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, $y$
\[
\begin{array}{ccccccc}
5 & 15 & 30 & 40 & 60 & 84 & 105 \\
69.0 & 67.8 & 66.3 & 65.6 & 64.1 & 62.9 & 60.6
\end{array}
\]
E. It is not appropriate to interpret the slope.
Interpret the $y$-intercept. 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 , on average. (Round to three decimal places as needed.)
B. For a commute time of zero minutes, the index score is predicted to be $68.995^{\prime}$ (Round to three decimal places as needed)
C. For an index score of zero, the commute time is predicted to be minutes (Round to three decimal places as needed)
D. For every unit increase in index score, the commute time falls by , on average. (Round to three decimal places as needed)
E. It is not appropriate to interpret the $y$-intercept because a commute time of zero minutes does not make sense and the value of zero minutes is much smaller than those observed in the data set (c) Predict the well-being index of a person whose commute time is 25 minutes
The predicted index score is (Round to one decimal place as needed )
Answer
Final Answer: The correct choice is B. For a commute time of zero minutes, the index score is predicted to be approximately \(\boxed{68.995}\). This is the y-intercept of the linear regression model.
Step 1 ：The given data represents commute times (in minutes) and scores on a well-being survey. The commute times are represented by the variable 'x' and the well-being index scores are represented by the variable 'y'. The data is as follows: \[x = [5, 15, 30, 40, 60, 84, 105]\] \[y = [69.0, 67.8, 66.3, 65.6, 64.1, 62.9, 60.6]\]
Step 2 ：We are asked to interpret the y-intercept of the linear regression model of this data. The y-intercept is the value of y when x is zero. In this context, it represents the predicted well-being index score for a person with a commute time of zero minutes.
Step 3 ：To find the y-intercept, we need to calculate the linear regression model of the given data. The linear regression model is given by the equation \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept.
Step 4 ：The y-intercept is the predicted well-being index score for a person with a commute time of zero minutes. This is represented by the variable 'b' in the linear regression model.
Step 5 ：Final Answer: The correct choice is B. For a commute time of zero minutes, the index score is predicted to be approximately \(\boxed{68.995}\). This is the y-intercept of the linear regression model.
