(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

(a) Using Karl Pearson's product-moment correlation coefficient, let X = test score and Y = sales ('000). For n = 10:
ΣX = 520, ΣY = 555, ΣX² = 28076, ΣY² = 31395, ΣXY = 29339.
Means: X̄ = 52, Ȳ = 55.5.
r = (nΣXY − ΣXΣY) / √[(nΣX² − (ΣX)²)(nΣY² − (ΣY)²)].
nΣXY = 293390, ΣXΣY = 288600, so numerator = 4790.
nΣX² − (ΣX)² = 280760 − 270400 = 10360.
nΣY² − (ΣY)² = 313950 − 308025 = 5925.
Denominator = √(10360 × 5925) = √61,383,000 = 7834.73.
r = 4790/7834.73 = 0.6114, i.e. r = 479/√613830 ≈ 0.611.
This is a moderate positive relationship. r² = 0.374, so test score explains about 37.4% of variation in sales. For n = 10, the 5% two-tail critical r ≈ 0.632; here r = 0.611, so it is not statistically significant at 5%. Also, a salesman with test score 33 had sales 58, showing poor test performance does not necessarily mean poor sales. Hence the company should not terminate any salesman solely because of poor test performance. The test may be used for training, counselling, probation, and re-testing, along with actual sales records.

(b) This is a binomial distribution problem. Let X = number of correct guesses. Here n = 9, p = 1/4, q = 3/4.
P(X = 6) = C(9,6)(1/4)⁶(3/4)³.
C(9,6) = C(9,3) = 84.
P = 84 × (1/4)⁶ × (3/4)³ = 84 × 27/4⁹ = 2268/262144 = 567/65536 ≈ 0.00865.
So the probability is 567/65536 ≈ 0.865%, assuming independent questions and constant p = 1/4.

(c)(i) Let x = number of boxes of type A and y = number of boxes of type B.
The LPP is:
Minimize C = 300x + 200y
subject to
3x + y ≥ 10 (chemical X)
2x + 2y ≥ 12, i.e. x + y ≥ 6 (chemical Y)
x + 2y ≥ 7 (chemical Z)
x ≥ 0, y ≥ 0; x, y integers.

Graphical method: the boundary lines are:
3x + y = 10: intercepts (10/3, 0), (0, 10)
x + y = 6: intercepts (6, 0), (0, 6)
x + 2y = 7: intercepts (7, 0), (0, 3.5).

The feasible region lies above all three lines. Key intersections:
3x + y = 10 and x + y = 6 give 2x = 4, so x = 2, y = 4. Check Z: 2 + 2(4) = 10 ≥ 7. Feasible. C = 300(2) + 200(4) = ₹1400.
3x + y = 10 and x + 2y = 7 give x = 13/5, y = 11/5, but x + y = 24/5 = 4.8 < 6, so not feasible.
x + y = 6 and x + 2y = 7 give y = 1, x = 5. Feasible. C = 300(5) + 200(1) = ₹1700.
Other boundary points: (0,10) gives C = ₹2000; (7,0) gives C = ₹2100.

The minimum occurs at (2, 4). Check: X = 3(2) + 1(4) = 10; Y = 2(2) + 2(4) = 12; Z = 1(2) + 2(4) = 10 ≥ 7. Therefore, the department should buy 2 boxes of A and 4 boxes of B, with minimum cost ₹1400.

(c)(ii) Sensitivity analysis studies how changes in costs or requirements affect the optimum. Here X and Y constraints are binding, while Z has surplus 10 − 7 = 3 units and zero shadow price. From dual/complementary slackness, with Z slack, shadow prices are ₹50 per extra unit of X and ₹75 per extra unit of Y. Thus small increases in X or Y raise minimum cost, but up to 3 extra units of Z cost nothing. If cost coefficients or requirements change beyond allowable ranges, the optimal mix may change.

(a) A company gives on-the-job training to its salesmen, which is followed by a test. The company is considering whether there is…

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