(a) Explain the concept of index number. Calculate the Fisher's ideal index number from the following data and verify that…

(a) An index number is a statistical device that measures relative change in a variable or group of variables, such as prices, quantities, or production, from a base period to another period, usually expressed as a percentage with base value 100. It is dimensionless, allows comparison over time, and is used to deflate monetary values, compare cost of living, and measure productivity. Index numbers may be simple or aggregate; here an aggregate index is needed because five commodities are combined. The Fisher index is preferred because it uses both base and current quantities and satisfies the two important tests. Here base year is 2006 and current year 2007.

For the data:

• Σp₀q₀ = 400+550+420+224+180 = 1774
• Σp₁q₀ = 480+550+510+280+195 = 2015
• Σp₀q₁ = 450+572+420+232+240 = 1914
• Σp₁q₁ = 540+572+510+290+260 = 2172

Laspeyres price index L₀₁ = (2015/1774)×100 = 113.585. Paasche price index P₀₁ = (2172/1914)×100 = 113.480. Fisher’s ideal index F₀₁ = √(L₀₁P₀₁) = 100√[(2015×2172)/(1774×1914)] = 100√(364715/282953) ≈ 113.53.

Time reversal test: F₀₁F₁₀ = 100². Here F₁₀ = 100√(282953/364715) ≈ 88.08, so 113.53×88.08 ≈ 10000. Hence it satisfies the time reversal test.

Factor reversal test: Fisher quantity index Fq₀₁ = 100√[(1914/1774)(2172/2015)] = 100√(2078604/1787305) ≈ 107.84. Value index V₀₁ = (2172/1774)×100 = 122.435. In percentage form, F₀₁×Fq₀₁ = 100×V₀₁; using unrounded values, 113.532×107.843 ≈ 12243.5. Hence it satisfies the factor reversal test.

(b) (i) Use the logistic model dP/dt = rP(1−P/K), K = 10000000 = 10000 thousand. Its solution is P(t)=K/(1+Ae^(−rt)), where A=(K−P(0))/P(0). Taking logs gives y=ln[(K−P)/P]=ln A−rt, a straight line in t. Estimate a=ln A and b=−r by least squares, minimizing squared residuals in y. This is valid because 0<P<K, so the logarithm is defined. The carrying capacity K is fixed, so only r and A are estimated.

For P in thousands:

• t=0, P=1000, y=ln(9000/1000)=2.197225
• t=2, P=1785, y=ln(8215/1785)=1.526543
• t=4, P=2575, y=ln(7425/2575)=1.059003
• t=6, P=3400, y=ln(6600/3400)=0.663294
• t=8, P=4900, y=ln(5100/4900)=0.040005
• t=10, P=6200, y=ln(3800/6200)=−0.489548

Sums: n=6, Σt=30, Σt²=220, Σy=4.996522, Σty=6.693425. The normal equations come from differentiating Σ(y−a−bt)² with respect to a and b: 6a+30b=4.996522 and 30a+220b=6.693425. Solving, b=(6×6.693425−30×4.996522)/(6×220−30²)=−109.735122/420=−0.261274. Then a=(4.996522−30b)/6=2.139124. The fitted line is y=2.139124−0.261274t. Thus r=0.261274 year⁻¹ and A=e^a=8.4920. The slope is negative, so r is positive, as required for growth toward K. Equivalently, fitted P(0)=10000/(1+8.4920)=1053.5 thousand. The fitted value at t=0 need not equal the observed value because both parameters are estimated from all six observations. At t=10, fitted y=−0.473617, close to observed −0.489548, and P≈6163 thousand, close to 6200 thousand. The estimate is sensitive to the log transformation, but it is the standard hand-calculation method for a fixed K.

(ii) For 2026, t=12. P(12)=10000/[1+8.4920e^(−0.261274×12)] thousand. Since e^(−3.135289)=0.043487, P(12)=10000/[1+8.4920×0.043487]=10000/1.369293=7303.04 thousand. Predicted population ≈ 7,303,000.

(c) Agricultural statistics in India relate to area and yield. Area statistics include net sown area, gross cropped area, area under individual crops, and area under irrigation; gross cropped area equals net sown area multiplied by multiple-cropping intensity, and the multiple-cropping index is gross cropped area divided by net sown area. Yield statistics measure output per unit area, usually quintals per hectare, and are obtained from crop-cutting experiments, sample surveys, administrative records, and remote sensing. In India, crop-wise area and yield for rice, wheat, pulses, oilseeds, cotton, and horticulture are tracked season-wise and region-wise to assess cropping pattern, multiple cropping, and regional balance. Production equals area multiplied by yield, so accuracy of both is essential. Major sources are Agricultural Statistics at a Glance, CSD, NSSO, state agriculture departments, and FCI records.

Agricultural statistics are collected for kharif, rabi, and summer seasons, and for major and minor crops. Area data show how much land is devoted to foodgrains, commercial crops, and plantation crops. Yield data show productivity per hectare and are affected by seed quality, irrigation, fertilizers, pesticides, weather, and extension services. In India, statistics are important because agriculture employs a large share of the labour force and provides raw material to industry. They help estimate rural income, poverty, and food availability. They also help compare states and districts, identify high-yield regions, and plan crop diversification. Satellite-based area assessment improves speed and reduces bias in administrative records. They are used by RBI, NABARD, ICAR, and state planning bodies for credit, research, and extension decisions. Timely statistics are especially important in a monsoon-dependent economy, where drought, excess rain, or pest attack can change area and yield sharply. They also support price policy, export-import decisions, and food processing industry planning.

The need and importance are wide. First, they estimate foodgrain availability and help plan food policy, procurement, buffer stocks, and minimum support prices. Second, they guide allocation of credit, crop insurance, subsidies, and disaster relief. Third, they identify regional and crop-wise deficits, support research on high-yielding varieties, irrigation, and climate adaptation, and inform trade, taxation, and income estimates. They also reveal the contribution of different crops to total output and help distinguish short-term weather shocks from long-term productivity trends. Governments use them for budgeting, procurement planning, and setting target prices. Reliable area-yield statistics are therefore necessary for monitoring agricultural growth, doubling farmers’ income, and ensuring food security. Because of underreporting, crop substitution, and seasonal variation, estimates should combine surveys, administrative data, and satellite-based area assessment.