Answer :
To solve this problem, we need to find a linear function [tex]\( S(t) \)[/tex] that models the given spending data over the years, where [tex]\( t \)[/tex] is the number of years since 2011. The data provided includes the amounts of federal education spending in billions of dollars from 2012 to 2017.
Here's how you can determine the function [tex]\( S(t) \)[/tex]:
1. Assign Values to [tex]\( t \)[/tex]:
- Since [tex]\( t = 1 \)[/tex] corresponds to 2011, [tex]\( t = 2 \)[/tex] corresponds to 2012, and so on. Therefore:
- 2012 is [tex]\( t = 2 \)[/tex]
- 2013 is [tex]\( t = 3 \)[/tex]
- 2014 is [tex]\( t = 4 \)[/tex]
- 2015 is [tex]\( t = 5 \)[/tex]
- 2016 is [tex]\( t = 6 \)[/tex]
- 2017 is [tex]\( t = 7 \)[/tex]
2. Create Data Points:
- Pair the values of [tex]\( t \)[/tex] with the corresponding spending amounts:
- [tex]\( (2, 109.3) \)[/tex]
- [tex]\( (3, 105.1) \)[/tex]
- [tex]\( (4, 104.5) \)[/tex]
- [tex]\( (5, 99.0) \)[/tex]
- [tex]\( (6, 99.3) \)[/tex]
- [tex]\( (7, 97.7) \)[/tex]
3. Perform Linear Regression:
- A linear regression will provide us with a line of best fit through these points. The equation of a line is typically given by:
[tex]\[ S(t) = mt + b \][/tex]
- Here, [tex]\( m \)[/tex] is the slope of the line, and [tex]\( b \)[/tex] is the y-intercept.
4. Results from Linear Regression:
- The calculations give us the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] as follows:
- Slope [tex]\( m = -2.31 \)[/tex]
- Y-intercept [tex]\( b = 112.88 \)[/tex]
5. Write the Function:
- Now, substitute [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the linear equation:
[tex]\[ S(t) = -2.31t + 112.88 \][/tex]
This equation [tex]\( S(t) = -2.31t + 112.88 \)[/tex] represents the model for the federal education spending, in billions of dollars, where [tex]\( t \)[/tex] is the number of years since 2011.
Here's how you can determine the function [tex]\( S(t) \)[/tex]:
1. Assign Values to [tex]\( t \)[/tex]:
- Since [tex]\( t = 1 \)[/tex] corresponds to 2011, [tex]\( t = 2 \)[/tex] corresponds to 2012, and so on. Therefore:
- 2012 is [tex]\( t = 2 \)[/tex]
- 2013 is [tex]\( t = 3 \)[/tex]
- 2014 is [tex]\( t = 4 \)[/tex]
- 2015 is [tex]\( t = 5 \)[/tex]
- 2016 is [tex]\( t = 6 \)[/tex]
- 2017 is [tex]\( t = 7 \)[/tex]
2. Create Data Points:
- Pair the values of [tex]\( t \)[/tex] with the corresponding spending amounts:
- [tex]\( (2, 109.3) \)[/tex]
- [tex]\( (3, 105.1) \)[/tex]
- [tex]\( (4, 104.5) \)[/tex]
- [tex]\( (5, 99.0) \)[/tex]
- [tex]\( (6, 99.3) \)[/tex]
- [tex]\( (7, 97.7) \)[/tex]
3. Perform Linear Regression:
- A linear regression will provide us with a line of best fit through these points. The equation of a line is typically given by:
[tex]\[ S(t) = mt + b \][/tex]
- Here, [tex]\( m \)[/tex] is the slope of the line, and [tex]\( b \)[/tex] is the y-intercept.
4. Results from Linear Regression:
- The calculations give us the slope [tex]\( m \)[/tex] and the y-intercept [tex]\( b \)[/tex] as follows:
- Slope [tex]\( m = -2.31 \)[/tex]
- Y-intercept [tex]\( b = 112.88 \)[/tex]
5. Write the Function:
- Now, substitute [tex]\( m \)[/tex] and [tex]\( b \)[/tex] into the linear equation:
[tex]\[ S(t) = -2.31t + 112.88 \][/tex]
This equation [tex]\( S(t) = -2.31t + 112.88 \)[/tex] represents the model for the federal education spending, in billions of dollars, where [tex]\( t \)[/tex] is the number of years since 2011.
