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Answer :
Sure! Let's go through this step by step to define the function model [tex]\( S \)[/tex] that represents the United States federal education spending over the given years.
The table provides spending in billions of dollars for each year from 2011 to 2017. We need to use linear regression to find a function model [tex]\( S \)[/tex].
Here's a breakdown of the procedure:
1. Convert Years to [tex]\( t \)[/tex]-values:
- For 2011, [tex]\( t = 1 \)[/tex]
- For 2012, [tex]\( t = 2 \)[/tex]
- And so on, until 2017, [tex]\( t = 7 \)[/tex]
2. Organize the Data:
- [tex]\( t = [1, 2, 3, 4, 5, 6, 7] \)[/tex]
- Spending [tex]\( S \)[/tex] in billions of dollars: [tex]\([112.8, 109.3, 105.1, 104.5, 99.0, 99.3, 97.7]\)[/tex]
3. Linear Regression Analysis:
- We find a linear relationship of the form [tex]\( S(t) = \text{slope} \times t + \text{intercept} \)[/tex].
4. Calculate the Slope and Intercept:
- The slope is given as [tex]\(-2.55\)[/tex].
- The intercept is given as [tex]\(114.16\)[/tex].
5. Define the Function [tex]\( S(t) \)[/tex]:
- The function model based on the linear regression analysis is:
[tex]\[
S(t) = -2.55 \times t + 114.16
\][/tex]
This function models the spending in billions of dollars over the given years, where [tex]\( t \)[/tex] represents the year with [tex]\( t = 1 \)[/tex] corresponding to 2011, [tex]\( t = 2 \)[/tex] corresponding to 2012, and so on.
Thus, the function [tex]\( S(t) \)[/tex] that models the spending is:
[tex]\[ S(t) = -2.55t + 114.16 \][/tex]
This represents the approximate federal education spending in billions of dollars for each year [tex]\( t \)[/tex] within the given range.
The table provides spending in billions of dollars for each year from 2011 to 2017. We need to use linear regression to find a function model [tex]\( S \)[/tex].
Here's a breakdown of the procedure:
1. Convert Years to [tex]\( t \)[/tex]-values:
- For 2011, [tex]\( t = 1 \)[/tex]
- For 2012, [tex]\( t = 2 \)[/tex]
- And so on, until 2017, [tex]\( t = 7 \)[/tex]
2. Organize the Data:
- [tex]\( t = [1, 2, 3, 4, 5, 6, 7] \)[/tex]
- Spending [tex]\( S \)[/tex] in billions of dollars: [tex]\([112.8, 109.3, 105.1, 104.5, 99.0, 99.3, 97.7]\)[/tex]
3. Linear Regression Analysis:
- We find a linear relationship of the form [tex]\( S(t) = \text{slope} \times t + \text{intercept} \)[/tex].
4. Calculate the Slope and Intercept:
- The slope is given as [tex]\(-2.55\)[/tex].
- The intercept is given as [tex]\(114.16\)[/tex].
5. Define the Function [tex]\( S(t) \)[/tex]:
- The function model based on the linear regression analysis is:
[tex]\[
S(t) = -2.55 \times t + 114.16
\][/tex]
This function models the spending in billions of dollars over the given years, where [tex]\( t \)[/tex] represents the year with [tex]\( t = 1 \)[/tex] corresponding to 2011, [tex]\( t = 2 \)[/tex] corresponding to 2012, and so on.
Thus, the function [tex]\( S(t) \)[/tex] that models the spending is:
[tex]\[ S(t) = -2.55t + 114.16 \][/tex]
This represents the approximate federal education spending in billions of dollars for each year [tex]\( t \)[/tex] within the given range.
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