State statistical services: Statistical Officer / Assistant Statistical Officer (State PSCs and selection boards) · India · State Public Service Commissions and Subordinate Service Selection Boards
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 State statistical services spread across the units of Sampling Theory I: Simple, Stratified, Systematic, Ratio & Regression Estimation. Difficulty: 23 easy, 19 medium, 2 hard.
| Simple random sampling with and without replacement | 8 | |
| Sampling versus complete enumeration; sampling and non-sampling errors | 6 | |
| Stratified sampling: estimation and allocation (proportional, Neyman, optimum) | 6 | |
| Estimation of proportions and sample-size determination | 5 | |
| Systematic sampling and its variants | 5 | |
| Product and difference estimators | 5 | |
| Probability sampling, sampling frames and inclusion probabilities | 4 | |
| Ratio estimators: bias, MSE and comparison | 3 | |
| Regression estimators | 2 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Why must crop yield be estimated from a sample rather than a complete enumeration of fields?
C. Measuring yield by harvesting plots is destructive and very costly
Crop-cutting involves harvesting, threshing and weighing a plot, which is expensive and destroys the crop on it; doing it on every field is impossible. A complete enumeration has no sampling error at all.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
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
A population of 10 units has S^2 = 20 (divisor N - 1). What is sigma^2 (divisor N)?
B. 18.0
sigma^2 = (N - 1) S^2 / N = 9 x 20 / 10 = 18.0. Multiplying by N/(N - 1) goes the wrong way (22.2).
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
Strata contain 600, 300, 100 holdings. A sample of 50 is allocated proportionally. How many holdings are sampled from each stratum?
A. 30, 15, 5
Proportional allocation: n_h = n N_h / N = 50 x (600, 300, 100)/1000 = 30, 15, 5. 17, 17, 16 is equal allocation; the other splits follow no rule.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Strata have sizes 600, 300, 100 and standard deviations 2, 4, 10. A total sample of 68 is allocated by Neyman allocation. What are the stratum sample sizes?
B. 24, 24, 20
n_h = n N_h S_h / sum N_h S_h; N_h S_h = [1200, 1200, 1000], sum 3400; n_h = 68 x (1200, 1200, 1000)/3400 = 24, 24, 20. Proportional allocation would give 41, 20, 7; allocating by S_h alone ignores stratum size.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
When does stratified random sampling with proportional allocation give the largest gain over simple random sampling of the same size?
C. When stratum means differ widely and units within strata are alike
The gain of proportional allocation over SRS is about [(1 - f)/n] sum W_h (Y-bar_h - Y-bar)^2, the between-stratum variation; it is largest when strata are internally homogeneous and differ from one another.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
A sample of villages gives total wheat production 1440 t on 1200 ha of wheat. The district has 5000 ha of wheat. What is the ratio estimate of district production?
A. 6000 t
R-hat = 1440/1200 = 1.2 t/ha; total = R-hat X = 1.2 x 5000 = 6000 t. Inverting the ratio gives 4167 t.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Using the difference estimator with k = 1, a sample gives y-bar = 40 and x-bar = 35, and the population mean of x is 38. What is the estimate of the mean of y?
A. 43.0
y-bar_D = y-bar + k(X-bar - x-bar) = 40 + (38 - 35) = 43.0. Ratio and product forms give 43.43 and 36.84.
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
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