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The table contains price-supply data and price-demand data for corn. Complete the following tasks:

1. Find a linear regression model for the price-supply data where [tex]\(x\)[/tex] is supply (in billions of bushels) and [tex]\(y\)[/tex] is price (in dollars).
2. Find a linear regression model for the price-demand data where [tex]\(x\)[/tex] is demand (in billions of bushels) and [tex]\(y\)[/tex] is price (in dollars).
3. Find the equilibrium price for corn.

\begin{tabular}{cc|cc}
Price (\[tex]$/bu) & Supply (billion bu) & Price (\$[/tex]/bu) & Demand (billion bu) \\
\hline
2.11 & 6.39 & 2.07 & 9.96 \\
2.22 & 7.28 & 2.15 & 9.42 \\
2.36 & 7.58 & 2.25 & 8.41 \\
2.46 & 7.77 & 2.34 & 8.08 \\
2.43 & 8.13 & 2.31 & 7.79 \\
2.55 & 8.38 & 2.44 & 6.82
\end{tabular}

### Tasks:

1. Linear Regression for Price-Supply Data:

Find a linear regression model where [tex]\(x\)[/tex] is supply (in billions of bushels) and [tex]\(y\)[/tex] is price (in dollars).

[tex]\( y = \square \)[/tex]

(Type an equation using [tex]\(x\)[/tex] as the variable. Round to two decimal places as needed.)

2. Linear Regression for Price-Demand Data:

Find a linear regression model where [tex]\(x\)[/tex] is demand (in billions of bushels) and [tex]\(y\)[/tex] is price (in dollars).

[tex]\( y = \square \)[/tex]

(Type an equation using [tex]\(x\)[/tex] as the variable. Round to two decimal places as needed.)

3. Equilibrium Price for Corn:

Select the correct choice and fill in any answer boxes present in your choice.

A. [tex]\( y = \$ \square \)[/tex]

(Round the final answer to two decimal places as needed. Round all intermediate values to two decimal places as needed.)

B. There is no solution.


Sagot :

To solve this problem, we need to perform linear regression on both the price-supply data and the price-demand data. After finding the linear equations for those relationships, we will solve these equations to find the equilibrium price.

### Step 1: Linear Regression for Price-Supply Data

Given price ([tex]$y$[/tex]) and supply ([tex]$x$[/tex]) data:
[tex]\[ \begin{array}{cc} \text{Price} (\$/\text{bu}) & \text{Supply (billion bu)} \\ \hline 2.11 & 6.39 \\ 2.22 & 7.28 \\ 2.36 & 7.58 \\ 2.46 & 7.77 \\ 2.43 & 8.13 \\ 2.55 & 8.38 \\ \end{array} \][/tex]

The linear regression model for this data yields the equation:
[tex]\[ y = 0.68 + 0.22x \][/tex]

### Step 2: Linear Regression for Price-Demand Data

Given price ([tex]$y$[/tex]) and demand ([tex]$x$[/tex]) data:
[tex]\[ \begin{array}{cc} \text{Price} (\$/\text{bu}) & \text{Demand (billion bu)} \\ \hline 2.07 & 9.96 \\ 2.15 & 9.42 \\ 2.25 & 8.41 \\ 2.34 & 8.08 \\ 2.31 & 7.79 \\ 2.44 & 6.82 \\ \end{array} \][/tex]

The linear regression model for this data yields the equation:
[tex]\[ y = 3.24 + (-0.12)x \][/tex]

### Step 3: Finding the Equilibrium Price

To find the equilibrium price, we need to set the supply linear equation equal to the demand linear equation and solve for [tex]$x$[/tex]:
[tex]\[ 0.68 + 0.22x = 3.24 - 0.12x \][/tex]

Solving for [tex]$x$[/tex]:
[tex]\[ 0.22x + 0.12x = 3.24 - 0.68 \\ 0.34x = 2.56 \\ x = \frac{2.56}{0.34} \\ x \approx 7.53 \][/tex]

Now, we substitute [tex]$x$[/tex] back into either linear equation to find the equilibrium price [tex]$y$[/tex]:
[tex]\[ y = 0.68 + 0.22 \times 7.53 \\ y = 0.68 + 1.66 \\ y \approx 2.34 \][/tex]

Therefore, the equilibrium price for corn is:
[tex]\[ \boxed{2.36} \][/tex]

So, the final answers are:

1. Linear regression model for price-supply data:
[tex]\[ y = 0.68 + 0.22x \][/tex]

2. Linear regression model for price-demand data:
[tex]\[ y = 3.24 + (-0.12)x \][/tex]

3. Equilibrium price:
[tex]\[ \text{A.} \quad y = \$2.36 \][/tex]