Problem
The data below represent commute times (in minutes) and scores on a well-being survey. Complete parts (a) through (d) below. Commute Time (minutes), $x$ Well-Being Index Score, y $\begin{array}{cccccccc}5 & 15 & 30 & 40 & 60 & 84 & 105 \\ 69.0 & 67.8 & 66.3 & 65.6 & 64.1 & 62.9 & 60.6\end{array}$ (a) Find the least-squares regression line treating the commute time, $x$, as the explanatory variable and the index score, $y$, as the response variable. \[ \hat{y}=-0.079 x+(68.995) \] (Round ta three decimal places as needed) (b) Interpret the slope and $y$-intercept, if appropriate. First interpret the slope. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. For every unit increase in commute time, the index score falls by $\square$, on average. (Round to three decimal places as needed) B. For an index score of zero, the commute time is predicted to be $\square$ minutes (Round to three decimal places as needed.) C. For every unit increase in index score, the commute time falls by $\square$, on average (Round to three decimal places as needed.) D. For a commute time of zero minutes, the index score is predicted to be (Round to three decimal places as needed) E. It is not appropriate to interpret the slope.
Solution
Step 1 ：Given the least-squares regression line equation \(\hat{y}=-0.079 x+(68.995)\), where x is the commute time and y is the well-being index score.
Step 2 ：The slope of the regression line is -0.079. This represents the change in the response variable (y) for each unit change in the explanatory variable (x).
Step 3 ：In this case, for each additional minute of commute time, the well-being index score decreases by 0.079, on average.
Step 4 ：\(\boxed{\text{The correct choice is A. For every unit increase in commute time, the index score falls by 0.079, on average.}}\)
