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Answer :
To find the cubic regression model for the given data and identify the coefficient of the [tex]\(x^2\)[/tex] term, we should go through the following steps:
1. Rescale the Data:
First, we need to adjust the years so that the year 2003 corresponds to [tex]\(x = 0\)[/tex]. This means we assign each year a value relative to the year 2003.
- Year 2003 becomes [tex]\(x = 0\)[/tex].
- Year 2004 becomes [tex]\(x = 1\)[/tex].
- Year 2005 becomes [tex]\(x = 2\)[/tex].
- Year 2006 becomes [tex]\(x = 3\)[/tex].
- Year 2007 becomes [tex]\(x = 4\)[/tex].
- Year 2008 becomes [tex]\(x = 5\)[/tex].
2. Record the Profits:
We have the profits corresponding to these years:
- 2003: 31.3 million dollars
- 2004: 32.7 million dollars
- 2005: 31.8 million dollars
- 2006: 33.7 million dollars
- 2007: 35.9 million dollars
- 2008: 36.1 million dollars
3. Fit a Cubic Regression Model:
The goal here is to fit a cubic polynomial model of the form:
[tex]\[
y = ax^3 + bx^2 + cx + d
\][/tex]
We would use the rescaled [tex]\(x\)[/tex] values (0, 1, 2, 3, 4, 5) and the corresponding profits to determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex].
4. Identify the Coefficient for [tex]\(x^2\)[/tex]:
Once the cubic regression is computed, we examine the coefficients to find the one associated with the [tex]\(x^2\)[/tex] term.
The coefficient of the [tex]\(x^2\)[/tex] term is [tex]\(0.5310\)[/tex].
Therefore, the answer to the question of which number is the coefficient of the [tex]\(x^2\)[/tex] term of the cubic regression model is [tex]\(0.5310\)[/tex].
1. Rescale the Data:
First, we need to adjust the years so that the year 2003 corresponds to [tex]\(x = 0\)[/tex]. This means we assign each year a value relative to the year 2003.
- Year 2003 becomes [tex]\(x = 0\)[/tex].
- Year 2004 becomes [tex]\(x = 1\)[/tex].
- Year 2005 becomes [tex]\(x = 2\)[/tex].
- Year 2006 becomes [tex]\(x = 3\)[/tex].
- Year 2007 becomes [tex]\(x = 4\)[/tex].
- Year 2008 becomes [tex]\(x = 5\)[/tex].
2. Record the Profits:
We have the profits corresponding to these years:
- 2003: 31.3 million dollars
- 2004: 32.7 million dollars
- 2005: 31.8 million dollars
- 2006: 33.7 million dollars
- 2007: 35.9 million dollars
- 2008: 36.1 million dollars
3. Fit a Cubic Regression Model:
The goal here is to fit a cubic polynomial model of the form:
[tex]\[
y = ax^3 + bx^2 + cx + d
\][/tex]
We would use the rescaled [tex]\(x\)[/tex] values (0, 1, 2, 3, 4, 5) and the corresponding profits to determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex].
4. Identify the Coefficient for [tex]\(x^2\)[/tex]:
Once the cubic regression is computed, we examine the coefficients to find the one associated with the [tex]\(x^2\)[/tex] term.
The coefficient of the [tex]\(x^2\)[/tex] term is [tex]\(0.5310\)[/tex].
Therefore, the answer to the question of which number is the coefficient of the [tex]\(x^2\)[/tex] term of the cubic regression model is [tex]\(0.5310\)[/tex].
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