An individual's income varies with his or her age. The...

An individual's income varies with his or her age. The following table shows the median income $I$ of males of different age groups within the United States for $2016 .$ For each age group, let the class midpoint represent the independent variable, $x$. For the class " 65 years and older," we will assume that the class midpoint is 69.5 . $$ \begin{array}{|lcc|} \hline \text { Age } & \begin{array}{c} \text { Class } \\ \text { Midpoint, } x \end{array} & \begin{array}{c} \text { Median } \\ \text { Income, } I \end{array} \\ \hline 15-24 \text { years } & 19.5 & \$ 12,396 \\ 25-34 \text { years } & 29.5 & \$ 38,152 \\ 35-44 \text { years } & 39.5 & \$ 51,443 \\ 45-54 \text { years } & 49.5 & \$ 51,476 \\ 55-64 \text { years } & 59.5 & \$ 47,001 \\ 65 \text { years and older } & 69.5 & \$ 31,618 \\ \hline \end{array} $$ (a) Use a graphing utility to draw a scatter plot of the data. Comment on the type of relation that may exist between the two variables. (b) Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income. (c) Use the function found in part (b) to determine the age at which an individual can expect to earn the most income. (d) Use the function found in part (b) to predict the peak income earned. (e) With a graphing utility, graph the quadratic function of best fit on the scatter plot

An individual's income varies with his or her age. The following table shows the median income $I$ of males of different age groups within the United States for $2016 .$ For each age group, let the class midpoint represent the independent variable, $x$. For the class " 65 years and older," we will assume that the class midpoint is 69.5 .
$$
\begin{array}{|lcc|}
\hline \text { Age } & \begin{array}{c}
\text { Class } \\
\text { Midpoint, } x
\end{array} & \begin{array}{c}
\text { Median } \\
\text { Income, } I
\end{array} \\
\hline 15-24 \text { years } & 19.5 & \$ 12,396 \\
25-34 \text { years } & 29.5 & \$ 38,152 \\
35-44 \text { years } & 39.5 & \$ 51,443 \\
45-54 \text { years } & 49.5 & \$ 51,476 \\
55-64 \text { years } & 59.5 & \$ 47,001 \\
65 \text { years and older } & 69.5 & \$ 31,618 \\
\hline
\end{array}
$$
(a) Use a graphing utility to draw a scatter plot of the data. Comment on the type of relation that may exist between the two variables.
(b) Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income.
(c) Use the function found in part (b) to determine the age at which an individual can expect to earn the most income.
(d) Use the function found in part (b) to predict the peak income earned.
(e) With a graphing utility, graph the quadratic function of best fit on the scatter plot

Key Concepts

Scatter Plot

A scatter plot is a type of graph used to display and visually assess the relationship between two variables by representing data points on a Cartesian coordinate system. It helps to identify trends, clusters, or any patterns in the data and can indicate if a certain type of mathematical model might be appropriate for further analysis.

Quadratic Regression

Quadratic regression is a statistical method used to model the relationship between an independent variable and a dependent variable by fitting a quadratic equation of the form ax² + bx + c to the data. This technique is particularly useful when the scatter plot shows a parabolic shape, allowing analysts to capture nonlinear trends and make predictions based on the fitted model.

Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the function reaches its maximum or minimum value, depending on the orientation of the parabola. This concept is critical for optimization problems, as it identifies the value of the independent variable that results in the extreme value (peak or trough) of the dependent variable.

Regression Analysis for Modeling Relationships

Regression analysis is a broader statistical framework that involves estimating relationships among variables. It includes methods like quadratic regression and is used to understand how the typical value of the dependent variable changes when any one of the independent variables is varied, thus offering a means for prediction and interpretation of real-world data.

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