Math Problem Statement

The following data represent commute times​ (in minutes) and scores on a​ well-being survey. Complete parts​ (a) through​ (d). Commute Time​ (minutes), x 5 15 30 40 50 72 105

​Well-Being Index​ Score, y 69.0 67.6 65.8 64.9 64.0 62.6 58.8 Question content area bottom Part 1 ​(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. ModifyingAbove y with caretequals    negative 0.097xplusleft parenthesis 69.074 right parenthesis    ​(Round to three decimal places as​ needed.) Part 2 ​(b) Interpret the slope and​ y-intercept, if appropriate. First interpret the slope. Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. For every unit increase in index​ score, the commute time falls by    enter your response here​, on average. ​(Round to three decimal places as​ needed.) B. For a commute time of zero​ minutes, the index score is predicted to be    enter your response here. ​(Round to three decimal places as​ needed.) C. For every unit increase in commute​ time, the index score falls by    enter your response here​, on average. ​(Round to three decimal places as​ needed.) D. For an index score of​ zero, the commute time is predicted to be    enter your response here minutes. ​(Round to three decimal places as​ needed.) E. It is not appropriate to interpret the slope.

Solution

Let's begin by calculating the least-squares regression line based on the provided data.

Step 1: Least-Squares Regression Line

The general form of the least-squares regression line is: y^=b0+b1x\hat{y} = b_0 + b_1 x

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Correlation

Formulas

Least-squares regression line formula: ŷ = b0 + b1x

Theorems

Least-Squares Theorem
Slope-Intercept Form

Suitable Grade Level

Grades 10-12