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 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) = 
  
Correct: Your answer is correct.
 .
The square of the correlation coefficient is r2 = 
  
Correct: Your answer is correct.
 .

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) = 
  
Correct: Your answer is correct.

The profit equation is P(x) = 
  
Incorrect: Your answer is incorrect.


The production will break even when 
  
Incorrect: Your answer is incorrect.
 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 profit 
Incorrect: Your answer is incorrect.
 (loss, profit) of $ 
183087
 
Incorrect: Your answer is incorrect.
 . When 4000 bales are produced, the gin experiences a profit 
Correct: Your answer is correct.
 (loss, profit) of $ 
244377
 
Incorrect: Your answer is incorrect.
 .

Let's walk through the problem step by step and solve each part carefully. We will start with Part A, move to Part B, and then tackle Part C.

### Part A: Determine the Best Fitting Line Using Least Squares

We are given the following data points:

- $ x $ (annual number of bales produced): 2000, 4000, 6000, 8000, 10000, 12000
- $ y $ (total operating cost in dollars): 163,200, 230,480, 301,500, 376,160, 454,400, 536,400

We are looking for the best-fitting linear model using least squares: $ C(x) = mx + b $, where $ C(x) $ is the cost as a function of $ x $.

#### 1. Finding the Line of Best Fit:
Using the least squares method, we can determine the slope $ m $ and intercept $ b $ of the best-fitting line. The formulas for these are:

\[
m = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}
\]

\[
b = \frac{\sum y_i - m \sum x_i}{n}
\]

Where:
- $ n $ is the number of data points,
- $ x_i $ and $ y_i $ are the given data points.

Let's calculate $ m $ and $ b $.

#### 2. Correlation Coefficient ($ r^2 $):
Once we have the line of best fit, we can calculate the square of the correlation coefficient $ r^2 $. The formula for $ r^2 $ is:

\[
r^2 = \frac{[\sum (x_i y_i) - (\sum x_i)(\sum y_i)/n]^2}{[\sum x_i^2 - (\sum x_i)^2/n][\sum y_i^2 - (\sum y_i)^2/n]}
\]

Let me calculate these values.The best-fitting line using the least squares method is:

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

The square of the correlation coefficient is:

\[
r^2 = 0.9987
\]

This high value of $ r^2 $ indicates an excellent fit between the data and the linear model.

### Part B: Break-even Point

Revenue per bale is given as $63.25. The revenue equation is:

\[
R(x) = 63.25x
\]

To find the break-even point, we need to find where the revenue equals the cost. This happens when:

\[
R(x) = C(x)
\]

Substitute the equations for $ R(x) $ and $ C(x) $:

\[
63.25x = 37.3203x + 82448
\]

Now, let's solve for $ x $ (the break-even number of bales).It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, if you'd like, I can guide you through the steps to solve the break-even point manually:

1. Start with the equation for the revenue and cost:
   \[
   63.25x = 37.3203x + 82448
   \]
   
2. Subtract $37.3203x$ from both sides:
   \[
   63.25x - 37.3203x = 82448
   \]
   
3. Simplify and divide to isolate $x$:
   \[
   (63.25 - 37.3203)x = 82448
   \]

   \[
   x = \frac{82448}{63.25 - 37.3203}
   \]

This will give you the break-even point.

Do you want further assistance, or would you like to ask questions on this?

Here are 5 related questions:
1. How does the correlation coefficient help in evaluating model accuracy?
2. What is the interpretation of the slope in the cost equation?
3. Can you calculate the profit margin for a specific production level?
4. How do changes in bale price affect the break-even point?
5. How would fixed costs impact the profit equation?

**Tip:** Always check the fit of a model using $ r^2 $ to ensure its reliability for prediction.

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 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. 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?

Explore how to calculate the least squares linear fit for cotton gin plant operating costs based on 1997 data. Learn how to determine the square of the correlation coefficient and calculate break-even points using a revenue equation. This analysis includes a step-by-step guide to solving for profits and losses at different production levels.

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