CUET-PG - Statistics · India · National Testing Agency
Original practice questions written to the published syllabus · not past papersMethods that hold without a normal model: the empirical distribution, goodness-of-fit, sign and rank tests for one, two and many samples, rank correlation, and tests used for scored or ranked agricultural data such as disease ratings and sensory panels.
In the exams: AICE-JRF Ag Stat Unit 2 (heavy: run, sign, rank, Wilcoxon, Kruskal-Wallis, Mann-Whitney, Cochran, Friedman named); GATE ST 'Non-parametric Statistics' (heavy); UPSC ISS Statistics II (present)
How our questions for CUET-PG spread across the units of Nonparametric & Rank-Based Methods. Difficulty: 31 easy, 29 medium, 9 hard.
| Mann-Whitney U test and two-sample Kolmogorov-Smirnov test | 9 | |
| Goodness of fit: chi-square and Kolmogorov-Smirnov tests | 7 | |
| Friedman's test for randomised blocks | 7 | |
| Asymptotic relative efficiency | 7 | |
| Empirical distribution function and its properties | 6 | |
| Sign test and Wilcoxon signed-rank test | 6 | |
| Kruskal-Wallis test | 6 | |
| Spearman's and Kendall's rank correlations | 6 | |
| Analysing ordinal scores: disease ratings and sensory panels | 6 | |
| Run test for randomness | 5 | |
| Cochran's Q test | 4 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Which statement compares Pearson's chi-square and the Kolmogorov-Smirnov goodness-of-fit tests correctly?
B. KS uses ungrouped continuous data; chi-square suits counts and grouped data
KS is exact and distribution-free for continuous, fully specified F₀ and avoids grouping. Chi-square handles discrete data and estimated parameters (with df adjustment) but loses information by grouping and ignores the order of classes.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
In a chi-square goodness-of-fit test of insect counts, several classes have expected frequencies below 5. What is the usual remedy?
B. Pool adjacent classes, then count df after pooling
Small expected counts make the χ² approximation unreliable; adjacent classes are pooled and the df counted on the pooled classes. Dropping classes discards data and biases the test.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
In a runs test for randomness, what do far too many runs suggest?
D. Alternation or negative dependence
Many short runs mean the two symbols alternate more than chance allows, as with regular spacing or negative serial correlation. Too few runs indicate clustering or trend.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What is a weakness of the Wald-Wolfowitz two-sample runs test compared with the Mann-Whitney test?
D. Low power against a pure location shift
The runs test reacts to any difference in distribution, but for a location shift Mann-Whitney is much more powerful. Neither needs normality, and the runs test does detect spread differences.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What does the Wilcoxon signed-rank test assume about the distribution of the paired differences that the sign test does not?
A. Symmetry of the differences
Both need independence and continuity; the signed-rank test also assumes symmetry, which is what lets each rank carry either sign with probability 1/2. Neither assumes normality.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Two independent samples have sizes 6 and 6, and sample 1's rank sum is W₁ = 25. What is the Mann-Whitney statistic U₁?
A. 4
U₁ = W₁ − n₁(n₁ + 1)/2 = 25 − 21 = 4. U₂ = n₁n₂ − U₁ = 32. 14 = 25 − 11 omits the correct offset.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Six treated and six untreated plants are scored for disease; the Mann-Whitney U for the treated group is 4. What is the normal-approximation Z, ignoring ties and continuity correction?
C. −2.24
E(U) = 36/2 = 18, Var(U) = 36 × 13/12 = 39, Z = (4 − 18)/√39 = −2.24. Using 36/4 as the variance gives −4.67.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
For k = 2 groups, how is the Kruskal-Wallis H related to the Mann-Whitney test?
D. H equals the square of the Mann-Whitney Z
With two groups H is the squared standardised rank-sum statistic, Z², and χ²₁ is the distribution of Z² for normal Z.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Which design is Friedman's test intended for?
C. An RCBD with one observation per treatment per block
Friedman ranks treatments within blocks, the rank analogue of the RCBD ANOVA. Binary blocked data use Cochran's Q.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
How is the Mann-Whitney test related to the proportional-odds model for comparing two groups on an ordinal scale?
A. It is closely related to the score test for the group effect in that model
For two groups, the score test of β = 0 in the cumulative logit model is essentially the Wilcoxon rank-sum test with mid-ranks, which is why the two usually agree; the model adds odds-ratio estimates and covariates.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
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