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.