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 $2006 .$ 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. CAN'T COPY THE FIGURE (a) Use a graphing utility to draw a scatter diagram 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. (c) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.

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 $2006 .$ 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.
CAN'T COPY THE FIGURE
(a) Use a graphing utility to draw a scatter diagram 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.
(c) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.

Key Concepts

Graphing Utility

Graphing utilities are computational tools that enable the visualization of data and mathematical functions. They are instrumental in plotting scatter diagrams, fitting regression models, and overlaying these models on raw data to assess the accuracy and appropriateness of the fit.

Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. In practical applications such as maximizing income, the vertex represents the point at which an individual can expect to achieve the highest income according to the quadratic model.

Least Squares Best Fit

The method of least squares is a standard approach in regression analysis for finding the best-fitting curve by minimizing the sum of the squares of the offsets (errors) between the data points and the model. This technique is used to determine the coefficients of the quadratic function that best model the given data.

Predictive Modeling

Predictive modeling involves using a fitted mathematical model to forecast or estimate unknown or future values. In this problem, the quadratic function is used to predict the age at which the maximum income is earned and to estimate what that peak income might be.

Regression Analysis

Regression analysis is a statistical process for estimating the relationships among variables. In the context of this problem, it involves fitting a curve—in this case, a quadratic function—to the data points to establish a mathematical model that best describes the underlying relationship between age and income.

Scatter Diagram

A scatter diagram, also known as a scatter plot, is a graphical representation of data points where each point's position is determined by two variables. This type of diagram is useful for visually determining the nature of the relationship (linear, quadratic, etc.) between the independent and dependent variables.

Quadratic Function as a Model

A quadratic function is a polynomial function of degree two, typically expressed in the form ax² + bx + c. Its parabolic graph can effectively model situations where the dependent variable increases up to a certain point and then decreases, making it suitable for modeling scenarios like income that may peak at a particular age.

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