery for use in a device three possible plate materials are tested at three temperature levels. The following table gives the life (in hours) of the plate materials :
Plate material type — Temperature (°F) 15 70 125
1 — 130, 155; 74, 180; 34, 40
2 — 80, 75; 70, 58; 159, 126
3 — 136, 115; 45, 138; 160, 150
— 139, 96
Considering the given data is the proportional data, assess the effect of temperature on the plate material type and present the ANOVA table. (Given : F_(4,11)(0-05) ≈ 3-36, F_(2,11)(0-05) = 3-98).
SECTION 'G'
(Computing with C and R)
13. Answer all of the following :
10×5=50
(a) Summarise the rules for type conversion in C when neither operand is unsigned. Explain conditional operator with a suitable example. 10
(b) Given a matrix A_(m×n) write a C function to find the product of A_(m×n) and its transpose and to print the result. 10
(c) Describe two different approaches to updating a data file. Write illustrative programs one each. 10
(d) Write R code to create a data frame with name, age and gender of 10 individuals.
(i) Extract age and gender of the 4th, 8th and 1st individual in this order.
(ii) Add three new records to the data frame created above. 10
(e) Write R code to find the median of the observations xᵢ (i = 1, 2, …, 10) without using the median function. 10
14. Answer any two from the following : 25×2=50
(a) Given the observed frequency distribution with the variate values xᵢ (i = 0,1,2,…,8) and the corresponding frequencies fᵢ (i = 0,1,2,…,8) write a C program to fit a binomial distribution and test for its goodness of fit and to print the result. 25
(b) Given class, section, name, date of birth, roll number and marks secured in four subjects for a group of 30 students write a C program to find the average mark secured by each student and to print roll number and average mark of each student. Make use of structure variables within the program. 25
(c) Write a C program to test for the independence of attributes in a given m × n contingency table and to print the result. 25
(d) (i) For a given observed frequency distribution with variate values xᵢ (i = 0,1,2,…,10) and the corresponding frequencies fᵢ (i = 0,1,2,…,10) write R code to find mean and variance without using R functions and compare them. Also print your comment. 15
(ii) Write R code to generate a random sample of size 15 from N(22, σ = 1.5) and another random sample of the same size from N(13, σ = 2) and to test for the equality of the means of the two populations at 1% level of significance and to print the model values of the samples. 10
1.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1319 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4665 0.4673 0.4681 0.4689 0.4696 0.4703 0.4711
2.0 0.4717 0.4726 0.4736 0.4744 0.4753 0.4761 0.4768 0.4774 0.4780 0.4788
2.1 0.4798 0.4808 0.4817 0.4826 0.4834 0.4842 0.4850 0.4857 0.4864
2.2 0.4861 0.4864 0.4868 0.4871 0.4874 0.4877 0.4880 0.4882 0.4888
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4958 0.4959 0.4960 0.4961 0.4962 0.4963
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4990 0.4990
3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993
3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995
3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4997 0.4997
3.4 0.4997 0.4997 0.4997 0.4997 0.4998 0.4998 0.4998 0.4998 0.4999
3.5 0.4998 0.4998 0.4998 0.4997 0.4998 0.4998 0.4998 0.4999 0.4999
3.6 0.4998 0.4998 0.4999 0.4998 0.4999 0.4999 0.4999 0.4999 0.4998
3.7 0.4999 0.4999 0.4999 0.4997 0.4999 0.4999 0.4999 0.5000 0.5000
3.8 0.4999 0.4999 0.4999 0.4999 0.4998 0.4999 0.4999 0.5000 0.5000
3.9 0.5000 0.5000 0.5000 0.5000 0.5001 0.5000 0.5000 0.5000 0.5002
Indian Economic Service - Indian Statistical Service Examination STATISTICS IV
Indian Economic Service - Indian Statistical Service Examination-STATISTICS III
Indian Economic Service - Indian Statistical Service Examination-STATISTICS II
Indian Economic Service - Indian Statistical Service Examination-STATISTICS I
