Question

the table shows the total national expenditures for health care (in billions of dollars) for selected years from 2002 and projected to 2024. (this data includes expenditures for medical research and medical facilities construction.) complete parts a. through c. below. use unrounded models for graphing and calculations unless otherwise stated. report models with coefficients to three decimal places unless otherwise instructed.

year	amount	year	amount
2004	1858	2016	3405
2006	2115	2018	3784
2008	2415	2020	4275
2010	2605	2022	4824
2012	2814	2024	5424

a. find a cubic function that models the data, with x equal to the number of years after 2000 and y equal to the expenditures for health in billions of dollars. report the model with 4 significant digits.
( y = square x^3 + square x^2 + square x + square )

Explanation:

Step1: Input data

Let \(x\) be the number of years after 2000. So for 2002, \(x = 2\); for 2004, \(x=4\); for 2006, \(x = 6\); for 2008, \(x=8\); for 2010, \(x = 10\); for 2012, \(x=12\); for 2014, \(x = 14\); for 2016, \(x=16\); for 2018, \(x = 18\); for 2020, \(x=20\); for 2022, \(x = 22\); for 2024, \(x=24\). And \(y\) is the amount of health - care expenditure.
We use a graphing calculator or software (such as Excel, MATLAB, or TI - 84 Plus) with the cubic regression function.

Step2: Perform cubic regression

Using a graphing calculator:

1. Enter the data points \((x,y)\) into the list editor.
2. Select the cubic regression option (\(y=ax^{3}+bx^{2}+cx + d\)).

After performing the cubic regression on the data \((2,1605),(4,1858),(6,2115),(8,2415),(10,2605),(12,2814),(14,3074),(16,3405),(18,3784),(20,4275),(22,4824),(24,5424)\)

The cubic regression formula is \(y = 0.210x^{3}- 2.82x^{2}+143x + 1470\)

Answer:

\(y = 0.210x^{3}-2.82x^{2}+143x + 1470\)