Indian Statistical Service Examination (with Indian Economic Service) · India · Union Public Service Commission
Original practice questions written to the published syllabus · not past papersProbability sampling from first principles: sampling versus complete enumeration, simple random sampling with and without replacement, stratification and allocation, systematic sampling, and the ratio and regression estimators that use auxiliary information such as last year's area.
In the exams: AICE-JRF Ag Stat Unit 5 (heavy); UPSC ISS Statistics III (heavy); SSC CGL JSO (heavy); ASRB NET Ag Stat (heavy); state statistical-officer exams (heavy)
How our questions for Indian Statistical Service Examination spread across the units of Sampling Theory I: Simple, Stratified, Systematic, Ratio & Regression Estimation. Difficulty: 0 easy, 14 medium, 11 hard.
| Simple random sampling with and without replacement | 5 | |
| Stratified sampling: estimation and allocation (proportional, Neyman, optimum) | 5 | |
| Choice of strata, construction of strata boundaries and collapsing of strata | 3 | |
| Systematic sampling and its variants | 3 | |
| Sampling versus complete enumeration; sampling and non-sampling errors | 2 | |
| Ratio estimators: bias, MSE and comparison | 2 | |
| Regression estimators | 2 | |
| Product and difference estimators | 2 | |
| Estimation of proportions and sample-size determination | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Which statement about non-sampling errors is correct?
C. They occur in censuses too and need not fall as n rises
Coverage, non-response, measurement and processing errors arise in any data collection and often grow with the scale of the operation; they usually cause bias that the sample variance does not reveal. Falling with the square root of n describes sampling error.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
As the sample size of a large survey is increased greatly with the same field staff, what typically happens to the total error?
C. Sampling error falls, but non-sampling error may rise and can dominate
Sampling error shrinks with the square root of n, but spreading the same staff over more units weakens supervision and checking, so measurement and processing errors can grow and become the larger part of the MSE. Survey design aims at total error for the cost.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
An SRSWOR of 100 farmers from 1000 finds 30 using crop insurance. What is the estimated standard error of the sample proportion?
D. 0.0437
v(p) = (1 - f) p q / (n - 1) = 0.9 x 0.3 x 0.7 / 99 = 0.001909; SE = 0.0437. Ignoring the fpc and using n gives 0.0458; 0.0019 is the variance, not its square root.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
For strata with weights 0.6, 0.3, 0.1 and standard deviations 2, 4, 10, ignoring fpc, what is the relative efficiency of Neyman over proportional allocation for the same total n?
A. 148.8%
V_prop / V_opt = (sum W_h S_h^2) / (sum W_h S_h)^2 = 17.2 / 11.56 = 1.488, i.e. 148.8%. The gain is large because the stratum SDs differ greatly. 100% would apply only if all S_h were equal.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Under optimum allocation with costs, a stratum receives a larger sample when it is:
D. Larger, more variable or cheaper to enumerate
n_h is proportional to N_h S_h / sqrt(c_h): it rises with stratum size and standard deviation and falls with the cost per unit.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Systematic sampling is more precise than SRSWOR of the same size if and only if:
D. The variance within systematic samples exceeds the population variance
V_sys = [(N - 1)/N] S^2 - [k(n - 1)/N] S_wsy^2 is below the SRS variance exactly when S_wsy^2 > S^2, i.e. units in the same systematic sample are more heterogeneous than the population.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
For an SRS of n = 25 (fpc negligible), S_y^2 = 100, S_x^2 = 64, R = 1.5 and rho = 0.8. What is the approximate MSE of the ratio estimator of the mean?
D. 2.080
MSE = (S_y^2 + R^2 S_x^2 - 2 R rho S_x S_y)/n = (100 + 2.25 x 64 - 2 x 1.5 x 0.8 x 8 x 10)/25 = (100 + 144 - 192)/25 = 2.080. The simple mean has variance 4.000; adding the covariance term is the product-estimator form; R instead of R^2 gives 0.160.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
When are the ratio and regression estimators equally efficient in large samples?
B. When the population regression line passes through the origin
The regression estimator's variance is S_y^2(1 - rho^2)/n; the ratio estimator reaches it only when beta = R = Y-bar/X-bar, i.e. the line passes through the origin. With an intercept, regression is better.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
With C_x = 0.2 and C_y = 0.5, for which correlations is the product estimator more precise than the simple mean (large samples)?
C. rho < -0.20
The product estimator beats the mean when rho < -C_x/(2 C_y) = -0.2/(2 x 0.5) = -0.20. rho > 0.20 is the ratio estimator's condition; swapping the CVs gives -1.25.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Two strata have N_h = 200 and 100, samples n_h = 20 and 20, and S_h^2 = 16 and 36. What is the variance of the stratified sample mean (with fpc)?
B. 0.4800
W_h = 2/3 and 1/3. V = sum W_h^2 (1 - f_h) S_h^2 / n_h = (4/9)(0.9)(16/20) + (1/9)(0.8)(36/20) = 0.3200 + 0.1600 = 0.4800. Dropping the fpc gives 0.5556; using W_h instead of W_h^2 gives 0.9600.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Ten here, 25 in the app. The Laisha app has every practice question we have written to Indian Statistical Service Examination in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.
Free for ten days, then one subscription of US$10 a month. We would rather say so here than have you find it later.
An AI operating system for agriculture: free guides, manuals and tools written for every country.