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The table contains price-supply data and price-demand data for corn.

\begin{tabular}{c|c|c|c}
& Price (\[tex]$/bu) & Supply (billion bu) & Demand (billion bu) \\
\hline
& 2.09 & 9.92 & \\
\hline
& 2.15 & 6.49 & \\
& 2.24 & 7.37 & \\
& 2.36 & 7.61 & \\
& 2.44 & 7.95 & \\
& 2.43 & 8.21 & \\
& 2.59 & 8.33 & \\
\end{tabular}

Find a linear regression model for the price-supply data where \( x \) is supply (in billions of bushels) and \( y \) is price (in dollars).
\[ y = \square \]
(Type an equation using \( x \) as the variable. Round to two decimal places as needed.)

Find a linear regression model for the price-demand data where \( x \) is demand (in billions of bushels) and \( y \) is price (in dollars).
\[ y = \square \]
(Type an equation using \( x \) as the variable. Round to two decimal places as needed.)

Find the equilibrium price for corn. Select the correct choice below and fill in any answer boxes present in your choice.

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

GinnyAnswer
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]
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