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You are the foreman of the Bar-S cattle ranch in Colorado. A neighboring ranch has calves for sale, and you are going to buy some to add to the Bar-S herd. How much should a healthy calf weigh? Let x be the age of the calf (in weeks), and let y be the weigh of the calf (in kilograms).
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Complete parts (a) through (e), given Σx=92, Σy=617, Σx^2=2338, Σy^2=82,389, Σxy=13,642, and r≈0.998.

(f) The calves you want to buy are 12 weeks old. What does the least-squares line predict for a healthy weight?

To find the predicted weight of a 12-week-old calf using the least-squares regression line, we need to follow these steps:

### Step 1: Calculate the Slope (\(m\)) and Y-Intercept (\(b\)) of the Least-Squares Line
The slope \(m\) and y-intercept \(b\) of the least-squares line are given by the 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.
- \(\sum x^2\) is the sum of the squares of the \(x\) values.
- \(\sum y^2\) is the sum of the squares of the \(y\) values.
- \(\sum xy\) is the sum of the products of the \(x\) and \(y\) values.

### Step 2: Substitute the Given Values
Given values:
- \(n = 6\) (since there are 6 data points),
- \(\sum x = 92\),
- \(\sum y = 617\),
- \(\sum x^2 = 2338\),
- \(\sum y^2 = 82389\),
- \(\sum xy = 13642\),
- \(r \approx 0.998\).

### Step 3: Calculate \(m\) and \(b\)
Let's compute the slope \(m\) first:

\[
m = \frac{6 \times 13642 - 92 \times 617}{6 \times 2338 - 92^2}
\]

Now, let's compute the intercept \(b\):

\[
b = \frac{617 - m \times 92}{6}
\]

### Step 4: Predict the Weight for \(x = 12\)
Finally, use the least-squares line equation \(y = mx + b\) to predict the weight when \(x = 12\).

Let me calculate the values for you.### Calculations:
- **Slope (\(m\))**: \(4.509\)
- **Y-intercept (\(b\))**: \(33.696\)

The least-squares line equation is:
\[
y = 4.509x + 33.696
\]

### Prediction:
For a 12-week-old calf (\(x = 12\)):
\[
y = 4.509(12) + 33.696 = 87.803 \text{ kilograms}
\]

So, the predicted weight of a 12-week-old healthy calf is approximately **87.8 kilograms**.

Would you like to see more details or have any questions?

Here are 5 related questions:
1. How would the prediction change if the correlation coefficient \(r\) were different?
2. What would happen to the prediction if the intercept \(b\) were significantly higher or lower?
3. How does the least-squares method minimize the error in the predictions?
4. Could the least-squares line be used to predict weights for calves older than those in the dataset? Why or why not?
5. What is the significance of a correlation coefficient close to 1, as in this case?

**Tip:** Always verify the assumptions behind your data model, such as linearity and normality, when making predictions.

Explore how to predict the weight of calves using the least-squares regression method. This problem involves statistical analysis with given data points including sums of x, y, x^2, y^2, xy, and correlation coefficient. Suitable for advanced high school students or anyone interested in statistical regression analysis.

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