n 1997 Fuller and coworkers at Texas A&M University estimated the operating costs of cotton gin plants of various sizes. The operating costs of the next to the smallest plant is shown in the following table.
x	
2000
4000
6000
8000
10000
12000
y	163,200	230,480	301,500	376,160	454,400	536,400

Here x is the annual number of bales produced, and y is the dollar total cost. (Use 4 decimal places in your answer.)
A. Determine the best fitting line using least squares. Also determine the square of the correlation coefficient.
The best fitting line is C(x) = 
  .
The square of the correlation coefficient is r2 = 
  .

B. The study noted that revenue was $63.25 per bale. At what level of production will this plant break even?
The revenue equation is R(x) = 
 
The profit equation is P(x) = 
 

The production will break even when 
  bales are produced.

C. What are the profits or losses when production is 3000 bales? 4000 bales? (Round to the nearest cent.)
When 3000 bales are produced, the gin experiences a 
 (loss, profit) of $ 
 . When 4000 bales are produced, the gin experiences a 
 (loss, profit) of $ 
 .


### A. Best Fitting Line using Least Squares and $r^2$

We are given the following data points:

\[
x = \{2000, 4000, 6000, 8000, 10000, 12000\}
\]
\[
y = \{163200, 230480, 301500, 376160, 454400, 536400\}
\]

To find the best fitting line $C(x) = mx + b$ using the least squares method, we need to calculate the slope $m$ and intercept $b$ of the linear equation. These are determined by the following formulas:

\[
m = \frac{n \sum{xy} - \sum{x}\sum{y}}{n \sum{x^2} - (\sum{x})^2}
\]
\[
b = \frac{\sum{y} - m \sum{x}}{n}
\]

where:
- $n$ is the number of data points
- $\sum{x}$ is the sum of the $x$-values
- $\sum{y}$ is the sum of the $y$-values
- $\sum{xy}$ is the sum of the product of corresponding $x$ and $y$ values
- $\sum{x^2}$ is the sum of the squares of the $x$-values

Also, the square of the correlation coefficient $r^2$ is given by:

\[
r^2 = \frac{(n \sum{xy} - \sum{x}\sum{y})^2}{(n \sum{x^2} - (\sum{x})^2)(n \sum{y^2} - (\sum{y})^2)}
\]

Let's compute these values.

### B. Break-even Production Level

Revenue per bale is given as $63.25. The revenue equation $R(x)$ is:

\[
R(x) = 63.25x
\]

The profit equation $P(x)$ is given by:

\[
P(x) = R(x) - C(x) = 63.25x - (mx + b)
\]

For break-even, profit equals zero, so:

\[
0 = 63.25x - (mx + b)
\]

Solve for $x$ to find the break-even production level.

### C. Profits or Losses at 3000 and 4000 Bales

Using the profit equation:

\[
P(x) = (63.25x - (mx + b))
\]

Substitute $x = 3000$ and $x = 4000$ to calculate the profit or loss at each production level.

I'll now calculate the necessary values.### A. Best Fitting Line and $ r^2 $

The best fitting line using least squares is:

\[
C(x) = 37.3203x + 82448
\]

The square of the correlation coefficient is:

\[
r^2 = 0.9492
\]

### B. Break-even Production Level

The revenue equation is:

\[
R(x) = 63.25x
\]

The profit equation is:

\[
P(x) = 63.25x - (37.3203x + 82448) = 25.9297x - 82448
\]

To break even, the production level is:

\[
x = 3179.67 \, \text{bales}
\]

### C. Profits or Losses at Different Production Levels

For 3000 bales:

\[
\text{Loss} = -\$4658.86
\]

For 4000 bales:

\[
\text{Profit} = \$21270.86
\]

Would you like further clarification on any part?

Here are 5 follow-up questions:
1. How is the correlation coefficient interpreted in this context?
2. What factors might affect the accuracy of the linear model?
3. How would profits change if the revenue per bale increases?
4. What is the importance of the break-even point for the business?
5. How would you calculate the margin of profit for different production levels?

**Tip:** In real-world applications, fixed costs (the intercept) can significantly impact profit at lower production levels.

In 1997, Fuller and coworkers at Texas A&M University estimated the operating costs of cotton gin plants of various sizes. The operating costs of the next to the smallest plant are shown in the following table.

x: {2000, 4000, 6000, 8000, 10000, 12000}

y: {163200, 230480, 301500, 376160, 454400, 536400}

Here x is the annual number of bales produced, and y is the total dollar cost.

A. Determine the best fitting line using least squares. Also, determine the square of the correlation coefficient.

B. The study noted that revenue was $63.25 per bale. At what level of production will this plant break even?

C. What are the profits or losses when production is 3000 bales? 4000 bales?

Learn how to apply the least squares method to estimate the operating costs of a cotton gin plant. This problem covers linear regression, the correlation coefficient, and break-even analysis for production levels. Includes step-by-step solutions for calculating the best fitting line and profits at different production levels.

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