Management 2022 Paper II 50 marks Solve

Q2

(a) The area sales manager of a company has compiled the following data for the eight territories under his/her supervision. All the territories have similar size and consumer characteristics. The sales manager believes that the number of selling agents assigned to a territory has an impact on its sales revenue: (i) Find the Pearson's correlation coefficient between the two variables mentioned above. Is the sales manager correct in his/her belief? (ii) Develop a linear regression model and using it, predict the sales in a territory if 16 selling agents are assigned to it. (15 marks) (b) A city administration conducted a study of the waiting time in the emergency wings of three hospitals. These hospitals are located in three zones of the city far away from each other. The administration is interested in reducing the waiting time at the emergency wings. To study this, a random sample of 10 emergency wing cases at each hospital was selected on a particular day and the waiting time was measured. The results are recorded in the following table. At 0·05 level of significance, is there evidence of a difference in the average waiting times in the three hospitals? (Relevant table is attached at the end of this Paper) (15 marks) (c) Agrofarms Ltd. is a company engaged in large-scale cultivation of organic vegetables, grains and cereals. In 200 acres of land, Agrofarms Ltd. grows only tomatoes and onions. For the upcoming season, it estimates that it can make a profit of ₹ 7,000 per acre of tomatoes and ₹ 2,000 per acre of onions. During this growing season, each acre of tomatoes will require 4 tons of fertilizers and 3 tons of pesticides, whereas each acre of onions will require 2 tons of fertilizers and 1 ton of pesticides. Agrofarms Ltd. has contracted for at most 600 tons of fertilizers and 330 tons of pesticides. (i) How many acres of land should be devoted to each crop to maximise its profit for the season? Is there any land that remains unfarmed? (10 marks) (ii) What is the minimum profit per acre of onions that would make it economically feasible for Agrofarms Ltd. to grow onions? (6 marks) (iii) If the profit per acre of onions were ₹ 2,500, how many acres of land should Agrofarms Ltd. plant of each crop to maximize the profit for the season? (4 marks)

हिंदी में प्रश्न पढ़ें

(a) एक कम्पनी के क्षेत्रीय विक्रय प्रबन्धक ने अपने पर्यवेक्षण के अन्तर्गत आने वाले आठ क्षेत्रों के लिए निम्नलिखित आँकड़े संकलित किए हैं। सभी क्षेत्रों का आकार एवं उपभोक्ता विशेषताएँ समान हैं। विक्रय प्रबन्धक मानता है कि एक क्षेत्र हेतु नियत विक्रय अभिकर्ताओं की संख्या ने उसकी विक्रय आमदनी को प्रभावित किया है: (i) उपर्युक्त दो चरों के बीच पियर्सन सहसम्बन्ध गुणांक निकालिए। क्या विक्रय प्रबन्धक का अपना विचार सही है? (ii) एक रेखीय प्रतिगमन मॉडल विकसित कीजिए और इसका प्रयोग करते हुए किसी क्षेत्र की बिक्री का पूर्वानुमान कीजिए, यदि 16 विक्रय अभिकर्ताओं को उस क्षेत्र में नियत किया जाय। (15 अंक) (b) एक शहर प्रशासन ने तीन अस्पतालों के आकस्मिक खंड में प्रतीक्षा समय का अध्ययन किया। ये अस्पताल एक-दूसरे से बहुत दूर शहर के तीन क्षेत्रों (जोन) में स्थित हैं। प्रशासन आकस्मिक खंडों में प्रतीक्षा समय कम करना चाहता है। इस अध्ययन हेतु एक विशेष दिन प्रत्येक अस्पताल के आकस्मिक खंड से 10 रोगियों को यादृच्छिक प्रतिदर्श के रूप में चुना गया और प्रतीक्षा समय को मापा गया। निम्नलिखित तालिका में परिणामों को अभिलिखित किया जाता है। 0·05 सार्थकता स्तर पर, क्या तीनों अस्पतालों में औसत प्रतीक्षा समय में अंतर का प्रमाण है? (उपयुक्त सारणी इस पत्र के अंत में संलग्न है) (15 अंक) (c) एग्रोफार्म्स लिमिटेड एक कम्पनी है जो बड़े पैमाने पर ऑर्गेनिक सब्जियों एवं अनाजों की खेती करती है। 200 एकड़ की जमीन पर एग्रोफार्म्स लिमिटेड केवल टमाटर और प्याज उगाती है। आने वाले मौसम हेतु यह अनुमान लगाती है कि टमाटर से प्रति एकड़ ₹ 7,000 और प्याज से प्रति एकड़ ₹ 2,000 लाभ कमा सकती है। इस फसली मौसम में टमाटर के लिए प्रति एकड़ 4 टन खादों एवं 3 टन कीटनाशक दवाओं की आवश्यकता है, जबकि प्याज के लिए प्रति एकड़ 2 टन खादों एवं 1 टन कीटनाशक दवाओं की आवश्यकता होगी। एग्रोफार्म्स लिमिटेड ने अधिक से अधिक 600 टन खादों और 330 टन कीटनाशक दवाओं के लिए अनुबन्ध किया है। (i) इस मौसम में अपने लाभ को अधिकतम करने के लिए प्रत्येक फसल के लिए कितने एकड़ निश्चित करना चाहिए? क्या कोई भूमि है जो अकृषित रही है? (10 अंक) (ii) प्रति एकड़ प्याज से न्यूनतम लाभ क्या है ताकि एग्रोफार्म्स लिमिटेड के लिए आर्थिक रूप से प्याज की खेती संभव हो? (6 अंक) (iii) यदि प्याज से प्रति एकड़ लाभ ₹ 2,500 हो, तो एग्रोफार्म्स लिमिटेड को इस मौसम में लाभ को अधिकतम करने के लिए प्रत्येक फसल की खेती कितने एकड़ जमीन में करने की आवश्यकता है? (4 अंक)

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Directive word: Solve

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How this answer will be evaluated

Approach

Solve this numerical problem by allocating time proportionally to marks: ~30% for part (a) correlation-regression (15 marks), ~30% for part (b) ANOVA (15 marks), and ~40% for part (c) linear programming (20 marks). Begin with clear problem identification, show all computational steps with formulas, present results in structured tables, and end with interpretation of findings for managerial decision-making.

Key points expected

  • Part (a)(i): Calculate Pearson's r using correct formula with covariance and standard deviations; interpret magnitude and direction of correlation; test significance to validate manager's belief
  • Part (a)(ii): Derive regression equation Y = a + bX with proper least-squares method; calculate intercept and slope accurately; predict sales for 16 agents with confidence
  • Part (b): Set up ANOVA table with correct degrees of freedom (between groups k-1=2, within groups N-k=27, total N-1=29); compute F-statistic; compare with critical F-value at α=0.05; draw conclusion about hospital waiting times
  • Part (c)(i): Formulate LP problem with objective function Z = 7000T + 2000O and constraints (land: T+O≤200, fertilizer: 4T+2O≤600, pesticide: 3T+O≤330, non-negativity); solve graphically or by corner-point method; identify optimal solution and check for slack in land constraint
  • Part (c)(ii): Calculate shadow price or perform sensitivity analysis to find minimum onion profit that changes optimal basis (breakeven where reduced cost becomes zero)
  • Part (c)(iii): Re-solve LP with revised objective Z = 7000T + 2500O; identify new optimal corner point and interpret changed resource allocation

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5All formulas applied correctly: Pearson's r with proper covariance calculation, regression coefficients via least squares, ANOVA with correct SS partitioning and F-ratio, LP with accurate constraint formulation and graphical/simplex solution; all arithmetic accurate to 2 decimal placesCorrect formulas identified but minor computational errors in 1-2 parts; correct final answers but missing intermediate steps; ANOVA degrees of freedom or LP constraint formulation slightly flawedMajor formula errors (e.g., using population instead of sample standard deviation, wrong ANOVA df, incorrect LP objective direction); computational mistakes affecting conclusions; missing critical steps
Framework citation15%7.5Explicitly names Karl Pearson for correlation, Gauss-Markov theorem assumptions for regression, R.A. Fisher's ANOVA framework, and George Dantzig's simplex method for LP; cites relevant statistical tables (t, F) with proper degrees of freedomMentions general statistical methods without specific attribution; uses correct table values but doesn't name sources; recognizes LP as operations research technique without naming DantzigNo framework attribution; confuses methods (e.g., calls ANOVA 'chi-square test'); uses wrong statistical tables or significance levels
Case / Indian example15%7.5Contextualizes findings with Indian agribusiness (e.g., comparing Agrofarms to ITC's e-Choupal or NABARD farm planning), cites Indian hospital waiting time studies (AIIMS, CMC Vellore benchmarks), or references NSSO agricultural data for validationBrief mention of Indian agricultural sector or public health context without specific data; generic reference to 'Indian conditions' without elaborationNo Indian context; purely abstract mathematical treatment; ignores managerial relevance to Indian business environment
Multi-perspective analysis25%12.5For (a): discusses correlation vs. causation, outliers, and confidence intervals; for (b): examines Type I/II errors, post-hoc tests need, and practical significance; for (c): analyzes sensitivity, shadow prices, and scenario planning with multiple constraint changesBasic interpretation of each result in isolation; limited discussion of assumptions or limitations; no integration across statistical methodsPure calculation with no interpretation; ignores assumptions (linearity, independence, normality); fails to discuss what results mean for managers
Conclusion & recommendation20%10Clear actionable recommendations: for sales manager on agent deployment with confidence bounds; for city administration on hospital resource reallocation with priority ranking; for Agrofarms on crop mix, unfarmed land utilization, and contingency plans for price volatilityRestates numerical findings without clear recommendations; generic advice not tied to calculated results; missing priority ranking or implementation sequenceNo conclusion section; answers end with final number; no managerial implications drawn; contradictory recommendations

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