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Sagot :
Let's solve the problem step-by-step.
### 1. Finding the linear regression model for the price-supply data
From the given data:
[tex]\[ \begin{array}{c|c} \text{Supply (billion bushels)} & \text{Price (\$/bu)} \\ \hline 6.49 & 2.15 \\ 7.37 & 2.24 \\ 7.61 & 2.36 \\ 7.95 & 2.44 \\ 8.21 & 2.43 \\ 8.33 & 2.59 \\ \end{array} \][/tex]
Using linear regression analysis, we can determine the relationship between supply ([tex]\(x\)[/tex]) and price ([tex]\(y\)[/tex]). The relationship can be expressed in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
From the analysis, we find that:
- Slope ([tex]\( m \)[/tex]) = 0.22
- Intercept ([tex]\( b \)[/tex]) = 0.71
Thus, the linear regression equation for the price-supply data is:
[tex]\[ y = 0.22x + 0.71 \][/tex]
### 2. Finding the linear regression model for the price-demand data
From the given data:
[tex]\[ \begin{array}{c|c} \text{Demand (billion bushels)} & \text{Price (\$/bu)} \\ \hline 9.92 & 2.09 \\ 9.42 & 2.11 \\ 8.49 & 2.26 \\ 8.08 & 2.38 \\ 7.78 & 2.36 \\ 6.89 & 2.49 \\ \end{array} \][/tex]
Using linear regression analysis, we can determine the relationship between demand ([tex]\(x\)[/tex]) and price ([tex]\(y\)[/tex]). The relationship can be expressed in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
From the analysis, we find that:
- Slope ([tex]\( m \)[/tex]) = -0.14
- Intercept ([tex]\( b \)[/tex]) = 3.47
Thus, the linear regression equation for the price-demand data is:
[tex]\[ y = -0.14x + 3.47 \][/tex]
### 3. Finding the equilibrium price for corn
The equilibrium price occurs where the price-supply and price-demand equations intersect.
Setting the two equations equal to each other:
[tex]\[ 0.22x + 0.71 = -0.14x + 3.47 \][/tex]
Combining like terms to solve for [tex]\( x \)[/tex]:
[tex]\[ 0.22x + 0.14x = 3.47 - 0.71 \][/tex]
[tex]\[ 0.36x = 2.76 \][/tex]
[tex]\[ x = \frac{2.76}{0.36} = 7.67 \][/tex]
Now, substituting [tex]\( x \)[/tex] back into either equation to find the equilibrium price [tex]\( y \)[/tex]:
[tex]\[ y = 0.22(7.67) + 0.71 \][/tex]
[tex]\[ y \approx 2.38 \][/tex]
So, the equilibrium price for corn is:
[tex]\[ y = \$2.38 \][/tex]
Therefore, the correct choice is:
[tex]\[ A. y = \$2.38 \][/tex]
### 1. Finding the linear regression model for the price-supply data
From the given data:
[tex]\[ \begin{array}{c|c} \text{Supply (billion bushels)} & \text{Price (\$/bu)} \\ \hline 6.49 & 2.15 \\ 7.37 & 2.24 \\ 7.61 & 2.36 \\ 7.95 & 2.44 \\ 8.21 & 2.43 \\ 8.33 & 2.59 \\ \end{array} \][/tex]
Using linear regression analysis, we can determine the relationship between supply ([tex]\(x\)[/tex]) and price ([tex]\(y\)[/tex]). The relationship can be expressed in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
From the analysis, we find that:
- Slope ([tex]\( m \)[/tex]) = 0.22
- Intercept ([tex]\( b \)[/tex]) = 0.71
Thus, the linear regression equation for the price-supply data is:
[tex]\[ y = 0.22x + 0.71 \][/tex]
### 2. Finding the linear regression model for the price-demand data
From the given data:
[tex]\[ \begin{array}{c|c} \text{Demand (billion bushels)} & \text{Price (\$/bu)} \\ \hline 9.92 & 2.09 \\ 9.42 & 2.11 \\ 8.49 & 2.26 \\ 8.08 & 2.38 \\ 7.78 & 2.36 \\ 6.89 & 2.49 \\ \end{array} \][/tex]
Using linear regression analysis, we can determine the relationship between demand ([tex]\(x\)[/tex]) and price ([tex]\(y\)[/tex]). The relationship can be expressed in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope of the line and [tex]\( b \)[/tex] is the y-intercept.
From the analysis, we find that:
- Slope ([tex]\( m \)[/tex]) = -0.14
- Intercept ([tex]\( b \)[/tex]) = 3.47
Thus, the linear regression equation for the price-demand data is:
[tex]\[ y = -0.14x + 3.47 \][/tex]
### 3. Finding the equilibrium price for corn
The equilibrium price occurs where the price-supply and price-demand equations intersect.
Setting the two equations equal to each other:
[tex]\[ 0.22x + 0.71 = -0.14x + 3.47 \][/tex]
Combining like terms to solve for [tex]\( x \)[/tex]:
[tex]\[ 0.22x + 0.14x = 3.47 - 0.71 \][/tex]
[tex]\[ 0.36x = 2.76 \][/tex]
[tex]\[ x = \frac{2.76}{0.36} = 7.67 \][/tex]
Now, substituting [tex]\( x \)[/tex] back into either equation to find the equilibrium price [tex]\( y \)[/tex]:
[tex]\[ y = 0.22(7.67) + 0.71 \][/tex]
[tex]\[ y \approx 2.38 \][/tex]
So, the equilibrium price for corn is:
[tex]\[ y = \$2.38 \][/tex]
Therefore, the correct choice is:
[tex]\[ A. y = \$2.38 \][/tex]
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