The data below represent commute times​ (in minutes) and scores on a​ well-being survey. Complete parts​ (a) through​ (d) below.
Commute Time​ (minutes), x
5
15
30
40
50
84
105
 
​Well-Being Index​ Score, y
69.0
67.8
66.2
65.5
64.7
62.6
60.3
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Part 1
​(a) Find the​ least-squares regression line treating the commute​ time, x, as the explanatory variable and the index​ score, y, as the response variable.

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

\[
\hat{y} = a + bx
\]

Where:
- \( \hat{y} \) is the predicted Well-Being Index score.
- \( x \) is the commute time.
- \( a \) is the y-intercept.
- \( b \) is the slope of the regression line.

The slope \( b \) and intercept \( a \) can be computed using the following formulas:

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

\[
a = \frac{\Sigma y - b(\Sigma x)}{n}
\]

Where:
- \( n \) is the number of data points,
- \( x \) and \( y \) are the individual data points for commute time and well-being index, respectively.

### Step 1: Compute the necessary sums.
Given:
- Commute times \( x = [5, 15, 30, 40, 50, 84, 105] \)
- Well-being scores \( y = [69.0, 67.8, 66.2, 65.5, 64.7, 62.6, 60.3] \)

We need to calculate the following:
- \( \Sigma x \) (sum of commute times)
- \( \Sigma y \) (sum of well-being index scores)
- \( \Sigma xy \) (sum of the product of commute times and well-being index scores)
- \( \Sigma x^2 \) (sum of squares of commute times)

I'll calculate these values now.Here are the computed values:
- \( \Sigma x = 329 \)
- \( \Sigma y = 456.1 \)
- \( \Sigma xy = 20,792.9 \)
- \( \Sigma x^2 = 23,331 \)
- \( n = 7 \) (number of data points)

### Step 2: Calculate the slope \( b \):

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

Substituting the values:

\[
b = \frac{7(20792.9) - (329)(456.1)}{7(23331) - (329)^2}
\]

I will calculate \( b \) now.The slope \( b \) of the least-squares regression line is approximately \( -0.0818 \).

### Step 3: Calculate the intercept \( a \):

\[
a = \frac{\Sigma y - b(\Sigma x)}{n}
\]

Substituting the values:

\[
a = \frac{456.1 - (-0.0818)(329)}{7}
\]

I will calculate \( a \) now.The y-intercept \( a \) of the least-squares regression line is approximately \( 69.00 \).

### Step 4: Write the equation of the least-squares regression line.

The equation is:

\[
\hat{y} = 69.00 - 0.0818x
\]

This is the equation of the regression line, where:
- \( x \) is the commute time in minutes,
- \( \hat{y} \) is the predicted well-being index score.

Would you like further explanation or calculations, such as predicting values or analyzing this regression?

Here are 5 related questions for practice:
1. How would you interpret the slope of the regression line in this context?
2. What would the predicted well-being score be for a commute time of 60 minutes?
3. What does the intercept \( a \) represent in the real-world context of this problem?
4. How can you check if this model fits the data well?
5. What is the residual for a commute time of 40 minutes based on this model?

**Tip:** The slope represents the change in the response variable (well-being score) for every unit increase in the explanatory variable (commute time).

The data below represent commute times​ (in minutes) and scores on a​ well-being survey. Complete parts​ (a) through​ (d) below. Commute Time​ (minutes), x: 5, 15, 30, 40, 50, 84, 105. ​Well-Being Index​ Score, y: 69.0, 67.8, 66.2, 65.5, 64.7, 62.6, 60.3. (a) Find the​ least-squares regression line treating the commute​ time, x, as the explanatory variable and the index​ score, y, as the response variable.

Learn how to find the least-squares regression line for commute time and well-being index score with a detailed step-by-step solution. This problem covers key concepts in statistics, including linear regression and the least-squares method. Suitable for college-level statistics students.

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