eight, x​ (inches)
Head​ circumference, y​ (inches)
​Height, x​ (inches)
Head​ circumference, y​ (inches)
27.7527.75
17.517.5
26.526.5
17.317.3
24.524.5
17.117.1
27.527.5
17.517.5
25.525.5
17.117.1
26.7526.75
17.317.3
25.525.5
17.317.3
26.7526.75
17.517.5
2525
16.916.9
27.527.5
17.517.5
27.7527.75
17.617.6Find the​ least-squares regression line treating height as the explanatory variable and head circumference as the response variable.
The​ least-squares regression line is ModifyingAbove y with caretyequals=enter your response herexplus+enter your response here.
​(Round to four decimal places as​ needed.)

To find the least-squares regression line, we need to compute the equation in the form:

\[
\hat{y} = b_0 + b_1x
\]

Where:
- \( \hat{y} \) is the predicted head circumference (response variable),
- \( x \) is the height (explanatory variable),
- \( b_1 \) is the slope, and
- \( b_0 \) is the y-intercept.

The formulas for the slope \( b_1 \) and intercept \( b_0 \) are:

\[
b_1 = \frac{n \sum{x_i y_i} - \sum{x_i} \sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}
\]

\[
b_0 = \frac{\sum{y_i} - b_1 \sum{x_i}}{n}
\]

We will need to calculate the following:
1. \( n \), the number of data points.
2. \( \sum{x_i} \), the sum of the heights.
3. \( \sum{y_i} \), the sum of the head circumferences.
4. \( \sum{x_i^2} \), the sum of the squares of the heights.
5. \( \sum{x_i y_i} \), the sum of the product of height and head circumference.

Let's calculate these values based on the provided dataTo find the least-squares regression line, we need to compute the equation in the form:

\[
\hat{y} = b_0 + b_1x
\]

Where:
- \( \hat{y} \) is the predicted head circumference (response variable),
- \( x \) is the height (explanatory variable),
- \( b_1 \) is the slope, and
- \( b_0 \) is the y-intercept.

The formulas for the slope \( b_1 \) and intercept \( b_0 \) are:

\[
b_1 = \frac{n \sum{x_i y_i} - \sum{x_i} \sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}
\]

\[
b_0 = \frac{\sum{y_i} - b_1 \sum{x_i}}{n}
\]

We will need to calculate the following:
1. \( n \), the number of data points.
2. \( \sum{x_i} \), the sum of the heights.
3. \( \sum{y_i} \), the sum of the head circumferences.
4. \( \sum{x_i^2} \), the sum of the squares of the heights.
5. \( \sum{x_i y_i} \), the sum of the product of height and head circumference.

Let's calculate these values based on the provided 

| Height (x) | Head Circumference (y) |
|------------|------------------------|
| 27.75      | 17.5                   |
| 26.5       | 17.3                   |
| 24.5       | 17.1                   |
| 27.5       | 17.5                   |
| 25.5       | 17.1                   |
| 26.75      | 17.3                   |
| 25.5       | 17.3                   |
| 26.75      | 17.5                   |
| 25.0       | 16.9                   |
| 27.5       | 17.5                   |
| 27.75      | 17.6                   |

Let me calculate these values for you.The least-squares regression line is:

\[
\hat{y} = 12.8349 + 0.1698x
\]

This equation predicts the head circumference (\(y\)) based on the height (\(x\)), rounded to four decimal places.

Do you want more details or have any questions?

Here are some related questions:
1. What does the slope \(b_1\) represent in the context of this regression line?
2. How do you interpret the y-intercept \(b_0\) in this scenario?
3. How can we assess the goodness of fit for this regression model?
4. How do we calculate the correlation coefficient between height and head circumference?
5. What are the assumptions of least-squares regression?

**Tip**: Always check the residuals of the regression to ensure the model fits the data well.

Find the least-squares regression line treating height as the explanatory variable and head circumference as the response variable. The least-squares regression line is ŷ = enter your response here x + enter your response here. (Round to four decimal places as needed.)

Learn how to find the least-squares regression line using height as the explanatory variable and head circumference as the response variable. This solution includes a detailed step-by-step calculation of the slope and y-intercept using provided height and head circumference data. Suitable for students in Grades 11-12 or introductory college-level statistics courses.

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