Answer :
To find the least-squares regression line using commute time as the explanatory variable (x) and the Well-Being Index Score as the response variable (y), follow these steps:
1. Understand the Variables:
- Commute Time (x): Represents the time taken to commute to work in minutes.
- Well-Being Index Score (y): Represents the index score based on the survey results.
2. Concept of Least-Squares Regression Line:
- The goal is to find a linear relationship between x and y. This relationship is represented by the regression line: [tex]\( \hat{y} = b_1x + b_0 \)[/tex], where:
- [tex]\( b_1 \)[/tex] is the slope of the line.
- [tex]\( b_0 \)[/tex] is the y-intercept.
3. Calculate the Means:
- Calculate the mean of x: [tex]\( \bar{x} \)[/tex].
- Calculate the mean of y: [tex]\( \bar{y} \)[/tex].
4. Determine Slope (b1):
- Use the formula for the slope [tex]\( b_1 \)[/tex]:
[tex]\[
b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\][/tex]
- This formula calculates how much y changes for a unit change in x.
5. Determine Intercept (b0):
- Use the formula for the intercept [tex]\( b_0 \)[/tex]:
[tex]\[
b_0 = \bar{y} - b_1 \bar{x}
\][/tex]
- This formula calculates where the line intercepts the y-axis.
6. Construct the Regression Line:
- Plug the calculated [tex]\( b_1 \)[/tex] and [tex]\( b_0 \)[/tex] values into the equation for the regression line:
[tex]\[
\hat{y} = b_1x + b_0
\][/tex]
Given the calculated values:
- Slope [tex]\( b_1 \)[/tex]: -0.095
- Intercept [tex]\( b_0 \)[/tex]: 69.090
The least-squares regression line is:
[tex]\[ \hat{y} = -0.095x + 69.090 \][/tex]
This equation suggests that for every additional minute of commute time, the Well-Being Index Score decreases by approximately 0.095 units.
1. Understand the Variables:
- Commute Time (x): Represents the time taken to commute to work in minutes.
- Well-Being Index Score (y): Represents the index score based on the survey results.
2. Concept of Least-Squares Regression Line:
- The goal is to find a linear relationship between x and y. This relationship is represented by the regression line: [tex]\( \hat{y} = b_1x + b_0 \)[/tex], where:
- [tex]\( b_1 \)[/tex] is the slope of the line.
- [tex]\( b_0 \)[/tex] is the y-intercept.
3. Calculate the Means:
- Calculate the mean of x: [tex]\( \bar{x} \)[/tex].
- Calculate the mean of y: [tex]\( \bar{y} \)[/tex].
4. Determine Slope (b1):
- Use the formula for the slope [tex]\( b_1 \)[/tex]:
[tex]\[
b_1 = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}
\][/tex]
- This formula calculates how much y changes for a unit change in x.
5. Determine Intercept (b0):
- Use the formula for the intercept [tex]\( b_0 \)[/tex]:
[tex]\[
b_0 = \bar{y} - b_1 \bar{x}
\][/tex]
- This formula calculates where the line intercepts the y-axis.
6. Construct the Regression Line:
- Plug the calculated [tex]\( b_1 \)[/tex] and [tex]\( b_0 \)[/tex] values into the equation for the regression line:
[tex]\[
\hat{y} = b_1x + b_0
\][/tex]
Given the calculated values:
- Slope [tex]\( b_1 \)[/tex]: -0.095
- Intercept [tex]\( b_0 \)[/tex]: 69.090
The least-squares regression line is:
[tex]\[ \hat{y} = -0.095x + 69.090 \][/tex]
This equation suggests that for every additional minute of commute time, the Well-Being Index Score decreases by approximately 0.095 units.
