Question

problems 12 - 14, use a graphing calculator to find a regression model to predict. 12. the table at right, shows the amount of money, in billions of dollars, spent on public education from 2010 to 2014. a. write the linear regression model for the data, where t = 0 represents 2010. b. what is the average rate of change of spending from 2011 to 2012? c. use the equation from part (a) to estimate projected spending for 2018. us education spending

year	$ in billions
2011	903.31
2012	920.57
2013	878.23
2014	883.00

source: www.usgovspending.com

Explanation:

Step1: Organize data points

Let \(t\) be the number of years since 2010. So the data points are \((0,919.67)\), \((1,903.31)\), \((2,920.57)\), \((3,878.23)\), \((4,883.00)\). Using a graph - ing calculator (or statistical software) to find the linear regression equation \(y = at + b\) where \(y\) is the spending in billions of dollars and \(t\) is the number of years since 2010. The general formulas for \(a\) and \(b\) in linear regression are \(a=\frac{n\sum_{i = 1}^{n}t_iy_i-\sum_{i = 1}^{n}t_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}t_i^{2}-(\sum_{i = 1}^{n}t_i)^{2}}\) and \(b=\overline{y}-a\overline{t}\). For our data \(n = 5\), \(\sum_{i=1}^{5}t_i=0 + 1+2 + 3+4=10\), \(\sum_{i = 1}^{5}y_i=919.67+903.31 + 920.57+878.23+883.00 = 4504.78\), \(\sum_{i=1}^{5}t_i^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}=30\), \(\sum_{i = 1}^{5}t_iy_i=0\times919.67+1\times903.31+2\times920.57+3\times878.23+4\times883.00=0 + 903.31+1841.14+2634.69+3532.00 = 8911.14\). Then \(a=\frac{5\times8911.14-10\times4504.78}{5\times30 - 10^{2}}=\frac{44555.7-45047.8}{150 - 100}=\frac{-492.1}{50}=-9.842\) and \(b=\frac{4504.78}{5}-(-9.842)\times\frac{10}{5}=900.956 + 19.684=920.64\). So the linear - regression model is \(y=-9.842t + 920.64\).

Step2: Calculate average rate of change

The average rate of change of a function \(y = f(t)\) from \(t_1\) to \(t_2\) is \(\frac{f(t_2)-f(t_1)}{t_2 - t_1}\). For 2011 (\(t = 1\)) and 2012 (\(t = 2\)), \(y_1 = 903.31\) and \(y_2 = 920.57\). The average rate of change is \(\frac{920.57 - 903.31}{2 - 1}=17.26\) billion dollars per year.

Step3: Estimate 2018 spending

For 2018, \(t=8\). Substitute \(t = 8\) into the equation \(y=-9.842t + 920.64\). Then \(y=-9.842\times8+920.64=-78.736+920.64 = 841.904\) billion dollars.

Answer:

A. \(y=-9.842t + 920.64\)
B. \(17.26\) billion dollars per year
C. \(841.904\) billion dollars