(a) The following table gives the production in three shifts and the corresponding number of defective goods that turned out in three weeks. Test at 5% level of significance whether weeks and shifts are independent.

Table :

| Shift | 1st week | 2nd

Question

(a) The following table gives the production in three shifts and the corresponding number of defective goods that turned out in three weeks. Test at 5% level of significance whether weeks and shifts are independent.

Table :

| Shift | 1st week | 2nd week | 3rd week | Total |
|-------|---------|---------|---------|-------|
| I | 20 | 20 | 20 | 60 |
| II | 5 | 10 | 15 | 30 |
| III | 15 | 20 | 25 | 60 |
| Total | 40 | 50 | 60 | 150 |

(15 marks)

(b) A random sample of 10 bags of urea are found to have the following weights (kg) : 40, 45, 49, 50, 55, 44, 52, 48, 50, 45

Test at 5% level of significance whether the average packing weight can be taken as 50 kg. (15 marks)

(c) A firm produces three products A, B and C which pass through three departments : Milling, Lathe and Grinder.

Each unit of Product A requires 8, 4 and 2 hours respectively in the three departments.

Each unit of Product B requires 2, 3 and 0 hours respectively in the three departments.

Each unit of Product C requires 3, 0 and 1 hours respectively in the three departments.

The available capacity on the machines which might limit output is given below.

Table : Capacity of the Machines for Production

| Machine Type | Available time (in machine hours per week) |
|---|---|
| Milling Machine | 250 |
| Lathe Machine | 150 |
| Grinder Machine | 50 |

The unit profit would be : ₹ 20, ₹ 6 and ₹ 8 for products A, B and C respectively.

(i) Formulate the problem in LPP.

(ii) How much of each product should the firm produce in order to maximize profit ? (8+12=20 marks)

UPSC Answer Check · Accepted Answer

The directive 'solve' demands precise numerical working with clear methodological steps. Allocate approximately 30% time to part (a) Chi-Square test (15 marks), 30% to part (b) t-test (15 marks), and 40% to part (c) LPP formulation and solution (20 marks). Structure each part with: null/alternative hypotheses or objective function, step-by-step calculations with formulae, critical value comparison, and final decision. No traditional introduction or conclusion needed; begin directly with part (a) and end with final LPP solution.
- Part (a): Calculate expected frequencies for all 9 cells using (Row Total × Column Total)/Grand Total; compute Chi-Square statistic Σ(O-E)²/E; compare with χ² critical value at df=4, α=0.05 (9.488); conclude independence or dependence
- Part (b): Calculate sample mean (47.8 kg) and sample standard deviation; compute t-statistic using formula (x̄-μ)/(s/√n); compare with t-critical at df=9, α=0.05 (two-tailed ±2.262); conclude whether population mean can be 50 kg
- Part (c)(i): Formulate correct LPP with decision variables X₁, X₂, X₃ for products A, B, C; objective function Max Z = 20X₁ + 6X₂ + 8X₃; constraints: 8X₁+2X₂+3X₃ ≤ 250, 4X₁+3X₂ ≤ 150, 2X₁+X₃ ≤ 50; non-negativity conditions
- Part (c)(ii): Solve using Simplex method or graphical/simplex approach; identify optimal solution with values of X₁, X₂, X₃ and maximum profit; verify slack variables and binding constraints
- Correct application of statistical tables and critical values; proper degrees of freedom calculation for each test
- Clear presentation of calculations with proper units and interpretation of results in management context

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	All three statistical tests and LPP solved with correct formulae, accurate arithmetic, proper degrees of freedom, and valid conclusions; Chi-Square expected frequencies and t-test standard deviation calculated precisely; LPP constraints correctly translated from problem data	Major concepts correct but minor calculation errors in 1-2 parts; correct hypotheses/formulation but arithmetic mistakes in final values; df or critical values occasionally wrong	Fundamental errors in formulae (e.g., wrong Chi-Square formula, incorrect t-test denominator); wrong LPP formulation with reversed inequalities or missing constraints; conclusions contradict calculations
Framework citation	15%	7.5	Explicitly states statistical frameworks: Pearson's Chi-Square test of independence, Student's t-test for single sample mean, and Simplex method/Linear Programming; cites appropriate tables (χ², t-distribution) with correct parameters	Uses correct methods implicitly without naming frameworks; mentions some tables but not systematically; framework awareness present but not articulated	No identification of statistical methods used; confuses frameworks (e.g., calls Chi-Square a t-test); no reference to distribution tables or decision rules
Case / Indian example	10%	5	Where applicable, contextualizes LPP solution with Indian manufacturing scenario (e.g., MSME production planning, Make in India efficiency) or notes relevance of quality control tests for Indian agri-input firms like IFFCO for urea bags	Brief mention of Indian context in conclusion or introduction without integration into numerical work; generic reference to quality control in Indian industry	No Indian context provided; purely mechanical solution without any application awareness
Multi-perspective analysis	25%	12.5	For (a), interprets what dependence between shifts and weeks means operationally; for (b), considers Type I/II error implications in quality control; for (c), analyzes shadow prices, sensitivity of solution, and practical constraints beyond mathematical optimum	Basic interpretation of each result without deeper operational analysis; mentions one limitation or assumption per part without elaboration	Pure calculation with zero interpretation; no discussion of what results mean for production decisions or quality management
Conclusion & recommendation	25%	12.5	Clear decision statements for each test: reject/fail to reject H₀ with justification; specific optimal production quantities for A, B, C with maximum profit figure; actionable recommendations (e.g., which shift needs monitoring, whether urea packing machine needs calibration, resource reallocation for LPP)	Conclusions present but vague or partially incorrect; optimal values stated without clear decision rule application; recommendations generic	Missing conclusions for one or more parts; conclusions contradict calculations; no recommendations; incomplete LPP solution without final profit figure

Q2

Directive word: Solve

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