https://file.aiquickdraw.com/mathbot/file/17265518643000ivwmj5c.jpg

To solve this problem, we need to compute the least-squares regression line, which has the general form:

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

Where:
- \( \hat{y} \) is the predicted value of the response variable (Well-Being Index Score),
- \( x \) is the explanatory variable (Commute Time),
- \( b_1 \) is the slope of the regression line,
- \( b_0 \) is the y-intercept.

The formulas to calculate the slope (\(b_1\)) and y-intercept (\(b_0\)) are as follows:

\[
b_1 = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]
\[
b_0 = \frac{\sum{y} - b_1\sum{x}}{n}
\]

Where:
- \( n \) is the number of data points,
- \( x \) is the Commute Time data,
- \( y \) is the Well-Being Index Score data.

I'll begin by calculating these values based on the data points provided in the image:

- Commute Time \(x\): 5, 20, 25, 40, 60, 84, 105
- Well-Being Index Score \(y\): 68.9, 67.2, 66.4, 65.1, 63.3, 61.8, 59.3

Let me compute these values.The least-squares regression line, treating the commute time (\(x\)) as the explanatory variable and the well-being index score (\(y\)) as the response variable, is:

\[
\hat{y} = -0.091x + 68.987
\]

This equation shows that for every additional minute of commute time, the well-being index score decreases by approximately 0.091 points.

Would you like more details or have any questions?

Here are five related questions to expand on this concept:
1. How do you interpret the y-intercept in the context of this problem?
2. How much does the well-being score change if the commute time increases by 10 minutes?
3. What would the predicted well-being score be for a commute time of 50 minutes?
4. How would you calculate the coefficient of determination (\(R^2\)) for this regression model?
5. What assumptions underlie the use of the least-squares regression method?

**Tip:** When analyzing regression results, it's important to check whether the data shows a linear relationship before fully trusting the regression line.

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 calculate the least-squares regression line with commute times (in minutes) and well-being index scores using linear regression. The problem involves key mathematical concepts such as algebra and the least-squares method. Suitable for Grades 10-12.

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