(a) A company gives on-the-job training to its salesmen, which is followed by a test. The company is considering whether there is any relationship of test scores and sales performance made by the ten salesmen during last one year, and also whether th

Question

(a) A company gives on-the-job training to its salesmen, which is followed by a test. The company is considering whether there is any relationship of test scores and sales performance made by the ten salesmen during last one year, and also whether the company should terminate the services of any salesman who does not perform well in the test:

| Test Score | 50 | 60 | 58 | 47 | 50 | 33 | 65 | 43 | 46 | 68 |
| Sales ('000) | 48 | 65 | 50 | 48 | 55 | 58 | 63 | 48 | 50 | 70 |

Give your comment whether the company should terminate the services of any salesman who does not perform well in the test. 15 marks
(b) A law graduate student randomly guesses at nine multiple-choice questions in All India Bar Examination. There are four possible answers for every question. However there is only one correct answer. Assuming that all questions are independent to each other, find the possibility that the student chooses six correct answers. 15 marks
(c) (i) For conducting Third Semester Practical Examination, the Chemistry department of a university affiliated college requires 10, 12 and 7 units of three chemicals X, Y and Z respectively. The chemicals are available in two boxes—box A and box B. Box A contains 3, 2 and 1 unit of X, Y and Z respectively and costs ₹ 300. Box B contains 1, 2 and 2 units of X, Y and Z respectively and costs ₹ 200.

Find how many boxes of each type should be purchased by the department so that the total cost is minimal by formulating the problem in linear programming problem and solve it through graphical method. 18 marks
(ii) Comment on the sensitivity analysis. 2 marks

UPSC Answer Check · Accepted Answer

Solve all four sub-parts systematically: for (a) calculate Karl Pearson's correlation coefficient and interpret its significance for HR decisions; for (b) apply binomial probability distribution with n=9, p=0.25; for (c)(i) formulate LPP with objective function Min Z=300A+200B subject to constraints 3A+B≥10, 2A+2B≥12, A+2B≥7, A,B≥0 and solve graphically; for (c)(ii) briefly explain sensitivity analysis regarding shadow prices and RHS changes. Allocate time proportionally: ~25% each for (a), (b), (c)(i), and ~5% for (c)(ii).
- For (a): Calculation of correlation coefficient (r ≈ 0.6-0.7), coefficient of determination, hypothesis testing at 5% significance, and reasoned recommendation on termination policy based on statistical vs. practical significance
- For (b): Identification of binomial distribution parameters (n=9, p=1/4, q=3/4), correct application of P(X=6) = ⁹C₆(1/4)⁶(3/4)³, and final probability calculation
- For (c)(i): Proper LPP formulation with decision variables, objective function, all three constraints, feasible region identification, corner point evaluation, and optimal solution (A=3, B=3, Z=₹1500 or equivalent)
- For (c)(ii): Definition of sensitivity analysis, explanation of shadow prices for constraints, and interpretation of allowable ranges for objective function coefficients
- Cross-part integration: Recognition that statistical tools in (a) and (b) inform managerial decisions, while (c) demonstrates operations research application in educational resource allocation

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	For (a): correct r calculation (~0.64), proper t-test for significance, r² interpretation; for (b): exact binomial formula application with correct arithmetic (⁹C₆=84); for (c): accurate constraint formulation, precise graphical solution with correct optimal point and Z-value; for (ii): correct shadow price concept	Minor arithmetic errors in correlation or probability; one constraint error in LPP; correct method but wrong final answer in 1-2 parts; vague sensitivity analysis	Wrong formula selection (e.g., Poisson instead of binomial); incorrect correlation method; major LPP formulation errors; missing or completely wrong sensitivity analysis
Framework citation	15%	7.5	Explicitly names Karl Pearson's correlation, coefficient of determination, Student's t-test for correlation significance; cites binomial probability distribution B(n,p); identifies standard LPP form, iso-cost lines, and corner point theorem; references sensitivity analysis as post-optimality analysis	Uses correct formulas without naming theorems; mentions correlation and binomial without formal framework; solves LPP without citing graphical method principles	No framework identification; confuses correlation with regression; uses normal approximation incorrectly; solves by trial without graphical method
Case / Indian example	10%	5	For (a): contextualizes to Indian HR practices like 'train-test-deploy' models in PSUs or IT sector; for (c): references similar resource allocation in Indian university systems (UGC-NET labs, NAAC accreditation requirements); mentions practical constraints in affiliated colleges under state universities	Brief mention of Indian educational/employment context without specific integration; generic references to 'organizations' or 'institutions'	No Indian context; purely abstract mathematical treatment; irrelevant international examples
Multi-perspective analysis	25%	12.5	For (a): balances statistical significance (r value) with practical HR considerations (experience, market conditions, sample size limitations); for (b): discusses assumption independence and real-world guessing patterns; for (c): considers integer constraints, alternative procurement scenarios, and trade-offs between cost and chemical surplus; for (ii): explains how sensitivity analysis guides re-optimization	Acknowledges one alternative perspective per part; mentions limitations without elaboration; basic sensitivity interpretation	Single perspective only; no discussion of assumptions, limitations, or alternatives; ignores managerial implications of statistical results
Conclusion & recommendation	25%	12.5	For (a): nuanced recommendation—do not terminate solely based on test (moderate correlation, other factors matter), suggest using test as one input in 360-degree evaluation; for (b): interprets probability meaningfully (very low chance, ~1%); for (c): clear procurement decision with cost justification; for (ii): practical implications of sensitivity for budget planning; integrated closing statement on quantitative methods in management	Direct answers without justification; generic recommendations; partial conclusions for some parts	No conclusions; contradictory recommendations; missing answers for sub-parts; purely mathematical ending without managerial interpretation

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