To find the least-squares regression line, we aim to fit a line in the form:

$y = mx + b$

where:

• $y$ is the response variable (Well-Being Index score),
• $x$ is the explanatory variable (Commute Time in minutes),
• $m$ is the slope, and
• $b$ is the y-intercept.

The formula for the slope $m$ and the intercept $b$ are given by:

$m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$ $b = \bar{y} - m \bar{x}$

where:

• $\bar{x}$ is the mean of the $x$ values (commute times),
• $\bar{y}$ is the mean of the $y$ values (Well-Being Index scores).

I'll calculate $m$ and $b$ using the provided data.The least-squares regression line is:

$y = -0.0527x + 69.05$

where:

• The slope $m = -0.0527$ suggests that for each additional minute of commute time, the Well-Being Index score decreases by approximately 0.0527 points.
• The y-intercept $b = 69.05$ represents the estimated Well-Being Index score when the commute time is zero.

Would you like a detailed step-by-step explanation of the calculations, or have any further questions?

Here are five related questions to explore further:

1. How can we interpret the significance of the slope in the context of well-being?
2. What does the y-intercept indicate about a hypothetical zero commute time?
3. How can we use this line to predict the well-being score for a different commute time, say 30 minutes?
4. What are some limitations of using a linear model in this case?
5. How would the regression line change if a new data point was added with a much higher commute time?

Tip: Regression lines help find trends in data, but individual predictions may vary due to factors outside the modeled relationship.