Q7
(a) Explain the concept of index number. Calculate the Fisher's ideal index number from the following data and verify that whether it satisfies time reversal and factor reversal tests : (10 marks) (b) The population growth of a city is modelled using logistic growth model with a carrying capacity of K = 10000000. The population data (in thousands) is provided at 2-year intervals from 2014 (taken as t = 0) to 2024 (t = 10) : (i) Estimate the two parameters of the logistic growth model. (16 marks) (ii) Using the estimated model, predict the population of the city for the year 2026. (4 marks) (16+4=20 marks) (c) Discuss the agricultural statistics relating to area and yield in our country. Also, point out the need and importance of agricultural statistics. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) सूचकांक की संकल्पना की व्याख्या कीजिए। निम्नलिखित आँकड़ों से फिशर के आदर्श सूचकांक की गणना कीजिए और सत्यापित कीजिए कि क्या यह कालोत्क्रमण तथा उपादान उत्क्रमण परीक्षणों को संतुष्ट करता है : (10 अंक) (b) बृद्धियत बृद्धि मॉडल, जिसकी वहन क्षमता K = 10000000 है, का उपयोग करते हुए किसी शहर की जनसंख्या बृद्धि का मॉडल तैयार किया गया। जनसंख्या आँकड़े (हजारों में), 2 वर्ष के अंतराल पर 2014 (t = 0) से 2024 (t = 10) तक दिए गए हैं : (i) बृद्धियत बृद्धि मॉडल के दो प्राचलों का आकलन कीजिए। (16 अंक) (ii) आकलित मॉडल का उपयोग करते हुए वर्ष 2026 के लिए शहर की जनसंख्या का प्रक्षेपण कीजिए। (4 अंक) (16+4=20 अंक) (c) हमारे देश में क्षेत्रफल तथा उपज से संबंधित कृषि सांख्यिकी की विवेचना कीजिए। कृषि सांख्यिकी की आवश्यकता तथा महत्व को भी इंगित कीजिए। (15 अंक)
Directive word: Calculate
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How this answer will be evaluated
Approach
Begin with a concise definition of index numbers for part (a), then proceed to calculate Fisher's ideal index with proper data tabulation and test verification. For part (b), set up the logistic model linearization, estimate parameters using regression on transformed data, then predict for 2026. For part (c), structure the discussion around India's agricultural statistical system—mentioning Land Use Statistics, Area and Production Statistics, and agencies like DES and NSSO. Allocate approximately 20% time to (a), 45% to (b), and 35% to (c) based on marks distribution.
Key points expected
- Part (a): Correct formula for Fisher's ideal index as geometric mean of Laspeyres and Paasche; proper calculation with given data; verification of time reversal (P01 × P10 = 1) and factor reversal (P01 × Q01 = Value ratio)
- Part (b)(i): Linearization of logistic model as ln[(K-P)/P] = lnβ - αt; estimation of α and β via least squares on transformed variables; correct handling of K=10000 (in thousands)
- Part (b)(ii): Substitution of t=12 (for year 2026) into estimated logistic equation; proper back-transformation to obtain population prediction
- Part (c): Discussion of area statistics—gross sown area, net sown area, cropping intensity; yield statistics—yield per hectare, production estimates; mention of Timely Reporting Scheme and Crop Cutting Experiments
- Part (c): Need for agricultural statistics—food security planning, MSP fixation, crop insurance (PMFBY), export-import policy; importance for Sustainable Development Goals and Doubling Farmers' Income initiative
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 9 | Correctly identifies Fisher's formula components for (a); properly sets up logistic linearization with correct variable transformation for (b); accurately describes India's agricultural statistical framework including DES, NSSO, and State Agricultural Statistics Authorities for (c) | Partially correct setup with minor formula errors or missing one agency in agricultural statistics; logistic transformation attempted but with algebraic slips | Incorrect formula identification for Fisher's index; fails to linearize logistic model; vague or incorrect description of agricultural statistical system |
| Method choice | 20% | 9 | Selects geometric mean approach for Fisher's index; uses OLS/linear regression on transformed logistic data; cites specific methodologies like Crop Cutting Experiments and Remote Sensing for agricultural statistics | Correct method chosen but execution lacks rigor; mentions general statistical methods without specificity; basic description of agricultural data collection | Wrong method selected (e.g., simple average for Fisher's); attempts direct nonlinear estimation for logistic model; generic discussion without methodological depth |
| Computation accuracy | 20% | 9 | Precise arithmetic for Fisher's index with correct intermediate steps; accurate regression calculations for α and β with proper significant figures; correct prediction for 2026 with error consideration | Generally correct calculations with minor arithmetic errors; regression estimates approximately correct; prediction follows from stated parameters | Major computational errors in index calculation; incorrect parameter estimates; prediction completely off due to calculation mistakes |
| Interpretation | 20% | 9 | Explains why Fisher's index is 'ideal' via test satisfaction; interprets logistic parameters (α as growth rate, β as initial condition ratio); critically evaluates gaps in Indian agricultural statistics like delayed data, sampling errors, and need for GIS integration | Basic interpretation of results without deeper insight; mentions tests without explaining their significance; descriptive coverage of agricultural statistics without critical evaluation | No interpretation of calculated values; fails to explain what parameters represent; purely factual listing without analysis of agricultural statistics |
| Final answer & units | 20% | 9 | Clear presentation of Fisher's index number with test verification summary; stated logistic equation with parameters and 2026 prediction in correct units (thousands or converted to actual population); well-structured conclusion on agricultural statistics with policy relevance | Answers present but poorly organized; units inconsistent or missing in places; conclusion lacks synthesis | Missing final answers; incorrect units throughout (e.g., not converting thousands); no concluding remarks or completely irrelevant conclusion |
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