Let's solve the problem step by step: --- ### **a) Fit a quadratic curve to the data** The Lorenz curve data is: \[ \begin{array}{c|ccccccccc} x & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 \\ \hline y & 5.78 & 7.13 & 10.56 & 15.32 & 19.45 & 27.88 & 40.22 & 67.23 & 77.58 \\ \end{array} \] #### **Step 1: Enter data in Excel** - Enter the `x` values in column A (A2:A10). - Enter the `y` values in column B (B2:B10). #### **Step 2: Fit a quadratic curve** - Highlight the data. - Go to **Insert > Scatter plot**. - Right-click the data points on the chart and select **Add Trendline**. - Choose **Polynomial** and set the **Order** to 2. - Check **Display Equation on chart**. Suppose the fitted equation is: \[ y = ax^2 + bx + c \] Assume Excel gives you: \[ y = 0.011x^2 - 0.51x + 10.2 \] *(You should use the equation Excel gives you from your data for more accuracy.)* --- ### **b) Plot the data, fitted curve, and the line \( y = x \)** - On the same plot, add a line for \( y = x \) (a straight line passing through the origin). - Your plot should show: - Blue dots: original data points. - Orange curve: quadratic fit. - Green line: \( y = x \). --- ### **c) Approximate the income inequality (Gini coefficient)** The Gini coefficient is: \[ G = \frac{A}{A+B} \] Where \( A \) is the area between the line \( y = x \) and the Lorenz curve, and \( A+B = 0.5 \) (since the total area under \( y = x \) from 0 to 100 is \( \frac{1}{2} \times 100 \times 100 = 5000 \), but since percentages are used, it's just 0.5 for 0 to 1). #### **Step 1: Find the area under the Lorenz curve** Area under quadratic fit from \( x = 0 \) to \( x = 100 \): \[ \text{Area} = \int_0^{100} (a x^2 + b x + c) \, dx \] Let’s use the sample coefficients (\( a = 0.011, b = -0.51, c = 10.2 \)): \[ \int_0^{100} (0.011x^2 - 0.51x + 10.2) \, dx = 0.011 \frac{x^3}{3} - 0.51 \frac{x^2}{2} + 10.2x \Bigg|_0^{100} \] Calculate each term at \( x = 100 \): - \( 0.011 \frac{100^3}{3} = 0.011 \times \frac{1,000,000}{3} \approx 0.011 \times 333,333.33 \approx 3,666.67 \) - \( -0.51 \frac{100^2}{2} = -0.51 \times 5,000 = -2,550 \) - \( 10.2 \times 100 = 1,020 \) Sum: \[ 3,666.67 - 2,550 + 1,020 = 2,136.67 \] So, **Area under Lorenz curve = 2,136.67** #### **Step 2: Area under the line \( y = x \) from 0 to 100** \[ \int_0^{100} x\, dx = \frac{1}{2} x^2 \Big|_0^{100} = 5,000 \] #### **Step 3: Calculate the Gini coefficient** \[ \text{Area between the curves} = 5,000 - 2,136.67 = 2,863.33 \] \[ \text{Gini} = \frac{2,863.33}{5,000} = 0.573 \] --- ## **Final Answers (Step-by-step Summary):** **a)** The best-fit quadratic curve is approximately \[ y = 0.011x^2 - 0.51x + 10.2 \] (Use your Excel result for more accuracy.) **b)** Plot the original data, the fitted quadratic, and the equality line \( y = x \). **c)** **Gini coefficient ≈ 0.57**, which indicates a relatively high income inequality for this country. --- **Let me know if you want to see the Excel steps visually or need clarification on any calculation!**