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:
Would you like more details or further clarifications?
Here are 5 follow-up questions to explore this further:
- How can we interpret the coefficients of this quadratic model?
- What does the vertex of this quadratic function represent in terms of income?
- How accurate is this model for predicting income beyond the provided age ranges?
- Can we use this model to estimate the income at a specific age (e.g., 50 years)?
- 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