ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR
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)
ICAR AICE-JRF/SRF Agricultural Statistics (Unit 5)
Sampling versus complete enumeration, probability sampling, SRS, stratified sampling with allocation, choice and construction of strata and collapsing of strata, ratio and regression estimation and systematic sampling: estimator properties, variance formulas and efficiency conditions.
How our questions for ICAR AICE-JRF/SRF spread across the units of Sampling Theory I: Simple, Stratified, Systematic, Ratio & Regression Estimation. Difficulty: 5 easy, 39 medium, 26 hard.
| Stratified sampling: estimation and allocation (proportional, Neyman, optimum) | 11 | |
| Choice of strata, construction of strata boundaries and collapsing of strata | 10 | |
| Ratio estimators: bias, MSE and comparison | 9 | |
| Regression estimators | 9 | |
| Product and difference estimators | 8 | |
| Systematic sampling and its variants | 7 | |
| Sampling versus complete enumeration; sampling and non-sampling errors | 5 | |
| Estimation of proportions and sample-size determination | 5 | |
| Probability sampling, sampling frames and inclusion probabilities | 3 | |
| Simple random sampling with and without replacement | 3 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Milk yield per animal has a coefficient of variation of 40%. How many animals must be sampled (fpc ignored) to estimate the mean within 5% of its value with 95% confidence (z = 1.96)?
C. 246
n0 = z^2 C^2 / r^2 = 3.8416 x 0.16 / 0.0025 = 245.86, rounded up to 246. Dropping z^2 gives 64; taking square roots throughout gives 16; not squaring C gives 615.
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
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
Why are strata sometimes collapsed in pairs at the analysis stage?
A. One unit per stratum leaves no within-stratum variance estimate
With a single unit per stratum, s_h^2 cannot be computed; pairing similar strata lets the variance be estimated from differences within pairs. The resulting estimate is conservative, not unbiased.
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
Under which model is the ratio estimator the best linear unbiased estimator?
D. A line through the origin with the variance of y proportional to x
The ratio estimator is BLUE when E(y) = beta x and V(y) is proportional to x, a model that often suits crop production against area. With a large intercept the regression estimator is better; with no relation the simple mean is.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
How is the variance of the regression estimator estimated from an SRS of n units?
A. [(1 - f)/n] times the residual sum of squares divided by n - 2
The residual mean square, sum of squared residuals over n - 2 (two parameters estimated), estimates S_y^2(1 - rho^2). The total SS over n - 1 estimates S_y^2 and ignores the regression; the regression SS is the explained part.
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
For the difference estimator y-bar + k(X-bar - x-bar), the population covariance S_yx = 36 and S_x^2 = 24. What value of k minimises the variance?
D. 1.50
V is minimised at k = S_yx / S_x^2 = 36/24 = 1.50, the population regression coefficient; the minimum equals the regression estimator's variance. k = 1 is a convenient choice, not the optimum here.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
Which statement about the product estimator y-bar x-bar / X-bar is correct?
C. Biased, with first-order bias proportional to S_yx / X-bar
E(y-bar x-bar) = Y-bar X-bar + Cov(y-bar, x-bar), so the bias is Cov(y-bar, x-bar)/X-bar = [(1 - f)/n] S_yx / X-bar. It is meant for negative correlation, where it reduces variance.
Further reading: Cochran, Sampling Techniques; Sukhatme et al., Sampling Theory of Surveys with Applications
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