Economists use Lorenz curves to illustrate the distribution...

Economists use Lorenz curves to illustrate the distribution of income in a country. A Lorenz curve, $y=f(x)$, represents the actual income distribution in the country. In this model, $x$ represents percents of families in the country and $y$ represents percents of total income. The model $y=x$ represents a country in which each family has the same income. The area between these two models, where $0 \leq x \leq 100$, indicates a country's "income inequality." The table lists percents of income $y$ for selected percents of families $x$ in a country. $$ \begin{aligned} &\begin{array}{|c|c|c|c|c|c|} \hline x & 10 & 20 & 30 & 40 & 50 \\ \hline y & 3.35 & 6.07 & 9.17 & 13.39 & 19.45 \\ \hline \end{array}\\ &\begin{array}{|c|c|c|c|c|} \hline x & 60 & 70 & 80 & 90 \\ \hline y & 28.03 & 39.77 & 55.28 & 75.12 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to find a quadratic model for the Lorenz curve. (b) Plot the data and graph the model. (c) Graph the model $y=x .$ How does this model compare with the model in part (a)? (d) Use the integration capabilities of a graphing utility to approximate the "income inequality."

Economists use Lorenz curves to illustrate the distribution of income in a country. A Lorenz curve, $y=f(x)$, represents the actual income distribution in the country. In this model, $x$ represents percents of families in the country and $y$ represents percents of total income. The model $y=x$ represents a country in which each family has the same income. The area between these two models, where $0 \leq x \leq 100$, indicates a country's "income inequality." The table lists percents of income $y$ for selected percents of families $x$ in a country.
$$
\begin{aligned}
&\begin{array}{|c|c|c|c|c|c|}
\hline x & 10 & 20 & 30 & 40 & 50 \\
\hline y & 3.35 & 6.07 & 9.17 & 13.39 & 19.45 \\
\hline
\end{array}\\
&\begin{array}{|c|c|c|c|c|}
\hline x & 60 & 70 & 80 & 90 \\
\hline y & 28.03 & 39.77 & 55.28 & 75.12 \\
\hline
\end{array}
\end{aligned}
$$
(a) Use a graphing utility to find a quadratic model for the Lorenz curve.
(b) Plot the data and graph the model.
(c) Graph the model $y=x .$ How does this model compare with the model in part (a)?
(d) Use the integration capabilities of a graphing utility to approximate the "income inequality."

Key Concepts

Lorenz Curve

A Lorenz curve is a graphical tool used to represent the distribution of a quantity—typically income or wealth—across a population. It plots the cumulative percentage of the population against the cumulative percentage of income earned, providing a visual indication of inequality within the distribution. Greater deviation from the line of perfect equality (y = x) indicates higher levels of inequality.

Quadratic Regression Model

Quadratic regression is a statistical method used to fit a second-degree polynomial (a quadratic function) to a set of data points. It finds the best-fitting curve by minimizing the differences between the observed data and the values predicted by the model. This technique is particularly useful when the relationship between variables exhibits curvature rather than linearity.

Graphical Analysis of Models

Graphical analysis involves plotting both the collected data and the corresponding fitted models on a coordinate system to assess and compare their behaviors. This method allows for visual examination of how well a model represents the underlying data and enables comparisons between different models, such as comparing a fitted curve with the line representing perfect equality.

Integration for Area Calculation

Integration in calculus is used to calculate the area under a curve. When applied to the analysis of income distribution, it helps measure the area between the Lorenz curve and the line of perfect equality. This area is a crucial component in quantifying inequality measures, such as the Gini coefficient, thereby providing a numerical assessment of income inequality.

Transcript

00:01 In this question we have a table as shown.

00:04 Here, x represents percents of families in the country.

00:09 Y represents total income.

00:15 Model y is equal to x represents a country in which each family has the same income.

00:26 The law range curve y is equals to fx represents the actual income distribution in the country and the area of the region between these two graphs or between these two models indicates the country's income inequality.

00:51 Here, x lies between 0 and 100.

01:00 We are required to find a quadratic model for the lawrence curve using graphing utility.

01:06 After that, we need to plot the data and graph the model.

01:11 Then we need to draw the graph for y is equals to x and we need to compare this with the model found in part a.

01:23 And finally, we need to approximate the income inequality with the help of integration capability of graphing utility.

01:32 So let's see how to solve this question.

01:37 When we apply the graphing utility, we found a quadratic model y is equals to 0 .012388 x squared minus 0 .396x plus 7 .846x plus 7 .8487.

02:06 So this is the final answer for part a.

02:09 And now let's move to part b.

02:16 The plot or the graph for this equation is shown below.

02:22 So this is the plot for y is equals to 0 .0123 -58x square minus 0 .3846x plus 7 .846x plus 7 .8487...

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