Question

The following graph represents the income distribution among the population of a country. The Gini Coefficient of the country (rounded off to three decimal places) is \(\underline{\hspace{1cm}}\). 

Hint:

The Gini coefficient is computed as $1 - 2 \times$ (area under Lorenz curve). Approximate the area using trapezoids when only key points are given.

Answer 1

Correct Answer: 0.24 - 0.27

From the Lorenz curve shown:
- At 30% population → cumulative income = 18%
- At 70% population → cumulative income = 45%
- At 100% population → cumulative income = 100%
We approximate the Lorenz curve as piecewise linear between these points: \[ (0,0),\ (0.3,0.18),\ (0.7,0.45),\ (1,1) \] The Gini coefficient is: \[ G = 1 - 2A \] where \(A\) is the area under the Lorenz curve. Compute area by trapezoidal rule: \[ A = \frac{1}{2}(0.3)(0+0.18) + \frac{1}{2}(0.4)(0.18+0.45) + \frac{1}{2}(0.3)(0.45+1) \] \[ A = 0.027 + 0.126 + 0.2175 \] \[ A = 0.3705 \] Then: \[ G = 1 - 2A = 1 - 2(0.3705) = 0.259 \] Thus, the Gini coefficient lies between: \[ \boxed{0.240\ \text{to}\ 0.270} \]