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Answer :
To solve the problem of finding a function model [tex]\( S \)[/tex] that describes U.S. federal education spending over the years 2011 to 2017, we will use a linear regression method. This approach helps us to find a linear relationship (straight line) that best fits the spending data over these years.
Here's the step-by-step process:
1. Identify the Variables:
- The variable [tex]\( t \)[/tex] represents the years, with [tex]\( t = 1 \)[/tex] for 2011, [tex]\( t = 2 \)[/tex] for 2012, continuing up to [tex]\( t = 7 \)[/tex] for 2017.
- The spending amounts in billions of dollars are given for each year: 112.8, 109.3, 105.1, 104.5, 99.0, 99.3, and 97.7.
2. Set Up Linear Regression:
- Linear regression is used to find the line of best fit. This line can be represented by the equation [tex]\( S(t) = mt + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Determine the Slope and Intercept:
- The slope ([tex]\( m \)[/tex]) represents the rate of change in spending per year.
- The intercept ([tex]\( b \)[/tex]) represents the expected spending when [tex]\( t = 0 \)[/tex].
- Through mathematical calculations, we find that the slope [tex]\( m \)[/tex] is approximately -2.55. This indicates that the spending decreases by about 2.55 billion dollars each year.
- The intercept [tex]\( b \)[/tex] is approximately 114.16. This would theoretically be the spending in billions at [tex]\( t = 0 \)[/tex], but since there is no year 0, it serves as a starting point for our model.
4. Write the Function:
- With the slope and intercept known, we can now write the function [tex]\( S(t) \)[/tex] as:
[tex]\[
S(t) = -2.55t + 114.16
\][/tex]
This function [tex]\( S(t) \)[/tex] provides an estimate of U.S. federal education spending for each year from 2011 to 2017, based on the trend described by the available data.
Here's the step-by-step process:
1. Identify the Variables:
- The variable [tex]\( t \)[/tex] represents the years, with [tex]\( t = 1 \)[/tex] for 2011, [tex]\( t = 2 \)[/tex] for 2012, continuing up to [tex]\( t = 7 \)[/tex] for 2017.
- The spending amounts in billions of dollars are given for each year: 112.8, 109.3, 105.1, 104.5, 99.0, 99.3, and 97.7.
2. Set Up Linear Regression:
- Linear regression is used to find the line of best fit. This line can be represented by the equation [tex]\( S(t) = mt + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
3. Determine the Slope and Intercept:
- The slope ([tex]\( m \)[/tex]) represents the rate of change in spending per year.
- The intercept ([tex]\( b \)[/tex]) represents the expected spending when [tex]\( t = 0 \)[/tex].
- Through mathematical calculations, we find that the slope [tex]\( m \)[/tex] is approximately -2.55. This indicates that the spending decreases by about 2.55 billion dollars each year.
- The intercept [tex]\( b \)[/tex] is approximately 114.16. This would theoretically be the spending in billions at [tex]\( t = 0 \)[/tex], but since there is no year 0, it serves as a starting point for our model.
4. Write the Function:
- With the slope and intercept known, we can now write the function [tex]\( S(t) \)[/tex] as:
[tex]\[
S(t) = -2.55t + 114.16
\][/tex]
This function [tex]\( S(t) \)[/tex] provides an estimate of U.S. federal education spending for each year from 2011 to 2017, based on the trend described by the available data.
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