Math Problem Statement

An​ individual's income varies with age. The table shows the median income I of individuals of different age groups within the United States for a certain year. For each age​ group, let the class midpoint represent the independent variable x. For the class​ "65 years and​ older," assume that the class midpoint is 69.5. Complete parts ​(a) through ​(e). Age Class ​Midpoint, x Median ​Income, I ​15-24 years 19.5 ​$10 comma 963 ​25-34 years 29.5 ​$32 comma 131 ​35-44 years 39.5 ​$43 comma 638 ​45-54 years 49.5 ​$46 comma 693 ​55-64 years 59.5 ​$42 comma 478 65 years and older 69.5 ​$23 comma 500

Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income. The quadratic function of best fit is yequals

enter your response herexsquaredplus

enter your response herexminus

enter your response here. ​(Type integers or decimals rounded to three decimal places as​ needed.)

Solution

The quadratic function of best fit that models the relationship between age and median income is:

y=47.075x2+4466.160x58560.637y = -47.075x^2 + 4466.160x - 58560.637

Would you like more details or further clarifications?

Here are 5 follow-up questions to explore this further:

  1. How can we interpret the coefficients of this quadratic model?
  2. What does the vertex of this quadratic function represent in terms of income?
  3. How accurate is this model for predicting income beyond the provided age ranges?
  4. Can we use this model to estimate the income at a specific age (e.g., 50 years)?
  5. What other types of models could fit this data, and how would they differ?

Tip: Quadratic models are useful for identifying maximum or minimum values in a given dataset, especially when the relationship is non-linear.

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Math Problem Analysis

Mathematical Concepts

Quadratic Modeling
Statistics
Regression Analysis

Formulas

Quadratic equation y = ax^2 + bx + c
Least squares regression formula

Theorems

Quadratic regression
Curve fitting using least squares

Suitable Grade Level

Grades 10-12