Step 1 :Given the commute times (in minutes) and scores on a well-being survey, we are to find the least-squares regression line treating the commute time, \(x\), as the explanatory variable and the index score, \(y\), as the response variable.
Step 2 :The least-squares regression line is given by the equation \(\hat{y} = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
Step 3 :The slope \(m\) is calculated as the covariance of \(x\) and \(y\) divided by the variance of \(x\).
Step 4 :The y-intercept \(c\) is calculated as the mean of \(y\) minus the slope times the mean of \(x\).
Step 5 :Given the values of \(x\) and \(y\), we calculate the mean of \(x\) as 45.286 and the mean of \(y\) as 64.629.
Step 6 :Calculating the covariance of \(x\) and \(y\) gives -693.057 and the variance of \(x\) gives 7103.429.
Step 7 :Dividing the covariance of \(x\) and \(y\) by the variance of \(x\) gives the slope \(m\) as -0.098.
Step 8 :Subtracting the product of the slope and the mean of \(x\) from the mean of \(y\) gives the y-intercept \(c\) as 69.047.
Step 9 :Substituting the values of \(m\) and \(c\) into the equation gives the least-squares regression line as \(\hat{y} = -0.098x + 69.047\).
Step 10 :\(\boxed{\text{Final Answer: The least-squares regression line is } \hat{y} = -0.098x + 69.047}\)