Answer :
To find the least-squares regression line, we need to determine the best-fitting line through the data points provided. The line can be expressed in the form:
[tex]\[
\hat{y} = mx + b
\][/tex]
where [tex]\( \hat{y} \)[/tex] is the predicted value, [tex]\( m \)[/tex] is the slope of the line, and [tex]\( b \)[/tex] is the y-intercept.
Step-by-step solution:
1. Identify the data sets:
- Commute Time (minutes) [tex]\( x = [5, 15, 25, 40, 50, 72, 105] \)[/tex]
- Well-Being Index Score [tex]\( y = [69.3, 68.3, 67.5, 66.9, 66.4, 66.1, 63.9] \)[/tex]
2. Calculate the slope ([tex]\( m \)[/tex]):
The formula for the slope [tex]\( m \)[/tex] in the least-squares regression line is given by:
[tex]\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\][/tex]
After calculations, we find [tex]\( m \approx -0.049 \)[/tex].
3. Calculate the y-intercept ([tex]\( b \)[/tex]):
The formula for the y-intercept [tex]\( b \)[/tex] is:
[tex]\[
b = \frac{\sum y - m(\sum x)}{n}
\][/tex]
After performing the necessary calculations, [tex]\( b \approx 69.083 \)[/tex].
4. Write the equation of the regression line:
Combine the calculated slope and y-intercept into the equation:
[tex]\[
\hat{y} = -0.049x + 69.083
\][/tex]
Thus, the least-squares regression line is [tex]\(\hat{y} = -0.049x + 69.083\)[/tex]. This equation allows us to predict the Well-Being Index Score based on the Commute Time.
[tex]\[
\hat{y} = mx + b
\][/tex]
where [tex]\( \hat{y} \)[/tex] is the predicted value, [tex]\( m \)[/tex] is the slope of the line, and [tex]\( b \)[/tex] is the y-intercept.
Step-by-step solution:
1. Identify the data sets:
- Commute Time (minutes) [tex]\( x = [5, 15, 25, 40, 50, 72, 105] \)[/tex]
- Well-Being Index Score [tex]\( y = [69.3, 68.3, 67.5, 66.9, 66.4, 66.1, 63.9] \)[/tex]
2. Calculate the slope ([tex]\( m \)[/tex]):
The formula for the slope [tex]\( m \)[/tex] in the least-squares regression line is given by:
[tex]\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\][/tex]
After calculations, we find [tex]\( m \approx -0.049 \)[/tex].
3. Calculate the y-intercept ([tex]\( b \)[/tex]):
The formula for the y-intercept [tex]\( b \)[/tex] is:
[tex]\[
b = \frac{\sum y - m(\sum x)}{n}
\][/tex]
After performing the necessary calculations, [tex]\( b \approx 69.083 \)[/tex].
4. Write the equation of the regression line:
Combine the calculated slope and y-intercept into the equation:
[tex]\[
\hat{y} = -0.049x + 69.083
\][/tex]
Thus, the least-squares regression line is [tex]\(\hat{y} = -0.049x + 69.083\)[/tex]. This equation allows us to predict the Well-Being Index Score based on the Commute Time.
